RINGS, MODULES, AND ALGEBRAS IN STABLE HOMOTOPY THEORY A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May Author addresses: Purdue University Calumet, Hammond IN 46323 E-mail address: aelmendo@math.purdue.edu The University of Michigan, Ann Arbor, MI 48109-1003 E-mail address: ikriz@math.lsa.umich.edu The University of Chicago, Chicago, IL 60637 E-mail address: mandell@math.uchicago.edu The University of Chicago, Chicago, IL 60637 E-mail address: may@math.uchicago.edu ii iii Abstract. Let S be the sphere spectrum. We construct an associative, com- mutative, and unital smash product in a complete and cocomplete category MS of "S-modules" whose derived category DS is equivalent to the classical stable homotopy category. This allows a simple and algebraically manageable definition of "S-algebras" and "commutative S-algebras" in terms of associative, or asso- ciative and commutative, products R ^S R -! R. These notions are essentially equivalent to the earlier notions of A1 and E1 ring spectra, and the older no- tions feed naturally into the new framework to provide plentiful examples. There is an equally simple definition of R-modules in terms of maps R ^S M -! M. When R is commutative, the category MR of R-modules also has an associa- tive, commutative, and unital smash product, and its derived category DR has properties just like the stable homotopy category. Working in the derived category DR, we construct spectral sequences that spe- cialize to give generalized universal coefficient and K"unneth spectral sequenc* *es. Classical torsion products and Ext groups are obtained by specializing our con- structions to Eilenberg-Mac Lane spectra and passing to homotopy groups, and the derived category of a discrete ring R is equivalent to the derived category* * of its associated Eilenberg-Mac Lane S-algebra. We also develop a homotopical theory of R-ring spectra in DR, analogous to the classical theory of ring spectra in the stable homotopy category, and we use it to give new constructions as MU-ring spectra of a host of fundamentally important spectra whose earlier constructions were both more difficult and less precise. Working in the module category MR, we show that the category of finite cell modules over an S-algebra R gives rise to an associated algebraic K-theory spectrum KR. Specialized to the Eilenberg-Mac Lane spectra of discrete rings, this recovers Quillen's algebraic K-theory of rings. Specialized to suspension spectra 1 (X)+ of loop spaces, it recovers Waldhausen's algebraic K-theory of spaces. Replacing our ground ring S by a commutative S-algebra R, we define R- algebras and commutative R-algebras in terms of maps A ^R A -! A, and we show that the categories of R-modules, R-algebras, and commutative R-algebras are all topological model categories. We use the model structures to study Bous- field localizations of R-modules and R-algebras. In particular, we prove that KO and KU are commutative ko and ku-algebras and therefore commutative S-algebras. We define the topological Hochschild homology R-module T HHR (A; M) of A with coefficients in an (A; A)-bimodule M and give spectral sequences for the calculation of its homotopy and homology groups. Again, classical Hochschild homology and cohomology groups are obtained by specializing the constructions to Eilenberg-Mac Lane spectra and passing to homotopy groups. iv Contents Introduction 1 Chapter I. Prologue: the category of L-spectra 9 1. Background on spectra and the stable homotopy category 9 2. External smash products and twisted half-smash products 11 3. The linear isometries operad and internal smash products 14 4. The category of L-spectra 18 5. The smash product of L-spectra 21 6. The equivalence of the old and new smash products 24 7. Function L-spectra 27 8. Unital properties of the smash product of L-spectra 30 Chapter II. Structured ring and module spectra 35 1. The category of S-modules 35 2. The mirror image to the category of S-modules 39 3. S-algebras and their modules 41 4. Free A1 and E1 ring spectra and comparisons of definitions 44 5. Free modules over A1 and E1 ring spectra 47 6. Composites of monads and monadic tensor products 50 7. Limits and colimits of S-algebras 52 Chapter III. The homotopy theory of R-modules 57 1. The category of R-modules; free and cofree R-modules 57 v vi CONTENTS 2. Cell and CW R-modules; the derived category of R-modules 60 3. The smash product of R-modules 65 4. Change of S-algebras; q-cofibrant S-algebras 68 5. Symmetric and extended powers of R-modules 71 6. Function R-modules 73 7. Commutative S-algebras and duality theory 77 Chapter IV. The algebraic theory of R-modules 81 1. Tor and Ext; homology and cohomology; duality 82 2. Eilenberg-Mac Lane spectra and derived categories 85 3. The Atiyah-Hirzebruch spectral sequence 89 4. Universal coefficient and K"unneth spectral sequences 92 5. The construction of the spectral sequences 94 6. Eilenberg-Moore type spectral sequences 97 7. The bar constructions B(M; R; N) and B(X; G; Y ) 99 Chapter V. R-ring spectra and the specialization to MU 103 1. Quotients by ideals and localizations 103 2. Localizations and quotients of R-ring spectra 107 3. The associativity and commutativity of R-ring spectra 111 4. The specialization to MU-modules and algebras 114 Chapter VI. Algebraic K-theory of S-algebras 117 1. Waldhausen categories and algebraic K-theory 117 2. Cylinders, homotopies, and approximation theorems 121 3. Application to categories of R-modules 124 4. Homotopy invariance and Quillen's algebraic K-theory of rings 128 5. Morita equivalence 130 6. Multiplicative structure in the commutative case 134 7. The plus construction description of KR 136 8. Comparison with Waldhausen's K-theory of spaces 141 Chapter VII. R-algebras and topological model categories 145 CONTENTS vii 1. R-algebras and their modules 146 2. Tensored and cotensored categories of structured spectra 149 3. Geometric realization and calculations of tensors 153 4. Model categories of ring, module, and algebra spectra 159 5. The proofs of the model structure theorems 163 6. The underlying R-modules of q-cofibrant R-algebras 167 Chapter VIII. Bousfield localizations of R-modules and algebras 173 1. Bousfield localizations of R-modules 174 2. Bousfield localizations of R-algebras 178 3. Categories of local modules 181 4. Periodicity and K-theory 184 Chapter IX. Topological Hochschild homology and cohomology 187 1. Topological Hochschild homology: first definition 188 2. Topological Hochschild homology: second definition 192 3. The isomorphism between thhR (A) and A S1 196 Chapter X. Some basic constructions on spectra 201 1. The geometric realization of simplicial spectra 201 2. Homotopical and homological properties of realization 204 3. Homotopy colimits and limits 209 4. -cofibrant, LEC, and CW prespectra 211 5. The cylinder construction 214 Chapter XI. Spaces of linear isometries and technical theorems 221 1. Spaces of linear isometries 221 2. Fine structure of the linear isometries operad 224 3. The unit equivalence for the smash product of L-spectra 230 4. Twisted half-smash products and shift desuspension 232 5. Twisted half-smash products and cofibrations 235 Chapter XII. The monadic bar construction 239 1. The bar construction and two deferred proofs 239 viii CONTENTS 2. Cofibrations and the bar construction 242 Chapter XIII. Epilogue: The category of L-spectra under S 247 1. The modified smash products CL, BL, and ?L 247 2. The modified smash products CR , BR , and ?R 251 Bibliography 257 Introduction The last thirty years have seen the importation of more and more algebraic tech- niques into stable homotopy theory. Throughout this period, most work in stable homotopy theory has taken place in Boardman's stable homotopy category [6], or in Adams' variant of it [2], or, more recently, in Lewis and May's variant [37]. That category is analogous to the derived category obtained from the category of chain complexes over a commutative ring k by inverting the quasi-isomorphisms. The sphere spectrum S plays the role of k, the smash product ^ plays the role of the tensor product, and weak equivalences play the role of quasi-isomorphism* *s. A fundamental difference between the two situations is that the smash product on the underlying category of spectra is not associative and commutative, whereas * *the tensor product between chain complexes of k-modules is associative and commu- tative. For this reason, topologists generally work with rings and modules in t* *he stable homotopy category, with their products and actions defined only up to ho- motopy. In contrast, of course, algebraists generally work with differential gr* *aded k-algebras that have associative point-set level multiplications. We here introduce a new approach to stable homotopy theory that allows one to do point-set level algebra. We construct a new category MS of S-modules that has an associative, commutative, and unital smash product ^S. Its derived category DS is obtained by inverting the weak equivalences; DS is equivalent to* * the classical stable homotopy category, and the equivalence preserves smash product* *s. This allows us to rethink all of stable homotopy theory: all previous work in t* *he subject might as well have been done in DS. Working on the point-set level, in MS, we define an S-algebra to be an S-module R with an associative and unital product R ^S R - ! R; if the product is also commutative, we call R a commutative S-algebra. Although the definitions are now very simple, these are not new notions: they are refinements of the A1 and E1 ring spectra that were introduced over twenty years ago by May, Quinn, and Ray [47]. In general, the 1 2 INTRODUCTION latter need not satisfy the precise unital property that is enjoyed by our new * *S- algebras, but it is a simple matter to construct a weakly equivalent S-algebra * *from an A1 ring spectrum and a weakly equivalent commutative S-algebra from an E1 ring spectrum. It is tempting to refer to (commutative) S-algebras as (commutative) ring spe* *c- tra. However, this would introduce confusion since the term "ring spectrum" has had a definite meaning for thirty years as a stable homotopy category level not* *ion. Ring spectra in the classical homotopical sense are not rendered obsolete by our theory since there are many examples that admit no S-algebra structure. In any case, the term S-algebra more accurately describes our new concept. With our theory, and the new possibilities that it opens up, it becomes vitally importan* *t to keep track of when one is working on the point-set level and when one is working up to homotopy. In the absence (or ignorance) of a good point-set level categor* *y of spectra, topologists have tended to be sloppy about this. The dichotomy will run through our work. The terms "ring spectrum" and "module spectrum" will always refer to the classical homotopical notions. The terms "S-algebra" and "S-module" will always refer to the strict point-set level notions. We define a (left) module M over an S-algebra R to be an S-module M with an action R ^S M - ! M such that the standard diagrams commute. We obtain a category MR of (left) R-modules and a derived category DR . There is a smash product M ^R N of a right R-module M and a left R-module N, which is an S- module. For left R-modules M and N, there is a function S-module FR (M; N) that enjoys properties just like modules of homomorphisms in algebra. Each FR (M; M) is an S-algebra. If R is commutative, then M ^R N and FR (M; N) are R-modules, and in this case MR and DR enjoy all of the properties of MS and DS. Thus each commutative S-algebra R determines a derived category of R-modules that has all of the structure that the stable homotopy category has. These new categories ar* *e of substantial intrinsic interest, and they give powerful new tools for the invest* *igation of the classical stable homotopy category. Upon restriction to Eilenberg-Mac Lane spectra, our topological theory subsum* *es a good deal of classical algebra. For a discrete ring R and R-modules M and N, we have TorRn(M; N) ~=ssn(HM ^HR HN) and Ext nR(M; N) ~=ss-nFHR (HM; HN): Here ^R and FR must be interpreted in the derived category; that is, HM must be a CW HR-module. Moreover, the algebraic derived category DR is equivalent to the topological derived category DHR . In general, for an S-algebra R, approximation of R-modules M by weakly equiv- alent cell R-modules is roughly analogous to forming projective resolutions in * *al- gebra. There is a much more precise analogy that involves developing the derived INTRODUCTION 3 categories of modules over rings or, more generally, DGA's in terms of cell mod* *ules. It is presented in [34], which gives an algebraic theory of A1 and E1 k-algeb* *ras that closely parallels the present topological theory. Upon restriction to the sphere spectrum S, the derived smash products M ^S N and function spectra FS(M; N) have as their homotopy groups the homology and cohomology groups N*(M) and N*(M). This suggests the alternative notations TorRn(M; N) = ssn(M ^R N) = NRn(M) and Ext nR(M; N) = ss-nFR (M; N) = NnR(M) for R-modules M and N. When R is connective, there are ordinary homology and cohomology theories on R-modules, represented by Eilenberg-Mac Lane spectra that are R-modules, and there are Atiyah-Hirzebruch spectral sequences for the computation of generalized homology and cohomology theories on R-modules. The realization of algebraic Tor and Ext groups via Eilenberg-Mac Lane spectra generalizes to spectral sequences E2p;q= TorR*p;q(M*; N*) =) TorRp+q(M; N) and Ep;q2= Extp;qR*(M*; N*) =) Extp+qR(M; N): These specialize to give K"unneth and universal coefficient spectral sequences * *in classical generalized homology and cohomology theories. There are also Eilenber* *g- Moore type spectral sequences for the calculation of E*(M ^R N) under appropria* *te hypotheses on R and E. Thinking of DR as a new stable homotopy category, where R is a commutative S-algebra, we can realize the action of an element x 2 Rn on an R-module M as a map of R-modules x : nM - ! M. We define M=xM to be the cofiber of x, and we define the localization M[x-1] to be the telescope of a countable iterat* *e of desuspensions of x, starting with M - ! -nM. By iteration, we can construct quotients by sequences of elements and localizations at sequences of elements. * *We define R-ring spectra, associative R-ring spectra, and commutative R-ring spect* *ra in the homotopical sense, with products A ^R A -! A defined via maps in the derived category DR , and it turns out to be quite simple to study when quotien* *ts and localizations of R-ring spectra are again R-ring spectra. When we take R = MU, we find easy direct constructions as MU-modules of all of the various spectra (MU=X)[Y -1] that are usually obtained by means of t* *he Baas-Sullivan theory of manifolds with singularities or the Landweber exact fun* *c- tor theorem. When their homotopy groups are integral domains concentrated in degrees congruent to zero mod 4, these MU-modules all admit canonical structures of associative and commutative MU-ring spectra. Remarkably, it is far simpler to 4 INTRODUCTION prove the sharper statements that apply in the derived category of MU-modules than the much weaker stable homotopy category level analogs that were obtainable before our theory. Thinking of MR as a new category of point-set level modules, where R is again a commutative S-algebra, we can define R-algebras A via point-set level prod- ucts A ^R A -! A such that the appropriate diagrams commute. For example, FR (M; M) is an R-algebra for any R-module M. These have all of the good formal properties of S-algebras. We repeat the dichotomy for emphasis: The terms "R- ring spectrum" and "R-module spectrum" will always refer to the homotopical no- tions defined in the derived category DR . The terms "R-algebra" and "R-module" will always refer to the strict, point-set, level notions. We shall construct Bousfield localizations of R-modules at a given R-module E. In principle, this is a derived category notion, but we shall obtain precise po* *int-set level constructions. Using different point-set level constructions, we shall p* *rove that the Bousfield localizations of R-algebras can be constructed to be R-algeb* *ras and the Bousfield localizations of commutative R-modules can be constructed to be commutative R-algebras. In particular, the localization RE of R at E is a commutative R-algebra, and we shall see that the category of RE -modules plays an intrinsically central role in the study of Bousfield localizations. As a very special case, this theory will imply that the spectra KO and KU that represent real and complex periodic K-theory can be constructed as commutative algebras over the S-algebras ko and ku that represent real and complex connecti* *ve K-theory. Therefore KO and KU are commutative S-algebras, as had long been conjectured in the earlier context of E1 ring spectra. Again, it is far simpl* *er to prove the sharper ko and ku-algebra statements than to construct S-algebra structures directly. For an R-algebra A, we define the enveloping R-algebra Ae = A ^R Aop, and we define the topological Hochschild homology of A with coefficients in an (A; * *A)- bimodule M to be the derived smash product T HHR (A; M) = M ^Ae A: This is the correct generalization from algebra to topology since, if R is a di* *screte commutative ring and A is an R-algebra that is flat as an R-module, then the algebraic and topological Hochschild homology are isomorphic: e HAe R HHRn(A; M) TorAn(M; A) ~=Torn (HM; HA) ssn(T HH (A; M)): In general, for a commutative S-algebra R, an R-algebra A, and an (A; A)-bimodu* *le M, there is a spectral sequence E2p;q= HHR*p;q(M*; A*) =) ssp+q(T HHR (A; M)) INTRODUCTION 5 under suitable flatness hypotheses. More generally, there are similar spectral* * se- quences converging to E*(T HHR (A; M)) for a commutative ring spectrum E. There is also a point-set level version thhR (A; M) of topological Hochschild homology. It is obtained by mimicking topologically the standard complex for the calculation of algebraic Hochschild homology. When M = A, this construction has particularly nice formal properties, as was observed in [52] and as we shall ex* *plain: it is isomorphic to the tensor A S1. A key technical point is that the derived category and point-set level definitions become equivalent after replacing R an* *d A by suitable weakly equivalent approximations. Our S-algebras and their modules are enough like ordinary rings and modules that we can construct the algebraic K-theory spectrum KR associated to an S- algebra R by applying Waldhausen's So-construction to the category of finite ce* *ll R-modules. Applied to the Eilenberg-Mac Lane spectrum HR of a discrete ring R, this gives a new construction of Quillen's algebraic K-theory. Applied to t* *he suspension spectrum 1 (X)+ , this gives a new construction of Waldhausen's algebraic K-theory of the space X. The resulting common framework for topolog- ical Hochschild homology and Quillen and Waldhausen algebraic K-theory opens up several new directions and appears to bring a number of standing conjectures within reach. We merely lay the foundations here. The technical heart of our theory is the problem of keeping our formal point-* *set level constructions under homotopical control. While we shall show by essential* *ly formal categorical arguments that our various categories of R-modules, R-algebr* *as, and commutative R-algebras are cocomplete and complete, tensored and coten- sored, topological model categories, this formal structure does not in itself a* *ddress the problem: forgetful functors from more to less structured spectra rarely pre* *serve cofibrant objects, and may well not do so even up to homotopy type. The problem requires deeper analysis, and a crucial aspect of our work is that our discussi* *on of model categories gives sufficient control on the underlying homotopy types of cofibrant R-algebras and cofibrant commutative R-algebras to allow the calcula- tional use of bar constructions and topological Hochschild homology complexes. This is also crucial to our proof that Bousfield localizations of R-algebras ca* *n be constructed as R-algebras. Another tool in keeping homotopical control is the category of "tame" spectra* *. It is an intermediate category between the ground category of spectra, which is we* *ll- designed for formal point-set level work but not for homotopical analysis, and * *the category of CW spectra, which is well-designed for homotopical analysis but not for formal work. Its homotopy category is symmetric monoidal under the smash product, and we can approximate any structured spectrum by a weakly equivalent tame structured spectrum by means of a "cylinder construction" defined using homotopy colimits. Actually, this tool will only be needed in Chapter I, since * *the 6 INTRODUCTION smash product of S-modules turns out to better behaved under weak equivalences than the smash product of spectra. The basic construction underlying all of our work is the "twisted half-smash product" A n E of a suitable space A and a spectrum E. This construction is defined with respect to a given map ff from A to an appropriate space of linear isometries. We prove that a homotopy equivalence A0 -! A, with homotopy inverse unrelated to ff, induces a homotopy equivalence A0n E -! A n E when E is tame. This invariance statement is the technical lynchpin of our theory. The construction of thh, of bar constructions needed in our work, and of func- torial homotopy colimits of spectra all require geometric realizations of simpl* *icial spectra. This raises another technical problem. To understand geometric real- ization homotopically, the given simplicial spectra must satisfy certain cofibr* *ation conditions, and it is hard to verify that a map of spectra is a cofibration (sa* *tisfies the homotopy extension property). The solution to this problem is basic to the homotopical understanding of cofibrant R-algebras and commutative R-algebras. The reader interested in using our theory need not be concerned with these matters, and most of the technical proofs are deferred until the last few chapt* *ers. The first three chapters explain the foundations needed for the applications of the next three, which are independent of one another. Chapter VII explains the foundations needed for Chapters VIII and IX, which are independent of each othe* *r. Each chapter has its own brief introduction. References within a chapter are of* * the form "Lemma 3.4"; references to results in other chapters are of the form "I.3.* *4". Our work is not independent of earlier work: the groundwork was laid in [37], and all of our ring, module, and algebra spectra are spectra in the sense of Le* *wis and May with additional structure. Moreover, the technical lynchpin referred to above depends on the first author's paper [19]. In [37], the focus was on equiv* *ariant stable homotopy theory, the study of spectra with actions by compact Lie groups G. We have chosen to write this book nonequivariantly in the interests of reada* *bil- ity. However, we have kept a close eye on the equivariant generalization, and we have been careful to use only arguments that directly generalize to the equivar* *iant setting. We state a metatheorem. Theorem 0.1. All of the definitions and all of the general theory in this p* *aper apply to G-spectra for any compact Lie group G. This has been used by Greenlees and May [27] to prove a completion theorem for the calculation of M*(BG) and M*(BG) for any MU-module spectrum M. Some of that work, together with some of ours, was described in the announcement [21] and in the series of expository papers [22, 25, 26]. The last two of those pap* *ers give both equivariant and non-equivariant applications of the present theory to localizations and completions of R-modules at ideals in ss*(R). INTRODUCTION 7 We warn the knowledgeable reader that this material has undergone several ma- jor revisions, and the final definitions and terminology are not those of earli* *er an- nouncements and drafts. In particular, our S-modules enjoy a unital property th* *at was not imposed on the S-modules, here called L-spectra, of the earlier versions written by Elmendorf, Kriz, and May alone. The fact that one can impose this unital property and still retain homotopical control is one of many new insights contributed by Mandell. This substantially sharpens and simplifies the theory. Paradoxically, however, one cannot impose such a unital property in the parallel algebraic theory of [34]. Therefore, to facilitate a comparison of the algebrai* *c and topological theories, we run through a little of the previous variant of our th* *eory in the last chapter. The chapter on algebraic K-theory has not been previously announced and is entirely work of Mandell: it is part of his Chicago PhD thesis in preparation. Two other Chicago students deserve thanks. Maria Basterra has carefully read several drafts and caught numerous soft spots of exposition. Jerome Wolbert has made many helpful comments, and his Chicago PhD thesis in preparation will analyze the new derived categories associated to the various K-theory spectra. It is a pleasure to thank Mike Hopkins, Gaunce Lewis, and Jim McClure for many helpful conversations and e-mails. We owe a critical lemma, namely I.3.4, to Hopkins [31]. Although trivial to prove, it broke a psychological barrier a* *nd played a pivotal role in our thinking. We should acknowledge the pioneering work on A1 ring and module spectra of Alan Robinson, which gave precursors of many of the results of Chapter IV. We learned the material of IXx3, on thh, from the* * ap- plication of our theory given in the paper [52] of McClure, Schw"anzl, and Vogt* *. We learned many of the results on cocomplete and complete, tensored and cotensored, topological model categories in IIx7 and VIIxx2,4 from Hopkins and McClure. Their foresight and insight have been inspirational. 8 INTRODUCTION CHAPTER I Prologue: the category of L-spectra In this prologue, we construct a category whose existence was previously though* *t to be impossible by at least two of the authors: a complete and cocomplete categor* *y of spectra, namely the L-spectra, with an associative and commutative smash produc* *t. This contrasts with the category constructed by Lewis and the fourth author in * *[47, 37], whose smash product is neither associative nor commutative (before passage to homotopy categories), and with the category constructed by the first author in [19], which is neither complete nor cocomplete. We will also give a function* * L- spectrum construction that is right adjoint to the new smash product. The categ* *ory of L-spectra has all of the properties that we desire except that its smash pro* *duct, denoted by ^L , is not unital. It has a natural unit map : S ^L M -! M, which is often an isomorphism and always a weak equivalence. The curtain will rise on our real focus of interest in the next chapter, wher* *e we will define an S-module to be an L-spectrum M such that : S ^L M -! M is an isomorphism. Restricting ^L to S-modules and renaming it ^S, this will give us a symmetric monoidal category in which to develop stable topological algebra. 1. Background on spectra and the stable homotopy category We begin by recalling the basic definitions in Lewis and May's approach to the stable category. We first recall the definition of a coordinate-free spectrum;* * see [37, Ix2] or [19, x2] for further details. A coordinate-free spectrum is a spec* *trum that takes as its indexing set, instead of the integers, the set of finite dime* *nsional subspaces of a "universe", namely a real inner product space U ~= R1 . Thus, a spectrum E assigns a based space EV to each finite dimensional subspace V of U, with (adjoint) structure maps ~=W-V "oeV;W: EV -! EW 9 10 I. PROLOGUE: THE CATEGORY OF L-SPECTRA when V W . Here W - V is the orthogonal complement of V in W and W X is the space of based maps F (SW ; X), where SW is the one-point compactificat* *ion of W . These maps are required to be homeomorphisms and to satisfy an evident associativity relation. A map of spectra f : E ! E0 is a collection of maps of based spaces fV : EV ! E0V for which each of the following diagrams commutes: fV EV ________________//_E0V o"eV;W|| |"oe0V;W| fflffl|W-V fW fflffl| W-V EW __________//W-V E0W: We obtain the category S U of spectra indexed on U. If we drop the requirement that the maps "oeV;Wbe homeomorphisms, we obtain the notion of a prespectrum and the category PU of prespectra. The forgetful functor S U - ! PU has a left adjoint L, details of which are given in [37, App]. Functors on prespectra* * that do not preserve spectra are extended to spectra by applying the functor L. For example, for a based space X and a prespectrum E, we have the prespectrum E^X specified by (E ^X)(V ) = EV ^X. When E is a spectrum, the structure maps for this prespectrum level smash product are not homeomorphisms, and we understand the smash product E ^X to be the spectrum L(E ^X). For example, E = E ^S1. Function spectra are easier. We set F (X; E)(V ) = F (X; EV ) and find that th* *is functor on prespectra preserves spectra. For example, E = F (S1; E). The following result is discussed in [37, p.13]. Proposition 1.1. The category S U is complete and cocomplete. Proof. Limits and colimits are computed on prespectra spacewise. Limits preserve spectra, and colimits of spectra are obtained by use of the functor L.* * __|_ | A homotopy in the category of spectra is a map E ^ I+ - ! E0, and we have cofiber and fiber sequences that behave exactly as in the category of spaces. T* *he cofiber Cf of a map f : E - ! E0 of spectra is the pushout E0 [f CE, where CE = E ^ I. A cofibration of spectra is a map i : E - ! E0 that satisfies the homotopy extension property (HEP: a homotopy h : E ^ I+ -! F of a restriction of a map f : E0 -! F extends to a homotopy "h: E ^ I+ -! F of f). The canonical maps E -! CE and E0 -! Cf are examples. The fiber F f of a map f : E0 -! E is the pullback E0 xf P E, where P E = F (I; E). A fibration of spectra is a map p : E -! E0that satisfies the covering homotopy property (CHP: a homotopy h : F ^ I+ - ! E0 of a projection p O f, f : F - ! E, is covered by a homotopy "h: F ^ I+ -! E of f). The canonical maps P E -! E and F f -! E0 are examples. 2. EXTERNAL SMASH PRODUCTS AND TWISTED HALF-SMASH PRODUCTS 11 A map f of spectra is a weak equivalence if each of its component maps fV is a weak equivalence of spaces. The stable homotopy category hS U is constructed from the homotopy category of spectra by adjoining formal inverses to the weak equivalences, a process that is made rigorous by CW approximation. The V th space functor from spectra to spaces has a left adjoint that we shall denote by 1V, or 1n when V = R n [37, Ix4]. Its definition will be recalled in X.4.5. When V = {0}, this is the suspension spectrum functor 1 . For n 0, the sphere spectrum Sn is the suspension spectrum 1 Sn of the sphere space Sn. For n > 0, the sphere spectrum S-n is 1nS0. There are canonical isomorphisms m Sn ~= Sm+n for m 0 and integers n and there are canonical isomorphisms 1mSn ~= Sn-m for m 0 and n 0. Sphere spectra are used to define the homotopy groups of spectra, ssn(E) = hS U(Sn; E), and a map of spectra is a weak equivalence if and only if it induces an isomorphism of spectrum-level homotopy groups. Although we shall not introduce different notations for space level and spect* *rum level spheres, we shall generally write S for the zero sphere spectrum, reservi* *ng the notation S0 for the two-point space. The theory of cell and CW spectra is developed by taking sphere spectra as the domains of attaching maps [37, Ix5]. The stable homotopy category hS U is equiv- alent to the homotopy category of CW spectra. It is important to remember that homotopy-preserving functors on spectra that do not preserve weak equivalences are transported to the stable category by first replacing their variables by we* *akly equivalent CW spectra. 2. External smash products and twisted half-smash products The construction of our new smash product will start from the external smash product of spectra. This is an associative and commutative pairing S U x S U0 ! S (U U0) for any pair of universes U and U0. It is constructed by starting with the pres* *pec- trum level definition (E ^ E0)(V V 0) = EV ^ E0V 0: The structure maps fail to be homeomorphisms when E and E0 are spectra, and we apply the spectrification functor L to obtain the desired spectrum level sma* *sh product. This external smash product is the one used in [19]. There is an associated function spectrum functor F : (S U0)opx S (U U0) -! S U 12 I. PROLOGUE: THE CATEGORY OF L-SPECTRA and an adjunction S (U U0)(E ^ E0; E00) ~=S U(E; F (E0; E00)) for E 2 S U, E02 S U0, and E002 S (U U0); see [37, p. 69]. Now let I denote the category whose objects are universes U and whose mor- phisms are linear isometries. Universes are topologized as the unions of their * *finite dimensional subspaces, and the set I (U; U0) of linear isometries U ! U0 is giv* *en the function space topology; it is a contractible space [37, II.1.5]. The categ* *ory S constructed in [19] augments to the category I . Since I fails to have limits a* *nd colimits (it even fails to have coproducts), S suffers from the same defects. In order to obtain smash products internal to a single universe U, we shall e* *xploit the "twisted half-smash product". The input data for this functor consist of two universes U and U0 (which may be the same), an unbased space A with a given structure map ff : A ! I (U; U0), and a spectrum E indexed on U. The output is the spectrum A n E, which is indexed on U0. It must be remembered that the construction depends on ff and not just on A, although different choices of ff lead to equivalent functors on the level of stable categories [37, VI.1.14].* * The intuition is that the twisted half-smash product is a generalization to spectra* * of the "untwisted" functor A+ ^ X on based spaces X. This intuition is made precise by the following "untwisting formula" relating twisted half-smash products and shi* *ft desuspensions. It is a substantial technical strengthening of results in [37] a* *nd will be proven in XIx4. Proposition 2.1. For any map A -! I (U; U0) and any n 0, there is an isomorphism of spectra A n 1nX ~=1n(A+ ^ X) that is natural in spaces A over I (U; U0) and based spaces X. Observe that the functor 1nimplicitly refers to the universe U on the left an* *d to the universe U0 on the right. The twisted-half smash product enjoys the followi* *ng formal properties, among others; see [37, VI.3.1 and VI.1.5] or [19, 3.18 and 5* *.1]. Proposition 2.2. The following statements hold. (i)There is a canonical isomorphism {idU} n E ~=E. (ii)Let A ! I (U; U0) and B ! I (U0; U00) be given; let B x A have the structure map given by the composite O 00 B x A ____//_I (U0; U00) x I (U;_U0)//_I (U; U ): Then there is a canonical isomorphism (B x A) n E ~=B n (A n E): 2. EXTERNAL SMASH PRODUCTS AND TWISTED HALF-SMASH PRODUCTS 13 (iii)Let A ! I (U1; U01) and B ! I (U2; U02) be given; let A x B have the structure map given by the composite 0 0 A x B _____//I (U1; U01) x I (U2;_U02)//_I (U1 U2; U1 U2): Let E1 and E2 be spectra indexed on U1 and U2 respectively. Then there is a canonical isomorphism (A x B) n (E1 ^ E2) ~=(A n E1) ^ (B n E2): (iv)For A ! I (U; U0), E 2 S U, and a based space X, there is a canonical isomorphism A n (E ^ X) ~=(A n E) ^ X: The functor A n (?) is a left adjoint [37, VI.1.1], and its right adjoint wil* *l be used in our construction of function S-modules. Proposition 2.3. For any space A over I (U; U0), the functor A n (?) has a right adjoint, which is denoted by F [A; ?) and called a twisted function spect* *rum. The functor A n E is homotopy-preserving in E, and it therefore preserves homotopy equivalences in the variable E. However, it only preserves homotopies over I (U; U0) in A. Nevertheless, it very often preserves homotopy equivalence* *s in the variable A. This fact will be essential in keeping control over the homotop* *ical behavior of our point-set level constructions. To state it in proper generality* *, we need the following notion of a well-behaved spectrum. Definition 2.4. A prespectrum D is -cofibrant if each of its structure maps oe : W DV -! D(V W ) is a cofibration of based spaces. A spectrum E is -cofibrant if it is isomorphic to one of the form LD, where D is a -cofibrant prespectrum. A spectrum E is tame if it is homotopy equivalent to a -cofibrant spectrum. We shall discuss such spectra in Xx4, where we shall see that all shift desus- pensions of based spaces are -cofibrant and that all CW spectra are tame. We shall show in Xx5 that structured ring or module spectra can be approximated functorially by weakly equivalent -cofibrant spectra with the same structure. Theorem 2.5. Let E 2 S U be tame and let A be a space over I (U; U0). If OE : A0 -! A is a homotopy equivalence, then OE n id : A0n E - ! A n E is a homotopy equivalence. 14 I. PROLOGUE: THE CATEGORY OF L-SPECTRA Proof. We may assume without loss of generality that E = LD for a - cofibrant prespectrum D. By the untwisting result in Proposition 2.1, the con- clusion certainly holds when E = 1nX for a based space X. By [37, I.4.7] (see X.4.4 below), LD ~= colim1nDn, where the colimit is taken over a sequence of cofibrations of spectra. Since the functor A n (?) has a right adjoint, it comm* *utes with colimits, and it also preserves cofibrations. The conclusion follows since* * the colimit of a sequence of homotopy equivalences is a homotopy equivalence when the source and target colimits are taken over sequences of cofibrations. __|_ | Corollary 2.6. Let E 2 S U be a spectrum that has the homotopy type of a CW spectrum and let A be a space over I (U; U0) that has the homotopy type of a CW complex. Then A n E has the homotopy type of a CW spectrum. Proof. We may assume without loss of generality that E is a CW spectrum and A is a CW complex, in which case A n E is a CW spectrum by [37, VI.1.11]. __|_* * | 3. The linear isometries operad and internal smash products For the rest of the paper, we restrict attention to a particular universe U; * *the reader is welcome to consider it as notation for R1 . We agree to write S inste* *ad of S U for the category of spectra indexed on U. Except where explicitly stated otherwise, all given spectra, whatever extra structure they may have, will be i* *n S . We are especially interested in twisted half smash products defined in terms of* * the following spaces of linear isometries. Notations 3.1. Let Uj be the direct sum of j copies of U and let L (j) = I (Uj; U). The space L (0) is the point i, where i : {0} ! U, and L (1) contains the identity map 1 = idU : U ! U. The left action of j on Uj by permutations induces a free right action of j on the contractible space L (j). Define maps fl : L (k) x L (j1) x . .x.L (jk) -! L (j1 + . .+.jk) by fl(g; f1; : :;:fk) = g O (f1 . . .fk): The spaces L (j) form an operad [44, p.1] with structural maps fl, called the linear isometries operad. Points f 2 L (j) give inclusions {f} -! L (j). The corresponding twisted half-smash product is denoted f*; it sends spectra indexe* *d on Uj to spectra indexed on U. Applied to a j-fold external smash product E1^. .^.* *Ej, it gives an internal smash product f*(E1 ^ . .^.Ej). All of these smash products become equivalent in the stable homotopy category hS , but none of them are associative or commutative on the point set level. In fact, the following shar* *per version of this assertion holds. 3. THE LINEAR ISOMETRIES OPERAD AND INTERNAL SMASH PRODUCTS 15 Theorem 3.2. Let St S be the full subcategory of tame spectra and let hSt be its homotopy category. The internal smash products f*(E ^ E0) determined by varying f 2 L (2) are canonically isomorphic in hSt, and hSt is symmetric monoidal under the internal smash product. For based spaces X and tame spectra E, there is a natural isomorphism E ^ X ' f*(E ^ 1 X) in hSt. Proof. The external and internal smash products of -cofibrant spectra are -cofibrant by results in Xx4. By Theorem 2.5, for any f 2 L (j) and any spectra Ei2 St, the map f*(E1 ^ . .^.Ej) -! L (j) n (E1 ^ . .^.Ej) induced by the inclusion {f} -! L (j) is a homotopy equivalence. Taking j = 2, this shows that the internal smash products obtained from varying f are homotopy equivalent. Replacing f by f O oe, where oe 2 2 is the transposition, we obtain* * a natural homotopy equivalence f*(E2 ^ E1) -! L (2) n E1 ^ E2; and this shows that the internal smash product is commutative up to homotopy. Similarly, for associativity, the inclusions of the points {f(1 f)} and {f(f * * 1)} in L (3) induce natural homotopy equivalences f*(E1 ^ f*(E2 ^ E3)) -! L (3) n (E1 ^ E2 ^ E3) - f*(f*(E1 ^ E2) ^ E3): It is natural to think of based spaces as spectra indexed on the universe {0}. * *Then i* and the suspension spectrum functor are both left adjoint to the zeroth space functor, hence i*X ~=1 X. The map L (2) -! L (1) that sends f to f O (1 i) and the inclusion {1} -! L (1) induce natural homotopy equivalences f*(E ^ 1 X) -! L (1) n (E ^ X) - E ^ X: Thus, up to natural isomorphisms, the internal smash product determined by f becomes commutative, associative, and unital with unit S = 1 S0 on passage to hSt. The commutativity of coherence diagrams that is required for the assertion that hStis symmetric monoidal (see [42, p. 180]) can be checked by an elaborati* *on of these arguments. __|_ | The following consequence strengthens the assertion [37, I.6.1] that the stab* *le homotopy category really is a stable category, in the sense that the suspension* * and loop functors and pass to inverse self-equivalences of hS . Corollary 3.3. For tame spectra E and f 2 L (2), there is a natural homo- topy equivalence between E and f*(E ^ S-1), and the unit j : E -! E and counit " : E -! E of the (; )-adjunction are homotopy equivalences. 16 I. PROLOGUE: THE CATEGORY OF L-SPECTRA Proof. For based spaces X, 1 X is naturally isomorphic to (11X) since the structural homeomorphism E0 -! E1 gives a natural isomorphism between their right adjoints. Therefore, for E 2 St, there is a natural homotopy equiva* *lence E = E ^ S0 ' f*(E ^ 1 S0) ~=f*(E ^ (11S0)) ~=(f*(E ^ S-1)); where the last isomorphism is given by Proposition 2.2(iv). It follows that, on hSt, the functor is an adjoint equivalence with inverse given by the functor f*(E ^ S-1). The rest is a formal consequence of the uniqueness of adjoints. _* *_|_ | Note that only actual homotopy equivalences, not weak ones, are relevant to these results. For this and other reasons, hSt will be a technically convenient halfway house between hS and the stable homotopy category hS , which is ob- tained from either of these homotopy categories by inverting the weak equivalen* *ces. We can deduce that cofiber sequences give rise to long exact sequences of ho- motopy groups. f 0 Corollary 3.4. Any cofiber sequence E- ! E -! Cf of tame spectra gives rise to a long exact sequence of homotopy groups . .-.! ssq(E) -! ssq(E0) -! ssq(Cf) -! ssq-1(E) -! . .:. Therefore the natural map F f -! Cf is a weak equivalence. Proof. Consider the diagram __id_ q____//q ____//_q __id_//q SqO //S CSO S O S -1fl O |ff| OfiO flO |ff| fflfflOfflffl|fifflffl fflfflOf fflffl| E _____//E0____//Cf_____//_E _____//E0: Here ff is given such that i O ff ' 0. A homotopy induces a map fi such that the second square commutes. The usual cofiber sequence argument gives fl such that the right two squares homotopy commute. Since jE : E -! E is a weak equivalence, there is a map -1fl : Sq -! E, unique up to homotopy, such that jE O -1fl ' fl O jSq: Therefore jE0 O f O -1fl = f O jE O -1fl ' f O fl O jSq ' ff O jSq = jE0 O ff: Since jE0 is a weak equivalence, this implies that f O -1fl ' ff. The long exact sequence follows by extending the given cofiber sequence to the right, as usual. The last statement follows by the five lemma and a comparison of our cofiber sequence with the fiber sequence associated to f. Details of this may be found * *in 3. THE LINEAR ISOMETRIES OPERAD AND INTERNAL SMASH PRODUCTS 17 [37, pp 128-130]. For later use, observe that we only used that the maps j are * *weak equivalences, not that they are homotopy equivalences, in this proof. __|_ | It follows that cofiber sequences are essentially equivalent to fiber sequenc* *es. More precisely, the cofibrations and fibrations give "triangulations" of the st* *able homotopy category such that the negative of a cofibration triangle is a fibrati* *on triangle, and conversely [37, pp 128-130]. Corollary 3.5. Pushouts of tame spectra along cofibrations preserve weak eq* *uiv- alences. That is, for a commutative diagram of tame spectra __i__ __f__// E oo D F fi|| ff|| |fl| fflffl| fflffl||fflffl E0ooi0_D0 __f0_//F 0 in which i and i0are cofibrations and ff, fi, and fl are weak equivalences, the* * induced map ffi : E [D F -! E0[D0 F 0of pushouts is a weak equivalence. Proof. As for spaces, Ci is homotopy equivalent to E=D, the induced map F - ! E [D F is a cofibration, and the induced map E=D -! E [D F=F is an isomorphism. The conclusion follows from the previous corollary by a diagram chase and the five lemma. __|_ | Proposition 3.6. If E is a CW spectrum and OE : F - ! F 0is a weak equiva- lence between tame spectra, then f*(id^OE) : f*(E ^ F ) -! f*(E ^ F 0) is a weak equivalence. Proof. The functor f*((?) ^ F ) preserves cofiber sequences. Therefore, by Corollary 3.5 and induction up the sequential filtration of E (see III.2.1), th* *e result will hold for general E if it holds for E = Sn. When E = S, the conclusion holds by the unit equivalence f*(S ^F ) ' F of Theorem 3.2. For n > 0, we easily dedu* *ce isomorphisms f*(Sn ^ F ) ~=nf*(S ^ F ) and nf*(S-n ^ F ) ~=f*(S ^ F ) from Proposition 2.2(iv). In view of Corollary 3.3, the result for E = S-n and E = Sn therefore follows from the result for S. __|_ | It follows that for general spectra E and tame spectra F , the smash product E ^ F in the stable homotopy category hS is represented by f*(E ^ F ), where E is a CW spectrum weakly equivalent to E. That is, we do not also have to apply CW approximation to F . The mild restriction to tame spectra serves to avoid pathological point-set behavior. 18 I. PROLOGUE: THE CATEGORY OF L-SPECTRA 4. The category of L-spectra We think of L (j)n(E1^. .^.Ej) as a canonical j-fold internal smash product. * *It is still not associative, but we shall construct a commutative and associative * *smash product by restricting to L-spectra and shrinking the fat out of the constructi* *on. To define L-spectra, we focus attention on a small part of the operad L . Recall the notion of a monad in a category from [42, ch.VI] or [44, 2.1]. Notations 4.1. Let L denote the monad in the category S that is specified by LE = L (1) n E; the product : LLE ~=(L (1) x L (1)) n E -! L (1) n E = LE is induced by the product fl : L (1) x L (1) -! L (1) and the unit j : E ~={1} n E -! L (1) n E = LE is induced by the inclusion {1} -! L (1) of the identity element. Definition 4.2. An L-spectrum is an L-algebra M, that is, a spectrum M together with an action : LM -! M by the monad L. Explicitly, the following diagrams are required to commute: j LLM _____//LM and M E___//_LM EE | L || || =EEEE | fflffl| fflffl| E""fflffl| LM _____//_M M: A map f : M ! N of L-spectra is a map of spectra such that the following diagram commutes: Lf LM ____//_LN M || |N| fflffl| fflffl| M __f__//_N: We let S [L] denote the category of L-spectra. There is a dual form of the definition that will occasionally be needed. It * *is based on the following standard categorical observation. Lemma 4.3. Let T be a monad in a category C , and suppose that the functor T has a right adjoint T# . Then T# is a comonad such that the categories of T-algebras and of T# -coalgebras are isomorphic. 4. THE CATEGORY OF L-SPECTRA 19 We shall consistently use the notation T# for the comonad associated to a mon* *ad T that has a right adjoint. In particular, by Proposition 2.3, we now have a comonad L# such that an L# -coalgebra is the same thing as an L-spectrum. This implies the following result. Proposition 4.4. The category of L-spectra is complete and cocomplete, with both limits and colimits created in the underlying category S . If X is a based space and M is an L-spectrum, then M ^ X and F (X; M) are L-spectra, and the spectrum level fiber and cofiber of a map of L-spectra are L-spectra. Proof. Since S [L] is the category of algebras over the monad L, the forgetf* *ul functor S [L] ! S creates limits [42, VI.2, ex. 2]. Since S is complete, this implies the statement about limits. The statement about colimits follows simila* *rly by use of the comonad L# . The last statement is immediate from the canonical isomorphism L (1) n (M ^ X) ~=(L (1) n M) ^ X of Proposition 2.2(iv) and its analog [37, VI.1.5] F [L (1); F (X; M)) ~=F (X; F [L (1); M)): __|_ | Lemma 4.5. The sphere spectrum S is an L-spectrum. More generally, for based spaces X, 1 X ~=S ^ X is naturally an L-spectrum. Proof. Recall from the proof of Theorem 3.2 that a based space X may be viewed as a spectrum indexed on {0} and that 1 X ~= i*X, i : {0} -! U. We may rewrite this as 1 X = L (0) n X. Then the structure map is given by fl n id: L (1) n (L (0) n X) ~=(L (1) x L (0)) n X -! L (0) n X: In the middle, L (1) x L (0) is regarded as a space over L (0) via fl, and the isomorphism is given by an instance of Proposition 2.2(ii). Of course, fl here* * is just the unique map from L (1) to the one-point space L (0), and our structure map is just the composite L (1) n 1 X ~=1 (L (1)+ ^ X) -! 1 (S0 ^ X) ~=1 X; where the first isomorphism is given by Proposition 2.1. __|_ | Warning 4.6. We issue a technical warning: it is neither necessary nor usef* *ul to consider possible L-spectrum structures on the shift desuspensions 1nX for n > 0. Any spectrum E is isomorphic to the colimit of the shift desuspensions 1nEn of its component spaces [22, I.4.7], and it is easy to construct a fallaci* *ous proof that every spectrum is an L-spectrum, the fallacy being that one cannot g* *ive shift desuspensions L-spectra structures that make the maps of the colimit syst* *em maps of L-spectra. 20 I. PROLOGUE: THE CATEGORY OF L-SPECTRA A homotopy in the category of L-spectra is a map M ^ I+ - ! N. A map of L-spectra is a weak equivalence if it is a weak equivalence as a map of spectra* *. The stable homotopy category hS [L] of L-spectra is constructed from the homotopy category hS [L] by adjoining formal inverses to the weak equivalences; again, t* *he process is made rigorous by CW approximation. Since the theory of cell and CW L-spectra is exactly like the theory of cell and CW spectra developed in [37, I* *x5], we shall not give details. The reader who would like to see an exposition is in* *vited to look ahead to IIIx2. The theory of cell R-modules to be presented there appl* *ies (with minor simplifications) to give what is needed. It is formal that the mon* *ad L may be viewed as specifying the free functor from spectra to L-spectra. The sphere L-spectra that we take as the domains of attaching maps when defining ce* *ll L-spectra are the free L-spectra LSn = L (1) n Sn. A weak equivalence of cell L- spectra is a homotopy equivalence, any L-spectrum is weakly equivalent to a CW L-spectrum, and hS [L] is equivalent to the homotopy category of CW L-spectra. We warn the reader that, although S itself is an L-spectrum, it does not have the homotopy type of a CW L-spectrum (see Warning 6.8 below). The following comparison between CW spectra and CW L-spectra establishes an equivalence between hS and hS [L]. Theorem 4.7. The following conclusions hold. (i)The free functor L : S - ! S [L] carries CW spectra to CW L-spectra. (ii)The forgetful functor S [L] -! S carries L-spectra of the homotopy types of CW L-spectra to spectra of the homotopy types of CW spectra. (iii)Every CW L-spectrum M is homotopy equivalent as an L-spectrum to LE for some CW spectrum E. (iv)The unit j : E -! LE of the adjunction S [L](LE; M) ~=S (E; M) is a homotopy equivalence if E 2 St, for example if E is a CW spectrum. (v)The counit : LM -! M of the adjunction is a homotopy equivalence of spectra if M is tame and is a homotopy equivalence of L-spectra if M has the homotopy type of a CW L-spectrum. The free and forgetful functors establish an adjoint equivalence between the st* *able homotopy categories hS and hS [L]. Proof. Part (i) is immediate by induction up the sequential filtration (see III.2.1). Part (iv) is immediate from Theorem 2.5 and, applied to sphere spec- tra, it implies (ii). Since O j = id: M -! M for any M, (iv) and the Whitehead theorem in the category of L-spectra imply (v). Part (iii) follows from (i) and* * (v) since there is a CW spectrum E and a homotopy equivalence of spectra E -! M. 5. THE SMASH PRODUCT OF L-SPECTRA 21 It is a formal consequence of (i) that we have an induced adjunction hS [L](LE; M) ~=hS (E; M) (see [37, I.5.13]), and its unit and counit are natural isomorphisms. __|_ | Observe that, dually, we can interpret L# as specifying the "cofree" functor * *from spectra to L-spectra. That is, we have an adjunction (4.8) S [L](M; L# E) ~=S (M; E): By part (ii) of the theorem and [37, I.5.13], there results an induced adjuncti* *on hS [L](M; L# E) ~=hS (M; E): It is an easy categorical observation that, in any adjoint equivalence of categ* *ories, the given left and right adjoints are also right and left adjoint to each other. Corollary 4.9. The functors L : hS - ! hS [L] and L# : hS - ! hS [L] are naturally isomorphic. 5. The smash product of L-spectra Via instances of the structural maps fl of the operad L , we have a left acti* *on of the monoid L (1) and a right action of the monoid L (1) x L (1) on L (2). These actions commute with each other. If M and N are L-spectra, then L (1) x L (1) acts from the left on the external smash product M ^ N via the map ^ : (L (1) x L (1)) n (M ^ N) ~=(L (1) n M) ^ (L (1) n N) ____//_M ^ N: To form the twisted half smash product on the left, we think of L (1) x L (1) as mapping to I (U2; U2) via direct sum of linear isometries. The smash product ov* *er L of M and N is simply the balanced product of the two L (1) x L (1)-actions. Definition 5.1. Let M and N be L-spectra. Define the operadic smash product M ^L N to be the coequalizer displayed in the diagram flnid_ (L (2) x L (1) x L (1)) n (M ^ N) ____////_L (2) n (M ^_N)__//M ^L N: idn Here we have implicitly used the isomorphism (L (2) x L (1) x L (1)) n (M ^ N) ~=L (2) n [(L (1) x L (1)) n (M ^ N)] given by Proposition 2.2(ii). The left action of L (1) on L (2) induces a left * *action of L (1) on M ^L N that gives it a structure of L-spectrum. 22 I. PROLOGUE: THE CATEGORY OF L-SPECTRA We may mimic tensor product notation and write M ^L N = L (2) nL (1)xL (1)(M ^ N): We will freely use such notations for coequalizers below. The commutativity of this smash product is immediate. Proposition 5.2. There is a natural commutativity isomorphism of L-spectra o : M ^L N -! N ^L M: Proof. The permutation oe 2 2 acts on L (2) by foe = f Ot, where t : U2 ! U2 is the transposition isomorphism. We may regard oe as a map of spaces over L (2) from id: L (2) -! L (2) to oe : L (2) -! L (2). We have an evident isomorphism : t*(M ^ N) ~=N ^ M on external smash products and, by Proposition 2.2(ii), there results a canonical isomorphism oe n : L (2) n M ^ N ~=L (2) n t*(M ^ N) ~=L (2) n N ^ M: There is an analogous isomorphism (oext)n : (L (2)xL (1)xL (1))n(M ^N) -! (L (2)xL (1)xL (1))n(N ^M): These maps induce an isomorphism of coequalizer diagrams flnid_ (L (2) x L (1) x L (1)) n (M ^ N) ____////_L (2) n (M ^_N)__//M ^L N idn (oext)n|| oen|| o|| fflffl| flnid fflffl| fflffl| (L (2) x L (1) x L (1)) n (N ^ M) ____//_//_L (2) n (N ^_M)_//N ^L M: idn __ |_ | To show that this smash product is associative, we need some preliminary ma- terial on coequalizers. We first recall a standard categorical definition [42, * *VI.6]. Definition 5.3. Working in an arbitrary category, suppose given a diagram __e__ g A ____////_B_//_C f in which ge = gf. The diagram is called a split coequalizer if there are maps h : C ! B and k : B ! A such that gh = idC, fk = idB, and ek = hg. It follows that g is the coequalizer* * of e and f. 5. THE SMASH PRODUCT OF L-SPECTRA 23 Observe that, while covariant functors need not preserve coequalizers in gene* *ral, they clearly do preserve split coequalizers. The next observation is crucial; * *we learned it from Hopkins [31]. Note that, via structural maps fl, L (1) acts fr* *om the left on any L (i), hence L (1) x L (1) acts from the left on L (i) x L (j). Lemma 5.4 (Hopkins). For i 1 and j 1, the diagram flxid_ fl L (2) x L (1) x L (1) x L (i) x L (j)___////L (2) x L (i) x L_(j)//_L (i + j) idxfl2 is a split coequalizer of spaces. Therefore, L (i + j) ~=L (2) xL (1)xL (1)L (i) x L (j): Proof. Choose isomorphisms s : Ui ! U and t : Uj ! U and define h(f) = (f O (s t)-1; s; t) and k(f; g; g0) = (f; g O s-1; g0O t-1; s; t): It is trivial to check the identities of Definition 5.3. __|_ | Theorem 5.5. There is a natural associativity isomorphism of L-spectra (M ^L N) ^L P ~=M ^L (N ^L P ): Proof. Note that, for any L-spectrum N, N ~=L (1)nL (1)N since L (1)nN = LN and, as with any monad [42, p. 148], we have a split coequalizer LLN ____//_//_LN_//_N: We have the isomorphisms (M ^L N) ^L P ~=L (2) nL (1)2(L (2) nL (1)2(M ^ N)) ^ (L (1) nL (1)P ) ~=(L (2) xL (1)2L (2) x L (1)) nL (1)3(M ^ N ^ P ) ~=L (3) nL (1)3M ^ N ^ P: The symmetric argument shows that this is also isomorphic to M ^L (N ^L P ). _* *_|_ | In view of the generality of Lemma 5.4, the argument iterates to prove the following statement. Theorem 5.6. For any j-tuple M1; : :;:Mj of L-spectra, there is a canonical isomorphism of L-spectra M1 ^L . .^.L Mj ~=L (j) nL (1)j(M1 ^ . .^.Mj); where the iterated smash product on the left is associated in any fashion. 24 I. PROLOGUE: THE CATEGORY OF L-SPECTRA 6. The equivalence of the old and new smash products We here show that the smash product ^L does in fact realize the classical sm* *ash product of spectra up to homotopy, in the sense that the equivalence between hS and hS [L] preserves smash products. Fix a linear isometric isomorphism f : U2 - ! U (not just an isometry) and use it to define the internal smash product of spectra in this section. We begi* *n the comparison of smash products of L-spectra with smash products of spectra with the following observation. Proposition 6.1. For spectra X and Y , there are isomorphisms of L-spectra LX ^L LY ~= L (2) n X ^ Y ~= Lf*(X ^ Y ): For CW L-spectra M and N, M ^L N is a CW L-spectrum with one (p + q)-cell for each p-cell of M and q-cell of N. Proof. The first isomorphism is immediate from the definition of ^L . Regard- ing f as a point in L (2), we see that fl : L (1)x{f} - ! L (2) is a homeomorph* *ism since f is an isomorphism. It follows from Proposition 2.2(ii) that Lf*(X ^ Y ) = L (1) n f*(X ^ Y ) ~=L (2) n (X ^ Y ): When X and Y are sphere spectra, so is f*(X ^ Y ) [37, II.1.4]. The second statement now follows exactly as for the smash product of CW complexes or CW spectra. __|_ | The crux of our comparison of smash products is the following proposition, wh* *ich implies that LS is the unit for the smash product in the stable homotopy catego* *ry hS [L]. We defer the proof to XIx3. Proposition 6.2. For L-spectra N, there is a natural weak equivalence of L- spectra ! : LS ^L N -! N, and : ssn(N) -! ssn+1(N) is an isomorphism for all integers n. If we knew a priori that preserved weak equivalences, we could derive the second clause from the first and the natural isomorphism of L-spectra LS ^L N ~=(LS-1 ^L N) by a formal uniqueness of adjoints argument (compare Corollary 3.3). It is a pl* *eas- ant and surprising technical feature of our theory, immediate from the proposit* *ion, that preserves weak equivalences of L-spectra. That is, the L structure some- how has the effect of eliminating point-set pathology. Since on homotopy groups is induced by j : N - ! N, the proposition also has the following immediate consequence. 6. THE EQUIVALENCE OF THE OLD AND NEW SMASH PRODUCTS 25 Corollary 6.3. For L-spectra N, the unit j : N - ! N and counit " : N -! N of the (; )-adjunction are weak equivalences. f 0 Corollary 6.4. Any cofiber sequence N- ! N -! Cf of L-spectra gives rise to a long exact sequence of homotopy groups . .-.! ssq(N) -! ssq(N0) -! ssq(Cf) -! ssq-1(N) -! . .:. Therefore the natural map F f -! Cf is a weak equivalence of L-spectra. Proof. This follows from Corollary 6.3 via the proof of Corollary 3.4. __|_* * | Corollary 6.5. Pushouts along cofibrations of L-spectra preserve weak equiv- alences. Proof. Since a cofibration of L-spectra is a cofibration of spectra, by the * *re- traction of mapping cylinders criterion, this follows from Corollary 6.4 via th* *e proof of Corollary 3.5. __|_ | Proposition 6.6. If M is a CW L-spectrum and OE : N -! N0 is a weak equivalence of L-spectra, then id^L OE : M ^L N -! M ^L N0 is a weak equivalence of L-spectra. Proof. The functor (?) ^L N preserves cofiber sequences, hence the result for general M follows from Corollary 6.4 and the result for M = LSn. Here the result for n = 0 follows from Proposition 6.2 and the result for n and -n, n > 0, foll* *ows from the result for n = 0 as in the proof of Proposition 3.6. __|_ | Thus, for L-spectra M and N, the smash product M ^L N in the stable homotopy category hS [L] is represented by M ^L N, where M is a CW L-spectrum weakly equivalent to M; here we do not need to assume that N is tame. This is analogous to the situation in algebra. When transporting tensor products to algebraic derived categories, we need only apply cell approximation to one of t* *he tensor factors, without any condition on the other [34]. Theorem 6.7. For L-spectra M and N, there is a natural map of spectra ff : f*(M ^N) -! M ^L N, and ff is a weak equivalence when M is a CW L-spectrum and N is a tame spectrum. For any L-spectrum N, the functor (?) ^L N from hS [L] to hS computes the derived internal smash product with N. Proof. Define ff to be the composite f*(M ^ N) -! L (2) n M ^ N -! M ^L N given by the the inclusion of {f} in L (2) and the definition of ^L . Let M be a CW L-spectrum throughout the proof. We first show that ff is an equivalence when N is also a CW L-spectrum. In this case, M and N have the homotopy types of 26 I. PROLOGUE: THE CATEGORY OF L-SPECTRA CW spectra by Theorem 4.7 and are therefore tame by X.4.3. Thus the first map is a homotopy equivalence by Theorem 2.5. By Theorem 4.7(iii), we may assume without loss of generality that M = L (1) n X and N = L (1) n Y for CW spectra X and Y . The second arrow then reduces to the homotopy equivalence L (2) n (L (1) n X) ^ (L (1) n Y ) -! L (2) n X ^ Y induced by the homotopy equivalence fl : L (2) x L (1) x L (1) - ! L (2) via Theorem 2.5. For a general L-spectrum N, choose a weak equivalence fl : N -! N, where N is a CW L-spectrum. If N is tame, then Propositions 3.6 and 6.6 imply that the vertical arrows are weak equivalences in the commutative diagram f*(M ^ N) __ff_//M ^L N id^fl|| |id^fl| fflffl| fflffl| f*(M ^ N) __ff_//_M ^L N: Thus the bottom arrow ff is a weak equivalence since the top one is. For the la* *st statement, simply note that the right-hand composite (id^fl) O ff : f*(M ^ N) -! M ^L N in the diagram is a weak equivalence even when N is not tame. __|_ | Warning 6.8. As said before, the sphere L-spectrum S does not have the ho- motopy type of a CW L-spectrum. To see this, assume that it did. Then the action : LS - ! S would be a homotopy equivalence of L-spectra, by the Whitehead theorem, and the 2-equivariant map ^L : LS ^L LS -! S ^L S would be a homotopy equivalence of L-spectra and thus of spectra. By Propositio* *ns 6.1 and 2.1, LS ^L LS is isomorphic to 1 (L (2)+ ), with 2-action induced by that on L (2). By Proposition 8.2 below, S ^L S is isomorphic to S = 1 S0 and has trivial action by 2. Under these isomorphisms, ^L coincides with 1 ss, where ss : L (2)+ -! S0 sends all of L (2) to the non-basepoint. Since L (2)=2 ' B(2), our assumption implies that we obtain a homotopy equivalence 1 B(2)+ - ! 1 S0 on passage to orbits from ^L , which is absurd. This argument also shows that our hypothesis that M be a CW L-spectrum is crucial in the previous two results. 7. FUNCTION L-SPECTRA 27 7. Function L-spectra We here construct a functor FL on L-spectra that is related to the smash pro* *duct ^L by an adjunction of the usual form and consider its homotopical behavior. Theorem 7.1. Let M, N, and P be L-spectra. There is a function L-spectrum functor FL (M; N), contravariant in M and covariant in N, such that S [L](M ^L N; P ) ~=S [L](M; FL (N; P )): Given the adjunction, we can deduce the homotopical behavior of FL from that of ^L . We run through this before turning to the construction. The following result is a formal consequence of Proposition 6.1; see [37, I.5.13]. Proposition 7.2. If M is a CW L-spectrum and OE : N ! N0 is a weak equiv- alence of L-spectra, then FL (id; OE) : FL (M; N) -! FL (M; N0) is a weak equivalence of L-spectra. There is an induced adjunction h S [L](M ^L N; P ) ~=hS [L](M; FL (N; P )): As in Section 6, we fix a linear isometric isomorphism f : U2 - ! U and use the isomorphism f* : S U2 -! S U to define internal smash products f*(M ^ N). Recall the external function spectrum F (M; ?) and the adjunction displayed for* * it at the start of Section 2. We use the inverse isomorphism f* = f-1*: S U -! S U2 to define internal function spectra F (M; f*N), as in [37, II.3.11]. Theorem 7.3. For L-spectra M and N, there is a natural map of spectra "ff: FL (M; N) -! F (M; f*N); and "ffis a weak equivalence if M is a CW L-spectrum. Therefore the equivalence* * of categories hS [L] -! hS induced by the forgetful functor from L-spectra to spec- tra carries the function L-spectrum functor FL to the internal function spectr* *um functor F . Proof. In the category hS [L], FL (M; N) means FL (M; N) where M is a CW L-spectrum weakly equivalent to M, hence the second statement will follow from the first. The desired map "ffis the adjoint of the composite f*(FL (M; N) ^ M)- ff!FL (M; N) ^L M- "!N; where ff is given by Theorem 6.7. By that result, if M is a CW L-spectrum and X is a CW spectrum, then ff : f*(LX ^ M) -! LX ^L M is a weak equivalence of spectra, and it induces a weak equivalence of L-spectra L(f*(LX ^ M)) -! LX ^L M: 28 I. PROLOGUE: THE CATEGORY OF L-SPECTRA Diagram chases show that the map "ff*: hS (X; FL (M; N)) -! hS (X; F (M; f*N)) coincides with the composite of the following chain of natural isomorphisms: hS (X; FL (M; N)) ~=hS [L](LX; FL (M; N)) ~=hS [L](LX ^L M; N) ~=hS [L](L(f*(LX ^ M)); N) ~=hS (f*(LX ^ M); N) ~=hS (LX ^ M; f*N) ~=hS (LX; F (M; f*N)) ~=hS (X; F (M; f*N)): __|_ | Lemma 7.4. The adjoint N - ! FL (LS; N) of the unit weak equivalence ! : LS ^L N -! N is a weak equivalence. Proof. This is immediate from the natural isomorphisms hS [L](M; N) ~=hS [L](LS ^L M; N) ~=hS [L](M; FL (LS; N)): __|_ | We must still prove Theorem 7.1. The desired adjunction dictates the definiti* *on of FL , and the reader is invited to skip to the next section. It will be simpl* *est to construct FL in two steps. Remember that M ^L N = L (2) nL (1)xL (1)M ^ N: In the first step we consider general spectra indexed on U2 and acted upon by L (1) x L (1), thought of as a space over I (U2; U2) via direct sum of isometri* *es. We call these L (1) x L (1)-spectra and denote the category of such spectra by S [L (1) x L (1)]. Of course, the examples we have in mind are of the form M ^ N. We use the twisted function spectrum construction F [A; ?) of Proposition 2.3. Lemma 7.5. Let N be an L-spectrum. There is an L (1) x L (1)-spectrum FL (1)[L (2); N) 2 S (U2) such that S [L](L (2) nL (1)xL (1)P; N) ~=S [L (1) x L (1)](P; FL (1)[L (2); N)) for L (1) x L (1)-spectra P . Proof. We construct FL (1)[L (2); N) as the equalizer of two maps F [L (2); N) F [L (1) x L (2); N): The first is induced by fl : L (1) x L (2) ! L (2). The second is the composite F[1;) F [L (2); N) -! F [L (1) x L (2); L (1) n N) ---! F [L (1) x L (2); N); here the unlabelled arrow is adjoint to (L (1) x L (2)) n F [L (2); N) ~=L (1) n L (2) n F [L (2); N) idn"--!L (1) n N; 7. FUNCTION L-SPECTRA 29 where " is the counit of the adjunction. The left action of L (1) x L (1) on FL (1)[L (2); N) is induced by its right action on L (2). __|_ | The second step lands us back in the category of L-spectra. Lemma 7.6. Let N be an L-spectrum and P be an L (1) x L (1)-spectrum. There is an L-spectrum ^F(N; P ) such that S [L (1) x L (1)](M ^ N; P ) ~=S [L](M; ^F(N; P )) for L-spectra M. Proof. Again, we construct ^F(N; P ) as an equalizer, this time of two maps F (N; P ) F (LN; P ): The first is induced by the structure map LN -! N. The second is the composite F (N; P ) -! F (LN; ({1} x L (1)) n P ) -! F (LN; P ); where the second arrow is induced by the structure map of P as an L (1) x L (1)- module and the first arrow is adjoint to F (N; P ) ^ LN ~=({1} x L (1)) n F (N; P ) ^ N -idn"-!({1} x L (1)) n P: The structure of ^F(N; P ) as an L-spectrum is induced by the action on P of th* *e first factor of L (1) in L (1) x L (1); more precisely, the action LF (N; P ) ! F (N;* * P ) is adjoint to the composite __ (LF (N; P ))^N ~=(L (1)x{1})n(F (N; P )^N) idn"--!(L (1)x{1})nP -! P: |_ | We combine these two functorial constructions to define FL . Definition 7.7. For L-spectra M and N, define FL (M; N) = ^F(M; FL (1)[L (2); N)): The adjunction of Theorem 7.1 is just the composite of the two adjunctions already obtained. 30 I. PROLOGUE: THE CATEGORY OF L-SPECTRA 8. Unital properties of the smash product of L-spectra As we have already seen, LS is a unit for the smash product ^L on hS [L]. However, for precision in the consideration of algebraic structures, we wish to work in a category of spectra that is actually symmetric monoidal under its sma* *sh product, with a point-set level unit isomorphism. The appropriate candidate for* * a unit object is not LS but S itself, and at this point another special, and surp* *rising, property of the linear isometries operad comes into play. Consider the diagram flxid_ fl L (2) x L (1) x L (1) x L (0) x L (0)____////L (2) x L (0) x L_(0)_//L (0): idxfl2 This is not a split coequalizer, but it turns out to be a coequalizer. The coeq* *ualizer of the parallel pair of arrows is the orbit space L (2)=L (1) x L (1). Lemma 8.1. The orbit space L (2)=L (1) x L (1) consists of a single point. This is far from obvious, and it is only possible because L (1) is a monoid b* *ut not a group. We defer its proof to XIx2. It has the following implication. Reca* *ll Lemma 4.5. Proposition 8.2. There is an isomorphism of L-spectra : S ^L S - ! S such that o = . For based spaces X and Y , there is a natural isomorphism of L-spectra : 1 X ^L 1 Y ~= 1 (X ^ Y ): Proof. The second statement follows from the first, or directly: fl induces * *the isomorphism L (2) nL (1)xL (1)(L (0) n X) ^ (L (0) n Y ) -! L (0) n X ^ Y: The relation o = : S ^L S -! S is clear since flo = fl. This formalizes our intuition that the smash product should be a stabilized g* *en- eralization of the smash product of based spaces. It is natural to try to gener* *alize the resulting isomorphism : S ^L 1 X ~=1 X to arbitrary L-spectra, and the map does generalize. Proposition 8.3. Let M and N be L-spectra. There is a natural map of L- spectra : S ^L N - ! N. The symmetrically defined map M ^L S - ! M coincides with the composite o. Moreover, under the associativity isomorphism, o ^L id= id^L : M ^L S ^L N -! M ^L N; and, under the commutativity isomorphism, these maps also agree with : S ^L (M ^L N) -! M ^L N: 8. UNITAL PROPERTIES OF THE SMASH PRODUCT OF L-SPECTRA 31 Proof. When N is the free L-spectrum LX = L (1) n X generated by a spectrum X, is given by the map S ^L LX = L (2) nL (1)xL (1)(L (0) n S0) ^ (L (1) n X) ~= (L (2) xL (1)xL (1)L (0) x L (1)) n (S0 ^ X) flnid - -! L (1) n X = LX: For general N, the map just constructed induces a map of coequalizer diagrams S ^L LLN ____//_//_S ^L__LN_//S ^L N | | | | | | fflffl|______//fflffl| fflffl| LLN __________//LN__________//N: The symmetry is clear when M is free and follows in general by an easy comparis* *on of coequalizer diagrams. Similarly, suppose that M = LX and N = LY for spectra X and Y . Then, under the associativity isomorphisms of their domains given in the proof of Theorem 5.5, the two unit maps defined on LX ^L S ^L LY agree with the map L (3) nL (1)3((L (1) n X) ^ (L (0) n S0) ^ (L (1) n Y )) ~=(L (3) xL (1)3L (1) x L (0) x L (1)) n (X ^ S0 ^ Y ) flnid - -! L (2) n (X ^ Y ) ~=LX ^L LY: The conclusion for general M and N follows by another comparison of coequalizer diagrams. The last statement can be proven similarly. __|_ | Any attempt to show that S is a strict unit for general L-spectra founders on the fact that Lemma 5.4 fails if i = 0 or j = 0 and i + j > 0. However, we shall prove the following up to homotopy version of that lemma in XI.2.2. Lemma 8.4. the space L^ (1) L (2) xL (1)xL (1)L (0) x L (1) is contractible. Therefore fl : ^L(1) -! L (1) is a homotopy equivalence. Again, this assertion is far from obvious. It leads us to the following cruc* *ial result. Theorem 8.5. Let M be an L-spectrum and consider : S ^L M -! M. (i)If M = LX for a tame spectrum X, then is a homotopy equivalence of spectra and thus a weak equivalence of L-spectra. (ii)If M is a CW L-spectrum, then is a homotopy equivalence of L-spectra. (iii)For any M, is a weak equivalence of L-spectra. 32 I. PROLOGUE: THE CATEGORY OF L-SPECTRA Proof. Since = fl n idon free L-spectra LX, Theorem 2.5 and the lemma give (i). By Theorem 4.7(iii), (i) applies to show that : S ^L M - ! M is a weak equivalence of L-spectra when M is a CW L-spectrum. By the Whitehead theorem for CW L-spectra, there is a map of L-spectra : M - ! S ^L M such that O ' id. To complete the proof of (ii), we must show that O ' id, and the following commutative diagram identifies this composite with id^L ( O ): id^ id^ S ^L M ____//_S ^L S ^L M___//_S ^L M nnnnnnnn || || nnnnnnnnnnn fflffl| fflffl|nnnnn M __________//S ^L M: The rectangle commutes by the naturality of and the triangle commutes by Proposition 8.3. For (iii), let M be arbitrary and consider the diagram * ssn(S ^L M) ~=hS [L](LSn; S ^L M) ____//_hS [L](S ^L LSn; S ^L M) * || |*| fflffl| fflffl| ssn(M) ~=hS [L](LSn; M) _____*_______//hS [L](S ^L LSn; M): By (ii), : S ^L LSn -! LSn is a homotopy equivalence of L-spectra, hence the horizontal arrows are isomorphisms. The right vertical arrow is an isomorphism since, for L-spectra K, * : S [L](S ^L K; S ^L M) -! S [L](S ^L K; M) is a natural isomorphism; its inverse sends f : S ^L K -! M to the composite -1^id id^f S ^L K ____//_S ^L S ^L _K___//S ^L M: (Compare II.1.3 below). Therefore the left vertical arrow is an isomorphism. _* *_|_ | Remark 8.6. The weak equivalence ! : LS ^L M -! M of Proposition 6.2 is just the composite ^id LS ^L M ____//_S ^L M____//M: Therefore ^ idis also a weak equivalence for all L-spectra M. Corollary 8.7. For any L-spectrum M, " : M -! FL (S; M) is a weak equiv- alence of L-spectra. 8. UNITAL PROPERTIES OF THE SMASH PRODUCT OF L-SPECTRA 33 Proof. For a spectrum X, "* : S (X; M) -! S (X; FL (S; M)) can be iden- tified with "* : S [L](LX; M) -! S [L](LX; FL (S; M)). In turn, by naturality and adjunction, this can be identified with * : S [L](LX; M) -! S [L](S ^L LX; M) ~=S [L](LX; FL (S; M)): If X is a CW spectrum, then : S ^L LX - ! LX is a homotopy equivalence of L-spectra, hence the displayed maps all induce isomorphisms on passage to homotopy classes of maps. The conclusion follows by letting X run through the sphere spectra. __|_ | 34 I. PROLOGUE: THE CATEGORY OF L-SPECTRA CHAPTER II Structured ring and module spectra We can now define and study our basic algebraic objects. We begin with the S- modules, which we think of as analogs of modules over a fixed commutative ring * *k. Since the category of S-modules is symmetric monoidal under its smash product, * *we can define S-algebras and commutative S-algebras exactly as we define (associat* *ive and unital) k-algebras and commutative k-algebras. Intuitively, S-algebras are * *as close as one can get to k-algebras in stable homotopy theory, and commutative S-algebras are as close as one can get to commutative k-algebras. By analyzing free objects, we demonstrate that these new definitions are unit* *al sharpenings of the definitions of A1 and E1 ring spectra that were first give* *n in [47]. This allows us to use [47, 49] to supply examples and is therefore funda- mentally important to the theory. We give a parallel analysis of the definition* *s of modules over S-algebras and commutative S-algebras and over A1 and E1 ring spectra. The new definitions drastically simplify the study of these algebraic * *struc- tures. For example, in a final categorical section, we prove that our new defin* *itions lead to elementary categorical proofs that the categories of S-algebras and of * *com- mutative S-algebras are cocomplete, as was first proven by Hopkins and McClure [31] for the categories of A1 and E1 ring spectra. 1. The category of S-modules Here, finally, is the promised definition of S-modules. Definition 1.1. Define an S-module to be an L-spectrum M which is unital in the sense that : S ^L M -! M is an isomorphism. Let MS denote the full subcategory of S [L] whose objects are the S-modules. For S-modules M and N, define M ^S N = M ^L N and FS(M; N) = S ^L FL (M; N): The justification for the name "S-module" is given by the commutative diagrams 35 36 II. STRUCTURED RING AND MODULE SPECTRA -1 S ^S S ^S M ^id_//_S ^S M and M ____//_HS ^S M HHH | id^ || || HHHHH | fflffl| fflffl| id $$Hfflffl| S ^S M _________//_M M: For the definition to be useful, we need examples, and I.8.2 and I.8.3 provide many. We consistently retain the notation M ^L N when the given L-spectra M and N are not restricted to be S-modules. Proposition 1.2. For any based space X, 1 X is an S-module, and 1 X ^S 1 Y ~= 1 (X ^ Y ): For any S-module M and any L-spectrum N, M ^L N is an S-module. In par- ticular, S ^L N is an S-module for any L-spectrum N. Proof. For the second statement, I.8.3 gives that for M ^L N is determined by for M and is therefore an isomorphism. __|_ | We have the following categorical relationship between S [L] and MS. Lemma 1.3. The functor S ^L (?) : S [L] -! MS is left adjoint to the functor FL (S; ?) : MS -! S [L] and right adjoint to the inclusion ` : MS -! S [L]. Proof. The first adjunction is immediate from I.7.1. For the second, let M be an S-module and N be an L-spectrum. A map f : M -! S ^L N of S-modules determines a map O f : M - ! N of L-spectra, and a map g : M - ! N of L-spectra determines a map (id^g) O -1 : M - ! S ^L N of S-modules. Using I.8.3, we see that these are inverse bijections. __|_ | This implies that to lift right adjoint functors from S [L] to MS, we must fi* *rst forget down to S [L], next apply the given functor, and then apply the functor S ^L (?). For example, limits in MS are created in this fashion. Proposition 1.4. The category of S-modules is complete and cocomplete. Its colimits are created in S [L]. Its limits are created by applying the functor S* * ^S (?) to limits in S [L]. If X is a based space and M is an S-module, then M ^ X is an S-module, and the spectrum level cofiber of a map of S-modules is an S-module. For a based space X and S-modules M and N, MS(M ^ X; N) ~=MS(M; S ^L F (X; N)): Moreover, M ^ X ~=M ^S 1 X and S ^L F (X; M) ~=FS(1 X; M): 1. THE CATEGORY OF S-MODULES 37 Remark 1.5. By the path S-module of an S-module N we must understand S ^L P N. By the fiber of a map f : M -! N of S-modules, we must understand S ^L F f. Lemma 1.3 implies that the following square of S-modules is a pullback and that its vertical arrows satisfy the CHP in the category of S-modules. S ^L F f____//_S ^L P N | | | | fflffl| fflffl| M _____f_____//N: The resulting fiber sequences of S-modules behave in exactly the same fashion as fiber sequences of spaces or spectra. Lemma 1.3 also explains our definition of function S-modules. Its second adju* *nc- tion and the adjunction of Theorem 7.1 compose to give the adjunction displayed in the following theorem. Theorem 1.6. The category MS is symmetric monoidal under ^S, and MS(M ^S N; P ) ~=MS(M; FS(N; P )) for S-modules M, N, and P . A homotopy in the category of S-modules is a map M ^ I+ - ! N. A map of S-modules is a weak equivalence if it is a weak equivalence as a map of spectra. The derived category DS of S-modules is constructed from the homotopy category hMS by adjoining formal inverses to the weak equivalences; again, the process is made rigorous by CW approximation. The free L-spectra LX are not S-modules, and we define sphere S-modules by (1.7) SnS S ^L LSn and use them as the domains of attaching maps when defining cell and CW S- modules. Observe that, by I.8.7 and Lemma 1.3, we have (1.8) ssn(M) hS (Sn; M) ~=hS [L](LSn; FL (S; M)) ~=hMS(SnS; M) for S-modules M. From here, the theory of cell and CW S-modules is exactly like the theory of cell and CW spectra and is obtained by specialization of the theory of cell R-modules to be presented in Chapter III. A weak equivalence of cell S-modules is a homotopy equivalence, any S-module is weakly equivalent to a CW S-module, and DS is equivalent to the homotopy category of CW S-modules. Again, as we shall explain in Remark 1.10, the S-module S does not have the homotopy type of a CW S-module. When working homotopically, we replace it with SS S0S. 38 II. STRUCTURED RING AND MODULE SPECTRA The following comparison between CW S-modules and CW L-spectra establishes an equivalence between DS and hS [L] and thus between DS and hS . Theorem 1.9. The following conclusions hold. (i)The functor S ^L (?) : S [L] - ! MS carries CW L-spectra to CW S- modules. (ii)The forgetful functor MS - ! S [L] carries S-modules of the homotopy types of CW S-modules to L-spectra of the homotopy types of CW L-spectra. (iii)Every CW S-module M is homotopy equivalent as an S-module to S ^S N for some CW L-spectrum N. (iv)The unit : S ^L M - ! M is a weak equivalence for all L-spectra M and is a homotopy equivalence of L-spectra if M has the homotopy type of a CW L-spectrum. The functors S ^L (?) and the forgetful functor establish an adjoint equivalen* *ce between the stable homotopy category hS [L] and the derived category DS. This equivalence of categories preserves smash products and function spectra. Proof. Part (i) is immediate by induction up the sequential filtration since* * the functor S^L (?) preserves spheres, cones, and colimits. Part (iv) is a recapitu* *lation of I.8.5 and, applied to sphere S-modules, it implies part (ii). Part (iii) fol* *lows from (i) and (iv) since there is a CW L-spectrum M0 and a homotopy equivalence of L-spectra M0 -! M. The claimed adjoint equivalence of categories is immediate from part (iv). For smash products, the last statement is clear from (ii) and t* *he fact that the smash product M ^S N of S-modules is their smash product as L-spectra. The statement for function spectra follows formally. __|_ | When doing classical homotopy theory, we can work interchangeably in hS , hS [L], or DS. These three categories are equivalent, and the equivalences pres* *erve all structure in sight. When working on the point set level, we have reached a nearly ideal situation with our construction of MS. We pause to comment on Lewis's observation [36] that there is no fully ideal situation. Remark 1.10. Suppose given a symmetric monoidal category of spectra with a suspension spectrum functor 1 such that S = 1 S0 is the unit for the smash product, denoted ^S, and there is a natural isomorphism 1 X ^S 1 Y ~= 1 (X ^ Y ) that is suitably compatible with the coherence isomorphisms for the unity, asso- ciativity, and commutativity of the respective smash products. Our category of S-modules satisfies all of these properties, and many other desiderata not incl* *uded among Lewis's axioms. Suppose further that 1 has a right adjoint "1 " and let 2. THE MIRROR IMAGE TO THE CATEGORY OF S-MODULES 39 QX = colimnnX. Then Lewis observes that there cannot be a natural weak equivalence : "1 "1 X -! QX such that O j : X - ! QX is the natural inclusion, where j is the unit of the adjunction. In our context, we have the two adjunction homeomorphisms MS(S ^L L1 X; M) ~=T (X; 1 FL(S; M)) and MS(1 X; M) ~=T (X; MS(S; M)); where T is the category of based spaces; see VIIx10 for discussion of these to* *pol- ogized Hom sets and of the second of these adjunctions. It is a standard proper* *ty of any symmetric monoidal category that the self-maps of the unit object form a commutative monoid under composition. In our situation MS(S; S) is therefore a commutative topological monoid. It cannot be weakly equivalent to QS0, and QS0 is weakly equivalent to 1 FL(S; S). Therefore the weak equivalence S^LLS -! S cannot be a homotopy equivalence of S-modules and S cannot be of the homotopy type of a CW S-module. 2. The mirror image to the category of S-modules The categorical picture becomes clearer when we realize that the category of S-modules has a "mirror image" category to which it is naturally equivalent. We find this material quite illuminating, but it will not be used until our discus* *sion of Quillen model categories. Definition 2.1. Define M Sto be the full subcategory of S [L] whose objects are those L-spectra N that are counital, in the sense that " : N -! FL (S; N) is an isomorphism. Looking through the mirror at Lemma 1.3 and noting that mirrors interchange left and right, we see the following reflection. Lemma 2.2. The functor FL (S; ?) : S [L] -! M Sis right adjoint to the func- tor S ^L (?) : M S- ! S [L] and left adjoint to the inclusion r : M S- ! S [L]. We agree to write (2.3) f = FL (S; ?) : S [L] -! M S and s = S ^L (?) : S [L] -! MS 40 II. STRUCTURED RING AND MODULE SPECTRA in the rest of this section. With this notation, Lemmas 1.3 and 2.2 give the fo* *llowing mirrored pairs of adjunctions, the upper arrow being left adjoint to the lower * *arrow in each case. __s__ __`_//_ __f__// _`sr//_ (2.4) S [L]oo__//MS_oo___S [L] and S [L]oo___M S oo___S [L] rf` s r f The display makes new information visible. The composite of the first two left adjoints is just the functor S ^L (?) and the composite of the second two right adjoints is just the functor FL (S; ?). Since these two endo-functors of S [L] * *are left and right adjoint, they must be equivalent to their displayed composite adjoint* *s. Lemma 2.5. For L-spectra M, the maps id^L " : S ^L M -! S ^L FL (S; M) and FL (id; ) : FL (S; S ^L M) -! FL (S; M) are natural isomorphisms. We now see that the reflection of a reflection is equivalent to the original. Proposition 2.6. The functors f` : MS -! M S and sr : M S- ! MS are inverse equivalences of categories. More precisely, " : srf`M = S ^L FL (S; M) -! M is an isomorphism for M 2 MS, and j : N -! FL (S; S ^L N) = f`srN is an isomorphism for N 2 M S, where " and j are the unit and counit of the (S ^L (?); FL (S; ?)) adjunction. Proof. The functor s` : MS -! MS is an equivalence, and it is left adjoint to the composite srf` : MS - ! MS. The functor fr : M S -! M S is an equivalence, and it is right adjoint to the composite f`sr. Therefore these two composites are natural equivalences. A little diagram chase from the previous lemma gives the more precise statement. __|_ | Proposition 2.7. The category M S, hence also the category MS, is equivalent to the category of algebras over the monad rf in S [L] determined by the adjunc* *tion (f; r). The category MS, hence also the category M S, is equivalent to the cate* *gory of coalgebras over the comonad `s in S [L] determined by the adjunction (`; s). 3. S-ALGEBRAS AND THEIR MODULES 41 Proof. The unit of the monad rf is " : M - ! FL (S; M) = rfM and its product is the natural isomorphism : rfrfM = FL (S; FL (S; M)) ~=FL (S; M) = rfM implied by the isomorphism S ^L S ~=S. Clearly, if " is an isomorphism, then M is an rf-algebra with action "-1. Conversely if : rfM -! M is an action, then O " = idand the following is a split coequalizer diagram in S [L]. _rf__ rfrfM _____////rfM___//_M: Applying f, we obtain a split coequalizer diagram in M S. Since the counit fr !* * id of the adjunction is an isomorphism, it induces an isomorphism of diagrams (frfrfM ____//_//_frfM) _____// (frfM _____////_fM): Applying r, rfM is the (split) coequalizer of the first and M is the (split) co* *equal- izer of the second. The resulting isomorphism rfM ! M is just the map , hence is an isomorphism of rf-algebras. __|_ | 3. S-algebras and their modules Let C be any symmetric monoidal category, with product and unit object I. Then a monoid in C is an object R together with maps j : I ! R and OE : R R ! R such that the evident associativity and unity diagrams commute; R is a commutative monoid if the evident commutativity diagram also commutes. A left R-object over a monoid R is an object M of C with a map : RM ! M such that the evident unity and associativity diagrams commute, and right R-objects * *are defined by symmetry. These definitions apply to our symmetric monoidal category MS. Definition 3.1. An S-algebra is a monoid in MS. A commutative S-algebra is a commutative monoid in MS. For an S-algebra or commutative S-algebra R, a left or right R-module is a left or right R-object in MS. Modules will mean left modules unless otherwise specified, and we let MR denote the category of l* *eft R-modules. Observe that if R is a commutative S-algebra, then an R-module is just a modu* *le over R regarded as an S-algebra, as in module theory in algebra. For this reaso* *n, even though our main interest is in the much richer commutative context, we work with general S-algebras wherever possible. We insert the following lemma for later reference. It records specialization* *s of observations that apply to monoids in any symmetric monoidal category. 42 II. STRUCTURED RING AND MODULE SPECTRA Lemma 3.2. Let R be an S-algebra and M be an R-module. Then the following diagrams of S-modules are split coequalizers: __OE^id___ OE R ^S R ^S R __________////R ^S_R___//R: id^OE and ___id^____ R ^S R ^S M __________////R ^S M___//_M: OE^id While we have given the most conceptual form of the definitions, it is worthw* *hile to write out the relevant diagrams explicitly. We find that they make perfect s* *ense for L-spectra that might not be S-modules, and this leads us back to the earlier notions of A1 and E1 ring spectra and their modules. Definition 3.3. An A1 ring spectrum is an L-spectrum R with a unit map j : S - ! R and a product OE : R ^L R ! R such that the following diagrams commute: j^id id^j S ^L R ____//_R ^L Roo__R ^L S MM qq MMMM OE| qqqq MMMM | qqqoq M&&Mfflffl|xxqq R and id^OE R ^L R ^L R _____//R ^L R OE^id|| |OE| fflffl| fflffl| R ^L R ____OE____//R; R is an E1 ring spectrum if the following diagram also commutes: R ^L RI _____o______//_R ^L R III uuuu III uuOEu OE II$$ zzuu R: A module over an A1 or E1 ring spectrum R is an L-spectrum M with a map : R ^L M ! M such that the following diagrams commute: j^id id^ S ^L M ____//_R ^L M and R ^L R ^L M ____//_R ^L M NN NNNN | OE^id| | NNNN | | | NN&&fflffl| fflffl| fflffl| M R ^L M __________//M: 3. S-ALGEBRAS AND THEIR MODULES 43 Lemma 3.4. An S-algebra or commutative S-algebra is an A1 or E1 ring spec- trum which is also an S-module. A module over an S-algebra or commutative S-algebra R is a module over R, regarded as an A1 or E1 ring spectrum, which is also an S-module. In view of Proposition 1.2, this leads to the following observations. Proposition 3.5. The following statements hold. (i)S is a commutative S-algebra with unit id and product . (ii)If R and R0are A1 or E1 ring spectra, then so is R ^L R0; if either R * *or R0 is an S-algebra, then so is R ^L R0. (iv)If R and R0 are A1 ring spectra, M is an R-module and M0 is an R0- module, then M ^L M0 is an R ^L R0-module. In particular, we have a functorial way to replace A1 and E1 ring spectra a* *nd their modules by S-algebras and commutative S-algebras and their modules. Corollary 3.6. For an A1 ring spectrum R, S ^L R is an S-algebra and : S ^L R - ! R is a weak equivalence of A1 ring spectra, and similarly in the E1 case. If M is an R-module, then S ^L M is an S ^L R-module and : S ^L M -! M is a weak equivalence of R-modules and of modules over S ^L R regarded as an A1 ring spectrum. Recall that the tensor product of commutative rings is their coproduct in the category of commutative rings. The proof consists of categorical diagram chases that apply to commutative monoids in any symmetric monoidal category. Proposition 3.7. If R and R0are commutative S-algebras, then R ^S R0is the coproduct of R and R0 in the category of commutative S-algebras. We shall construct coproducts in the category of S-algebras in Section 7, whe* *re we show more generally that the categories of S-algebras and of commutative S- algebras are cocomplete. There is a version of the proposition that is true for E1 ring spectra, but * *this is not obvious. We shall return to this point in Chapter XIII, where we show that the category of L-spectra under S is symmetric monoidal under a modified smash product ?S and that A1 and E1 ring spectra are exactly the monoids and commutative monoids in that symmetric monoidal category. This was the starting point for the earlier version of the present theory announced in [22]. 44 II. STRUCTURED RING AND MODULE SPECTRA 4. Free A1 and E1 ring spectra and comparisons of definitions We focus on A1 and E1 ring spectra here. It was proven in [47, 49] that var* *ious Thom spectra, Eilenberg-Mac Lane spectra, and connective algebraic and topolog- ical K-theory spectra are E1 ring spectra. Using the results stated in the pre* *vious section, we can convert these E1 ring spectra to weakly equivalent commutative S-algebras. However, on the face of it, the original definitions of A1 and E1 * * ring spectra appear to be different from those that we have given here. As in algebr* *a, it is important to understand free A1 and E1 ring spectra, and we shall use t* *his understanding to verify that our present definitions agree with the original on* *es. There is no difficulty in constructing the relevant monads. In fact, we shall construct two pairs of monads and then relate them. The first is defined on the ground category of spectra and is transparently related to the earlier definiti* *ons. The second is defined on the ground category of S-modules and is transparently related to the present definitions. The connection between them will establish * *the required equivalence of definitions. In effect, our new definition of E1 ring * *spectra is obtained from the old one simply by factoring the original defining monad C * *in S through a new defining monad P in the more highly structured category S [L]. Construction 4.1. Construct monads B and C in the category of spectra as follows. Let X be a spectrum and let Xj be its j-fold external smash power, with X0 = S0. Define _ BX ~= L (j) n Xj j0 and _ C X ~= L (j) nj Xj; j0 where L (j) nj Xj is the orbit spectrum (L (j) n Xj)=j . The units of these monads are induced by the unit maps X ~={1} n X ! L (1) n X. Their products are induced by wedge sums of maps induced by the structure maps fl of the linear isometries operad L . The notion of an L -spectrum was defined in [37, VII.2.1]. The definition used permutations, and there is a corresponding notion of a non- L -spectrum. An immediate comparison of definitions gives the following result. Proposition 4.2. The category of B-algebras is isomorphic to the category of non- L -spectra. The category of C -algebras is isomorphic to the category of L -spectra. Actually, O-spectra were defined in [37, VII.2.1] for any operad O that is au* *g- mented over L . An E1 operad is one such each O(j) is j-free and contractible. In earlier work, E1 ring spectra were understood to mean O-spectra for any 4. FREE A1 AND E1 RING SPECTRA AND COMPARISONS OF DEFINITIONS 45 E1 operad O augmented over L . The present theory is based on properties that are special to L . The following result, which will be proven in XIIx1, shows t* *hat restriction to L results in no loss of generality. There is an analogue for A1 * * ring spectra that is obtained by forgetting about permutations. Proposition 4.3. Let O be an E1 operad over L . There is a functor V that assigns a weakly equivalent L -spectrum V R to an O-spectrum R. Construction 4.4. Construct monads T and P in the category of L-spectra as follows. Let M be an L-spectrum and let Mj be its j-fold power with respect to ^L , with M0 = S. Define _ TM ~= Mj j0 and _ PM ~= Mj=j: j0 Here passage to orbits preserves L-spectra since it is a finite colimit. The un* *it is the inclusion of M = M1. The product is induced by the maps Mj1^L . .^.L Mjk -! Mj1+...+jk that are given by the evident identifications if each jr 1 and by use of the u* *nit map if any jr = 0. Observe that T and P restrict to monads in the category of S-modules. The letters T and P are mnemonic for "tensor algebra" and "polynomial" (or symmetric) algebra. As is clear for S-modules and will be made explicit in Defi* *ni- tion 7.1, the definitions fit into a general categorical framework that include* *s those constructions. The following result is an easy direct consequence of our defini* *tions. Proposition 4.5. The categories of A1 ring spectra and of S-algebras are is* *o- morphic to the categories of T-algebras in S [L] and of T-algebras in MS. The categories of E1 ring spectra and of commutative S-algebras are isomorphic to * *the categories of P-algebras in S [L] and of P-algebras in MS. To relate the monads B and C to the monads T and P, recall from I.4.2 that the category of L-spectra is the category of L-algebras in S . Together with Propos* *i- tions 4.2 and 4.5, the following result gives the promised comparison between t* *he old and new definitions of A1 and E1 ring spectra. Proposition 4.6. The monads B and TL are isomorphic, hence the categories of non- L -spectra and of A1 ring spectra are isomorphic. The monads C and PL are isomorphic, hence the categories of L -spectra and of E1 ring spectra a* *re isomorphic. 46 II. STRUCTURED RING AND MODULE SPECTRA Proof. The isomorphisms on objects are immediate from I.5.6 applied to L- spectra Mi= LXi. Since these isomorphisms are induced from the structure maps fl of L , the comparison of monad structures is immediate. In both statements, * *the second clause is a categorical consequence of the first, as we shall show in Le* *mma 6.1 below. __|_ | Remark 4.7. Observe that we have quotient maps of monads B - ! C and T - ! P. In Section 6, we shall give categorical definitions that show how to exploit these maps to construct an E1 ring spectrum C B R (or P T R) from an A1 ring spectrum R by "passage to quotients", just as we construct commutative algebras as quotients of associative algebras; see Lemma 6.7 and Corollary 7.3. Formally, C B R is a coequalizer of a right action of B on C and the given acti* *on of B on R. Remark 4.8. Passage to orbits and passage to coequalizers are often hard to analyze homotopically. We show how to deal with the first difficulty in IIIx5, where we show that symmetric powers and extended powers of S-modules (and, more generally, R-modules) are essentially equivalent. One often circumvents the second difficulty by replacing a construction like C B R with its associated bar construction B(C ; B; R), which we shall introduce in XIIx1. Remark 4.9. There are reduced monads "Band "Cin the category S \S of spec- tra under S and "T and "Pin the category S [L]\S of L-spectra under S. They are constructed from the unreduced monads by unit map identifications similar to the basepoint identifications in the James construction or the infinite symmetr* *ic product. Observe that S \S is the category of algebras over the monad U that is specified by UX = X _ S, with product given by the folding map S _ S -! S, and similarly for S [L]\S. In all four cases, the unreduced monad is the compos* *ite of the reduced monad with U, hence, by Lemma 6.1 below, the reduced and unre- duced monads have the same algebras. The difference is that, when considering the reduced monad, one is considering the unit map S ! R as preassigned and then ensuring that the unit map created by the monad action coincides with it. * *It follows that the monad U acts from the right on the unreduced monads, and it is easy to write down this action directly. The reduced monad "Ccan then be con- structed from C by setting "CX = C U X for a spectrum X under S, with structure maps induced by passage to coequalizers, and similarly for our other monads. A more explicit description is given in [37, VIIx3], where "Cis denoted by C . Wh* *ile the monad C is more convenient for formal work, the monad "C is of far greater homotopical interest. 5. FREE MODULES OVER A1 AND E1 RING SPECTRA 47 5. Free modules over A1 and E1 ring spectra There is an analogue for modules of the original explicit definition of A1 a* *nd E1 ring spectra in terms of twisted half-smash products, and there is an analo* *gous comparison of definitions. Proposition 5.1. The category of modules over an L -spectrum R is isomor- phic to the category of spectra M together with associative, unital, and, in the E1 context, equivariant systems of action maps L (j) n (Rj-1 ^ M) -! M: Since we shall not need the details, we shall not write out the relevant diag* *rams. They make sense for any operad O augmented over L , and they are exact analogs of diagrams that are written out in the context of algebraic operads in [34, I.* *4.1]. Remarkably, with this alternative form of the definition, it is far from obvious that a module over an E1 ring spectrum R is the same thing as a module over R regarded as an A1 ring spectrum. In fact, this appears to be false in the cont* *ext of modules over an O-spectrum R for a general E1 operad O augmented over L . However, we have the following analogue of Proposition 4.3, which will be proven in XIIx1. Again, there is an analogue for A1 ring spectra and modules that is obtained by forgetting about permutations. Proposition 5.2. Let O be an E1 operad over L and R be an O-spectrum. There is a functor V that assigns a weakly equivalent V R-module to an R-module M, where V R is the L -spectrum of Proposition 4.3. There is a conceptual monadic proof of Proposition 5.1 that is based on analo* *gs of Propositions 4.2, 4.5, and 4.6. To carry out this argument, we need to know * *that there is a free R-module functor. This is obvious enough when we are considering S-modules: R ^S M is then the free R-module generated by an S-module M. For a general A1 ring spectrum R and an L-spectrum M, R ^L M is an R-module but, since M need not be isomorphic to S ^L M, it is not the free R-module generated by M. Definition 5.3. For an A1 ring spectrum R and an L-spectrum M, define an L-spectrum RM and maps of L-spectra ss : R ^L M -! RM and j : M -! RM by the pushout diagram j^id S ^L M _____//R ^L M || |ss| fflffl| fflffl| M ____j____//_RM: 48 II. STRUCTURED RING AND MODULE SPECTRA Dually, define an L-spectrum R #M by the pullback diagram R #M __________//_M | | | |" fflffl| fflffl| FL (R; M) F(j;i//d)_FL (S; M): These are special cases of general constructions to be studied in Chapter XII* *I. Such constructions permeated earlier versions of the present theory. Of course,* * ss is an isomorphism if M is an S-module. As will be generalized in XIII.1.4, we deduce the following homotopical property by applying the functor S ^L (?) to t* *he defining pushout diagram. Proposition 5.4. The map ss : R ^L M -! RM is a weak equivalence for any L-spectrum M. The unit diagram of an R-module M ensures that its product factors through a map R M - ! M. More formally, elementary inspections of definitions give the following result. Proposition 5.5. Let R be an A1 ring spectrum. Then R is a monad in S [L] with unit j : M ! RM and product induced from the product OE : R ^L R -! R. A left R-module is an algebra over the monad R and, for an L-spectrum M, R M is the free R-module generated by M. The functor R # is right adjoint to R and * *is therefore a comonad in S [L] such that an R-module is a coalgebra over R #. It is logical to denote the category of R-modules by S [L][R ], reserving the notation MR for the case when R is an S-algebra and R-modules are required to be S-modules. We have freeness and cofreeness adjunctions S [L][R ](R M; N) ~=S [L](M; N) and S [L][R ](N; R# M) ~=S [L](N; M) for L-spectra M and R-modules N. Clearly there results a composite adjunction that starts with spectra. Proposition 5.6. For a spectrum X, define FX = RLX. Then FX is the free R-module generated by X. Thus S [L][R ](FX; N) ~=S (X; N) for an R-module N. Dually, define F# X = R #L# X. Then F# X is the cofree R-module generated by X, so that S [L][R ](N; F# X) ~=S (N; X): 5. FREE MODULES OVER A1 AND E1 RING SPECTRA 49 In Construction 6.2, we shall show how to combine the monads of the previous section with these free module constructions to obtain monads B[1] and C[1] in * *the category of pairs of spectra such that a B[1]-algebra or C [1]-algebra (R; M) i* *s an A1 or E1 ring spectrum R together with an R-module M in the alternative operad action sense described in Proposition 5.1. The construction will also give mona* *ds T[1] and P[1] in the category of pairs of L-spectra such that a T[1]-algebra or* * P[1]- algebra (R; M) is an A1 or E1 ring spectrum R together with an R-algebra M in the sense of Definition 3.3. The monad B[1] has the general form B[1](X; Y ) = (BX; B(X; Y )); and similarly in the other three cases. Propositions 4.6 and 5.5, together with* * in- spection of the cited construction, directly imply the following analogue of Pr* *opo- sition 4.6. By Lemma 6.1, this in turn implies Proposition 5.1. Proposition 5.7. The monads B[1] and T[1]O(L; L) are isomorphic. The mon- ads C [1] and P[1] O (L; L) are isomorphic. The second coordinates of the four monads are given explicitly as follows. Applied to a pair of spectra (X; Y ), _ B(X; Y ) = L (j) n (Xj-1 ^ Y ) j1 and _ C(X; Y ) = L (j) nj-1 (Xj-1 ^ Y ): j1 Applied to a pair of L-spectra (M; N), _ T(M; N) = Mj-1 ^S N j1 and _ P(M; N) = (Mj-1=j-1) ^S N: j1 If N is an S-module, then so are T(M; N) and P(M; N). Remark 5.8. Construction 6.2 applies equally well to give reduced versions * *of our four monads, giving monads in the category of pairs (of spectra or L-spectr* *a), the first coordinate of which lies under S. The monad "B[1] has the form B"[1](X; Y ) = ("BX; "B(X; Y )) and similarly in the other three cases. Inspection of definitions shows that "BS = "CS = S and B"(S; Y ) = "C(S; Y ) = L (1) n Y: This fact dictates our original definition of L-spectra and is thus the concept* *ual starting point of our entire theory. 50 II. STRUCTURED RING AND MODULE SPECTRA 6. Composites of monads and monadic tensor products In this section and the next, we collect a number of purely categorical obser- vations and constructions that are needed in our work. We shall return to these topics in Chapter VII, but we shall make no further use of this material until * *then. The reader may prefer to skip these sections on a first reading. We here give t* *he description of algebras over composite monads that was at the heart of our com- parisons of definitions and formalize the tensor product construction that appe* *ared briefly in Section 4. Lemma 6.1. Let S be a monad in a category C and let T be a monad in the category C [S] of S-algebras. Then the category C [S][T] of T-algebras in C [S* *] is isomorphic to the category C [TS] of algebras over the composite monad TS in C . Moreover, the unit of T defines a map S -! TS of monads in C . An analogous assertion holds for comonads. Proof. Strictly speaking, in constructing TS, we are regarding S as the free* * S- algebra functor C ! C [S], applying the functor T, and then applying the forget* *ful functor back to C . We continue to neglect notation for forgetful functors and to write S and T ambiguously for both the given monads and the resulting free functors. The unit of TS is given by the composite of unit maps X -! SX -! TSX: The product of TS is given by the composite maps TSTSX -! TTSX -! TSX; where the second arrow is given by the product of T and the first is obtained by application of T to the action STSX - ! TSX given by the fact that T takes S- algebras to S-algebras. If R is a T-algebra in C [S], with action by S and act* *ion O by T, then it is a TS-algebra with action the composite T O TSR ____//_TR____//_R: If Q is a TS-algebra with action !, then Q is a T-algebra in C [S] with actions* * the composites j ! Tj ! SQ ____//_TSQ____//_Q and TQ ____//_TSQ____//_Q: These correspondences establish the required isomorphism of categories. Easy diagram chases show that S -! TS is a map of monads. __|_ | When applying this to modules, we used the following construction. 6. COMPOSITES OF MONADS AND MONADIC TENSOR PRODUCTS 51 Construction 6.2. For a category C , let C [1] be the category of pairs (X;* * Y ) in C and pairs of maps. Let S be any of the monads B, C , T, or P, and let C be its ground category S , S [L], or MS. Construct a monad S[1] in C [1] as follow* *s. On a pair (X; Y ), the functor S[1] is given by S[1](X; Y ) = (SX; S(X; Y )); where S(X; Y ) is the free SX-module generated by Y. This functor factors throu* *gh the evident category of pairs (S-algebra; object ofC ) as the composite of (S; id) and (id; free module), where the free module functor is that associated to the algebra in the first variable. Since the identity fu* *nctor is a monad in a trivial way, each of these functors is a monad. Therefore, by Lemma 6.1, their composite S[1] is a monad such that an S[1]-algebra (R; M) is an S-algebra R together with an R-module M. We used the following definition in our construction of E1 ring spectra from A1 ring spectra. Definition 6.3. Let (S; ; j) be a monad in a cocomplete category C . A (righ* *t) S-functor in a category C 0is a functor F : C ! C 0together with a natural transformation : F S ! F such that the following diagrams commute: Fj S F S oo___F and F SS ____//_F S __ | | ||____ F | | fflffl|""id__ fflffl| fflffl| F F S _____//_F: Given an S-algebra (R; ), define F S R to be the coequalizer displayed in the diagram _____ F SR ____////_F_R_//F S R: F Given a monad S0 in C 0and a left action : S0F ! F , we say that F is an (S0; S)-bifunctor if the following diagram commutes: S____ S0F S //F S S0 || || fflffl| fflffl| S0F _____//_F: Example 6.4. The functor S is an (S; S)-bifunctor, with both left and right action . If ss : S ! S0is a map of monads in C , then S0is an (S0; S)-bifunctor* * with right action = 0O S0ss : S0S -! S0. Observe that, for X 2 C , S0S SX ~=S0X: 52 II. STRUCTURED RING AND MODULE SPECTRA When C 0in Definition 6.3 has a forgetful functor to the category of spectra, we shall construct a bar construction B(F; S; R) that will give the appropriate homotopical version of F S R in XIIx1. Assuming that F is an (S0; S)-bifunctor for one of the monads constructed earlier in this chapter, we will find that B(* *F; S; R) is an S0-algebra. It is natural to ask whether or not F S R is itself an S0-alg* *ebra. To answer this, we need another categorical definition. Definition 6.5. In any category C , a coequalizer diagram __e__ g A _____////B__//_C: f is said to be a reflexive coequalizer if there is a map h : B -! A such that eO* *h = id and f O h = id. The following categorical observation is standard and easy. Although their st* *ated hypotheses are different, the proofs of similar results in [42, p. 147] and [4,* * pp. 106- 108] apply to give the first statement, and the second statement follows. Lemma 6.6. Let S be a monad in C such that S preserves reflexive coequalizer* *s. If __e__ g A ____////_B_//_C f is a reflexive coequalizer in C such that A and B are S-algebras and e and f are maps of S-algebras, then C has a unique structure of S-algebra such that g is a map of S-algebras, and g is the coequalizer of e and f in the category C [S]. * *If, further, T is a monad in C [S] such that T preserves reflexive coequalizers, th* *en T O S also preserves reflexive coequalizers. Since the coequalizer diagram used to define F S R is reflexive, via the map F j : F R -! F SR, the first statement implies an answer to the question we ask* *ed originally. Lemma 6.7. Let S be a monad in C , S0 be a monad in C 0, R be an S-algebra, and F : C -! C 0be an (S0; S)-bifunctor. If S0 preserves reflexive coequalize* *rs, then F S R is an S0-algebra. 7. Limits and colimits of S-algebras We here prove that the categories of A1 and E1 ring spectra and of S-algebr* *as and commutative S-algebras are complete and cocomplete. In fact, completeness follows immediately from Proposition 4.5. All four of our categories are catego* *ries of algebras over a monad in a complete category, and it follows that they are complete, with their limits created in their respective ground categories [42, * *VI.2, ex. 2]. The first statement of Lemma 6.6 applies to construct colimits, but to 7. LIMITS AND COLIMITS OF S-ALGEBRAS 53 explain this properly we need some preliminary definitions that put our definit* *ions of A1 and E1 ring spectra in perspective. Definition 7.1. A weak symmetric monoidal category C with product and unit object I is defined in exactly the same way as a symmetric monoidal catego* *ry [42, p. 180], except that its unit map : I X - ! X is not required to be an isomorphism; C is said to be closed if the functor (?) Y has a right adjoint Hom (Y; ?) for each Y 2 C . Monoids and commutative monoids in C are defined in terms of diagrams of the form displayed in Definition 3.3. As in Construction 4.4 and Proposition 4.5, if C is cocomplete, then there are monads T and P in C whose algebras are the monoids and commutative monoids in C . For X 2 C , a a TX ~= Xj and PX ~= Xj=j: j0 j0 The proof of the following result is abstracted from an argument that Hopkins gave for the monad C [31]. He proceeded by reduction to a proof that the j-fold symmetric powers of based spaces preserve reflexive coequalizers. With our new associative smash products, an abstraction of the latter proof makes the reduct* *ion unnecessary. Proposition 7.2. Let C be any cocomplete closed weak symmetric monoidal category. Then the monads T and P in C preserve reflexive coequalizers. Proof. For T, it suffices to prove that the j-fold product X1. .X.jpreserves reflexive coequalizers. Thus let _ei__ gi Xi ____////_Yi_//Zi fi be reflexive coequalizer diagrams in C , 1 i j, and let hi : Yi -! Xi satisfy eiO hi= idand fiO hi= id. Let " = e1 . . .ej; OE = f1 . . .fj; and fl = g1 . . .gj: Let fi : Y1 . . .Yj -! Z be the coequalizer of " and OE. Since fl" = flOE, the* *re is a unique map : Z -! Z1 . . .Zj such that O fi = fl. We claim that is an isomorphism, and we proceed by induction on j. Let "i= (id)i-1 ei (id)j-i: Y1 . . .Yi-1 Xi Yi+1 . . .Yj -! Y1 . . .Yj and, similarly, define OEi= (id)i-1 fi (id)j-i. We observe first that Z1 . . .Zj is the colimit of the diagram given by the j pairs of maps {"i; OEi}. Indeed, f* *or any map ff : Y1 . . .Yj -! W such that ff O "i = ff O OEi for 1 i j, we obtain 54 II. STRUCTURED RING AND MODULE SPECTRA unique maps ^ffand "ffthat make the following diagram commute by the induction hypothesis and the fact that the -product preserves colimits and epimorphisms: g1...gj-1id Y1 . . .Yj-1 Xj __________//_Z1 . . .Zj-1 Xj idej ||idfj|| idej||idfj|| fflffl|fflffl|g1...gj-1id fflffl|fflffl| Y1 . . .Yj-1 Yj ___________//Z1 . . .Zj-1 Yj gg g | ff|| g gg^ffgg idgj| fflffl|ssggggg "ff fflffl| W oo____ _ __ _ _ __ Z1 . . .Zj-1 Zj: Now let ki= h1 . . .hi-1 idhi+1 . . .hj. Visibly "i= " O ki and OEi= OE O ki: Since fi" = fiOE, fi"i = fiOEi for 1 i j and the universal property gives a m* *ap i : Z1 . . .Zj -! Z. It is easy to check from the universal properties that i and are inverse isomorphisms. In the symmetric case, we may take our j given coequalizer diagrams to be the same and compose the j-fold power, regarded as a functor to the category of j-objects in C , with the orbit functor. The latte* *r is constructed as a coequalizer in C and is a left adjoint, so preserves coequaliz* *ers. __|_ | Corollary 7.3. The functors T and P on S [L], their restrictions to functors T and P on MS, and the functors B and C on S preserve reflexive coequalizers. Proof. This is immediate since B = TL, C = PL, the functor L : S - ! S [L] preserves colimits, and colimits in S [L] and in MS are created in S . __|_ | Our claim that the categories of A1 and E1 ring spectra and of S-algebras and commutative S-algebras are cocomplete is now an immediate corollary of the following known result, which we also learned from Hopkins. Proposition 7.4. Let S be a monad in a cocomplete category C . If S preserves reflexive coequalizers, then C [S] is cocomplete. Proof. Consider a diagram {Ri} of S-algebras. Let colimRi be its colimit in C and let i: Ri- ! colimRi be the natural maps. Let ff : colimSRi- ! S colimRi be the unique map in C whose composite with the natural map SRi- ! colimSRi is Si for each i. Define colimSRi by the following coequalizer diagram in C : S(colimi) S(colim SRi)__________////_S(colim_Ri)//_colimSRi: OSff 7. LIMITS AND COLIMITS OF S-ALGEBRAS 55 This is a reflexive coequalizer, via S(colim ji). Thus, by Lemma 6.6, colimSRi * *is an S-algebra such that the displayed diagram is a coequalizer in C [S]. It foll* *ows easily that colimSRi is the colimit of {Ri} in C [S]. __|_ | For later reference, we observe that this result is closely related to the fo* *llowing result of Linton [40] (see also [4, Thm 2, p. 319]). Theorem 7.5 (Linton). Let S be a monad in a cocomplete category C . If C [* *S] has coequalizers, then C [S] is cocomplete. 56 II. STRUCTURED RING AND MODULE SPECTRA CHAPTER III The homotopy theory of R-modules We here develop the homotopy theory of modules over an S-algebra R. The classic* *al theory of cell spectra generalizes to give a theory of cell modules over R. We * *define the smash product over R, ^R , and the function R-module functor, FR , by direct mimicry of the definitions of tensor product and Hom functors for modules over an algebra. When specialized to commutative S-algebras, our smash product of R-modules is again an R-module, and similarly for FR . Here the category of R- modules has structure precisely like the category of S-modules, and duality the* *ory works exactly as it does for spectra. We assume familiarity with IIxx1,3 and wo* *rk in the ground category MS of S-modules. 1. The category of R-modules; free and cofree R-modules Fix an S-algebra R. We understand R-modules to be left R-modules unless otherwise specified. We first observe that the category MR of R-modules is clos* *ed under various constructions in the underlying categories of spectra and S-modul* *es. As in algebra, an R-module is the same thing as an algebra over the monad R^S (* *?) in MS or, equivalently, a coalgebra over the adjoint comonad FS(R; ?) in MS. The functors R ^S (?) and FS(R; ?) from MS to MR are left and right adjoint to the forgetful functor. That is, R ^S (?) and FS(R; ?) are the free and cofree funct* *ors from S-modules to R-modules. Together with II.1.4 and formal arguments exactly like those in algebra, this leads to the following result. Theorem 1.1. The category of R-modules is complete and cocomplete, with both limits and colimits created in the underlying category MS. Let X be a based space, K be an S-module, and M and N be R-modules. Then the following con- clusions hold, where the displayed isomorphisms are obtained by restriction of * *the corresponding isomorphisms for S-modules. (i)M ^X is an R-module and the spectrum level cofiber of a map of R-modules is an R-module. 57 58 III. THE HOMOTOPY THEORY OF R-MODULES (ii)S ^L F (X; N) is an R-module and MR (M ^ X; N) ~=MR (M; S ^L F (X; N)): (iii)M ^S K and FS(K; N) are R-modules and MR (M ^S K; N) ~=MR (M; FS(K; N)): (iv)FS(M; K) is a right R-module. (v)As R-modules, M ^ X ~=M ^S 1 X and S ^L F (X; N) ~=FS(1 X; N): The cofiber and fiber of a map of R-modules are R-modules, where the fiber is understood to be obtained by application of the functor S ^L (?) to the fiber * *con- structed in the category of spectra. Proof. The only point that might need comment is the R-module structure on S ^L F (X; N). The evaluation map " : F (X; N) ^ X -! N is a map of L-spectra. The adjoint of R ^L " is a map of L-spectra "": R ^L F (X; N) -! F (X; R ^S N); and we obtain the desired action upon applying S ^L ""and using the given action of R on N. This leads to the R-module structure on the specified fiber of a map of R-modules; compare II.1.5. __|_ | The free R-module functor on spectra is the starting point of cellular theory. Definition 1.2. Define the free R-module generated by a spectrum X to be FR X = R ^S FSX; where FSX = S ^L LX. Equivalently, since R ^S S ~=R, FR X = R ^L LX: We abbreviate FX = FR X when R is clear from the context. The term "free" is technically a misnomer, since F is not left adjoint to the forgetful functor. However, it is nearly so. Proposition 1.3. The functor F : S - ! MR is left adjoint to the functor that sends an R-module M to the spectrum FL (S; M), and there is a natural map of R-modules : FM -! M whose adjoint M -! FL (S; M) is a weak equivalence of spectra. Therefore ssn(M) ~=hMR (FSn; M): 1. THE CATEGORY OF R-MODULES; FREE AND COFREE R-MODULES 59 Proof. In view of II.1.3, we have the chain of isomorphisms MR (FR X; N) ~=MS(FSX; N) ~=S [L](LX; FL (S; N)) ~=S (X; FL (S; N)): By I.8.7, we have a natural weak equivalence " : M -! FL (S; M) of S [L]-spectr* *a. Thought of as a map of spectra, its adjoint is the required R-map . The stateme* *nt about the homotopy groups ssn(M) = hS (Sn; M) is clear; compare II.1.8. __|_ | The following central theorem shows that we have homotopical control on FX without any hypotheses (such as tameness or CW homotopy type) on R. Theorem 1.4. In the stable homotopy category hS , FX is naturally isomor- phic to the internal smash product R ^ X. Moreover, the composite Fj i : FS- ! FR- ! R is a weak equivalence of R-modules. Proof. The first statement is clear from II.1.9 and I.6.7, but we point out a variant proof that makes clear that the weak equivalence is one of R-module spe* *ctra (in the homotopical sense). In Xx5, we shall construct a tame A1 ring spectrum KR and a weak equivalence of A1 ring spectra r : KR - ! R. Since we are working in the stable homotopy category, we may take X to be a CW spectrum. Then, by I.4.7 and I.6.6, r ^L id: KR ^L LX -! R ^L LX = FX is a weak equivalence. By I.4.7 and I.6.7, there are natural weak equivalences KR ^ X -! KR ^ LX -! KR ^L LX: For the second statement, observe that i is the common composite in the diagram id^ R ^L LS _____//R ^LH S HHH~= id^Lj|| id^j|| HHH fflffl| fflffl|HH$$ R ^L LR id^_//_R ^L ROE_//_R: By I.8.6, the top map id^ is a weak equivalence. __|_ | Corollary 1.5. If X is a wedge of sphere spectra, then ss*(FX) is the free ss*(R)-module with one generator of degree n for each wedge summand Sn. We shall need one further fundamental property of free R-modules. Definition 1.6. A compact spectrum is one of the form 1VX for a compact space X and an indexing space V U. A compact R-module is one of the form FK for a compact spectrum K. 60 III. THE HOMOTOPY THEORY OF R-MODULES Proposition 1.7. Let L be a finite colimit of compact R-modules and let {Mi} be a sequence of R-modules and (spacewise) inclusions Mi- ! Mi+1. Then MR (L; colimMi) ~=colimMR (L; Mi): The generalization from compact R-modules to their finite colimits is immedia* *te. The compact case would be elementary if the free functor were left adjoint to t* *he forgetful functor, and we shall show in XIx2 that this is near enough to being * *true to give the conclusion. While they play a less central role, we shall also make use of cofree R-modul* *es. Recall from Ix4 that L# : S - ! S is the right adjoint of L and gives a comonad whose coalgebras are the L-spectra. In particular, L# X is an L-spectrum for any spectrum X. Definition 1.8. Define the cofree S-module generated by a spectrum X to be F#SX = S ^L L# X. Then define the cofree R-module generated by X to be F#RX = FS(R; F#SX) with left action of R induced by the right action of R on itself. We abbreviate F# X = F#RX when R is clear from the context. The term "cofree" is not a misnomer, since here we do have the expected ad- junction. Proposition 1.9. The functor F#R: S - ! MR is right adjoint to the forgetful functor MR -! S . Proof. Let M be an R-module and X be a spectrum. Lemma 5.5(ii) below gives the first of the following isomorphisms, and II.1.3 and I.4.7 give the ot* *hers: MR (M; F#RX) ~=MS(M; F#SX) ~=S [L](M; L# X) ~=S (M; X): __|_ | Theorem 1.10. In the stable homotopy category hS , F# X is naturally isomor- phic to the internal function spectrum F (R; X). Proof. This is immediate from II.1.9, I.7.3, and I.4.9. __|_ | 2. Cell and CW R-modules; the derived category of R-modules To develop cell and CW theories for R-modules, we think of the free R-modules SnR FSn as "sphere R-modules". This is consistent with the sphere S-modules of II.1.7. For cells, we note that the cone functor CX = X ^ I commutes with F, so that CFSn ~=FCSn. Since F has a right adjoint, maps out of sphere R-modules and their cones are induced by maps on the spectrum level; the fact that the ri* *ght adjoint is not the obvious forgetful functor will create no difficulties. In fa* *ct, we 2. CELL AND CW R-MODULES; THE DERIVED CATEGORY OF R-MODULES 61 can simply parrot the cell theory of spectra from [37, Ix5], reducing proofs to* * those given there via adjunction. Definitions 2.1. We define cell and relative cell R-modules. (i)A cell R-module M is the union of an expanding sequence of sub R-modules Mn such that M0 = * and Mn+1 is the cofiber of a map OEn : Fn -! Mn, where Fn is a (possibly empty) wedge of sphere modules SqR(of varying dimensions). The restriction of OEn to a wedge summand SqRis called an attaching map. The induced map CSqR-! Mn+1 M is called a cell. The sequence {Mn} is called the sequential filtration o* *f M. (ii)For an R-module L, a relative cell R-module (M; L) is an R-module M specified as in (i), but with M0 = L. (ii)A map f : M -! N between cell R-modules is sequentially cellular if f(Mn) Nn for all n. (iii)A submodule L of a cell R-module M is a cell submodule if L is a cell R-module such that Ln Mn and the composite of each attaching map SqR- ! Ln of L with the inclusion Ln -! Mn is an attaching map of M. Thus every cell of L is a cell of M. Observe that (M; L) may be viewed as a relative cell R-module. (iv)A cell R-module is finite dimensional if it has cells in finitely many di* *men- sions. It is finite if it has finitely many cells. The sequential filtration is essential for inductive arguments, but it should* * be regarded as flexible and subject to change whenever convenient. It merely recor* *ds the order in which cells are attached and, as long as the cells to which new ce* *lls are attached are already present, it doesn't matter in what order cells are attache* *d. Lemma 2.2. Let f : M - ! N be an R-map between cell R-modules. Then M admits a new sequential filtration with respect to which f is sequentially cell* *ular. Proof. Assume inductively that Mn has been given a filtration as a cell R- module Mn = [M0qsuch that f(M0q) Nq for all q. Let O : SrR- ! Mn be an attaching map for the construction of Mn+1 from Mn and let "O: CSrR- ! Mn+1 be the corresponding cell. By Proposition 1.7, there is a minimal q such that b* *oth Im (O) M0qand Im(f O "O) Nq+1: Extend the filtration of Mn to Mn+1 by taking O to be a typical attaching map of a cell of M0q+1. __|_ | 62 III. THE HOMOTOPY THEORY OF R-MODULES We shall occasionally need the following two reassuring results. Their proofs* * are similar to those of their spectrum level analogs [37, pp 494-495] and depend on Proposition 1.7 and its proof. Lemma 2.3. A map from a compact R-module to a cell R-module has image contained in a finite subcomplex, and a cell R-module is the colimit of its fin* *ite subcomplexes. If K and L are subcomplexes of a cell R-module M, then we understand their intersection and union in the combinatorial sense. That is, K \ L is the cell * *R- module constructed from the attaching maps and cells that are in both K and L and K [ L is the cell R-module constructed from the attaching maps and cells that are in either K or L. However, we also have their categorical intersection* * and union, namely the pullback of the inclusions of K and L in M and the pushout of the resulting maps from the categorical intersection to K and to L. Lemma 2.4. For subcomplexes K and L of a cell R-module M, the canonical map from the combinatorial intersection to the categorical intersection and from the categorical union to the combinatorial union of K and L are isomorphisms of R-modules. Definition 2.5. A cell R-module M is said to be a CW R-module if each cell is attached only to cells of lower dimension. The n-skeleton Mn of a CW R-module is the union of its cells of dimension at most n. A map f : M - ! N between CW R-modules is cellular if f(Mn) Nn for all n. We do not require that f also be sequentially cellular but, by Lemma 2.2, that can always be arranged by changing the order in which cells are attached. Relative CW R-modules (M; L) are defined similarly, with each cell attached only to the union of L and the c* *ells of lower dimension. Proposition 2.6. The collection of cell R-modules enjoys the following closu* *re properties. (i)A wedge of cell R-modules is a cell R-module. (ii)The pushout of a map along the inclusion of a cell submodule is a cell R-module. (iii)The union of a sequence of inclusions of cell submodules is a cell R-mod* *ule. (iv)The smash product of a cell R-module and a based cell space (with based attaching maps) is a cell R-module. (v)The smash product over S of a cell R-module and a cell S-module is a cell R-module. The same statements hold with "cell" replaced by "CW", provided that, in (ii), * *the given map is cellular. 2. CELL AND CW R-MODULES; THE DERIVED CATEGORY OF R-MODULES 63 Proof. In (ii), we apply Lemma 2.2 to ensure that the given map is sequentia* *lly cellular. Part (v) follows from I.6.1, which implies that the smash product of* * a sphere R-module and a sphere S-module is a sphere R-module. Otherwise the proofs are the same as for cell and CW spectra [37, Ix5]. __|_ | The following result is the "Homotopy Extension and Lifting Property". Theorem 2.7 (HELP). Let (M; L) be a relative cell R-module and let e : N - ! P be a weak equivalence of R-modules. Then, given maps f : M - ! P , g : L - ! N, and h : L ^ I+ - ! P such that f|L = hi0 and eg = hi1 in the following diagram, there are maps "gand "hthat make the entire diagram commute. i0 i1 L _____________//L ^ I+oo__________ L | h ww | g """| | www | "" | | www | "" | | --w |e """ | | P oo______|________cNcG | | ">> G | ``A | | f """ G | A | | "" G G | A | fflffl|"" "h fflffl| "g A fflffl| M ______i0___//_M ^ I+oo__i1______M Proof. This is proven for (M; L) = (CSqR; SqR) by reduction to the spectrum level analog. Technically, we use that the fact that our spheres are obtained f* *rom sphere spectra by applying a functor that is left adjoint to a functor that pre* *serves weak equivalences (even though it is not the obvious forgetful functor). The ge* *neral case follows by induction up the sequential filtration, and the inductive step * *reduces directly to the case of (CSqR; SqR) already handled. __|_ | The Whitehead theorem is a formal consequence. Theorem 2.8 (Whitehead). If M is a cell R-module and e : N -! P is a weak equivalence of R-modules, then e* : hMR (M; N) - ! hMR (M; P ) is an isomorphism. Therefore a weak equivalence between cell R-modules is a homotopy equivalence. Recall that a spectrum is "connective" if it is (-1)-connected. When R is con- nective, ssq(N=Nq) = 0 for any CW R-module and we can prove the following cellular approximation theorem exactly as in [37, I.5.8]. For non-connective R,* * this result fails and we must content ourselves with cell R-modules. For connective * *R, there is no significant loss of information if we restrict attention to CW R-mo* *dules. Theorem 2.9 (Cellular approximation). Assume that R is connective and let (M; L) and (M0; L0) be relative CW R-modules. Then any map f : (M; L) -! (M0; L0) is homotopic relative to L to a cellular map. Therefore, for cell R-mo* *dules 64 III. THE HOMOTOPY THEORY OF R-MODULES M and M0, any map M -! M0 is homotopic to a cellular map, and any two ho- motopic cellular maps are cellularly homotopic. Theorem 2.10 (Approximation by cell modules). For any R-module M, there is a cell R-module M and a weak equivalence fl : M -! M. If R is connective, then M can be chosen to be a CW R-module. Proof. Choose a wedge of sphere R-modules N0 and a map fl0 : N0 -! M that induces an epimorphism on homotopy groups. Given fln : Nn -! M, we construct Nn+1 from Nn as a homotopy coequalizer of pairs of representative maps for all pairs of unequal elements of any ssq(Nn) that map to the same element in ssq(M). We have homotopies that allow us to extend fln to fln+1. We let M be the union of the Nn, and the fln give a map fl : M - ! M. We deduce from Proposition 1.7 that fl is a weak equivalence, and we deduce from Proposition 2.6 that M is a cell R-module. If R is connective, we may take our representative maps to be cellular, and N is then a CW R-module. __|_ | Construction 2.11. For each R-module M, choose a cell R-module M and a weak equivalence fl : M - ! M. By the Whitehead theorem, for a map f : M -! N, there is a map f : M -! N, unique up to homotopy, such that the following diagram is homotopy commutative: f M ____//_N fl|| |fl| fflffl| fflffl| M __f__//_N: Thus is a functor hMR - ! hMR , and fl is a natural transformation from to the identity. The derived category DR can be described as the category whose objects are the R-modules and whose morphisms are specified by DR (M; N) = hMR (M; N); with the evident composition. When M is a cell R-module, DR (M; N) ~=hMR (M; N): Using the identity function on objects and on morphisms, we obtain a functor i : hMR - ! DR that sends weak equivalences to isomorphisms and is universal with this property. Let CR be the full subcategory of MR whose objects are the * *cell R-modules. Then the functor induces an equivalence of categories DR -! hCR with inverse the composite of i and the inclusion of hCR in hMR . 3. THE SMASH PRODUCT OF R-MODULES 65 Therefore the derived category and the homotopy category of cell R-modules can be used interchangeably. Homotopy-preserving functors on R-modules that do not preserve weak equivalences are transported to the derived category by fi* *rst applying , then the given functor. The category DR has all homotopy limits and colimits. They are created as the corresponding constructions on the underlying diagrams of S-modules; equiva- lently, homotopy colimits are created on the spectrum level and homotopy limits* * are created from spectrum level homotopy limits, which are S [L]-spectra, by applyi* *ng the functor S ^L (?). Explicit functorial constructions will be given in Xx3. * *We have enough information to quote the categorical form of Brown's representabili* *ty theorem given in [13]. Adams' analogue [3] for functors defined only on finite * *CW spectra also applies in our context, with the same proof. Theorem 2.12 (Brown). A contravariant functor k : DR ! Sets is repre- sentable in the form k(M) ~= DR (M; N) for some R-module N if and only if k converts wedges to products and converts homotopy pushouts to weak pullbacks. Theorem 2.13 (Adams). A contravariant group-valued functor k defined on the homotopy category of finite cell R-modules is representable in the form k(M* *) ~= DR (M; N) for some R-module N if and only if k converts finite wedges to direct products and converts homotopy pushouts to weak pullbacks of underlying sets. 3. The smash product of R-modules We mimic the definition of tensor products of modules over algebras. Definition 3.1. Let R be an S-algebra and let M be a right and N be a left R-module. Define M ^R N to be the coequalizer displayed in the following diagram of S-modules: __^Sid____ M ^S R ^S N __________////M ^S N____//M ^R N; id^S where and are the given actions of R on M and N. When R = S, we are coequalizing the same isomorphism (see I.8.3). Therefore our new M ^S N coincides with our old M ^S N. We shall shortly construct function R-modules satisfying the usual adjunction. It will follow that the functor ^R preserves colimits in each of its variables.* * It is clear that smash products with spaces commute with ^R , in the sense that (X ^ M) ^R N ~=X ^ (M ^R N) ~=(M ^R N) ^ X ~=M ^R (N ^ X): Therefore the functor ^R commutes with cofiber sequences in each of its variabl* *es. We also have the following adjunction, which complements Theorem 1.1(iv). 66 III. THE HOMOTOPY THEORY OF R-MODULES Lemma 3.2. For an S-module K, MS(M ^R N; K) ~=MR (N; FS(M; K)): The commutativity, associativity, and unity properties of the smash product o* *ver S and comparisons of coequalizer diagrams give commutativity, associativity, and unity properties of the smash product over R, exactly as in algebra. We state t* *hese properties in the generality of their algebraic counterparts. An S-algebra R with product OE : R ^S R ! R has an opposite S-algebra Rop with product OE O o, and a left R-module with action is a right Rop-module with action O o. Lemma 3.3. For a right R-module M and left R-module N, M ^R N ~=N ^RopM: For S-algebras R and R0, we define an (R; R0)-bimodule to be a left R and rig* *ht R0-module M such that the evident diagram commutes: R ^S M ^S R0 ____//_M ^S R0 | | | | fflffl| fflffl| R ^S M __________//M: As in algebra, an (R; R0)-bimodule is the same thing as an (R ^S (R0)op)-module. Proposition 3.4. Let M be an (R; R0)-bimodule, N be an (R0; R00)-bimodule, and P be an (R00; R000)-bimodule. Then M ^R0N is an (R; R00)-bimodule and (M ^R0N) ^R00P ~=M ^R0(N ^R00P ) as (R; R000)-bimodules. The unity isomorphism has already been displayed, in the guise of a split co- equalizer diagram, in II.3.2. We restate the conclusion. Lemma 3.5. The action : R ^S N - ! N of an R-module N factors through an isomorphism of R-modules : R ^R N -! N: For an S-module K, R ^S K ~= K ^S R is an (R; R)-bimodule. In particular, this applies to the free left R-module FR X = R ^S FSX generated by a spectrum X, which may be identified with the free right R-module generated by X. The following instances of the isomorphisms above will be used in conjunction with the weak equivalences of I.6.7 and II.1.9. They allow us to deduce homotopical properties of ^R from corresponding properties of ^S. 3. THE SMASH PRODUCT OF R-MODULES 67 Proposition 3.6. Let K and L be S-modules and let N be an R-module. There is a natural isomorphism of R-modules (K ^S R) ^R N ~=K ^S N: There is also a natural isomorphism of (R; R)-bimodules (K ^S R) ^R (R ^S L) ~=R ^S (K ^S L): Using I.6.1, we obtain the following consequence, in which we use an isomor- phism of universes f : U U ! U to define the internal smash product f*(X ^ Y ). Corollary 3.7. Let X and Y be spectra and let N be an R-module. There is a natural isomorphism of R-modules FR X ^R N ~=FSX ^S N: There is also a natural isomorphism of (R; R)-bimodules FR X ^R FR Y ~= FR f*(X ^ Y ): Theorem 3.8. If M is a cell R-module and OE : N -! N0 is a weak equivalence of R-modules, then id^R OE : M ^R N -! M ^R N0 is a weak equivalence of S-modules. Proof. When M = FR X for a CW spectrum X, the conclusion is immediate from the corollary and I.6.6. The general case follows from the case of sphere R-modules by induction up the sequential filtration and passage to colimits. _* *_|_ | We construct ^R as a functor rDR x `DR ! DS by approximating one of the variables by a cell R-module; here "r" and "`" indi* *cate right and left R-modules. That is, the derived smash product of M and N is represented by M ^R N. The following technical sharpening of Corollary 1.5 will be the starting poin* *t for our later construction of a spectral sequence for the computation of ss*(M ^R N* *). Proposition 3.9. Let X be a wedge of sphere spectra and let N be a cell R- module. Then there is an isomorphism ss*(FR X ^R N) ~=(ss*(R) H*(X)) ss*(R)ss*(N) that is natural in the R-modules FR X and N. 68 III. THE HOMOTOPY THEORY OF R-MODULES Proof. The point is that naturality on general maps g : FR X -! FR X0, with their induced maps ss*(R) H*(X) -! ss*(R) H*(X0), and not just on maps of the form g = FR f, f : X - ! X0, will be essential in the cited application. T* *he diagram FR X ^S R ^S N ____//_//_FR X ^S_N_//_FR X ^R N is a split coequalizer in MS and thus in S , and it is visibly natural in both FR X and N. It remains a coequalizer on applying ss*, and the required naturali* *ty follows. __|_ | Finally, we record an analogue of the behavior of tensor products of modules with respect to tensor products of algebras. Proposition 3.10. Let R and R0 be S-algebras, M and N be right and left R-modules, and M0 and N0 be right and left R0-modules. Then there is a natural isomorphism of S-modules (M ^S M0) ^R^SR0 (N ^S N0) ~=(M ^R N) ^S (M0^R0N0): If M is a cell R-module and N0 is a cell R0-module, then M ^S N0 is a cell R^S * *R0- module. Proof. The first statement is a comparison of coequalizer diagrams. The sec- ond statement holds since, on spheres, I.6.1 implies isomorphisms (LSq ^L R) ^S (R0^L LSr) ~=(R ^S R0) ^L LSq+r: __|_ | 4. Change of S-algebras; q-cofibrant S-algebras In this section, we assume given a map of S-algebras OE : R -! R0; and we study the relationship between the categories of R-modules and of R0- modules. By pullback along OE, we obtain a functor OE* : MR0 -! MR . It preserv* *es weak equivalences and thus induces a functor OE* : DR0 -! DR . It is vital to t* *he theory that this functor is an equivalence of categories when OE is a weak equi* *valence. As we explain, this allows us to replace general S-algebras by better behaved "* *q- cofibrant" ones whenever convenient, without changing the derived category. Regard R0as a right R-module via the composite id^OE 0 0 0 R0^S R ____//_R ^S R_____//R0: Observe that R0is an (R0; R)-bimodule with the evident left action by R0and tha* *t, for an R-module M, R0^R M is an R0-module. 4. CHANGE OF S-ALGEBRAS; q-COFIBRANT S-ALGEBRAS 69 Proposition 4.1. Define OE* : MR - ! MR0 by OE*M = R0^R M. Then OE* is left adjoint to OE*, and the adjunction induces a derived adjunction DR0(OE*M; M0) ~=DR (M; OE*M0): Moreover, the functor OE* preserves cell modules. Proof. The required isomorphism MR0(OE*M; M0) ~=MR (M; OE*M0) is proven exactly as in algebra. It sends an R-map M ! M0 to the induced composite R0^R M -! R0^R M0 -! R0^R0M0 ~=M0; and it sends an R0-map R0^R M -! M0 to its restriction along the canonical map M - ! R0^R M. Since the functor OE* preserves weak equivalences, it is formal that the functor OE* carries R-modules of the homotopy types of cell modules to R0-modules of the homotopy types of cell modules and induces an adjunction on derived categories [37, I.5.13]. Clearly R0^R (R ^S L) ~=R0^S L for an S-module L. Therefore the functor OE* carries sphere R-modules to sphere R0-modules. Since, as a left adjoint, OE* preserves colimits, this implies tha* *t OE* preserves cell modules and not just homotopy types of cell modules. __|_ | Theorem 4.2. Let OE : R -! R0be a weak equivalence of S-algebras. Then OE* : DR -! DR0 and OE* : DR0 -! DR are inverse adjoint equivalences of categories. Proof. If M is a cell R-module, then the unit OE ^R id: M ~=R ^R M -! R0^R M of the adjunction is a weak equivalence by Theorem 3.8. Now let M0 be an R0- module. In the derived category, the composite OE*OE*M0 means R0^R M0, where M0 is a cell R-module for which there is a weak equivalence of R-modules fl : M0 -! OE*M0. The counit of the adjunction is given by id^OEfl : R0^R M0 -! R0^R0M0 ~=M0: An easy diagram chase shows that the composite map of R-modules OE^Rid 0 0 id^OEfl 0 0 0 M0 ~=R ^R M0 ____//_R ^R M ____//_R ^R0M ~=M coincides with fl. Since OE ^R idis a weak equivalence, so is id^OEfl. __|_ | 70 III. THE HOMOTOPY THEORY OF R-MODULES We shall give the category of S-algebras a Quillen (closed) model category st* *ruc- ture in Chapter VII. We will then have the notion of a "q-cofibrant S-algebra", which is a retract of a "cell S-algebra". For any S-algebra R, there is a weak equivalence : R -! R, where R is a cell S-algebra. By the previous result, induces an adjoint equivalence between the categories DR and DR . Actually, we will have two quite different model categories, one for S-algebras and another * *for commutative S-algebras. The comments that we have just made apply in either context. As we shall explain in VIIx6, the forgetful functor from R-algebras to R-modules is better behaved homotopically in the non-commutative case than in the commutative case. In fact, VII.6.2 will give the following result. Theorem 4.3. If R is a q-cofibrant S-algebra, then (R; S) is a relative cel* *l S- module, the inclusion S - ! R being the unit of R. Therefore (R; S) has the homotopy type of a relative CW S-module. Since we can approximate a commutative S-algebra by a non-commutative cell S-algebra without changing the derived category of modules (up to equivalence), we can use the previous result to obtain homotopical information about the deri* *ved categories of commutative S-algebras. We illustrate the force of these ideas by using them to obtain a complementary adjunction to the case of Proposition 4.1 that is obtained by specializing to t* *he unit j : S -! R of an R-algebra: DR (R ^S M; N) ~=DS(M; j*N) for S-modules M and R-modules N. Proposition 4.4. The forgetful functor j* : DR - ! DS has a right adjoint j# : DS -! DR , so that DR (N; j# M) ~=DS(j*N; M) for S-modules M and R-modules N. Proof. On the point set level, we have the adjunction MR (j*N; M) ~=MS(N; FS(R; M)): Here we regard R as an (S; R)-bimodule, and the right action of R on itself ind* *uces a left action of R on FS(R; M) (as with Hom functors in algebra). However, ther* *e is no reason to believe that the functor FS(R; M) of M preserves weak equivalences, so that it is not clear how to pass to derived categories. Let : R -! R be a weak equivalence of S-algebras, where R is a cell S-algebra. It follows easily from the previous theorem that the functor FS(R; M) : MS -! MR 5. SYMMETRIC AND EXTENDED POWERS OF R-MODULES 71 of M does preserve weak equivalences. We therefore have an adjunction DS((j0)*N0; M) ~=DR (N; FS(R; M)) for R-modules N0 and S-modules M, where j0 is the unit of R. Theorem 4.2 implies that we also have an adjunction DR (*N; N0) ~=DR (N; *N0): Since j = O j0 : S -! R and these forgetful functors all preserve weak equiva- lences, j* = (j0)* O * : DR -! DS. We define j# (M) = *FS(R; M) and obtain the desired adjunction as the composite of the adjunctions just given. __|_ | 5. Symmetric and extended powers of R-modules Let R be an S-algebra and M be an R-module. The jth symmetric power of M is defined to be Mj=j and the jth extended power of M is defined to be DjM = (Ej)+ ^j Mj: In both notions, Mj denotes the j-th power of M with respect to ^R . One of the most striking features of our smash product of R-modules is that, in the derived category DR , these are essentially equivalent notions. This fact will give us* * ho- motopical control on free R-algebras and will play an important role in our stu* *dy of Bousfield localizations of R-algebras in Chapter VIII, but it will not be ne* *eded before then. To explain this fact, observe that I.5.6 implies that, for a spectrum K, we h* *ave an equivariant isomorphism (LK)j ~=L (j) n Kj; where the j-th power is taken with respect to ^L on the left and with respect * *to the external smash product on the right. Therefore (LK)j=j ~=L (j) nj Kj: Behavior like this propagates through our constructions. However, to retain suf* *ffi- cient homotopical control on our constructions to prove this, we must assume th* *at R is a q-cofibrant S-algebra (in either the non-commutative or the commutative sense) and apply results to be proven in VIIx6. Note that S itself is q-cofibra* *nt in both senses. Theorem 5.1. Let R be a q-cofibrant S-algebra or commutative S-algebra. If M is a cell R-module, then the projection ss : (Ej)+ ^j Mj -! Mj=j is a homotopy equivalence of spectra. 72 III. THE HOMOTOPY THEORY OF R-MODULES Proof. The conclusion is trivial for j = 1 and we may assume inductively that it holds for i < j. We first prove the result for any j when M is the free R-module generated by a CW-spectrum X. Expanding definitions and commuting the smash product with (Ej)+ through our constructions, we find that Mj ~=R ^S S ^L (L (j) n Xj); (Ej)+ ^ Mj ~=R ^S S ^L ((Ej x L (j)) n Xj); and ss is induced from the j-equivalence Ej x L (j) -! L (j) by passage to orbits. When R = S, ss is a homotopy equivalence of spectra by I.2.5 and I.8.5. For general R, VII.6.5 and VII.6.7 imply that the functor R ^S (?) carries this homotopy equivalence to a weak equivalence. However, the domain and target have the homotopy types of CW spectra, by VII.6.6. Next, let M be a subcomplex of a cell R-module N and assume that the con- clusion holds for M and N=M. As explained for the (external) smash power of spectra in [14, pp 37-38] and works equally well for the (internal) smash power* * of R-modules, we have a filtration of Nj by j-cofibrations of R-modules Mj = FjNj Fj-1Nj . . .F1Nj F0Nj = Nj: Here FiNj is the union of the subcomplexes M1 ^R . .^.RMj, where each Mk is either M or N and i of the Mk are M. The subquotients can be identified equivariantly as FiNj=Fi+1Nj ~=j xixj-i (Mi ^R (N=M)j-i): As a (ix j-i)-space, Ej is homotopy equivalent to Eix Ej-i, and there result homotopy equivalences (Ej)+ ^j FiNj=Fi+1Nj ' ((Ei)+ ^i Mi) ^R ((Ej-i)+ ^j-i (N=M)j-i): Applying the original induction hypothesis on j and inducting up the filtration* *, we deduce the conclusion for N. Finally, turning to the general case, let {Mn} be the sequential filtration * *of M, with M0 = *. By the first step, the conclusion holds for each Mn+1=Mn. By the second step, the conclusion for Mn implies the conclusion for Mn+1. Since Mj is the colimit of the sequence of j-cofibrations of R-modules (Mn)j -! (Mn+1)j, the conclusion for M follows. __|_ | 6. FUNCTION R-MODULES 73 6. Function R-modules Let R be an S-algebra. We have a function R-module functor FR to go with our smash product. Its definition is dictated by the expected adjunction. Definition 6.1. Let M and N be (left) R-modules. Define FR (M; N) to be the equalizer displayed in the following diagram of S-modules: _*___ FR (M; N) _____//FS(M; N)_!__////_FS(R ^S M; N): Here * = FS(; id) and ! is the adjoint of the composite R ^S (M ^S FS(M; N)) _id^//"_R ^S_N___//N: When R = S, our new and old function S-modules FS(M; N) are identical. We state the expected adjunction in a general form, but we are most interested in * *the case R0= S. Lemma 6.2. Let M be an (R; R0)-bimodule, N be an R0-module, and P be an R-module. Then MR (M ^R0N; P ) ~=MR0(N; FR (M; P )): Proof. The general case follows from the case R = S of Lemma 3.2 by use of the coequalizer definition of ^R0 and the equalizer definition of FR . __|_ | As in algebra, this leads to a function module analogue of Proposition 3.4. Proposition 6.3. Let M be an (R; R0)-bimodule, N be an (R0; R00)-bimodule, and P be an (R; R000)-bimodule. Then FR (M; P ) is an (R0; R000)-bimodule, and FR (M ^R0N; P ) ~=FR0(N; FR (M; P )) as (R00; R000)-bimodules. Similarly, the unit isomorphism of Lemma 3.5 implies a counit isomorphism. Lemma 6.4. The adjoint " : M ! FR (R; M) is an isomorphism. We also have analogs of Proposition 3.6 and Corollary 3.7. While we are inter- ested primarily in the versions relating FR to the functor ^S, there are also v* *ersions relating FR to the functor FS. The following lemma is needed for the latter ve* *r- sions. Its algebraic analogue is proven by a formal argument that applies equal* *ly well in topology. Lemma 6.5. Let R and R0 be S-algebras. 74 III. THE HOMOTOPY THEORY OF R-MODULES (i)Let M be an R-module, M0 be an R0-module and P be an R ^S R0-module. Then there is a natural bijection MR (M; FR0(M0; P )) ~=MR^SR0(M ^S M0; P ): (ii)Let M be a left R-module, N be a right R-module, and K be an S-module. Then there is a natural bijection MR (M; FS(N; K)) ~=MS(N ^R M; K): Proof. It suffices to check (i) when M = R ^S L and M0 = R0^S L0 are the free modules generated by S-modules L and L0. Similarly, it suffices to check (* *ii) when M = R ^S L. These cases are easy consequences of our adjunctions. __|_ | Proposition 6.6. Let K be an S-module and M be a left R-module. There is a natural isomorphism of left R-modules FR (K ^S R; M) ~=FS(K; M) and a natural isomorphism of right R-modules FR (M; FS(R; K)) ~=FS(M; K): Proof. The first isomorphism is immediate from the following chain of isomor- phisms of represented functors on left R-modules N, which result from Propositi* *on 6.3, Proposition 3.6, and Theorem 1.1(iii), respectively. MR (N; FR (K ^S R; M)) ~=MR ((K ^S R) ^R N; M) ~=MR (K ^S N; M) ~=MR (N; FS(K; M)): The second isomorphism results from the following chain of isomorphisms of rep- resented functors on right R-modules N: MRop(N; FR (M; FS(R; K))) ~=MR^SRop(M ^S N; FS(R; K)) ~=MS(R ^R^SRop (M ^S N); K) ~=MS(M ^RopN; K) ~=MRop(N; FS(M; K)): The first two isomorphisms are instances of isomorphisms of the lemma. The third follows from the fact that there is a natural isomorphism R ^R^SRop (M ^S N) ~=M ^RopN; as is easily checked when M and N are free R-modules and follows in general. _* *_|_ | 6. FUNCTION R-MODULES 75 Corollary 6.7. Let X be a spectrum and M be an R-module. There is a natural isomorphism of left R-modules FR (FR X; M) ~=FS(FSX; M) and a natural isomorphism of right R-modules FR (M; F#RX) ~=FS(M; F#SX)): The functor FR (M; N) converts colimits and cofiber sequences in M to limits and fiber sequences and it preserves limits and fiber sequences in N, as we see formally on the spectrum level (compare [37, III.2.5]) and deduce in order on t* *he levels of L-spectra, S-modules, and R-modules (compare II.1.5 and Theorem 1.1). Using the previous corollary to deal with sphere R-modules and proceeding by induction up the sequential filtration of M, we obtain the analogue of Theorem 3.8. Theorem 6.8. If M is a cell R-module and OE : N -! N0 is a weak equivalence of R-modules, then FR (id; OE) : FR (M; N) -! FR (M; N0) is a weak equivalence. In the derived category DR , FR (M; N) means FR (M; N), where M is a cell approximation of M. We are entitled to conclude that DR (M ^S N; P ) ~=DS(N; FR (M; P )): As in Proposition 3.9, we have the following calculational sharpening of Coro* *llary 6.7. It will be the starting point for our later construction of a spectral seq* *uence for the calculation of ss*(FR (M; N)). Corollary 6.9. Let X be a wedge of sphere spectra and N be an R-module. Then there is an isomorphism ss*(FR (FX; N)) ~=Hom ss*(R)(ss*(R) H*(X); ss*(N)) that is natural in the R-modules FX and N. We end this section by recording a composition pairing that is a formal im- plication of Lemma 6.2 and Proposition 6.3. This works exactly as with tensor products and Hom in algebra and, as there, it is convenient for this purpose to use the commutativity of the smash product over S to rewrite our adjunctions and isomorphisms with their variables occurring in the same order on both sides, 76 III. THE HOMOTOPY THEORY OF R-MODULES returning to our original conventions of Ix7. Thus, for an S-module L and for R-modules M and N, we have the natural isomorphism of S-modules (6.10) FR (L ^S M; N) ~=FS(L; FR (M; N)): Let P be another R-module. Using the evaluation R-map " : FR (M; N) ^S M -! N; we obtain a composite R-map id^S" " FR (N; P ) ^S FR (M; N) ^S M_____//FR (N; P ) ^S_N__//_P: Its adjoint is a composition pairing of S-modules (6.11) ss : FR (N; P ) ^S FR (M; N) -! FR (M; P ): This pairing is unital and associative in the sense that the following diagrams commute; let j : S -! FR (M; M) be the adjoint of : S ^S M -! M: FR (N; P ) ^SSS SSSS id^Sj|| SoSSSSSS fflffl| S))S FR (N; P ) ^S FR (N; N)ss_//FR (N; P ); S ^S FR (M; N) SSSSS j ^Sid|| SSSSSSS fflffl| SSS)) FR (N; N) ^S FR (M; N)__ss_//FR (M; N); and, for another R-module L, id^Sss FR (N; P ) ^S FR (M; N) ^S FR (L; M)___//_FR (N; P ) ^S FR (L; N) ss ^Sid|| |ss| fflffl| fflffl| FR (M; P ) ^S FR (L; M)_______ss______//_FR (L; P ): This leads to a host of examples of S-algebras and their modules. Proposition 6.12. Let R be an S-algebra and let M and N be (left) R-modules. Then FR (N; N) is an S-algebra with product ss and unit j. Moreover, FR (M; N) is an (FR (N; N); FR (M; M))-bimodule with left and right actions given by ss. 7. COMMUTATIVE S-ALGEBRAS AND DUALITY THEORY 77 7. Commutative S-algebras and duality theory We assume that R is a commutative S-algebra in this section, and we show that the study of modules over R works in exactly the same way as the study of modul* *es over commutative algebras. If : R^S M ! M gives M a left R-module structure, then O o : M ^S R ! M gives M a right R-module structure such that M is an (R; R)-bimodule. As in the study of modules over commutative algebras, this leads to the following important conclusion. Theorem 7.1. If M and N are R-modules, then M ^R N and FR (M; N) have canonical R-module structures induced from the R-module structure of M or, equi* *v- alently, N. The smash product over R is commutative, associative and unital. There is an adjunction (7.2) MR (L ^R M; N) ~=MR (L; FR (M; N)): Moreover, the adjunction passes to derived categories. We have the following consequence of Corollary 3.7. Proposition 7.3. If M and M0 are cell R-modules, then M ^R M0 is a cell R-module with one (p + q)-cell for each p-cell of M and q-cell of M0. For R-modules L, M and N, we have a natural isomorphism of R-modules (7.4) FR (L ^R M; N) ~=FR (L; FR (M; N)) because both sides represent the same functor. Exactly as in the previous secti* *on, but working entirely with R-modules, we obtain a natural associative and R-unit* *al composition pairing (7.5) ss : FR (M; N) ^R FR (L; M) -! FR (L; N): The formal duality theory explained in [37, Ch. III] applies to the stable ca* *tegory of R-modules. We define the dual of an R-module M to be DR M = FR (M; R). We have an evaluation map " : DR M ^R M -! R and a map j : R ! FR (M; M), namely the adjoint of : R ^R M -! M. There is also a natural map (7.6) : FR (L; M) ^R N -! FR (L; M ^R N): By composition with the isomorphism FR (id; ), specializes to a map (7.7) : DR M ^R M -! FR (M; M): 78 III. THE HOMOTOPY THEORY OF R-MODULES We say that M is "strongly dualizable", if it has a coevaluation map j : R -! M ^R DR M such that the following diagram commutes in DR : j R _________//M ^R DR M (7.8) j || |o| fflffl| fflffl| FR (M; M) oo___DR M ^R M: The definition has many purely formal implications. The map of (7.6) is an isomorphism in DR if either L or N is strongly dualizable. The map of (7.7) is an isomorphism in DR if and only if M is strongly dualizable, and the coevaluat* *ion map j is then the composite o-1j in (7.8). The natural map ae : M -! DR DR M is an isomorphism in DR if M is strongly dualizable. The natural map ^ : FR (M; N) ^R FR (M0; N0) -! FR (M ^R M0; N ^R N0) is an isomorphism in DR if M and M0 are strongly dualizable or if M is strongly dualizable and N = R. Say that a cell R-module N is a wedge summand up to homotopy of a cell R- module M if there is a homotopy equivalence of R-modules between M and N _N0 for some cell R-module N0. In contrast with the usual stable homotopy category, if M is finite it does not follow that N must have the homotopy type of a finit* *e cell R-module. Via Eilenberg-Mac Lane spectra, finitely generated projective modules that are not free give rise to explicit counterexamples. Define a semi-finite * *R- module to be an R-module that is a wedge summand up to homotopy of a finite cell R-module, and note for use in Chapter VI that this notion makes sense even when R is not commutative. Theorem 7.9. A cell R-module is strongly dualizable if and only if it is se* *mi- finite. Proof. Observe first that SqRis strongly dualizable with dual S-qR, hence any finite wedge of sphere R-modules is strongly dualizable. Observe next that the cofiber of a map between strongly dualizable R-modules is strongly dualizable. * *In fact, the evaluation map " induces a natural map "# : DR (L; N ^R DR M) ! DR (L ^R M; N); and M is strongly dualizable if and only if "# is an isomorphism for all L and N [37, III.1.6]. Since both sides convert cofiber sequences in the variable M int* *o long exact sequences, the five lemma gives the observation. We conclude by induction on the number of cells that a finite cell R-module is strongly dualizable. It * *is 7. COMMUTATIVE S-ALGEBRAS AND DUALITY THEORY 79 formal that a wedge summand in DR of a strongly dualizable cell R-module is strongly dualizable. For the converse, let N be a cell R-module that is strong* *ly dualizable with coevaluation map j : R ! N ^R DR N. Since j is determined by its restriction to S and S is compact, j factors through M ^R DR N for some fin* *ite cell subcomplex M of N. By [37, III.1.2], the bottom composite in the following commutative diagram is the identity (in DR ): M ^R5DR5N ^R N id^"//_M ^R R_'__//_M kkk kkkk | | | kkkk | | | kkk fflffl| fflffl|' fflffl| N ~=R ^R N j^id//_N ^R DR N ^R N id^"//_N ^R R____//_N: Therefore N is a retract up to homotopy and thus, by a comparison of exact triangles, a wedge summand up to homotopy of M: retractions split in triangulat* *ed categories. __|_ | 80 III. THE HOMOTOPY THEORY OF R-MODULES CHAPTER IV The algebraic theory of R-modules We define generalized Tor and Ext groups as the homotopy groups of derived smash product and function modules, and we interpret these groups in terms of general- ized homology and cohomology theories on R-modules. Specializing to Eilenberg- Mac Lane spectra, these groups give the classical Tor and Ext groups, and we show how to topologically realize classical algebraic derived categories of com- plexes of modules over a ring. Starting with a connective S-algebra R, rather t* *han an Eilenberg-Mac Lane spectrum, the discussion generalizes to give ordinary ho- mology and cohomology theories on R-modules, together with Atiyah-Hirzebruch spectral sequences for the computation of generalized homology and cohomology theories on R-modules. In Sections 4 and 5, we construct "hyperhomology" spectral sequences for the calculation of our generalized Tor and Ext groups in terms of ordinary Tor and * *Ext groups, and we show that these specialize to give universal coefficient and K"u* *nneth spectral sequences for homology and cohomology theories defined on spectra. In Sections 6 and 7, we generalize to Eilenberg-Moore spectral sequences for the computation of E*(M ^R N) under varying hypotheses on R and E. In particular, we give a bar construction approximation to M ^R N that allows us to view the classical space level Eilenberg-Moore-Rothenberg-Steenrod spectral sequence as a special case. Except that his theory was intrinsically restricted to the A1 context, Robin* *son's series of papers [56, 57, 58, 59, 60] gave earlier versions of many of the resu* *lts of this chapter. Of course, with the earlier technology, the proofs were substanti* *ally more difficult. As usual, for a spectrum E, we shall often abbreviate notations by setting En = ssn(E) = E-n : 81 82 IV. THE ALGEBRAIC THEORY OF R-MODULES 1. Tor and Ext; homology and cohomology; duality Definition 1.1. Let R be an S-algebra. For a right R-module M and a left R-module N, define Tor Rn(M; N) = ssn(M ^R N): For left R-modules M and N, define ExtnR(M; N) = ss-n(FR (M; N)): Here the smash product and function modules are understood to be taken in the derived category DR . For Tor, this means that M or N must be replaced by a weakly equivalent cell R-module before applying the module level functor ^R . F* *or Ext, this means that M must be approximated by a cell R-module before applying FR . At this point in our work, however, we act as traditional topologists, tak* *ing it for granted that all spectra and modules are to be approximated as cell modules, without change of notation, whenever necessary. We will point out explicitly any places where this gives rise to mathematical issues. Clearly TorR*(M; N) and Ext*R(M; N) are R*-modules when R is commutative. Various properties reminiscent of those of the classical Tor and Ext functors f* *ollow directly from the definition and the results of the previous chapters. The intu* *ition is that the definition gives an analogue of the differential Tor and Ext funct* *ors (alias hyperhomology and cohomology functors) in the context of differential gr* *aded modules over differential graded algebras. In particular, the grading should no* *t be thought of as the resolution grading of the classical torsion product, but rath* *er as a total grading that sums a resolution degree and an internal degree; this idea* * will be made precise by the grading of the spectral sequences that we shall construct for the calculation of these functors. Proposition 1.2. Tor R*(M; N) satisfies the following properties. (i)If R, M, and N are connective, then TorRn(M; N) = 0 for n < 0. (ii)A cofiber sequence N0 ! N ! N00gives rise to a long exact sequence . .!.TorRn(M; N0) ! TorRn(M; N) ! TorRn(M; N00) ! TorRn-1(M; N0) ! . .:. (iii)TorR*(M; R) ~=ss*(M) and, for a spectrum X, TorR*(M; FX) ~=ss*(M ^ X): (iv)The functor TorR*(M; ?) carries wedges to direct sums. Proof. In (i), M and N can be approximated by CW R-modules with cells of non-negative dimension, hence it suffices to check the conclusion for N = Sr* *R, r 0, in which case it is immediate from (iii). Part (iii) follows from III.1.4* * and III.3.7. __|_ | 1. TOR AND EXT; HOMOLOGY AND COHOMOLOGY; DUALITY 83 The commutativity and associativity relations for the smash product imply var- ious further properties. We content ourselves with the following specialization. Proposition 1.3. If R is commutative, then TorR*(M; N) ~=TorR*(N; M) and Tor R*(M ^R N; P ) ~=TorR*(M; N ^R P ): Say that a spectrum N is coconnective if ssqN = 0 for q > 0. Proposition 1.4. Ext *R(M; N) satisfies the following properties. (i)If R and M are connective and N is coconnective, then ExtnR(M; N) = 0 for n < 0. (ii)Fiber sequences N0 ! N ! N00and cofiber sequences M0 ! M ! M00give rise to long exact sequences . .!.ExtnR(M; N0) ! ExtnR(M; N) ! ExtnR(M; N00) ! Extn+1R(M; N0) ! . . . and . .!.ExtnR(M00; N) ! ExtnR(M; N) ! ExtnR(M0; N) ! Extn+1R(M00; N) ! . .:. (iii)Ext*R(R; N) ~=ss-*(N) and, for a spectrum X, Ext*R(FX; N) ~=ss-*(F (X; N)) and Ext *R(M; F# X) ~=ss-*(F (M; X)): (iv)The functor Ext*R(?; N) carries wedges to products and the funct* *or Ext *R(M; ?) carries products to products. Proof. It suffices to check (i) for M = SrR, r 0, in which case the conclus* *ion is immediate from (iii). Part (iii) follows from III.1.4, III.1.10, and III.6.7* *. __|_ | Passing to homotopy groups from the pairings of III.6.11 and III.7.5, we obta* *in the following further property. Proposition 1.5. There is a natural, associative, and unital system of pairi* *ngs ss : Ext*R(M; N) ss*(S)Ext*R(L; M) -! Ext*R(L; N): If R is commutative, then these are pairings of R*-modules, and the tensor prod* *uct may be taken over R*. 84 IV. THE ALGEBRAIC THEORY OF R-MODULES Proof. The first statement is clear. The second uses the the fact that ssq(M* *) = DR (SqR; M) and the associative and unital system of equivalences of R-modules SqR^R SrR~=Sq+rR given by III.3.7. __|_ | The formal duality theory of IIIx6 implies the following result, together with various other such isomorphisms. Proposition 1.6. Let R be commutative. For a finite cell R-module M and any R-module N, TorRn(DR M; N) ~=Ext -nR(M; N): We think of the derived category DR as a stable homotopy category. Changing notations, we may reinterpret the functors Tor and Ext as prescribing homology and cohomology theories in this category. Definition 1.7. Let E0 be a right and E a left R-module. For left R-modules M and N, define E0Rn(M) = ssn(E0^R M) and EnR(M) = ss-n(FR (M; E)): The properties of Tor and Ext translate directly to statements about homology and cohomology. All of the standard homotopical machinery is available to us, a* *nd the previous result now takes the form of Spanier-Whitehead duality. Corollary 1.8. Let R be commutative. For a finite cell R-module M and any R-module E, ERn(DR M) ~=E-nR(M): Since the equivalence between the classical stable homotopy category and the derived category of S-modules preserves smash products and function spectra, we obtain versions of all of the usual homology and cohomology theories on spectra* * by taking R = S. Moreover the following reinterpretation of Propositions 1.2(iii) * *and 1.4(iii) shows that the specializations to R-modules of all of the usual homolo* *gy and cohomology theories on spectra are given by instances of our new homology and cohomology theories on R-modules. Corollary 1.9. For a spectrum E and a (left) R-module M, E*(M) ~=(FE)R*(M) and E*(M) ~=(F# E)*R(M): 2. EILENBERG-MAC LANE SPECTRA AND DERIVED CATEGORIES 85 2. Eilenberg-Mac Lane spectra and derived categories In this section, we change notation and let R denote a discrete ring. Applying multiplicative infinite loop space theory [49] to obtain an A1 ring spectrum a* *nd then applying the functor S ^L (?), we obtain an Eilenberg-Mac Lane spectrum HR = K(R; 0) that is an S-algebra and is a commutative S-algebra if R is com- mutative. An elaboration of multiplicative infinite loop space theory, followed* * by application of the functor S ^L (?), can be used to realize passage to Eilenbe* *rg- Mac Lane spectra as a point-set level functor H from R-modules in the sense of algebra to HR-modules. We shall shortly use the present theory to give two diff* *er- ent homotopical constructions of such Eilenberg-Mac Lane HR-modules. Granting this for the moment, we have the following result. Theorem 2.1. For a ring R and R-modules M and N, Tor R*(M; N) ~=TorHR*(HM; HN) and Ext*R(M; N) ~=Ext *HR(HM; HN): If R is commutative, then these are isomorphisms of R-modules. Under the second isomorphism, the topologically defined pairing Ext*HR(HM; HN) Ext*HR(HL; HM) -! Ext*HR(HL; HN) coincides with the algebraic Yoneda product. Proof. If 0 ! N0 ! N ! N00! 0 is a short exact sequence of R-modules, then HN0 ! HN ! HN00is equivalent to a cofiber sequence. The conclusion is now immediate from Propositions 1.2 and 1.4, together with the axioms for algebraic Tor and Ext functors. It should be noted that right exactness and pro* *per behavior on free modules together imply algebraically that TorR0(M; N) ~=M R N and Ext 0R(M; N) ~=Hom R(M; N): It is important to remember that the axioms for Ext require verifications about free or injective modules, but not both. The last statement follows from Yoneda* *'s axiomatization [70], which only requires proper behavior in degree zero and pro* *per behavior relating connecting homomorphisms to products. The last follows topo- logically from commutation with cofiber sequences, which is easily derived from the adjoint construction of our pairings in IIIx5. __|_ | We can elaborate this result to an equivalence of derived categories. We sha* *ll restrict attention to morphisms of degree zero since the extension to graded mo* *r- phisms is formal. Recall from [66] or [34, Ch.III] that the derived category DR* * is obtained from the homotopy category of chain complexes over R by localizing at 86 IV. THE ALGEBRAIC THEORY OF R-MODULES the quasi-isomorphisms, exactly as we obtained the category DHR from the homo- topy category of HR-modules by localizing at the weak equivalences. The algebra* *ic theory of cell and CW chain complexes over R in [34, Ch.III] makes the analogy especially close. The proof of the equivalence is quite simple. The category DHR is equivalent to the homotopy category of CW HR-modules and cellular maps. We will see that CW HR-modules have associated chain complexes. This gives a functor DHR - ! DR , and we will obtain an inverse functor directly from Brown's representability theorem. Definition 2.2. Let M be a CW HR-module. Define the associated chain complex C*(M) of R-modules by letting Cn(M) = ssn(Mn; Mn-1) and letting the differential dn : Cn(M) -! Cn-1(M) be the connecting homorphism of the triple (Mn; Mn-1; Mn-2). Observe that a cellular map of HR-modules induces a map of chain complexes and that a cellular homotopy induces a chain homotopy. Observe too that, since Mn=Mn-1 is a wedge of free modules SnHR ' HR ^ Sn, Cn(M) is a free R-module. Lemma 2.3. For CW HR-modules M, the homology groups H*(C*(M)) are nat- urally isomorphic to the homotopy groups of M. Proof. Since HR is connective, the inclusion Mn - ! M induces a bijection on ssq for q < n and a surjection on ssn. By induction up the sequential filtra* *tion of Mn-1, ssq(Mn-1) = 0 for q n. Therefore the quotient map M - ! M=Mn-1 induces a monomorphism on ssn. The conclusion follows by a simple diagram chase. __|_ | Theorem 2.4. The cellular chain functor C* on HR-modules induces an equiv- alence of categories DHR - ! DR . The inverse equivalence satisfies H*(X) ~=ss*((X)): Proof. The functor C* carries wedges to direct sums and carries homotopy colimits of cellular diagrams to chain level homotopy colimits. For a fixed ch* *ain complex X, the functor k on DHR specified by k(M) = DR (C*(M); X) there- fore satisfies the wedge and Mayer-Vietoris axioms. By Brown's representability theorem, III.2.12, k is represented by an HR-module spectrum (X). By the functoriality of the representation, this gives a functor : DR - ! DHR and an adjunction DR (C*(M); X) ~=DHR (M; (X)): Since Hn(X) ~=DR (nR; X), where nR is the free R-module on one generator of degree n and C*(SnHR) = nR, this implies that H*(X) ~=ss*((X)). We claim that 2. EILENBERG-MAC LANE SPECTRA AND DERIVED CATEGORIES 87 the unit j : M - ! (C*(M)) and counit " : C*((X)) -! X of the adjunction are natural isomorphisms. On hom sets, the functor C* coincides with j* : DHR (L; M) -! DHR (L; (C*(M))) ~=DR (C*(L); C*(M)): As L runs through the SnHR, j* runs through the isomorphisms ssn(M) -! Hn(C*(M)) of the previous lemma. Therefore j is an isomorphism in DHR for all M. Since the composite j " X -! C*(X) -! X is the identity, it follows that " is an isomorphism in DHR for all X. The foll* *owing natural diagram commutes: DHR (L; C*(X)) ~=DR (C*(L); C*(X)) (")*|| |"*| fflffl| fflffl| DHR (L; X) ~=DR (C*(L); X): As L runs through the sphere modules SnHR, the resulting isomorphisms "* show that " induces an isomorphism on all homology groups and is therefore an isomor- phism in DR . __|_ | In the commutative case, we have the following important addendum to the theorem. See [34, III] for a discussion of tensor product and Hom functors in t* *he derived category DR . As in topology, they are constructed by first applying CW approximation of R-modules and then taking the point-set level functor. Proposition 2.5. Assume that R is commutative. If M and N are CW HR- modules, then M ^HR N is a CW HR-module such that C*(M ^HR N) ~=C*(M) R C*(N): Therefore such an isomorphism holds in the derived category DR for general HR- modules M and N. Moreover, in DR , C*(FHR (M; N)) ~=Hom R(C*(M); C*(N)): If X and Y are chain complexes, then (X R Y ) ~=X ^HR Y and Hom R(X; Y ) ~=FHR (X; Y ) in DHR . 88 IV. THE ALGEBRAIC THEORY OF R-MODULES Proof. The first statement is implied by III.7.3, and the last three derived category level isomorphisms are all formal consequences of the first. __|_ | Regarding R-modules as chain complexes concentrated in degree zero, we see that the functor restricts to a functor H that assigns an Eilenberg-Mac Lane H* *R- module spectrum HM to an R-module M. We give a more explicit construction. Construction 2.6. (i) For an R-module M, we construct HM = K(M; 0) as a CW module L with sequential filtration {Ln} and skeletal filtration {Ln} rela* *ted by Ln-1 = Ln. Choose a free resolution . .-.! Fn-dn!Fn-1 -! . .-.! F0-"! M -! 0 of M. Let K0 be a wedge of 0-spheres, with one sphere for each basis element of F0. For n 1, let Kn be a wedge of (n - 1)-spheres, with one sphere for each basis element of Fn. Define L1 = FK0. For n 2, Ln will have two non-vanishing homotopy groups, namely ss0(Ln) = M and ssn-1(Ln) = Im dn, and the inclusion in : Ln -! Ln+1 will induce an isomorphism on ss0. By freeness, we can realize * *d1 by a map of HR-modules FK1 -! FK0. Let L2 be its cofiber. Then the resulting map FK0 -! L2 realizes " on ss0 and the resulting map L2 -! FK1 realizes the inclusion Im d2 F1 on ss1. Inductively, given Ln, we can realize dn : Fn -! Im* * dn on the (n - 1)st homotopy group by a map of HR-modules FKn - ! Ln. We let Ln+1 be its cofiber. The claimed properties follow immediately. The union L = [Ln is the desired CW HR-module HM. (ii) Given a map f : M - ! M0 of R-modules, we construct a cellular map Hf : HM - ! HM0 of CW HR-modules that realizes f on ss0. Construct L0 = HM0 as above, writing Fn0, etc. As usual, we can construct a sequence of R-maps fn : Fn -! Fn0that give a map of resolutions. We can realize fn on homotopy groups by an HR-map FKn -! FK0n. Starting with L1 = FK0 and proceeding inductively, we can use a standard cofibration sequence argument, carried out in the category of HR-modules, to construct HR-maps Ln - ! L0nsuch that the middle squares commute and the left and right squares commute up to homotopy in the following diagrams of HR-modules: FKn+1 _____//Ln____//Ln+1____//FKn+1 | | | | | | | | fflffl| |fflffl fflffl| |fflffl FK0n+1_____//L0n___//L0n+1___//FK0n+1: On passage to unions, we obtain the desired cellular map Hf : HM - ! HM0. A similar argument works to show that if we choose another map g* : F* -! F*0 of resolutions over f and repeat the construction, then the resulting HR-maps a* *re homotopic. 3. THE ATIYAH-HIRZEBRUCH SPECTRAL SEQUENCE 89 Remark 2.7. Since they are HZ -module spectra, the underlying spectra of the HR-module spectra studied in this section all have the homotopy types of Eilenberg-Mac Lane spectra. 3. The Atiyah-Hirzebruch spectral sequence We assume given a connective S-algebra R in this section, and we let k = ss0(* *R). Since R is connective, its derived category DR is equivalent to the homotopy ca* *te- gory hC WR of CW R-modules and cellular maps. We shall see that the Eilenberg- Mac Lane spectrum Hk is an R-module that plays a role in the study of R-modules analogous to the role played by HZ in the category of spectra. We use this insi* *ght to construct Atiyah-Hirzebruch spectral sequences and prove a Hurewicz theorem in the category of R-modules. Although we shall not assume this, the theory is * *most useful when R is commutative; of course, k may well be commutative even when R is not. Remember that modules mean left modules unless otherwise specified. Proposition 3.1. There is a map of S-algebras ss : R -! Hk that realizes the identity homomorphism on ss0(R) = k. Proof. We sketch two proofs. The first is an application of multiplicative infinite loop space theory. By [47, VII.2.4], the zeroth space R0 of R is an A1* * ring space. Modulo some point-set care to ensure continuity (e.g, we could replace R by a weakly equivalent "q-cofibrant" S-algebra, which is of the homotopy type o* *f a CW spectrum by VII.6.5 and VII.6.6), we obtain a discretization map ffi : R0 -!* * k, and it is immediate from the definitions that it is an A1 ring map. By [47, VI* *Ix4], there is a functor E from A1 ring spaces to A1 ring spectra, hence there resu* *lts a map of A1 ring spectra ER0 -! Ek. By [47, VII.3.2 and 4.3] and the connectivity of R, there is a natural weak equivalence of A1 ring spectra between ER0 and R, and the homotopical properties of E immediately imply that Ek is an Hk. Now apply the functor S ^L (?) to replace A1 ring spectra by S-algebras, and repl* *ace R by the weakly equivalent S-algebra S ^L ER0. The second proof makes more serious use of the Quillen model category structu* *re on the category of S-algebras that we construct in VIIxx4,5. Using it, we can m* *imic the classical space level argument of killing higher homotopy groups, successiv* *ely attaching cell S-algebras to kill the higher homotopy groups of R. __|_ | It follows that Hk is an (R; R)-bimodule. If R and therefore also Hk are com- mutative S-algebras, then Hk is a commutative R-algebra in the sense of VIIx1 below. If j is a k-module, then the Hk-module Hj is an R-module by pullback along ss. We consider the homology and cohomology theories represented by the Hj as ordinary homology and cohomology theories defined on the derived category of R-modules: they clearly satisfy the analogs of the Eilenberg-Steenrod axioms 90 IV. THE ALGEBRAIC THEORY OF R-MODULES for ordinary homology and cohomology theories. We agree to alter the notations of Definition 1.7 by setting (3.2) HR*(M; j0) = (Hj0)R*(M) and H*R(M; j) = (Hj)*R(M) for a left R-module M, a right k-module j0 and a left k-module j. We have symmetric definitions with left and right reversed. These theories can be calculated as the homology and cohomology of the cellul* *ar chain complexes of CW R-modules M. In fact, the definition of the associated chain complex of a CW R-module M is formally identical to Definition 2.2. Definition 3.3. Let M be a CW R-module. Define the associated chain com- plex CR*(M) of k-modules by letting CRn(M) = ssn(Mn; Mn-1) and letting the differential dn : CRn(M) -! CRn-1(M) be the connecting homorphism of the triple (Mn; Mn-1; Mn-2). Observe that a cellular map of R-modules induces a map of chain complexes and that a cellular homotopy induces a chain homotopy. Observe too that, since Mn=Mn-1 is a wedge of free modules SnR' R ^ Sn, CRn(M) is a free k-module. For right and left k-modules j0 and j, define chain and cochain complexes of abelian groups CR*(M; j0) = j0k CR*(M) and C*R(M; j) = Hom k(CR*(M); j): Clearly these chain and cochain functors induce covariant and contravariant functors from the derived category DR to the derived category DZ of chain com- plexes over Z, interpreted as homologically or cohomologically graded, respecti* *vely. When k is commutative, these functors take values in Dk. We have the following analogue of Proposition 2.5. Proposition 3.4. If R is a commutative S-algebra and M and N are CW R- modules, then M ^R N is a CW R-module such that CR*(M ^R N) ~=CR*(M) k CR*(N): Therefore such an isomorphism holds in the derived category Dk for general R- modules M and N. Moreover, in Dk, there is a natural map "": CR*(FR (M; N)) -! Hom k(CR*(M); CR*(N)); and ""is an isomorphism if M is a finite CW R-module. Proof. The first statement is implied by III.7.3. For the second, the evalua* *tion map FR (M; N) ^R M -! N induces a map CR*(FR (M; N)) k CR*(M) ~=CR*(FR (M; N) ^R M) -! C*(N) in Dk, and its adjoint is the required map "". Clearly ""is an isomorphism when* * M is a sphere R-module. It is therefore an isomorphism for all finite CW R-modules 3. THE ATIYAH-HIRZEBRUCH SPECTRAL SEQUENCE 91 since the functors FR and Hom kboth convert cofibration sequences in the first variable to fibration sequences. __|_ | We cannot expect the derived chain complex functor to preserve function objec* *ts in general, as the case R = S makes clear. By checking axioms, as in the classical case, we reach the following conclusi* *on. Theorem 3.5. For R-modules M and right and left k-modules j0 and j, there are natural isomorphisms HR*(M; j0) = H(CR*(M; j0)) and H*R(M; j) = H(C*R(M; j)): The map ss ^ id: M ~=R ^R M -! Hk ^R M induces the Hurewicz homomor- phism h : ss*(M) -! HR*(M; k), and the proof of the Hurewicz theorem is exactly the same induction over skeleta as in the classical case. Theorem 3.6 (Hurewicz). Let M be an (n - 1)-connected R-module. Then HRi(M; k) = 0 for i < n and h : ssn(M) -! HRn(M; k) is an isomorphism. Applying a generalized homology or cohomology theory to the skeletal filtrati* *on of a CW R-module M, we obtain an exact couple and thus a spectral sequence that generalizes the chain and cochain description of the ordinary represented homol* *ogy and cohomology of M. Theorem 3.7 (Atiyah-Hirzebruch Spectral Sequence). For a homology theory ER*and a cohomology theory E*Ron R-modules, there are natural spectral sequences of the form E2p;q= HRp(M; ERq) =) ERp+q(M) and Ep;q2= HpR(M; EqR) =) Ep+qR(M): Convergence is as in the classical case, and we refer the reader to Boardman [7, x14] (see also [24, App B]) for discussion. If M is bounded below, then the homology spectral sequence converges strongly to ER*(M) and the cohomology spectral sequence converges conditionally to E*R(M). If, further, for each fix* *ed (p; q) there are only finitely many r such that dr is non-zero on Erp;q, then t* *he cohomology spectral sequence converges strongly. The multiplicative properties of the spectral sequences are as one would expe* *ct from Proposition 3.4. 92 IV. THE ALGEBRAIC THEORY OF R-MODULES 4. Universal coefficient and K"unneth spectral sequences There are spectral sequences for the calculation of our Tor and Ext groups th* *at are analogous to the Eilenberg-Moore (or hyperhomology) spectral sequences in differential homological algebra. Compare [18, 28, 34]. They specialize to give* * uni- versal coefficient and K"unneth spectral sequences in the homology and cohomolo* *gy theory of spectra. We state our results in this section and give the constructi* *on in the next. Fix an S-algebra R. Theorem 4.1. For right R-modules M and left R-modules N, there is a natural spectral sequence of the form (4.2) E2p;q= TorR*p;q(M*; N*) =) TorRp+q(M; N): For left R-modules M and N, there is a natural spectral sequence of the form (4.3) Ep;q2= Extp;qR*(M*; N*) =) Extp+qR(M; N): If R is commutative, then these are spectral sequences of differential R*-modul* *es. The Tor spectral sequence is of standard homological type, with drp;q: Erp;q-! Erp-r;q+r-1: It lies in the right half-plane and converges strongly. The Ext spectral sequen* *ce is of standard cohomological type, with dr : Ep;qr! Ep+r;q-r+1r: It lies in the right half plane and converges conditionally. We have the follo* *wing addendum. Proposition 4.4. The pairing FR (M; N) ^S FR (L; M) ! FR (L; N) induces a pairing of spectral sequences that coincides with the algebraic Yoneda pairing Ext*;*R*(M*; N*) S* Ext*;*R*(L*; M*) -! Ext*;*R*(L*; N*) on the E2-level and that converges to the induced pairing of Ext groups. The rest of the results of this section are corollaries of the results alread* *y stated. With the specializations of variables that we cite, the conclusions are immedia* *te from the properties of our free and cofree functors and properties of Tor and E* *xt recorded in Section 1. Recalling Definition 1.7, we see that our spectral seque* *nces can be viewed as universal coefficient spectral sequences for the computation of homology and cohomology theories on R-modules. Via Corollary 1.9, they special- ize to give universal coefficient spectral sequences for the computation of hom* *ology and cohomology theories on spectra. Thus, setting M = FX in the two spectral 4. UNIVERSAL COEFFICIENT AND K"UNNETH SPECTRAL SEQUENCES 93 sequences of Theorem 4.1, we obtain the following result. We have written the stars to indicate the way the grading is usually thought of in cohomology. Theorem 4.5 (Universal coefficient). For an R-module N and any spec- trum X, there are spectral sequences of the form TorR**;*(R*(X); N*) =) N*(X) and Ext*;*R*(R-*(X); N*) =) N*(X): Of course, replacing R and N by Eilenberg-Mac Lane spectra HR and HN for a ring R and R-module N, we obtain the classical universal coefficient theorem. H* *ere we are thinking of the module N as defining theories acting on general spectra.* * By instead taking N = FE and N = F# E in the two spectral sequences of Theorem 4.1, we obtain spectral sequences that are suitable for calculating the E-homol* *ogy and cohomology of M. Theorem 4.6. For an R-module M and any spectrum E, there are spectral sequences of the form TorR**;*(M*; E*(R)) =) E*(M) and Ext *;*R*(M*; E*(R)) =) E*(M): When E is also an R-module, we can take M = E and so obtain spectral sequences that converge to the E-Steenrod algebra E*(E) and its dual E*(E). For example, when R = S and M = E = HZ p, the cohomology spectral sequence is a backwards Adams spectral sequence that converges from Ext *;*S*(Z p; Zp) to t* *he mod p Steenrod algebra A. Such a spectral sequence was first studied in [39]. Replacing N by FY and by FR (FY; R) in the two universal coefficient spectral sequences, we arrive at K"unneth spectral sequences. Theorem 4.7 (K"unneth). For any spectra X and Y , there are spectral se- quences of the form Tor R**;*(R*(X); R*(Y )) =) R*(X ^ Y ) and Ext *;*R*(R-*(X); R*(Y )) =) R*(X ^ Y ): 94 IV. THE ALGEBRAIC THEORY OF R-MODULES A reference to Adams [1] is mandatory. He was the first to observe that one c* *an derive K"unneth spectral sequences from universal coefficient spectral sequence* *s, and he observed that, by duality, the four spectral sequences of Theorems 4.5 and 4.7 imply two more universal coefficient and two more K"unneth spectral se- quences. He derived spectral sequences of this sort under the hypothesis that h* *is given ring spectrum E is the colimit of finite subspectra Effsuch that H*(Eff; * *E*) is E*-projective and the Atiyah-Hirzebruch spectral sequence converging from H*(Eff; E*) to E*(Eff) satisfies E2 = E1 . Of course, this is an ad hoc calcu- lational hypothesis that requires case-by-case verification. It covers some ca* *ses that are not covered by our results, and conversely. 5. The construction of the spectral sequences The construction is similar to the construction of Eilenberg-Mac Lane spectra* * at the end of Section 2. For a right R-module M, we choose an R*-free resolution dp " (5.1) . .-.! Fp-! Fp-1 -! . .-.! F0-! M* -! 0: Let Q0 = ker" and Qp = kerdp for p 1, so that dp defines an epimorphism Fp ! Qp-1. For p 0, let Kp be the wedge of one (p + s)-sphere for each basis element of Fp of degree s and let M0 = M. Proceeding inductively, we can use freeness to construct cofiber sequences of right R-modules kp ip jp+1 (5.2) FKp-! Mp-! Mp+1-! FKp for p 0 that satisfy the following properties: (i)k0 realizes " on ss*. (ii)ss*(Mp) = pQp-1 for p 1. (iii)kp realizes pdp : pFp -! pQp-1 on ss* for p 1. (iv)ip induces the zero homomorphism on ss* for p 0. (v)jp+1 realizes the inclusion p+1Qp -! p+1Fp on ss* for p 0. Observe that (iii) implies the case p + 1 of (ii) together with (iv) and (v). To obtain the spectral sequence for Tor, we define (5.3) D1p;q= ssp+q+1(Mp+1 ^R N) and E1p;q= ssp+q(FKp ^R N): The maps displayed in (5.2) give maps i (ip)* : D1p-1;q+1-! D1p;q j (jp+1)* : D1p;q-! E1p;q k (kp)* : E1p;q-! D1p-1;q: 5. THE CONSTRUCTION OF THE SPECTRAL SEQUENCES 95 These display an exact couple in standard homological form. We see from III.3.9 that E1p;q~=(Fp R* N*)q and that d1 agrees under the isomorphism with d 1. This proves that E2p;q= TorR*p;q(M*; N*): Observe that k : E10;q-! D1-1;qcan and must be interpreted as ssq(FK0 ^R N) -! ssq(M ^R N): On passage to E2, it induces the edge homomorphism (5.4) E20;q= M* R* N* -! ss*(M ^R N): The convergence is standard, although it appears a bit differently than in mo* *st spectral sequences in current use. Write i0;pfor both the evident composite map M -! Mp and its smash product with N. We filter ss*(M ^R N) by letting Fpss*(M ^R N) be the kernel of (i0;p+1)* : ss*(M ^R N) -! ss*(Mp+1 ^R N): By (iv) above, we see that the telescope telMp is trivial. Since the functor (?* *)^R N commutes with telescopes, tel(Mp ^R N) is also trivial. This implies that the filtration is exhaustive. Consider the (p; q)th term of the associated bigraded* * group of the filtration. It is defined as usual by E0p;qss*(M ^R N) = Fpssp+q(M ^R N)=Fp-1ssp+q(M ^R N); and the definition of the filtration immediately implies that this group is iso* *morphic to the image of (i0;p)* : ssp+q(M ^R N) -! ssp+q(Mp ^R N): The target of (i0;p)* is D1p-1;q, and of course E1p;q= ssp+q(FKp ^R N) also maps into D1p-1;q, via k. It is a routine exercise in the definition of a spectral s* *equence to check that k induces an isomorphism E1p;q-! Im(i0;p)*: (We know of no published source, but this verification is given in [7, x6].) To see the functoriality of the spectral sequence, suppose given a map f : M ! M0 of R-modules and apply the constructions above to M0, writing Fp0, etc. Con- struct a sequence of maps of R*-modules fp : Fp ! Fp0that give a map of res- olutions. We can realize the maps fp on homotopy groups by R-module maps FKp ! FK0p. Starting with f = f0 and proceeding inductively, a standard cofiber 96 IV. THE ALGEBRAIC THEORY OF R-MODULES sequence argument allows us to construct a map Mp+1 ! M0p+1such that the following diagram of R-modules commutes up to homotopy: F Kp ____//_Mp____//_Mp+1____//FKp | | | | | | | | fflffl| fflffl| fflffl| fflffl| F K0p____//_M0p___//_M0p+1__//_FK0p: There results a map of spectral sequences that realizes the induced map TorR**;*(M*; N*) -! TorR**;*(M0*; N*) on E2 and converges to (f ^R id)*. Functoriality in N is obvious. To obtain the analogous spectral sequence for Ext, we switch from right to le* *ft modules in our resolution (5.1) of M* and its realization by R-modules. We defi* *ne (5.5) Dp;q1= ss-p-q(FR (Mp; N)) and Ep;q1= ss-p-q(FR (FKp; N)): The maps displayed in (5.1) give maps i (ip)* : Dp+1;q-11-! Dp;q1 j (kp)* : Dp;q1-! Ep;q1 k (jp+1)* : Ep;q1-! Dp+1;q1: These display an exact couple in standard cohomological form. We see by III.6.9 that Ep;q1~=Hom qR*(Fp; N*), where Fp is regraded cohomologically, and that d1 agrees with Hom (d; 1) under the isomorphism. This proves that Ep;q2= Extp;qR*(M*; N*): Observe that j : D0;q1! E0;q1can and must be interpreted as ss-q(FR (M; N)) -! ss-q(FR (FK0; N)): On passage to E2, it induces the edge homomorphism (5.6) ss-q(FR (M; N)) -! Hom qR*(M*; N*) = E0;q2: To see the convergence, let 0;p: FR (Mp; N) -! FR (M; N) be the map induced by the evident iterate M ! Mp and filter ss*(FR (M; N)) by letting F pss*(FR (M; N)) be the image of (0;p)* : ss*(FR (Mp; N)) -! ss*(FR (M; N)): The (p; q)th term of the associated bigraded group of the filtration is Ep;q0ss*(FR (M; N)) = F pss-p-q(FR (M; N))=F p+1ss-p-q(FR (M; N)): 6. EILENBERG-MOORE TYPE SPECTRAL SEQUENCES 97 The group Ep;q1is defined as the subquotient Zp;q1=Bp;q1of Ep;q1, where Bp;q1= j(ker(0;p)*); and a routine exercise in the definition of a spectral sequence shows that the * *additive relation (0;p)* O j-1 induces an isomorphism Ep;q1~=Ep;q0ss*(FR (M; N)): Since telMp is trivial, so is the homotopy limit, or "microscope", micFR (Mp; N) ~=FR (telMp; N): By the lim1exact sequence for the computation of ss*(mic FR (Mp; N)), we conclu* *de that lim ss*(FR (Mp; N)) = 0 and lim 1ss*(FR (Mp; N)) = 0: This means that the spectral sequence {Ep;qr} is conditionally convergent. The functoriality of the spectral sequence is clear from the argument for torsion p* *roducts already given. Finally, turning to the proof of Proposition 4.4, consider the pairing FR (M; N) ^R FR (L; M) ! FR (L; N): Construct a sequence {Lp} as in (5.2). Then the maps M ! Mp induce a compat- ible system of pairings FR (Mp; N) ^R FR (Lp0; M)____//_FR (M; N) ^R FR (Lp0; M)__//_FR (Lp0; N): These induce the required pairing of spectral sequences. The convergence is cle* *ar, and the behavior on E2 terms is correct by comparison with the axioms or by comparison with the usual construction of Yoneda products. 6. Eilenberg-Moore type spectral sequences Let R be an S-algebra and let M be a right and N a left R-module. Let E be an associative ring spectrum in the sense of classical stable homotopy theory. By * *I.6.7 and II.1.9, we may assume without loss of generality that E is an associative S- ring spectrum (in the sense to be defined formally in Vx2). Under several diffe* *rent further hypotheses, we shall construct a spectral sequence of the form (6.1) TorE*(R)p;q(E*(M); E*(N)) =) Ep+q(M ^R N): The simplest version of this spectral sequence is the following one. Theorem 6.2. A spectral sequence of the form (6.1) exists if E*(R) is a flat right R*-module. 98 IV. THE ALGEBRAIC THEORY OF R-MODULES Proof. By a standard comparison of homology theories argument, the flatness hypothesis implies that, for left R-modules N, the natural map E*(R) R* N* -! ss*((E ^S R) ^R N) ~=ss*(E ^S N) = E*(N) is an isomorphism. It also ensures that the functor E*(R) R* (?) carries the ex* *act sequence (5.1) to an exact sequence of E*(R)-modules. We now apply the functor E*(?) = ss*(E^S?), rather than the functor ss*, to the sequence of cofibrations obtained from (5.2) by smashing over R with N and find that the rest of the proof of Theorem 4.1 carries over verbatim. In fact, if R is commutative, then * *the spectral sequence (6.1) results from the first spectral sequence of Theorem 4.1* * by applying the exact functor E*(R) R* (?). __|_ | This flatness hypothesis is generally unrealistic. By assuming that E is also* * an S-algebra and exploiting the S-algebra E ^S R, we can obtain a theorem like this without flatness hypotheses. We need a lemma. Lemma 6.3. Let R be an S-algebra such that (R; S) has the homotopy type of a relative CW S-module and let M and N be right and left cell R-modules. Then M, N, and M ^R N have the homotopy types of cell S-modules. Proof. Up to homotopy, S -! R is a cofibration of S-modules and R=S is a CW S-module. Since FR X = R ^S FSX, it follows from the cofiber sequence FSX -! R ^S FSX -! R=S ^S FSX that FR X has the homotopy type of a CW S-module if X has the homotopy type of a CW spectrum. Therefore sphere R-modules and, by III.3.7, their smash products are of the homotopy types of CW S-modules. The conclusion follows. __|_ | Theorem 6.4. Let E and R be S-algebras and assume that (R; S) is of the homotopy type of a relative CW S-module. Let M be a right and N a left R- module. Then there is a spectral sequence of the form TorE*(R)p;q(E*(M); E*(N)) =) Ep+q(M ^R N): Proof. Replace the triple (R; M; N) in Theorem 4.1 by the triple (E ^S R; E ^S M; E ^S N): The E2-term of the resulting spectral sequence is Tor (E^SR)**;*((E ^S M)*; (E ^S N)*): It converges to ss*((E ^S M) ^E^SR (E ^S N)) and, by III.3.10, we have (E ^S M) ^E^SR (E ^S N) ~=E ^S (M ^R N): 7. THE BAR CONSTRUCTIONS B(M; R; N) AND B(X; G; Y ) 99 Since we are working in derived categories, we may assume that M and N are cell R-modules. Then M, N, and M ^R N are of the homotopy types of CW S-modules, and I.6.7 and II.1.9 imply that their smash products over S with E a* *re isomorphic in hS to the corresponding internal smash products. This is also true for R=S, and of course E ^S S ~=E ' E ^ S. We conclude that (E ^S R)* ~=E*(R); (E ^S M)* ~=E*(M) and (E ^S N)* ~=E*(N); so that the E2 term of the spectral sequence is as stated, and ss*((E ^S M) ^E^SR (E ^S N)) ~=E*(M ^R N); so that the target of the spectral sequence is also as stated. __|_ | The hypothesis that (R; S) is of the homotopy type of a relative CW S-module results in no loss of generality since, as discussed in IIIx4, the model catego* *ry theory of Chapter VII implies that, for any S-algebra R, there is a q-cofibrant S-algebra R and a weak equivalence : R - ! R. The map induces an equivalence of categories DR DR , and (R; S) is of the homotopy type of a relative CW S-module. Remark 6.5. To deal with multiplicative structures, it is important to work with commutative S-algebras. As we shall see in Chapter VII, the category of commutative S-algebas also admits a model category structure. However, we do not believe that its q-cofibrant objects are of the homotopy types of relative * *CW S-modules. This is a significant technical difference between the theories of * *S- algebras and of commutative S-algebras. One way of getting around this diffi- culty is to approximate commutative S-algebras by q-cofibrant non-commutative S-algebras. We shall find a more satisfactory solution in VIIx6, where we exam- ine the homotopical properties of q-cofibrant commutative S-algebras. The resul* *ts there show that the proofs of Theorem 6.4 and of Theorem 7.7 below work in the commutative context provided that we assume that our given commutative S-algebras are q-cofibrant. 7. The bar constructions B(M; R; N) and B(X; G; Y ) The spectral sequence (6.1) is reminiscent of the Rothenberg-Steenrod-Eilen- berg-Moore spectral sequence (7.1) TorE*(G)*;*(E*(X); E*(Y )) =) E*B(X; G; Y ); where G is a topological monoid, X is a right G-space, Y is a left G-space, and B(X; G; Y ) is the two-sided bar construction [50]. We here describe a spectrum level two-sided bar construction B(M; R; N) that explains the analogy. We will use the bar construction to derive a version of (6.1) for general commutative r* *ing 100 IV. THE ALGEBRAIC THEORY OF R-MODULES spectra E that applies under different, and more realistic, flatness hypotheses* * than those of Theorem 6.2, and we will show that the classical spectral sequence (7.* *1) is a special case. Definition 7.2. For an S-algebra (R; OE; j), a right R-module (M; ), and a l* *eft R-module (N; ), define a simplicial S-module B*(M; R; N) by setting Bp(M; R; N) = M ^S Rp ^S N; where Rp is the p-fold ^S-power, interpreted as S if p = 0. The face and degene* *racy operators on Bp(M; R; N) are 8 >>< ^ (idR)p-1 ^ idN if i = 0 di= >idM ^(idR)i-1^ OE ^ (idR)p-i-1^ idN if 0 < i < p >: p-1 idM ^(idR) ^ if i = p and si= idM ^(idR)i^ j ^ (idR)p-i^ idNif 0 i p. If M is an (R0; R)-bimodule, then B*(M; R; N) is a simplicial R0-module. We will discuss the geometric realization of simplicial spectra in Xxx1-2, an* *d we agree to write B(M; R; N) = |B*(M; R; N)|: By X.1.5, geometric realization carries simplicial R-modules to R-modules. By XII.1.2 and X.1.2, there is a natural map (7.3) : B(R; R; N) -! N of R-modules that is a homotopy equivalence of S-modules. More generally, by use of the product on R and its action on the given modules, we obtain a natural map of S-modules (7.4) : B(M; R; N) -! M ^R N: Proposition 7.5. For a cell R-module M and any R-module N, : B(M; R; N) -! M ^R N is a weak equivalence of S-modules. Proof. If M__is the constant simplicial R-module at M, then, by X.1.3 and the isomorphism M ^R R ~=M, we have M ^R B(R; R; N) ~=|M__^R B*(R; R; N)| ~=B(M; R; N): Moreover, under this isomorphism, id^R agrees with . Since of (7.3) is a weak equivalence of R-modules, the conclusion follows from III.3.8. __|_ | 7. THE BAR CONSTRUCTIONS B(M; R; N) AND B(X; G; Y ) 101 For the bar construction to be useful calculationally, the simplicial spectrum B*(M; R; N) must be proper, in the sense of X.2.1 and X.2.2. By the following result, which is part of IX.2.7, we lose no generality by assuming this. Proposition 7.6. If R is a q-cofibrant S-algebra, then B*(M; R; N) is a prop* *er simplicial S-module. By X.2.9, when B*(M; R; N) is proper, we can use the simplicial filtration of B(M; R; N) to construct a well-behaved spectral sequence that converges to E*B(M; R; N) for any spectrum E. When E is a commutative ring spectrum, we can use flatness hypotheses to identify the E2-term. Recall that, in algebra, i* *f A is an algebra over a commutative ring k, then there is a notion of a relatively* * flat A-module F , for which the functor (?) A F is exact when applied to k-split exa* *ct sequences. The obvious examples are the relatively free A-modules A k L for k-modules L. There is a concomitant relative torsion product Tor(A;k)*(M; N), a* *nd similarly for graded algebras over commutative graded rings. When k is a field, these reduce to ordinary absolute torsion products. Theorem 7.7. Let E be a commutative ring spectrum. Let R be an S-algebra such that (R; S) is of the homotopy type of a relative CW S-module. Let M be a right and N a left cell R-module such that B*(M; R; N) is proper. If E*(R) and either E*(M) or E*(N) is E*-flat, then the bar construction spectral sequen* *ce converging to E*B(M; R; N) ~=E*(M ^R N) satisfies E2p;q= Tor(E*(R);E*)p;q(E*(M); E*(N)) Proof. Our hypotheses ensure that we can use smash products over S and internal smash products interchangeably when computing homology and homotopy groups. Our flatness hypotheses ensure that E*(M ^S Rp ^S N) ~=E*(M) E* E*(R)p E* E*(N); where the p-fold tensor power is taken over E*. This determines the E-homology of the spectrum of p-simplices of B*(M; R; N). Since B*(M; R; N) is proper, it follows that the complex that computes E2 (see X.2.9) is the standard bar compl* *ex for the computation of the relative torsion product. __|_ | Intuitively, interchanging the roles of M and N in the proof of Proposition 7* *.5, we see that the filtration quotients FpB(M; R; R)=Fp-1B(M; R; R) play a role similar to that played by the terms FKp in the construction of the spectral sequence of Theorem 6.2. 102 IV. THE ALGEBRAIC THEORY OF R-MODULES As promised, we have the following result, which shows that the spectral sequ* *ence of (7.2) is a special case. Theorem 7.8. Let G be a topological monoid, X a right G-space, and Y a left G-space. Then 1 G+ is an S-algebra, 1 X+ is a right 1 G+ -module, and 1 Y+ is a left 1 G+ -module. Moreover, there is a natural isomorphism of S-modules 1 B(X; G; Y )+ ~=B(1 X+ ; 1 G+ ; 1 Y+ ); and B*(1 X+ ; 1 G+ ; 1 Y+ ) is proper if G is nondegenerately based. Proof. The first statement is immediate from I.8.2 and II.1.2, together with the obvious identification X+ ^ Y+ ~=(X x Y )+ for spaces X and Y . The product on 1 G+ is induced from the product on G, 1 G+ ^S 1 G+ ~=1 (G x G)+ -! 1 G+ ; and similarly for the actions on 1 X+ and 1 Y+ . The second statement fol- lows from the fact that the functors 1 and geometric realization commute, by X.1.3, and that 1 preserves properness; see X.2.1. We obtain an identification* * of simplicial spectra 1 B*(X; G; Y )+ ~=B*(1 X+ ; 1 G+ ; 1 Y+ ) by applying 1 to the spaces (X x Gp x Y )+ ~=X+ ^ (G+ )p ^ Y; where (G+ )p is the p-fold smash power. If G is non-degenerately based, then B*(X; G; Y )+ is a proper simplicial based space. __|_ | CHAPTER V R-ring spectra and the specialization to M U In this chapter, we think of the derived category of R-modules as an analog of * *the stable homotopy category. From this point of view, we have the notion of an R-r* *ing spectrum, which is just like the classical notion of a ring spectrum in the sta* *ble homotopy category. We shall study such homotopical structures in this chapter. We first show how to construct quotients M=IM and localizations M[X-1] of modules over a commutative S-algebra R. We then study when these constructions inherit a structure of R-ring spectrum from an R-ring spectrum structure on M. When specialized to MU, our results give more highly structured versions of spectra that in the past have been constructed by means of the Baas-Sullivan theory of manifolds with singularities or the Landweber exact functor theorem. * *At least at odd primes, we obtain an entirely satisfactory, and surprisingly simpl* *e, treatment of MU-ring structures on the resulting MU-modules. 1. Quotients by ideals and localizations Let R be a commutative S-algebra. We assume that all given R-modules M are of the homotopy types of cell R-modules, but we must keep in mind that R itself will not be of the homotopy type of a cell R-module. By III.1.4, we have* * a canonical weak equivalence of R-modules i : SR -! R, where the sphere R-module SR = FR S is the free R-module generated by S, and we implicitly replace R by SR when performing constructions on R regarded as an R-module. Concomitantly, we must sometimes replace the unit isomorphism R ^R M ~= M by its composite with the weak equivalence i ^ id. This is consistent with the standard practice* * of replacing spectra by weakly equivalent CW spectra without change of notation. We shall work throughout in the derived category DR of R-modules. To deduce S-module or spectrum level conclusions from our R-module level arguments, we must apply the forgetful functors DR - ! DS and DS - ! hS . The process is routine, but it does entail reapproximating cell R-modules by CW S-modules or 103 104 V. R-RING SPECTRA AND THE SPECIALIZATION TO MU CW spectra since, in general, cell R-modules need not be of the homotopy types of CW S-modules or of CW spectra. We are interested in homotopy groups, and we make use of the isomorphisms (1.1) ssn(M) = hS (Sn; M) ~=hMS(SnS; M) ~=hMR (SnR; M) to represent elements as maps of R-modules, where, as usual, SnR= FR Sn. We write M* = ss*(M), and we do not distinguish notationally between a representat* *ive map of spectra Sn -! M and a representative map of R-modules SnR-! M. By III.3.7 and standard properties of spectrum level spheres ([37, pp 386-389* *]), we have a system of equivalences of R-modules (1.2) SqR^R SrR' Sq+rR that is associative, commutative, and unital up to a coherent system of homotopy equivalences and is compatible with suspension as q and r vary. For a pairing of R-modules L ^R M -! N, we therefore obtain a pairing of homotopy groups L* R* M* -! N*: Of course, L ^R M is an R-module since R is commutative. For x 2 Rn, thought of as an R-map SnR-! R, we have the R-map (1.3) SnR^R M-x^id!R ^R M ~=M: This map of R-modules realizes multiplication by x on M*. We agree to write nM for SnR^R M and to write x : nM -! M for the map (1.3) throughout this chapter. By III.3.7, SnR^R M is isomorphic as an R-module to SnS^S M and, by I.6.7 and II.1.9, SnS^S M is weakly equivalent as a spectrum to Sn ^ M. Therefo* *re, on passage to hS , the R-module nM is a model for the spectrum level suspension of M. Definition 1.4. Define M=xM to be the cofiber of the map (1.3) and let ae : M -! M=xM be the canonical map. Inductively, for a finite sequence {x1; : :;:xn} of elements of R*, define M=(x1; : :;:xn)M = N=xnN; where N = M=(x1; : :;:xn-1)M: For a (countably) infinite sequence X = {xi}, define M=XM to be the telescope of the M=(x1; : :;:xn)M, where the telescope is taken with respect to the successi* *ve canonical maps ae. We have the following analogue of the universal property of quotients by prin* *cipal ideals in algebra. 1. QUOTIENTS BY IDEALS AND LOCALIZATIONS 105 Lemma 1.5. Let N be an R-module such that x : nN -! N is zero and let ff : M - ! N be a map of R-modules. Then there is a map of R-modules "ff: M=xM -! N such that "ffO ae = ff, and "ffis unique if [n+1M; N]R = 0. Proof. This is obvious from the diagram _x___ __ae_// ____//_ nM //M M=xM n+1M ff|| |ff| fflffl|x fflffl| nN _____//N; in which the row is the cofiber sequence that defines M=xM. __|_ | Clearly we have a long exact sequence ae* (1.6) . .-.! ssq-n(M)- x!ssq(M)- ! ssq(M=xM) -! ssq-n-1(M) -! . .:. If x is not a zero divisor for ss*(M), then ae* induces an isomorphism of R*-mo* *dules (1.7) ss*(M)=xss*(M) ~=ss*(M=xM): If {x1; : :;:xn} is a regular sequence for ss*(M), in the sense that xi is not * *a zero divisor for ss*(M)=(x1; : :;:xi-1)ss*(M) for 1 i n, then (1.8) ss*(M)=(x1; : :;:xn)ss*(M) ~=ss*(M=(x1; : :;:xn)M); and similarly for a possibly infinite regular sequence X = {xi}. We shall see i* *n a moment that M=XM is independent of the ordering of the elements of the set X. If I denotes the ideal generated by a regular sequence X, then, by Corollary 2.* *10 below, M=XM is independent of the choice of regular sequence (under reasonable hypotheses) and it is reasonable to define (1.9) M=IM = M=XM: However, this notation must be used with caution since, if we fail to restrict * *at- tention to regular sequences X, the homotopy type of M=XM will depend on the set X and not just on the ideal it generates. For example, quite different modu* *les are obtained if we repeat a generator xi of I in our construction. As in algebra, we can describe the construction on general R-modules M as the smash product of M with the construction on R regarded as an R-module. We write R=X or R=I instead of R=XR or R=IR. Lemma 1.10. For a sequence X of elements of R*, there is a natural isomor- phism in DR (R=X) ^R M -! M=XM: 106 V. R-RING SPECTRA AND THE SPECIALIZATION TO MU In particular, for a finite sequence X = {x1; : :;:xn}, R=(x1; : :;:xn) ' (R=x1) ^R . .^.R(R=xn); and R=X is therefore independent of the ordering of the elements of X. Proof. Working on the point-set level, we have an isomorphism of cofiber se- quences : ae^id SnR^R R ^R M _x^id//_R ^R M____//(R=xR) ^R M id^ || || || fflffl| x fflffl|ae fflffl| SnR^R M _________//_M__________//M=xM: We only claim an isomorphism in DR since, working homotopically, we should replace R by SR and use the weak equivalence i^id : SR ^R M -! R^R M to obtain a composite weak equivalence (SR =xSR ) ^R M - ! (R=xR) ^R M - ! M=xM. The rest follows by iteration and use of the commutativity of ^R . __|_ | We turn next to localizations of R-modules at subsets X = {xi} of R*. We restrict attention to countable sets for notational convenience, but this restr* *iction can easily be removed. Let {yi} be any cofinal sequence of products of the xi, * *such as that specified inductively by y1 = x1 and yi= x1. .x.iyi-1. If yi2 Rni, we m* *ay represent yi by an R-map S0R- ! S-niR, which we also denote by yi. Let q0 = 0 and, inductively, qi= qi-1+ ni. The map of R-modules yi^id-ni S0R^R M -! SR ^R M represents yi. Smashing over R with S-qi-1Rand using equivalences (1.2), we obt* *ain a sequence of maps of R-modules (1.11) S-qi-1R^R M -! S-qiR^R M: Definition 1.12. Define the localization of M at X, denoted M[X-1], to be the telescope of the sequence of maps (1.11). Since M ' S0R^R M in DR , we may regard the inclusion of the initial stage S0R^R M of the telescope as a natural* * map : M -! M[X-1]. Again, we have an analogue of the standard universal property of localization* * in algebra. Lemma 1.13. Let N be an R-module such that xi : kiN - ! N, deg xi = ki, is an equivalence for each i and let ff : M - ! N be a map of R-modules. Then there is a unique map of R-modules "ff: M[X-1] -! N such that "ffO = ff. 2. LOCALIZATIONS AND QUOTIENTS OF R-RING SPECTRA 107 Proof. Passage to telescopes gives "ff: M[X-1] -! N[X-1] ' N. The lim1 term is zero in the exact sequence 0 -! lim1[1-qiM; N]R -! [M[X-1]; N]R -! lim[-qiM; N]R -! 0 since the maps of the inverse system are isomorphisms. Therefore "ffis unique. * * __|_ | Since homotopy groups commute with localization, by III.1.7, we see immediate* *ly that induces an isomorphism of R*-modules (1.14) ss*(M[X-1]) ~=ss*(M)[X-1]: Arguing as in Lemma 1.10, we see that the localization of M is the smash prod* *uct of M with the localization of R. Lemma 1.15. For a set X of elements of R*, there is a natural weak equivale* *nce R[X-1] ^R M -! M[X-1]: Moreover, R[X-1] is independent of the ordering of the elements of X. For sets X and Y , R[(X [ Y )-1] is equivalent to the composite localization R[X-1][Y -1* *]. Proof. The independence of ordering is shown by use of the union of any two given cofinal sequences. The last statement is shown by use of the usual Fubini type theorem for iterated homotopy colimits. __|_ | 2. Localizations and quotients of R-ring spectra Again, fix a commutative S-algebra R. Since DR is a symmetric monoidal cat- egory under ^R with unit R, we have the notion of a monoid or a commutative monoid in DR . These are the analogs of associative or of associative and commu- tative ring spectra in classical stable homotopy theory. As there, we must all* *ow weaker structures. Definition 2.1. An R-ring spectrum A is an R-module A with unit j : R -! A and product OE : A ^R A -! A in DR such that the following left and right unit diagram commutes in DR : j^id id^j R ^R A _____//A ^R Aoo___A ^R R LLL rrr LLL OE| rrr LLL | rrro LL&&fflffl|xxrr A: 108 V. R-RING SPECTRA AND THE SPECIALIZATION TO MU Of course, by neglect of structure, an R-ring spectrum A is a ring spectrum in the sense of classical stable homotopy theory; its unit is the composite of the* * unit of R and the unit of A and its product is the composite of the product of A and the canonical map A ^ A ' A ^S A -! A ^R A: Similarly, for an R-ring spectrum A, we have the evident homotopical notion of * *an A-module spectrum M. Here, in conformity with the definition just given, we only require that the action : A^R M -! M satisfy the unit condition O(j ^id) = id in DR . When A is associative, it is conventional to insist that M also satisfy* * the evident associativity condition. These structures play a role in the study of * *our new derived categories of R-modules that is analogous to the role played by ring spectra and their module spectra in classical stable homotopy theory. When R = * *S, I.6.7 and II.1.9 imply that S-ring spectra and their module spectra are equival* *ent to classical ring spectra and their module spectra. Lemma 2.2. If A and B are R-ring spectra, then so is A ^R B. If A and B are associative or commutative, then so is A ^R B. We ask the behavior of quotients and localizations with respect to R-ring str* *uc- tures. For localizations, the answer is immediate, and we shall give a sharper point-set level analogue in VIIIx3. Proposition 2.3. Let X be a set of elements of R*. If A is an R-ring spectru* *m, then A[X-1] is an R-ring spectrum such that : A -! A[X-1] is a map of R-ring spectra. If A is associative or commutative, then so is A[X-1]. Proof. Since A[X-1] ' R[X-1] ^R A, it suffices to prove that R[X-1] is an associative and commutative R-ring spectrum with unit . Lemma 1.15 gives an equivalence R[X-1] ^R R[X-1] ' R[X-1][X-1] ' R[X-1] under R, and this equivalence is the required product. __|_ | This doesn't work for quotients since (R=X)=X is not equivalent to R=X. How- ever, we can analyze the problem by analyzing the deviation, and, by Lemma 1.10, we may as well work one element at a time. We have a necessary condition for R=x to be an R-ring spectrum that will be familiar from classical stable homoto* *py theory. We generally write j and OE for the units and products of R-ring spectr* *a; as stated before, we write n for the module theoretic suspension functor SnR^R (?). Lemma 2.4. Let A be an R-ring spectrum. If A=xA admits an R-ring spectrum structure such that ae : A -! A=xA is a map of R-ring spectra, then x : A=xA -! A=xA is null homotopic as a map of R-modules. 2. LOCALIZATIONS AND QUOTIENTS OF R-RING SPECTRA 109 Proof. We have the following commutative diagram (where we omit suspension coordinates from the labels of maps): j^id n x^id nR ^R (A=xA) ____//_ (A=xA) ^R (A=xA) ____//_(A=xA) ^R (A=xA) TT TTTTT | | TTTTT |OE |OE TTT**T fflffl| fflffl| n(A=xA) ______x________//_A=xA: In view of the following commutative diagram, its top composite is null homotop* *ic: ____x_____ 9nA9r //A rr jrrrrr ae|| |ae| rrr fflffl| fflffl| nR __j_//_n(A=xA) __x_//_A=xA: __ |_ | Thus, for example, the Moore spectrum S=2 is not an S-algebra since the map 2 : S=2 -! S=2 is not null homotopic. We need a lemma in order to obtain an R-ring spectrum structure on R=x in appropriate generality. Lemma 2.5. Let ae : R -! M be any map of R-modules. Then (ae ^ id) O ae = (id^ae) O ae : R -! M ^R M: Proof. Since = O o : R ^R R -! R, the following diagram commutes: ppRONNNO aeppppp || NNNNaeN ppp | NNNNN wwpppp | N'' MOO || MOO || | || | | | | | | | R ^R RN |Oo || id^aeppppp NNNae^idNN|| | pp NNNN | | xxpppp &&N| R ^R M N M ^R R NNN pppp NNNN pppp ae^idNN&&N xxppid^aep __ M ^R M: |_ | 110 V. R-RING SPECTRA AND THE SPECIALIZATION TO MU Theorem 2.6. Let x 2 Rn, where ssn+1(R=x) = 0 and ss2n+1(R=x) = 0. Then R=x admits a structure of R-ring spectrum with unit ae : R -! R=x. Therefore A=XA admits a structure of R-ring spectrum such that ae : A -! A=XA is a map of R-ring spectra for every R-ring spectrum A and every sequence X of elements * *of R* such that ssn+1(R=x) = 0 and ss2n+1(R=x) = 0 for all x 2 X, where deg x = n. Proof. Consider the following diagram in the derived category DR : (2.7) 2n+1R |x| fflffl| x n+1R ________//_R m m | m mm |ae| |ae x ae^id vvmm ss fflffl|x fflffl| n(R=x) ____//_R=x____//_(R=x)o^Ro(R=x)___//____n+1(R=x)__//_oo___(R=x) OE oe |ss0| fflffl| 2n+2R: The map x is that specified by (1.3). The bottom row is the cofiber sequence th* *at results from the equivalence (R=x) ^R (R=x) ' (R=x)=x of Lemma 1.10, and the column is also a cofiber sequence. The composite x O ae * *is null homotopic since ae O x is null homotopic and the square commutes. Therefore there is a map such that ss O = ae, and is unique since ssn+1(R=x) = 0. Since ss O O x = ae O x = 0, O x factors through a map 2n+1R - ! R=x. Since ss2n+1(R=x) = 0, such maps are null homotopic. Thus O x is null homotopic. Therefore there is a map oe such that oe O ae = . Now ss O oe O ae = ss O = ae* *, hence (ssOoe-id)ae = 0. Therefore ssOoe-id factors through a map 2n+2R -! n+1(R=x). Again, such maps are null homotopic. Therefore ssOoe = id. Thus the bottom cofi* *ber sequence splits (proving in passing that x : n(R=x) -! R=x is null homotopic, as it must be). A choice OE of a splitting gives a product on R=x. The unit condit* *ion OE O (ae ^ id) = idis automatic. To see that OE O (id^ae) = id, we observe that* *, by the lemma, (OE O (id^ae) - id) O ae = OE O (id^ae - ae ^ id) O ae = 0: Therefore OE O (id^ae) - idfactors through a map n+1R -! R=x. Again, such maps are null homotopic, hence OE O (id^ae) = id. This completes the proof that R=x is an R-ring spectrum with unit ae. The rest follows from Lemmas 1.10 and 2.2. __|_ | 3. THE ASSOCIATIVITY AND COMMUTATIVITY OF R-RING SPECTRA 111 The product on R=x can be described a little more concretely. The wedge sum (2.8) (ae ^ id) _ oe : (R=x) _ n+1(R=x) -! (R=x) ^R (R=x) is an equivalence. The product OE restricts to the identity on the first wedge * *sum- mand and to the trivial map on the second wedge summand. Thus the product is determined by the choice of oe, and two choices of oe differ by a composite 0 (2.9) n+1(R=x) _ss_//_2n+2R____//_(R=x) ^R (R=x): By the splitting (2.8) and the assumption that ssn+1(R=x) = 0, we can view the second map as an element of ss2n+2(R=x). If x is not a zero divisor, then ss0*=* * 0 on homotopy groups and any two products have the same effect on homotopy groups. Before continuing our discussion of these R-ring spectra, we insert the follo* *wing easy consequence of the mere existence of the R-ring structure. Corollary 2.10. Assume that Ri = 0 if i is odd. Let X and Y be regular sequences in R* that generate the same ideal. Then there is an equivalence of R-modules : R=X -! R=Y under R. Proof. It suffices to construct a map under R since it will automatically induce an isomorphism on homotopy groups. Each xi is an R*-linear combination of the yj and each yj : R=Y - ! R=Y is zero. By Lemma 1.5, we obtain a unique map i : R=xi -! R=Y under R. Since R=Y is an R-ring spectrum, we may first take the smash product of these maps and then use the product (associated conveniently) on R=Y and passage to telescopes (if X is infinite) to obtain . * *__|_ | 3. The associativity and commutativity of R-ring spectra We assume given an R-ring spectrum A. For x 2 Rn as in Theorem 2.6, we give A=xA ' (R=x) ^R A the product induced by one of our constructed products on R=x and the given product on A. We refer to any such product as a "canonical" product on A=xA. Of course, we do not claim that every product on A=xA is canonical. Observe that, by first using the product on A, the product on A=xA can be factored through OE ^R id: (R=x) ^R (R=x) ^R A -! (R=x) ^R A: This allows us to smash any diagram giving information about the product on R=x with A and so obtain information about the product on A=xA. Obviously any diagram so constructed is a diagram of right A-module spectra via the product action of A on itself. This smashing with A can kill obstructions. Clearly, a m* *ap of A-modules qA -! M is determined by its restriction Sq -! M along the unit of A regarded as a map of spectra (or S-modules), which is just an element of ssq(M). These considerations lead to the following result. 112 V. R-RING SPECTRA AND THE SPECIALIZATION TO MU Theorem 3.1. Let x 2 Rn, where ssn+1(R=x) = 0 and ss2n+1(R=x) = 0. Let A be an R-ring spectrum and assume that ss2n+2(A=xA) = 0. Then there is a unique canonical product on A=xA. If A is commutative, then A=xA is commutative. If A is associative and ss3n+3(A=xA) = 0, then A=xA is associative. Proof. The second arrow of (2.9) becomes zero after smashing with A since it is then given by an element of ss2n+2(A=xA) = 0. This proves the uniqueness statement. The commutativity statement follows since if OE is a canonical produ* *ct on A=xA, then so is OEo. However, it may be worth displaying the obstruction th* *at lies in ss2n+2(A=xA). Looking at the splitting (2.8), we see that OE is commuta* *tive on the wedge summand R=x by the unit property. For the summand n+1R, consider the diagram ae n+1 (OE-OEOo)Ooe n+1R ____//_ (R=x)__________//R=x:66l l l ss0|| l lfll fflffl|ll 2n+2R The horizontal composite is null homotopic since ssn+1(R=x) = 0. Thus there exi* *sts fl such that the triangle commutes. It is the obstruction to the commutativity * *of R=x, and smashing with A gives the obstruction to the commutativity of A=xA. For the associativity, consider the splitting displayed in the following diag* *ram: (R=x) _ n+1(R=x) _ n+1(R=x) _ 2n+2(R=x) '|| fflffl| [(R=x) _ n+1(R=x)] _ n+1[(R=x) _ n+1(R=x)] [(ae^id)_oe]_n+1[(ae^id)_oe]|| fflffl| [(R=x) ^R (R=x)] _ n+1[(R=x) ^R (R=x)] '|| fflffl| (R=x) ^R [(R=x) _ n+1(R=x)] id^[(ae^id)_oe]|| fflffl| (R=x) ^R (R=x) ^R (R=x): The question of associativity can be considered separately on the restrictions * *of the iterated product to the four wedge summands. Via easy diagram chases, we see that, under the splitting (2.8) and unit isomorphisms, the natural maps ae ^ id^ id: R ^R (R=x) ^R (R=x) -! (R=x) ^R (R=x) ^R (R=x) 3. THE ASSOCIATIVITY AND COMMUTATIVITY OF R-RING SPECTRA 113 and id^ae ^ id: (R=x) ^R R ^R (R=x) -! (R=x) ^R (R=x) ^R (R=x) correspond to the inclusions of the first and third and first and second wedge summands, respectively. Therefore the unital property of OE and the unital and associativity properties of ^R imply that the restriction of OE to the first t* *hree wedge summands is associative. Let ! denote the displayed inclusion of the four* *th wedge summand and consider the diagram ae 2n+2 [OEO(OE^id)-OEO(id^OE)]O! 2n+2R _____// (R=x) _____________________//_R=x ss0|| fflffl| 3n+3R Call the horizontal composite ff. If ff is nullhomotopic then the deviation fr* *om associativity [OE O (OE ^ id) - OE O (id^OE)] O ! factors through a map 3n+3R -* *! R=x. Thus if ss3n+3(R=x) = 0, then the element ff 2 ss2n+2(R=x) is the obstruction t* *o the associativity of R=x. If both relevant homotopy groups become zero after smashi* *ng with A, we can conclude that A=xA is associative if A is associative. __|_ | We can iterate the argument to arrive at the following fundamental conclusion. Theorem 3.2. Assume that Ri = 0 if i is odd and let X be a sequence of non zero divisors in R* such that ss*(R=X) is concentrated in degrees congruent to * *zero mod 4. Then R=X has a unique canonical structure of R-ring spectrum, and it is commutative and associative. Proof. We first observe that for an element x 2 ssq(R), an R-module M, and an R-module N such that x : qN - ! N is null homotopic, the map ae : M - ! M=xM induces an epimorphism ae* : [M=xM; N]R -! [M; N]R since the action x* : [M; N]R - ! [qM; N]R can be computed from the action on N and is therefore zero. Let Xn be the subsequence consisting of the first n elements of the sequence X. Then R=X is defined to be the telescope of the R=Xn, and Lemma 2.4 implies that multiplication by xn is null homotopic on R=X for each n. Since R=X ^R R=X is equivalent to the telescope of the R=Xn ^R R=Xn, we obtain a product on R=X from a canonical product on the R=Xn by passage to telescopes. Moreover, by the Mittag-Leffler criterion, our first observation im* *plies that all relevant lim1terms are zero. Thus it suffices to show that any two pro* *ducts on Xn become equal and the commutativity and associativity diagrams for R=Xn become commutative upon composition with the map R=Xn - ! R=X, and we may proceed by induction on n. The conclusion follows since the obstructions to 114 V. R-RING SPECTRA AND THE SPECIALIZATION TO MU uniqueness, commutativity, and associativity of each R=xn become trivial when we map to R=X. __|_ | 4. The specialization to MU-modules and algebras It was observed in [47] that MU can be constructed as an E1 ring spectrum, and we apply S ^L (?) to convert it to a commutative S-algebra. Of course, its homotopy groups are concentrated in even degrees, and every non-zero element is a non zero divisor. Thus Proposition 2.3, Theorem 2.6, and Theorem 3.2 combine to give the following result. Theorem 4.1. Let X be a regular sequence in MU*, let I be the ideal generat* *ed by X, and let Y be any sequence in MU*. Then there is an MU-ring spectrum (MU=X)[Y -1] and a natural map of MU-ring spectra (the unit map) j : MU -! (MU=X)[Y -1] such that j* : MU* -! ss*((MU=X)[Y -1]) realizes the natural homomorphism of MU*-algebras MU* -! (MU*=I)[Y -1]: If MU*=I is concentrated in degrees congruent to zero mod 4, then there is a unique canonical product on (MU=X)[Y -1], and this product is commutative and associative. In comparison with constructions of this sort based on the Baas-Sullivan theo* *ry of manifolds with singularities or on Landweber's exact functor theorem (where * *it applies), we have obtained a simpler proof of a substantially stronger result. * *We emphasize that an MU-ring spectrum is a much richer structure than just a ring spectrum and that commutativity and associativity in the MU-ring spectrum sense are much more stringent conditions than mere commutativity and associativity of the underlying ring spectrum. Observe that the assumption that X is regular is used only to obtain the calculational description of j*. We illustrate by explaining how BP appears in this context. Fix a prime p and write (?)p for localization at p. Let BP be the Brown-Peterson spectrum at p. We are thinking of Quillen's idempotent construction, and we have the splitting ma* *ps i : BP -! MUp and e : MUp -! BP . These are maps of commutative and associative ring spectra such that e O i = id. Let I be the kernel of the compo* *site MU* -! MUp*- ! BP*: 4. THE SPECIALIZATION TO MU-MODULES AND ALGEBRAS 115 Then I is generated by a regular sequence X, and our MU=X is a canonical integr* *al version of BP . For the moment, let BP 0= (MU=X)p. Let : BP - ! BP 0be the composite __i__ _ip_//_ 0 BP //MUp BP : It is immediate that is an equivalence. In effect, since we have arranged that ip has the same effect on homotopy groups as e, induces the identity map of (MU*=I)p on homotopy groups. By the splitting of MUp and the fact that self- maps of MUp are determined by their effect on homotopy groups [2, II.9.3], maps MUp -! BP are determined by their effect on homotopy groups. This implies that O e = ip : MUp -! BP 0. The product on BP is the composite __i^i_ _OE_//_ __e_//_ BP ^ BP //MUp ^ MUp MUp BP: Since ip is a map of MU-ring spectra and thus of ring spectra, a trivial diagram chase now shows that the equivalence : BP -! BP 0is a map of ring spectra. We conclude that our BP 0is a model for BP that is an MU-ring spectrum, commutative and associative if p > 2. The situation for p = 2 is interesting. We conclude from the equivalence that BP 0is commutative and associative as a ring spectrum, although we do not know that it is commutative or associative as an MU-algebra. Recall that ss*(BP ) = Z(p)[vi|deg(vi) = 2(pi- 1)], where the generators vi c* *ome from ss*(MU) (provided that we use the Hazewinkel generators). We list a few of the spectra derived from BP , with their coefficient rings. Let Fp denote the f* *ield with p elements. BP Z(p)[v1; : :;:vn] E(n) Z(p)[v1; : :;:vn; v-1n] P (n) Fp[vn; vn+1; : :]:B(n) Fp[v-1n; vn; vn+1; : :]: k(n) Fp[vn] K(n) Fp[vn; v-1n] By the method just illustrated, we can construct canonical integral versions of the BP and E(n). All of these spectra fit into the context of Theorem 4.1. If p > 2, they all have unique canonical commutative and associative MU-ring spectra structures. Further study is needed when p = 2, but we leave that to the interested reader. In any case, our theory makes it unnecessary to appeal to Baas-Sullivan theory or to Landweber's exact functor theorem for the constructi* *on and analysis of spectra such as these. We have started with MU because it appears in nature with a canonical structu* *re as a commutative S-algebra. However, it is also possible to start with BP , usi* *ng the second author's result that BP admits a commutative S-algebra structure; in fact, it admits uncountably many different ones [33]. 116 V. R-RING SPECTRA AND THE SPECIALIZATION TO MU CHAPTER VI Algebraic K-theory of S-algebras In this chapter we apply the basic constructions of algebraic K-theory to the n* *ew categories of module over S-algebras. We show how to construct a K-theory spec- trum KR for each S-algebra R in such a way that K becomes a functor from the category of S-algebras to the stable category. When R is a connective commutati* *ve S-algebra, so is KR. We prove that weakly equivalent S-algebras have equivalent K-theory, and we prove a Morita invariance result. When R is connective we are able to give an alternate description of this K-theory in terms of Quillen's pl* *us con- struction, a "plus equals So" theorem. When R = Hk is an Eilenberg-Mac Lane S-algebra, this K-theory is essentially Quillen's algebraic K-theory of the rin* *g k. When R = 1 |GSX |+ is the suspension spectrum of a a disjoint basepoint plus the geometric realization of the loop group of the singular complex of a topolo* *gical space X, this K-theory is Waldhausen's algebraic K-theory of the space X. 1. Waldhausen categories and algebraic K-theory We first review the basic definitions of Waldhausen [68] that we shall use. Definition 1.1. A category with w-cofibrations C is a (small) category with preferred zero object "*", together with a chosen subcategory co(C ) that satis* *fies the following three axioms: (i)Any isomorphism in C is a morphism in co(C ); in particular, co(C ) conta* *ins all the objects of C . (ii)For every object A in C , the unique map * -! A is in co(C ). (iii)If A -! B is a map in co(C ), and A -! C is a map in C , then the pushout B qA C exists in C and the canonical map C -! B qA C is in co(C ); in particular, C has finite coproducts. We call the morphisms in co(C ) w-cofibrations, and often use the feathered arrow "ae" to denote them in diagrams. Although Waldhausen called these arrows 117 118 VI. ALGEBRAIC K-THEORY OF S-ALGEBRAS "cofibrations" we will consistently use the term "w-cofibration" so that there * *will be no confusion with our standard use of the word "cofibration" to mean those maps that satisfy the homotopy extension property (HEP). Definition 1.2. A Waldhausen category (in [68], "a category with cofibrations and weak equivalences") is a category with w-cofibrations C and a chosen subcat- egory w (C ) of C that satisfies the following axioms: (i)Any isomorphism in C is a morphism in w(C ); in particular, w(C ) contains all the objects of C . (ii)Given any commutative diagram in C C oo___A _//__//_B | | | | | | fflffl| fflffl| fflffl| C0 oo___A0 //__//B0 in which the vertical maps are in w (C ) and the feathered arrows are in co(C ), the induced map B qA C -! B0qA0C0 is in w (C ). We call the morphisms in w (C ) weak equivalences, and often use the arrows "-~! " to denote them. We say that the weak equivalences are saturated or that C is a saturated Waldhausen category if whenever f and g are composable arrows in C and any two of f, g, and gf are weak equivalences then so is the third. Definition 1.3. A functor between Waldhausen categories is exact if it pre- serves all of the above structure; i.e. it must send w-cofibrations to w-cofibr* *ations, weak equivalences to weak equivalences, the preferred zero object to the prefer* *red zero object, and it must preserve the pushouts along a w-cofibration. We now have all the necessary ingredients to describe Waldhausen's So constru* *c- tion [68]. Let C be a Waldhausen category. For each n 0, define a category SnC as follows. An object of SnC consists of n + 1 composible arrows in co(C ) star* *ting from the preferred zero object *, ff0 ff1 ffn * = A0 _//_//_A1//_//_._././//_An; together with objects Ai;jfor 0 i j n and maps ai;j:Aj -! Ai;jsuch that Ai;i= *, Aj = A0;jwith a0;jthe identity map, and the diagrams ffj-1O...Offi Ai_//________//Aj | | ai;j| | | fflffl| fflffl| * //________//Ai;j 1. WALDHAUSEN CATEGORIES AND ALGEBRAIC K-THEORY 119 are pushouts for 0 i < j n. A morphism of SnC from {Aj; Ai;j; ffj; ai;j} to {A0j; A0i;j; ff0j; a0i;j} is a sequence of maps fj: Aj -! A0jsuch that the diag* *ram ff0 ff1 ffn A0 _//__//A1//__//._././//_An f0|| |f1| fn|| fflffl| fflffl| fflffl| A00_//ff//0_A01////0__./././/0__A0n 0 ff1 ffn commutes. Observe that by the universal property of pushouts, we have induced maps Ai;j-! A0i;jmaking all the appropriate diagrams commute. We give SnC the structure of a Waldhausen category by defining a map {f0; : :;:fn} to be a w-cofibration (resp. weak equivalence) if each fj is a w-cofibration (resp. w* *eak equivalence) of C . Observe that when {f0; : :;:fn} is a w-cofibration (resp. w* *eak equivalence) all the induced maps Ai;j-! A0i;jare w-cofibrations (resp. weak equivalences). Notice that S0C is the trivial category and that S1C is isomorph* *ic to C . For 0 i n, define dk: SnC -! Sn-1C to be the functor that drops the k-th row and k-th column from the matrix {Ai;j}. More precisely, d0 sends the object {Aj; Ai;j; ffj; ai;j} of SnC to the object {Bj; Bi;j; fij; bi;j} of Sn-* *1C where Bj = A1;j+1, Bi;j= Ai+1;j+1, and the maps fij and bi;jare the maps induced from ffj+1 and ai+1;j+1by the universal property of the pushout. For k > 0, the func* *tor dk is defined similarly. For 0 k n, define sk: SnC - ! Sn+1C to be the functor that repeats the k-th row and k-th column in the matrix {Ai;j}. More precisely,* * sk sends the object {Aj; Ai;j; ffj; ai;j} of SnC to the object {Bj; Bi;j; fij; bi;* *j} of Sn+1C where ae Aj if j k Bj = A j-1 if j > k 8 < Ai;j if j k Bi;j = : Ai;j-1 if j > k and i k Ai-1;j-1if i > k 8 < ffj if j < k fij = : id if j = k ffj-1 if j > k 8 < ai;j if j k bi;j = : ai;j-1 if j > k and i k ai-1;j-1if i > k. Observe that the functors dk and sk satisfy the simplicial identities and the collection {SnC } assembles into a simplicial category, which we denote SoC . F* *ur- thermore, the functors dk and sk are exact and So has the structure of a simpli* *cial 120 VI. ALGEBRAIC K-THEORY OF S-ALGEBRAS Waldhausen category. In particular, we can iterate this construction to form t* *he bisimplicial Waldhausen category S(2)oC = SoSoC , and the polysimplicial Wald- hausen categories S(n)oC = So. .S.oC . We abbreviate the notation for the categ* *ory of weak equivalences in S(n)oC to wS(n)oC . We are interested in the classify* *ing spaces of these categories, the spaces |wS(n)oC|. Since S0C is the trivial cate* *gory, we see that |wSoC | is connected, and it is not much harder to see that in gen- eral |wS(n)oC| is (n - 1)-connected. Observe that the identifications of C wi* *th S1C and more generally S(n)oC with S1S(n)oC induce maps |wC | -! |wSoC | and |wS(n)oC| -! |wS(n+1)oC| which are inclusions of subcomplexes. It is a funda- mental observation of [68, x1.3, 1.5.3] that in the sequence |wC | -! |wSoC | -! 2|wS(2)oC| -! . . . all maps beyond the first are homotopy equivalences. This motivates the followi* *ng definition. Definition 1.4. The algebraic K-theory of the Waldhausen category C is the spectrification of the -cofibrant prespectrum {|wC |; |wSoC |; |wS(2)oC|; : :* *}:. We denote this spectrum by the symbol KC . The algebraic K-groups of C are the homotopy groups of this spectrum, KnC = ssnKC = ssn+1|wSoC |. In particular, KnC = 0 for n < 0. Waldhausen observes that in the special case when C is an exact category (whe* *re the w-cofibrations are the admissible monos, and the weak equivalences are the isomorphisms) the algebraic K-groups defined above agree with those defined by Quillen [55]. In fact, the basic properties of the Q-construction are all easi* *ly provable in terms of the So construction [64] (see also [51]). Observe that an exact functor C - ! D induces an exact functor SoC - ! SoD and hence exact functors S(n)oC -! S(n)oD. This induces a map of prespectra {wS(n)oC } -! {wS(n)oD} and hence a map of spectra KC - ! KD. If the map |wSoC | -! |wSoD | is a weak equivalence, then the maps |wS(n)oC| -! |wS(n)oD| are weak equivalences and therefore homotopy equivalences. Since these prespect* *ra are -cofibrant, maps between them that are spacewise homotopy equivalences induce homotopy equivalences of their spectrifications. In other words, an exa* *ct functor that induces a weak equivalence on |wSo- | induces a homotopy equivale* *nce of K-theory spectra. For this reason, although Waldhausen defines the algebraic K-theory of a Waldhausen category C to be the space |wSoC |, all the results we use from [68] apply to the K-theory spectra, even when they are stated only for the K-theory spaces, and we will use them this way without further comment; moreover, whenever we shall assert a result about K-theory, we shall mean the result about the K-theory spectra unless otherwise noted. 2. CYLINDERS, HOMOTOPIES, AND APPROXIMATION THEOREMS 121 2. Cylinders, homotopies, and approximation theorems Let MOR C be the category whose objects are the morphisms of C and whose morphisms are the commutative diagrams. For A, A0, B, B0 objects of C and a: A - ! A0 and b: B - ! B0 maps of C , a and b are objects of MOR C . If f :A -! B and f0: A0- ! B0 are maps in C that make the diagram f A _____//B a || b|| fflffl| fflffl| A0__f0_//B0 commute, then (f; f0) is a map in MOR C . Whenever C is a Waldhausen category, we can give MOR C the structure of a Waldhausen category by saying that a map (f; f0) is a w-cofibration (resp. weak equivalence) of MOR C if both f and f0 * *are w-cofibrations (resp weak equivalences) of C . Definition 2.1. [68, 1.6] Let C be a Waldhausen category. A cylinder functor is a functor T :MOR C - ! C together with natural transformations i1, i2, and p that make the following diagram commute for a morphism f :A -! B in C : i1 i2 A _//_//_ATofoooB__ AA | """""" AAA |p """""" f AA|fflffl"""" B and that satisfies the following properties: (i)i1 q i2: A q B ae T f is in co(C ). (ii)The functor (A -! B) 7! (A q B i1qi2-!T f) is an exact functor MOR C - ! MOR C . (iii)T (* -! B) = B, with p and i2 the identity map. We say that the cylinder functor satisfies the cylinder axiom if in addition p: T f - ! B is in w (C ) for all morphisms f. We will often refer to i1 and i2 as face maps and to p as the collapse map. Theorem 2.2. (Waldhausen, [68, 1.6.7], "The Approximation Theorem") Let A and B be saturated Waldhausen categories, where A has a cylinder functor that satisfies the cylinder axiom. Let F :A - ! B be an exact functor such that (i)If f is a morphism in A such that F (f) is in w (B), then f is in w (A ). (ii)For any object A 2 A and any map f :F A - ! B, there exists a map g :A -! A0 in A and a weak equivalence h: F A0 -! B in w (B) such 122 VI. ALGEBRAIC K-THEORY OF S-ALGEBRAS that f = h O F (g): f F (A) __________//_B:77pp ppp F(g)|| ppppp fflffl|hppp F (A0) Then F induces a homotopy equivalence KA - ! KB. Remark 2.3. In [68] it is required that g be a w-cofibration, but [65] poin* *ts out that this requirement is unnecesary since we can use cylinders to replace an arbitrary map with a w-cofibration. Often, we have the situation where it is easy to make the diagram in 2.2(ii) commute up to some kind of homotopy. By integrating the idea of homotopy into Waldhausen's language of K-theory, we can prove two easy but extremely useful corollaries of the approximation theorem. To this end, we offer the foll* *owing definitions. Definitions 2.4. Let C be a Waldhausen category with cylinder functor T . Observe that T gives an exact functor I :C - ! C by restriction along the exact functor 1: C - ! MOR C that sends an object to its identity morphism. We call (W; j1; j2; q) a cylinder object of the object X if W = IX, q = p (the collapse* * map) and either j1 = i1 and j2 = i2, or j1=i2 and j2 = i1. We say that (W; j1; j2; q* *) is a generalized cylinder object of the object X, if W is the pushout over alternate* * face maps of a sequence of cylinder objects, j1; j2: X ae W are the two unused face maps, and q :W - ! X is the gluing of the collapse maps; in particular observe that q O ji= 1X for i = 1; 2. We call two maps f1; f2: X -! Y homotopic if there exists a generalized cylinder object W of X and a map OE: W - ! Y such that OE O ji= fi for i = 1; 2. It is easy to see that this specifies an equivalence * *relation. Let us say that an exact functor F :C - ! D between Waldhausen categories with cylinder functors preserves cylinder objects if there is a natural isomorp* *hism ff :F IC ~= ID F such that ff O F (ik) = ik and p O ff = F (p): wF;IA|;ccGG F IA|HH F(i1)www| GGF(i2)G | HHF(p)HH www | GGG | HHH w | | ## F A ff| F A ff| F A: GG | w | v;; GGG | www | vvv GG | ww | vvpv i1 G##fflffl|--i2ww fflffl|vv IF A IF A Observe that a functor that preserves cylinder objects also preserves generaliz* *ed cylinder objects, in the sense that ff gives an isomorphism of F W to a general* *ized cylinder object W 0with ff O F (jk) = j0kand q0O ff = F (q). It is easy to see * *that 2. CYLINDERS, HOMOTOPIES, AND APPROXIMATION THEOREMS 123 when F preserves cylinder objects, F also preserves the relation of homotopy of morphisms. Theorem 2.5. (Homotopy Approximation Theorem) Let A and B be small saturated Waldhausen categories with cylinder functors satisfying the cylinder * *ax- iom. Let F :A - ! B be an exact functor that preserves cylinder objects and such that (i)If f is a morphism in A such that F (f) is in w (B), then f is in w (A ). (ii)For any object A 2 A and any map f :F A - ! B, there exists a map a: A -! X in A and a weak equivalence e: F X -! B in w (B) such that f is homotopic to e O F (a). Then F induces a homotopy equivalence KA - ! KB. Proof. We produce an object A0 and maps g, h that satisfy condition (ii) of the Approximation Theorem. We have assumed that f is homotopic to e O F (a), so there exists a generaliz* *ed cylinder object (W 0; j01; j02; q0) of F A and :W 0-! B with O j01= f, O * *j02= e O F a. We can construct in A the generalized cylinder object W with the same gluings of faces; then we have ff :F W ~=W 0with ff O F (ji) = j0i, since F pre* *serves cylinder objects. Let A0= W qa X, and let g be the evident map A -! A0induced by j1: A -! W . Then ff induces an isomorphism F A0- ! W 0qF(a)F X, which we will denote by "ff. Consider the map "h = qF(a)e: W 0qF(a)F X -! B. The inclusion F X - ! W 0qF(a)F X is a weak equivalence by property 1.2.(ii), since it is the pushout of the following weak equivalence of diagrams F(a) id F X oo___F A //__//F A ~|id| ~ id|| ~|j01| |fflffl fflffl| |fflffl F X ooF(aF)A_//j0//__W 0: 1 The composite of this inclusion with "his the weak equivalence e, so we concluse that "his a weak equivalence since B is saturated by assumption. Choosing h to be the weak equivalence "hO "ffmakes diagram 2.2(ii) commute. __|_ | The next corollary of Waldhausen's Approximation Theorem requires some pre- liminary definitions. Definitions 2.6. We say that a map f :X - ! Y is a homotopy equivalence if there exists a morphism g :Y - ! X so that f O g and g O f are homotopic to the respective identity morphisms. In this case, we call g a homotopy inverse * *to f. We say that C is a category with w-cofibrations and homotopy equivalences 124 VI. ALGEBRAIC K-THEORY OF S-ALGEBRAS or a Waldhausen homotopy category (WH category for short) if C is a saturated Waldhausen category with cylinder functor satisfying the cylinder axiom such th* *at the weak equivalences are the homotopy equivalences. We need to make an observation about the "derived" category of a WH category, the category formed by inverting the homotopy equivalences. First note that if f is a homotopy equivalence and g a homotopy inverse to f, then g O f and f O g a* *re both identity morphisms in the derived category. From this, it is easy to see t* *hat every map in the derived category of C is represented by a map in C . One can ask when two maps give the same map in the derived category. First observe that for homotopic maps f1; f2 and homotopic maps g1; g2, the compositi* *ons f1O g1 and f2O g2 are homotopic; so we can form the homotopy category, hC whose objects are the objects of C and whose maps are homotopy classes of maps. It is straightforward to verify that homotopic maps represent the same map in the derived category, and that if two maps represent the same map in the derived category then they are homotopic. We conclude that the natural map from C to its derived category factors through hC , and that the map from hC to the deriv* *ed category is actually an isomorphism (not merely equivalence) of categories. The next result is now an immediate corollary to Theorem 2.5. Corollary 2.7. Let A and B be WH categories. Suppose F :A - ! B is an exact functor that preserves cylinder objects and that passes to an equivalence* * on the derived categories. Then F induces a homotopy equivalence KA - ! KB. Proof. We reduce to Theorem 2.5: Condition (i) is clear. For condition (ii), choose X 2 A so that F X is isomorphic to B in the derived category of B. Since every map in the derived category of B is represented by an actual map in B, we can choose e: F X -! B that represents this isomorphism. Then e is a homotopy equivalence; let e0be a homotopy inverse. Now there exists a map a: A -! X so that F a: F A -! F X represents the same map as e0O f in the derived category of B. We conclude that f is homotopic to e O F a. __|_ | 3. Application to categories of R-modules For an S-algebra R, let CR be the full subcategory MR consisting of the cell R-modules, and C W R the category of CW R-modules and cellular maps. We denote by fCR the full subcategory of CR of finite cell R-modules and fC W R the full subcategory of C W R of finite CW R-modules; more precisely, we must choose small full subcategories containing at least one object of each isomorph* *ism class, but the fact that the category of spectra has canonical colimits allows a strict interpretation of the definition of cell and CW R-modules under which the categories fCR and fC W R are already small. When C is one of the categories fCR or fC W R , we can give C the structure of a WH category as follows. We 3. APPLICATION TO CATEGORIES OF R-MODULES 125 define the category of w-cofibrations, co(C ), to consist of those maps which a* *re isomorphic (in MOR C ) to the inclusion of a subcomplex, and the category of w* *eak equivalences w(C ) to consist of those maps in C which are homotopy equivalence* *s. We take as our cylinder functor the ordinary mapping cylinder. Proposition 3.1. These definitions specify the structure of WH-categories on fCR and fC W R and the inclusion fC W R -! fCR is an exact functor which pre- serves cylinder objects. Furthermore, when R is connective, this inclusion indu* *ces a homotopy equivalence of K-theory spectra. Proof. We check the definitions directly. Let C be either fCR or fC WR with co(C ) and w (C ) as above. First we check that co(C ) is a category. Let f :A -! B and g :B -! C be arrows in C . These are isomorphic to inclusions of subcomplexes, and without l* *oss of generality we can assume that f is the inclusion of a subcomplex and that g * *is isomorphic to an inclusion g0:B0- ! C with the isomorphism on the codomain C the identity: __b__ 0 B //B g || |g0| fflffl|fflffl| C ______C: By choosing different sequential filtrations if necessary we can assume that th* *e map B -! C is sequentially cellular and therefore so is the map B -! B0 (we adjust the sequential filtration on A as well if necessary so that it remains a subcom* *plex of B). Since C is built from B0 by attaching cells, we can form an isomorphic complex D by attaching the same cells to B via b-1. In the CW case, D is CW and the isomorphisms to and from C are cellular because we have assumed that the isomorphisms b and b-1 are cellular. Now A is a subcomplex of B, which is a subcomplex of D and the map A -! D is isomorphic to the map A -! C. Properties 1.1(i) and (ii) and 1.2(i) are clear as will be 1.2(ii) once we sh* *ow 1.1(iii). Given a diagram in C , f A //__//B g|| fflffl| C with f an arrow in co(C ), we show that we can find a pushout in C so that the * *map from C is the inclusion of a subcomplex. We can assume without loss of generali* *ty that f is the inclusion of a subcomplex and that the map g is sequentially cell* *ular. Since B is built from A by attaching cells, we can form a cellular complex D 126 VI. ALGEBRAIC K-THEORY OF S-ALGEBRAS by attaching the cells to C via g :A -! C. In the CW case, D will be a CW complex since we have assumed that g is cellular. For categorical reasons, D mu* *st be a pushout of the above diagram, and by construction the map from C is the inclusion of a subcomplex. If R is connective, we can approximate any finite cellular R-module by a fini* *te CW R-module, and the last statement follows by Theorem 2.5. __|_ | Definition 3.2. We define the algebraic K-theory of the S-algebra R to be the spectrum KfCR , and we denote it by KR. We define the algebraic K-groups of R to be the homotopy groups of this spectrum, KnR = ssnKR = KnfCR . Although the categories fCR and fC W R seem the most natural choices for K- theory, there are many other possibilities. Indeed, since pushouts along cofibr* *ations in MR preserve weak equivalences, it is easy to see that any subcategory of MR * *that is a category with w-cofibrations such that all of the w-cofibrations are cofib* *rations becomes a Waldhausen category by taking the weak equivalences to be the ordinary weak equivalences. In particular, when X is small full subcategory of MR that contains the trivial R-module, and is closed under pushouts along cofibrations,* * then X is a category with w-cofibrations the set of all cofibrations in X and in thi* *s way becomes a Waldhausen category. We shall call the resulting Waldhausen category structure on X the standard Waldhausen structure. If C is a full subcategory of CR that is small, contains the trivial R-module and is closed under pushouts along maps isomorphic to inclusions of subcomplexes, then C is a category with * *w- cofibrations the set of maps in C isomorphic to the inclusion of a subcomplex a* *nd in this way becomes a Waldhausen category. We shall call the resulting Waldhausen category structure on C the standard cellular Waldhausen structure. If X or C is closed under smashing with I+ , then the mapping cylinder gives a cylinder functor satisfying the cylinder axiom, which we shall call the standard cylinder functor. Since a standard cellular Waldhausen category with the the standard cylinder functor is a WH category, we shall call such a category a standard WH category. It might at first appear that the standard Waldhausen structures are somewhat rare, but the following remark demonstrates that they are actually qui* *te common. Remark 3.3. (Smallest Standard Waldhausen Categories) Given a set of ob- jects A MR that is not necessarily closed under pushouts along cofibrations, we can form a small category B containing A that is. We let B be the union of an expanding sequence of small categories A0 -! A1 -! A2 -! . .,.where Ob (A0) = A and An+1 is the full subcategory of MR of objects that are pushouts of diagrams (with one leg a cofibration) in An (one choice of object for each s* *uch diagram). Since the set of all maps in An is small (by induction), An+1 is a sm* *all category. It is easy to see that B has a kind of universal property: whenever a 3. APPLICATION TO CATEGORIES OF R-MODULES 127 standard Waldhausen category contains a full subcategory equivalent to A (re- garded as a full subcategory of MR ), it must contain a full subcategory compat* *ibly equivalent to B. For this reason, we will refer to B as the smallest standard Waldhausen category containing A . Observe that when all the objects of A have the weak homotopy types of finite cell complexes then so do all the objects of B (by Corollary I.6.5). Often we will want our standard Waldhausen categories to have the standard cylinder functor. In forming An+1 from An above we could also include X ^ I+ for each X 2 An. In this case, An+1 will still be small, but now B will be clos* *ed under smashing with I+ , and hence have the standard cylinder functor. It is ea* *sy to see that B will now have a similar universal property with respect to standa* *rd Waldhausen categories with the standard cylinder functor. For this reason we wi* *ll refer to the category constructed in this way as the smallest standard Waldhaus* *en category with standard cylinder functor containing A . Again, when all the obje* *cts of A have the weak homotopy types of finite cell complexes then so do all the objects of B. If A CR , we can do a similar construction but using maps isomorphic to inclusions of subcomplexes in place of cofibrations. Then the resulting catego* *ry B CR is a standard cellular Waldhausen category (a WH category if we include smashes with I+ ) and has a similar universal property with respect to standard cellular Waldhausen categories. We shall not actually use this construction, b* *ut we could call B in this case the smallest standard cellular Waldhausen category containing A (or if we include smashes with I+ , smallest standard WH category containing A ). Furthermore, observe that if all the objects of A have the homo* *topy types of finite cell complexes, then so do all the objects of B. One advantage of the standard Waldhausen structures is that inclusions of sub- categories are exact functors. Proposition 3.4. Suppose X is a subcategory of Y . If X and Y are both standard Waldhausen categories or both standard cellular Waldhausen categories, then the inclusion X -! Y is an exact functor. If X is a standard cellular Waldhausen category and Y is a standard Waldhausen category, then the inclusion X - ! Y is an exact functor. Many standard Waldhausen categories have K-theory equivalent to fCR . The following proposition follows directly from Theorem 2.5 (and the Whitehead The- orem) and will often apply to the smallest standard categories constructed abov* *e. Proposition 3.5. Let X be a standard Waldhausen category with standard cylinder functor or a standard WH category. If X contains fCR , and if fur- thermore each object of X is weakly equivalent to a finite cell complex, then * *the induced map of K-theory spectra is a homotopy equivalence. 128 VI. ALGEBRAIC K-THEORY OF S-ALGEBRAS The K-theory we consider, the K-theory of fCR , is best thought of as analogo* *us to the K-theory of finitely generated free modules: indeed, since all objects * *are constructed from the sphere R-modules by a finite number of extensions by spher* *es, it follows immediately that the obvious homomorphism Kf0ss0R -! K0R (induced by [n] 7! _ni=1SR ) is surjective, where "Kf0ss0R" denotes K0 of the finitely g* *enerated free modules of the ring ss0R. When R is connective this homomorphism is an isomorphism whose inverse is given by the Euler characteristic of a CW object X, the alternating sum of the classes of Cn(X), where C* is the chain functor of I* *V.3. For this reason, categories that could be reasonable alternatives to the catego* *ries fCR and fC W R would be those small categories of semi-finite cell R-modules th* *at are standard WH categories. When such a category contains fCR , it follows from [65, 1.10.1], [68, 1.5.9] and the argument of [23, x1] (as observed in [65, 1.1* *0.2]) that the inclusion will induce an isomorphism of homotopy groups of K-theory spectra except in dimension zero. Intuitively, whereas the K-theory of fCR or fC W R is like the K-theory of the finitely generated free modules, we might think of the* * K- theory of semi-finite objects as analogous to the K-theory of the finitely gene* *rated projective modules. We conclude this section by remarking that when R is an A1 ring spectrum but not an S-algebra, we can make analogous observations about the K-theory of categories of its modules. However the functor S ^L (?) is an exact functor that converts such a category to the corresponding category of S ^L R-modules and induces a homotopy equivalence of K-theory spectra by Theorem 2.5. Thus, results about the K-theory of A1 ring spectra follow from results about the K- theory of S-algebras. 4. Homotopy invariance and Quillen's algebraic K-theory of rings In this section we prove some properties of the K-theory of the category fCR and compare with the K-theory of (discrete) rings. We observe that K-theory as defined above gives a functor from the category of S-algebras to the stable category which has nice homotopical properties. Proposition 4.1. If OE: A - ! B is a map of S-algebras, then the functor B ^A (-): fCA - ! fCB (or fC W A -! fC W B ) is exact and preserves cylinder objects. This association makes K into a functor from the category of S-algebras to the stable category. Proof. The first statement follows from III.4.1, the second from the isomor- phisms C ^B (B ^A (-)) ~=C ^A (-) and A ^A (-) ~=id. __|_ | Proposition 4.2. If OE: A -! B is a map of S-algebras that is a weak equiva- lence, then KOE is a homotopy equivalence. 4. HOMOTOPY INVARIANCE AND QUILLEN'S ALGEBRAIC K-THEORY 129 Proof. From III.4.2, B ^A (-) induces an equivalence of derived categories DA - ! DB , which restricts to an equivalence of the derived categories of fini* *te cell complexes by an easy application of the Whitehead theorem. The result foll* *ows from Corollary 2.7. __|_ | We compare this K-theory with Quillen's algebraic K-theory. Let k be a ring, and let Hk denote the Eilenberg-Mac Lane S-algebra of k. We shall use the symbol Kfk for the algebraic K-theory of the finitely generated free modules of* * k, a covering spectrum of Kk. Theorem 4.3. KHk is homotopy equivalent to Kfk, naturally in k. Proof. We can identify Kfk with the K-theory of the WH category of finite free k-chain complexes with w-cofibrations the split monics, weak equivalences * *the quasi-isomorphisms, and the cylinder functor given by the usual mapping cylinde* *r. (see, for example, [65, 1.11.7].) We will denote this WH category as fC W k. The functor C*: fC W Hk -! fC W k of IV.2 is exact and preserves cylinder objects. By the Hurewicz theorem IV.3.6, a map between finite CW modules is a weak equivalence if and only if its image under C* is a quasi-isomorphism, hence the theorem will follow from Theorem 2.2 if we can show that condition (ii) hol* *ds. Given a finite free chain complex M*, we can actually construct a CW Hk- module X whose cellular chain complex C*(X) is isomorphic to M*. We proceed by induction. Since M* is finite, Mi is zero below some m, and we take the i- skeleton of X, Xi to be the trivial Hk-module for i < m. Now assume that we have constructed Xn and an isomorphism C*(Xn) -! Mn* , where Mn* denotes the brutal n-truncation of M*, i.e. Mni = Mi for i n and Mni = 0 for i > n. By IV.2.3, ssn(Xn) ~=Hn(Mn* ), which is the kernel of the differential dn-1, i.* *e. the cycles of Mn. Via this isomorphism and a choice of basis for Mn+1, the differen* *tial dn specifies a homotopy class of maps from a wedge of SnHk to Xn. Choose a representative of this homotopy class, and let Xn+1 be the CW complex formed by attaching (n + 1)-cells along this map. By construction Cn+1(Xn+1) ~=Mn+1, compatibly with the differentials. Given an Hk-module A and a map f :C*(A) -! M*, we show that we can find a map a: A -! X such that C*(a) agrees with f via the isomorphism contructed above. Assume we have constructed this map as far as the n-skeleton of A, i.e. we have an : An -! X such that C*(an) = fn . Now An+1 is formed from An by attaching a finite wedge _CSnHkalong a map ff : _ SnHk -! An. The map fn+1: Cn+1(A) -! Mn+1 specifies a homotopy class of maps (_CSnHk; _SnHk) -! (Xn+1; Xn) -! (X; Xn), whose class on _SnHkagrees with [an O ff], since ssn(Xn) coincides with the cycles of Mn. We choose a representative in the homotopy cla* *ss whose restriction to SnHkis anOff. This extends to a map an+1: An+1 -! Xn+1 -! X, and by construction C*(an+1) agrees with fn+1 . __|_ | 130 VI. ALGEBRAIC K-THEORY OF S-ALGEBRAS Remark 4.4. For R a connective S-algebra, the functor C* of IV.3 is an exact functor fC W R - ! fC W k, for k = ss0R. The induced map of K-theory can be thought of as "discretization" and factors as KR -! KHk -! Kfk. Remark 4.5. Another question one may ask is how the K-theory of k compares with the K-theory of k regarded as an A1 ring, i.e. the K-theory of the catego* *ry of finite cell A1 k-modules (as constructed in [34]). In fact, Propositions 4.* *1 and 4.2 have exact analogs in the theory of discrete A1 rings (with close analogs * *of the proofs). In particular, it follows from Corollary 2.7 that the natural qua* *si- isomorphism of the ring k with its A1 enveloping algebra induces a homotopy equivalence from the free K-theory of k to its A1 -K-theory. 5. Morita equivalence Next, we discuss Morita equivalence, the relationship of the category of R- modules to the category of modules over (the analogue of) a matrix ring of R. We introduce the shorthand notation _nX for _ni=1X, and we (temporarily) define MnRQ= FR (_nR; _nR), Mn1 = FR (R; _nR) ~= _nR, and M1n = FR (_nR; R) ~= nR. By III.6.12, we see that MnR is an S-algebra, Mn1 an (MnR; R)-bimodule, and M1n an (R; MnR)-bimodule. Classical Morita equivalence is the theorem that for a (discrete) ring R, tensoring with these two bimodules gives an equivalence between the category of R-modules and the category of MnR-modules. The ob- servation that this restricts to an equivalence between the categories of finit* *ely generated projective modules proves that Quillen's algebraic K-theory is Morita invariant. In the case we consider, it is unreasonable to hope for an equivalence between MR and MMnR since products and coproducts are not isomorphic, but we can ask for an equivalence of DR and DMnR . Furthermore, because our K-theory is really the K-theory of free modules, we cannot expect the induced map of K-theo* *ry to give an isomorphism on K0 in general (since for a (non-zero) discrete ring, * *the image of the free module of rank one is a projective but not free module for n * *> 1), but we can ask for an isomorphism of the higher K-groups. In this section we fi* *nd affirmative answers to each of these questions in the following theorems. Theorem 5.1. (Morita Equivalence) The derived functors of Mn1 ^R (?) and M1n ^MnR (?) give an equivalence of categories DR ' DMnR , which restricts to an equivalence of the derived categories of semi-finite objects. Theorem 5.2. (Morita Invariance of K-Theory) The functor M1n^MnR (?) in- duces a map of K-theory KMnR -! KR, which on homotopy groups (K-groups) sends a generator in dimension zero to n times a generator, and gives an isomor- phism on the higher groups. 5. MORITA EQUIVALENCE 131 We prove Theorem 5.1 by imitating as much as possible the proof of classical Morita equivalence. The following lemma gives a good start in this direction. Lemma 5.3. The (R; R)-bimodules R and M1n ^MnR Mn1 are isomorphic. Proof. By a comparison of colimits, using the map of S-algebras R -! MnR, it is not hard to see that the following diagram is a coequalizer (cf. VII.1.9) M1n ^R MnR ^R Mn1 ____//_//_M1n ^R Mn1_//_M1n ^MnR Mn1: The evaluation map M1n^R Mn1 -! R coequalizes this diagram, so induces a map M1n ^MnR Mn1 -! R, which is evidently an (R; R)-bimodule map. We show that this is an isomorphism by observing that M1n ^R MnR ^R Mn1 ____//_//_M1n ^R Mn1_//_R is a split coequalizer of S-modules. The splitting is given by maps analogous * *to those in the discrete case: The map R ~= R ^R R -! M1n ^R Mn1 is the smash over R of the map R -! _nR that includes it as the first wedge-summand with the map R -! FR (_nR; R) induced by the map _nR -! R which collapses onto the first summand. The map M1n ^R Mn1 ~=M1n ^R Mn1 ^R R -! M1n ^R MnR ^R Mn1 is induced by the identity on M1n smashed over R with the map Mn1 -! MnR (FR (R; _nR) -! FR (_nR; _nR)) induced by the map _nR -! R collapsing onto the first summand in the first variable smashed over R with the inclusion of R * *as the first summand in Mn1. It is straight-forward to verify that the composites * *are as required to split the diagram. __|_ | Proof of Theorem 5.1. We verify that the composite DR -! DMnR - ! DR is naturally isomorphic to the identity. Let X be a cellular R-module, and let * *Y be a cellular MnR-module approximation to Mn1 ^R X; we must show that the map M1n ^MnR Y - ! M1n ^MnR Mn1 ^R X ~=X is a weak equivalence. Observe that the obvious map _nMn1 -! MnR is a weak equivalence and a map of (MnR; R)-bimodules. Since X is a cellular R-module, the map _nMn1 ^R X - ! MnR ^R X is a weak equivalence and the composite map _nY -! MnR ^R X is a homotopy equivalence. Now we conclude that M1nQ^MnR _nY -! _nX must be a weak equivalence, since the map _nX - ! ( nR) ^R X ~=M1n^R X is, but the induced map on homotopy groups is just the direct sum of n copies of the map we are interested in, so this map must also b* *e a weak equivalence. The reverse composite DMnR -! DR - ! DMnR is simpler. Let X be a cellu- lar MnR-module. Since Mn1 ^R (?) preserves weak equivalences, the composite 132 VI. ALGEBRAIC K-THEORY OF S-ALGEBRAS functor can be represented by X 7! Mn1^R M1n^MnR X. Observe that the evalua- tion map Mn1^R M1n -! MnR is a weak equivalenceQof (MnR;QMnR)-bimodules. On the underlyingQR-modulesQit is a map _n n R -! n_nR inducing an iso- morphsim R* -! R*. This induces the natural isomorphism (in DMnR ) to the identity. Since the derived categories of semi-finite objects are full subcategories of* * the derived categories DR and DMnR , we see that this equivalence restricts, since * *both functors send wedge summands of finite objects to wedge summands of finite ob- jects. __|_ | Let C be the smallest standard Waldhausen category with standard cylinder functor containing fCR and the image of CMnR . By Proposition 3.5, the K-theory of C is homotopy equivalent to KR. Let I be the full subcategory of C of objects weakly equivalent to objects in the image of fCMnR . Since pushouts alo* *ng cofibrations are homotopy equivalent to homotopy pushouts, which M1n ^MnR (?) preserves, it is easy to check that I is closed under pushouts along cofibratio* *ns and therefore is a standard Waldhausen category with standard cylinder functor; moreover, the functor M1n ^MnR (?): fCMnR - ! I is exact. Lemma 5.4. The exact functor M1n ^MnR (?): fCMnR -! I induces a homo- topy equivalence of K-theory. Proof. We apply Theorem 2.5: for A 2 fCMnR , B 2 I , and f :M1n ^MnR A -! B, we find X 2 fCMnR , a weak equivalence e: M1n ^MnR X - ! B and a: A -! X, such that eOM1n^MnR a is homotopic to f. By assumption, B is weakly equivalent to the image of some X 2 fCMnR , so Mn1^R B is an MnR-module weakly equivalent to Mn1^R M1n^MnR X, which in turn is weakly equivalent to X. Thus, by the Whitehead Theorem, there exists a weak equivalence ffl: X -! Mn1 ^R B. Since the natural map i: Mn1 ^R M1n ^MnR A -! MnR ^MnR A ~= A is a weak equivalence, it has a homotopy retraction r, and there exists a map a: A -! X such that ffl O a is homotopic to (Mn1 ^R f) O r, again by the Whitehead theore* *m. Thus the solid line part of the following diagram commutes up to homotopy. Mn1 ^R M1n ^MnR A ____OOVVVVVMn1^fVVV| i _______r| VVVVVVV fflffl_____| VV**V A h4Mn14^R B: hhhh a|| hhhh'fflhhhhh fflffl|hhhhhhhh X We apply the functor M1n ^MnR (?). The isomorphism constructed in Lemma 5.3 induces a natural transformation : id -! M1n ^MnR Mn1 ^R (?), from which we 5. MORITA EQUIVALENCE 133 get the diagram M1n ^MnR A ____RRRRRfROO| (M1n^i)O_______M1RRRRRn^r| fflffl_____| RRRR((R M1n ^MnR A llB:66 llll M1n^a|| ll'lllll fflffl|-1O(M1n^ffl)lll M1n ^MnR X By the associativity of the multiplication pairing, the diagram ~= M1n ^MnR Mn1 ^R M1n _____//R ^R M1n M1n^i|| |~=| fflffl| fflffl| M1n ^MnR MnR _____~=___//_M1n must commute, and we conclude that (M1n^MnR i) O is the identity. Now, letting e = -1 O M1n^MnR ffl, we see that e O M1n^MnR a is homotopic to f as required. * * __|_ | Lemma 5.5. I is closed under extensions in C . Proof. We need to show that for a cofibration sequence A ae B i C in C , if A and C are in I , then B is also in I . It suffices to consider the case when * *A, B, and C are celluar R-modules, since any cofibration sequence can be replaced by a weakly equivalent one of this form. Using the proof of the last lemma, the map * -! C allows us to find X in fCMnR and a weak equivalence e: M1n^MnR X -! C. Composing with the map c: C - ! A implied by the cofibration sequence, and applying once again the proof of the last lemma, we find an object Y in fCM* *nR , a map a: X - ! Y , and a weak equivalence f :M1n ^MnR Y -! A that make the following square homotopy commute: M1n^a M1n ^MnR X ____//_M1n ^MnR Y e|'| ' |f| fflffl| fflffl| C _______c______//A: We conclude that the induced map on cofibers C(M1n^MnR a) -! C(c) is a weak equivalence. The functor M1n ^MnR (?) commutes with smashing over S on the right with S-1Sand smashing on the right with the space S1, both composites of which are homotopic to the identity in fCMnR . We conclude that the map C(M1n^MnR a) ^S S-1S-! C(c) ^S S-1Sis a weak equivalence. But C(c) ^S S-1Sis homotopy equivalent to B and C(M1n^MnR a) ^S S-1S~=M1n^MnR (C(a) ^S S-1S) is in the image of fCMnR , hence B is in I . __|_ | 134 VI. ALGEBRAIC K-THEORY OF S-ALGEBRAS Proof of Theorem 5.2. The inclusion I -! C is an exact functor. It is easy to see that on K0 it sends a generator (SMnR ) to n times a generator (M1n ^MnR MnR ^S SS ~=M1n ^S SS ' _nSR ). We want to see that it induces an isomorphism of the higher K-groups. Let J be the full subcategory of C of all objects whose class in K0C is in the image of K0I (so in particular I J ). It follows from the relations that define K0 that J inherits the structure of a standard Waldhausen category with standard cylinder functor. By Proposition 3.4, the inclusions I - ! J and J - ! C are exact functors. We use the argument of [23, x1] to show that I is strictly cofinal in J (in the sense of [68, 1.5.9]). We define an equivalence relation on the objects of * *C by letting A and A0be equivalent if there exists some X 2 I such that A_X is weakly equivalent to A0_ X. Let G be the set of equivalence classes under this relatio* *n. Then G is a group under the operation "_" with the inverse of A represented by _n-1A. We have an obvious homomorphism G -! K0C =K0I ; we construct an inverse to this homomorphism. If A ae B i C is a cofibration sequence in C , then _nA ae B _ (_n-1A) _ (_n-1C) i _nC is a cofibration sequence. But _nA and _nC are in I , so B _ (_n-1A) _ (_n-1C) is in I since I is closed under extensions in C ; therefore, B _ (_n-1A) _ (_n-1C) represents the identity in G* * and hence B represents the same element as A _ C in G. If A is weakly equivalent to A0then they represent the same element in G. We see that G satisfies the univer* *sal relations that define K0C , and so specifies a map K0C - ! G. This map clearly factors through a map K0C =K0I -! G that is evidently inverse to the map above. Now we see that J consists of the objects whose class in G is the identi* *ty, so we conclude that for any X 2 J , there exists Y 2 I so that X _ Y is weakly equivalent to an object of I and hence X _ Y is an object of I . Now by [68, 1.5.9], I - ! J induces a homotopy equivalence of K-theory, but by [65, 1.10.1], KiJ - ! KiC is an isomorphism for i > 0. __|_ | 6. Multiplicative structure in the commutative case In this section, we prove the following theorem (cf. [62]). Theorem 6.1. If R is a connective commutative S-algebra then KR is homo- topy equivalent to an E1 ring spectrum and therefore weakly equivalent to a co* *m- mutative S-algebra. This result and the results of the next section depend on the following techn* *ical lemma, which the reader may recognize as a simple application of the theory of [68, x1.6-1.8] to our new categories. Although we believe that Theorem 6.1 may generalize to non-connective commutative S-algebras, this lemma is peculiar to * *the connective case and relies on the existence of the ordinary homology theories of 6. MULTIPLICATIVE STRUCTURE IN THE COMMUTATIVE CASE 135 IV.3. For this lemma, we need R to be a connective but not necessarily commu- tative S-algebra. Let C be a standard Waldhausen category of R-modules that contains fCR and that only contains objects of the weak homotopy type of finite cell R-modules. We see by Proposition 3.5 that KC is homotopy equivalent to KR. We denote by C m the full subcategory of objects weakly equivalent to a finite wedge of SmR. Observe that for each m, the category of weak equivalences* * of C m is a symmetric monoidal category under the operation of wedge, and denote the associated spectrum as kC m. Suspension induces a system of maps of spectra kC m -! kC m+1. Lemma 6.2. The homotopy colimit of the system {kC m} is homotopy equivalent to KC . Proof. The Hurewicz theorem IV.3.6 allows us to identify C m with the full subcategory of C of objects whose ordinary homology HR*is zero in all dimensions except m, and in dimension m is a finitely generated free module. Let "C mbe th* *e full subcategory of C of objects whose ordinary homology HR*is zero in all dimensions except m and in dimension m is a finitely generated stably free module, i.e. is isomorphic to the kernel of a surjective map of finitely generated free modules* *. Let C n be the full subcategory of C of those objects which are (n - 1)-connected. * *By the Hurewicz theorem IV.3.6 these are exactly the objects whose homology is zer* *o in dimensions less than n. We give the categories C m and "C mWaldhausen structures by defining the w-cofibrations to be the w-cofibrations of C whose quotients li* *e in the subcategory in question. The categories C n have the structure of standard Waldhausen categories with the standard cylinder functor. Suspension is an exact functor "C m-! "C m+1and C n - ! C n . The inclusion of "C min C n is an exact functor, and induces a map hocolimm!1|wSo"C m| -! hocolim |wSoC n | for each n. Next observe that ordinary homology HR*restricted to C n is a homol- ogy theory in the sense of [68, x1.7] (at least after shifting the indexing), a* *nd that the categories "C mform categories of "spherical objects" for C n for the clas* *s of finitely generated stably free modules. Since this theory satisfies the "Hypoth* *esis" of [68, 1.7.1], we conclude that the map above is a weak equivalence. On the ot* *her hand the inclusions C n -! C n+1 are exact functors which induce cofibrations |wSoC n | -! |wSoC n+1 |, whose colimit is |wSoC |. Taking the colimit (over* * n) of the homotopy equivalence above, we get a homotopy equivalence hocolimm!1|wSo"C m| -! hocolim |wSoC |: 136 VI. ALGEBRAIC K-THEORY OF S-ALGEBRAS The maps on the right are all homotopy equivalences (by [68, 1.6.2]), so we con* *clude that there exists a homotopy equivalence hocolimKC"m - ! KC . We apply the Strict Cofinality Theorem [68, 1.5.9] to conclude that KC"m is homotopy equivalent to KC m. Now we are reduced to comparing KC m with kC m. According to [68, 1.8.1], it suffices to observe that cofibrations in C * *m are "splittable up to weak equivalence". Given a cofibration A ae B, we can find a basis of the free module Hm (B) that represents the union of bases for Hm A and Hm (B=A). The Hurewicz theorem IV.3.6 now specifies a homotopy class of weak equivalence from the wedge of A and a wedge of spheres to both B and A _ B=A, relative to the maps from A. __|_ | Proof of Theorem 6.1 Let C be the smallest standard Waldhausen category with standard cylinder functor containing R, SnR(all n) and all finite smash pr* *oducts over R of these. It is easy to check that the bifunctor (?)^R (?) restricts to * *C (up to equivalence), so C is a symmetric bimonoidal category under coproduct and smash product over R. Let C 0be as in the lemma above. Then C is the full subcategory of C of objects weakly equivalent to a finite wedge of SR . Since smash product over R with R and with SnRpreserve weak equivalences, so do smash products over R with any object of C , and the smash product over R of objects in C 0is weakly equivalent to a finite wedge of SR and therefore is an object of C 0. Thus the smash product over R restricts to a bifunctor on C 0that makes C 0a symmetric bimonoidal category. By the work of [49], we can construct kC 0functorially as * *an E1 ring spectrum. Next observe that suspension and S-1R^R (?) give functors C m -! C m+1 and C m+1 -! C m for which both composites are weakly equivalent to the identity. We conclude that suspension gives a homotopy equivalence kC m -! kC m+1, and that kC 0is homotopy equivalent to KC by the previous lemma. __|_ | 7. The plus construction description of KR We have observed that the category fCR gives a K-theory KfCR that has some right to be called the algebraic K-theory of R. This section is devoted to a co* *m- parison with another possible definition, based on Quillen's plus construction.* * In what follows, R is a fixed connective S-algebra, and k = ss0R. We shall make us* *e of classifying spaces of the topological monoids MR (X; X). Unfortunately even when X = SR , we cannot guarantee that the inclusion of the identity element is a co* *fi- bration. There are well-known ways of overcoming this difficulty, i.e. whiskeri* *ng the monoids [44] or using thickened realizations [63]. In this and the next sec* *tion, we shall take advantage of such techniques implicitly wherever necessary without further comment. 7. THE PLUS CONSTRUCTION DESCRIPTION OF KR 137 W W Let M"nR be the topological space fCR ( nSR ; nSR ); then ss0M"nR ~= Mn(k), the ordinary matrix ring of the ring k. Let GgLnR be the space consisting of those connected components of M"nR whose image in Mn(k) is invertible. Then gGLnR is a topological monoid; indeed, it is the monoid of homotopy equivalences in M"nR. We can consider its classifying space BGgLnR. We have the inclusion in :gGLnR -! gGLn+1R obtained by sending the last wedge summand to the last wedge summand via the identity map, and it induces Bin :BGgLnR -! BGgLn+1R. Let BGgLR be the telescope of these maps. Now ss1BGgLR ~=GL(k) has a perfect normal subgroup, so we can form BGgLR+ (Quillen's plus construction). We shall see in a moment that Kf0k x BGgLR+ is an infinite loop space. Define K+ R to be the connective spectrum obtained by delooping Kf0k x BGgLR+ . We prove the following "plus equals So" theorem. Theorem 7.1. K+ R is weakly equivalent to KR. First we need to specify the infinite loop space structure on Kf0k x BGgLR+ .* * For this, we observe that Kf0kxBGgLR+ is the group completion of the classifying sp* *ace of the topological category W whose (discrete) set of objects is the finite wed* *ges of SR and whose space of morphisms is the set of homotopy equivalences topologized as a subspace of the space of morphisms of MR . Call this group completion B. In the case when Kf0k is the integers, the classifying space of W is the disjo* *int union of the BGgLnR and we may apply the remarks of [63, x4] to conclude that we ` have a homology isomorphism to B from the telescope of maps BGgLnR to itself induced by the maps Bin :BGgLnR -! BGgLn+1R. This telescope is easily seen to be Kf0k x hocolimnBGgLnR. We conclude that B ' Kf0k x (hocolimn BGgLnR)+ . In the pathological case when Kf0k is not the integers, i.e. when there exist* *s a ho- motopy equivalence _jSR ' _kSR for j 6= k, we still have a homology isomorphism to the group completion B from the telescope T of maps from BW to itself induced by addition of an identity map on the wedges of sphere R-modules. Proposition 7.2 below allows us to see that BW is homotopy equivalent to a disjoint union * *of of some of the BGgLnR, one choice for each isomorphism class of finitely genera* *ted free ss0R-modules. Now we see that the telescope T is homotopy equivalent to Kf0k x hocolimnBGgLnR, and we conclude that B ' Kf0k x (hocolimn BGgLnR)+ . To identify the homotopy type of BW in the pathological case above, we need the following proposition. We will need a similar result again later, and we ha* *ve written this proposition in the minimal possible generality necessary to handle both cases. The proposition says essentially that if the morphisms in a catego* *ry are all homotopy equivalences (in a certain sense), then the classifying space * *of the monoid of endomorphisms of any object is homotopy equivalent to its connected component in the classifying space of the category. Because this proposition h* *as 138 VI. ALGEBRAIC K-THEORY OF S-ALGEBRAS obvious generalizations with more general scope than its use in this section, we break our rule of not mentioning the necessary cofibration assumptions. As al- ways the reader has the choice of deleting the cofibration assumption by using a whiskering technique or employing the thickened realization. Proposition 7.2. (cf. [68, 2.2.7]) Let C be a topological category with disc* *rete set of objects such that the identity morphism (from objects to morphisms) is a cofibration. Let X be an object of C and denote by CX the full subcategory of C containing X. Suppose that for each morphism f :Y -! Z in C , there is some f0: Z -! Y so that f0 O f and f O f0 each lie in the same path component of C (Y; Y ) and C (Z; Z) as the respective identity elements. Then the inclus* *ion CX - ! C induces a homotopy equivalence of the classifying space of CX with the connected component of its image in the classifying space of the category C . Proof. First observe that Quillen's "Theorem A" [55] holds with essentially the same proof for continuous functors between topological categories with disc* *rete object sets whose identity map (objects to morphisms) is a cofibration. Since the connected component of the image of CX in the classifying space of C is the classifying space of the connected component (as a graph) of C that cont* *ains X, we can reduce to this smaller category and assume without loss of generality* * that C is connected (as a graph). Applying Quillen's Theorem A (dual formulation), we are reduced to showing that for every Y in C , the topological category CX =* *Y is contractible. But if f :Y - ! Z is morphism in C , then we have f0: Z -! Y and paths fl :f0Of 1Y and fl0:f Of0 1Z. We can interpret the morphisms f and f0 as continuous functors CX =Y - ! CX =Z, CX =Z -! CX =Y , and the paths fl and f* *l0 as continuous functors CX =Y x I -! CX =Y , CX =Z x I -! CX =Z. Passing to the classifying spaces we see that the paths represent homotopies B(CX =Y ) x I - ! B(CX =Y ) and B(CX =Z) x I -! B(CX =Z) from the compositions Bf0 O Bf and Bf O Bf0 to the repective identities. In short, CX =Y and CX =Z are homotopy equivalent. Since we have reduced to the case when C is connected (as a graph), we see that CX =Y is homotopy equivalent to CX =X. The lemma is established by the observation that CX =X has a final object and therefore is contractible. _* *_|_ | We begin to compare K+ R to KfCR . One obvious obstacle is that we have defined K+ R in terms of a topological category and KfCR in terms of a discrete one. Let wC 0denote the (discrete) full subcategory of w (fCR ) whose objects a* *re homotopy equivalent to wedges of SR ; the set of morphisms is the set of homoto* *py equivalences. Using arguments similar to [68, 2.2], we relate wC 0to both W and wSofCR . Lemma 7.3. (cf. [68, 2.2.5]) There is a chain of weak equivalences relating * *the classifying spaces of the categories W and wC 0. Each map in the chain is a map 7. THE PLUS CONSTRUCTION DESCRIPTION OF KR 139 of E1 spaces. Proof. For each k, let W k be the (discrete) category whose objects are the objects of wC 0and whose morphisms W k(X; Y ) consist of the set of continu- ous maps [k] - ! fCR (X; Y ) whose image lands in the component of a weak equivalence, where [k] denotes the standard topological k-simplex. In light of the adjunction T ([k]+ ; fCR (X; Y )) ~= fCR (X ^ [k]+ ; Y ), we see that this * *is the same as the set of weak equivalences X ^ [k]+ - ! Y . This is a simplicial category. Let Nj;kbe the nerve of this category. If we realize Nj;kin the k direction, we obtain a simplicial set that is the * *nerve of a topological category with a discrete set of objects. We denote this categ* *ory as |W |. In particular, the objects of |W | are the objects of wC 0and the * *mor- phism space |W |(X; Y ) is the geometric realization of the total singular co* *mplex of the subspace of fCR (X; Y ) consisting of those components which contain ho- motopy equivalences. For each X 2 W , let |W |X be the full subcategory of |W | consisting of the single object X. By the previous proposition, the incl* *usion |W |X -! |W | induces a homotopy equivalence from the classifying space of |W |X to its connected component in the classifying space of |W |. On the o* *ther hand we have a natural weak equivalence of monoids |W |(X; X) -! W (X; X), giving a weak equivalence of their classifying spaces. Let |W SR | be the fu* *ll subcategory of |W | consisting of the finite wedges of SR . Then we have weak equivalences kNj;kk -~ kW SR k -~! |W |. Next we produce a weak equivalence between wC 0and W *. The map [k] -! S0 induces a functor F :wC 0 -! W k that is the identity on objects. Let G: W k -! wC 0be the functor induced by the map S0 -! [k]+ that sends the non-basepoint to the zeroth vertex of [k]. Then GF is the identity functor on wC 0. We show that F G is homotopic to the identity. Let H :W k -! W k be the functor that takes X to X ^ I+ and that on morphisms is induced by a map I x [k] -! [k] that is the identity on the bottom face and sends the whole top face to the zeroth vertex. There are obvious natural transformations id- ! H and F G -! H given by the inclusion of bottom face and the inclusion of top face, f* *rom which we conclude that F G is homotopic to the identity. We may regard wC 0as a simplicial category constant in the k direction. The functors F are compatible * *with the faces and degeneracies (in k), and therefore assemble to a simplicial funct* *or wC 0- ! W * that induces a homotopy equivalence upon passage to classifying spaces. It is easy to see that the simplicial maps above realize to maps of E1 space* *s as they are induced by functors that preserve wedges. __|_ | Proof of Theorem 7.1 If we let C be the category fCR , then wC 0is exactly the subcategory of weak equivalences of the category C 0defined above Lemma 140 VI. ALGEBRAIC K-THEORY OF S-ALGEBRAS 6.2, the associated spectrum of which we denoted kC 0. Again suspension and S-1R^R (?) give functors C m -! C m+1 and C m+1 -! C m whose composites are weakly equivalent to the identity. We conclude that the maps in the homotopy colimit are homotopy equivalences and that kC 0is homotopy equivalent to KR. On the other hand, the previous proposition shows that K+ R is weakly equivalent to kC 0. __|_ | Remark 7.4. Note that we only needed the connectivity hypothesis to show the relationship between kC 0and KfCR . More generally we do have a homotopy equivalence kC 0' K+ R (the spectrum whose zeroth space is Kf0k x BGgLR+ ), but there is no reason to expect that the map kC 0-! KfCR will be a homotopy equivalence. In particular kC 0cannot see any relationships between spheres of different dimensions. For example, if Kf0k = Z, but SR ' S1R, then |wSofCR | * *is contractible but |wNoC 0 | is not. Remark 7.5. We should also observe that this allows another interpretation * *of the discretization map: ss0 applied to the simplicial space NW gives an E1 map KR(0) ' Kf0k xBGgLR+ -! Kf0k xBGLk+ ' Kfk(0), which evidently coincides with the discretization map and is a weak equivalence in the case when R = Hk. Remark 7.6. (Monomial Matrices) Let V be the subcategory of W of those maps _nSR -! _nSR that are wedges of n maps SR -! SR in any order. Thinking of W (_nSR ; _nSR ) as analogous to GLnR, then V (_nSR ; _nSR ) is analogous to the subgroup of monomial matrices, those matrices with a single non-zero entry * *in each row and column. Let Rx denote the monoid VR(SR ; SR ) = W (SR ; SR ). Then V (_nSR ; _nSR ) is isomorphic to the monoid n Rx and the classifying space of V is isomorphic to the disjoint union of the classifying spaces of these monoi* *ds; moreover, under this isomorphism the E1 space structure induced by wedge sums becomes the E1 space structure induced by block sums. We conclude that the group completion of the classifying space of V is homotopic to QBRx +, and th* *at V - ! W induces a map of spectra 1 BRx +- ! KR. Remark 7.7. (Naturality) Let A -! B be a map of S-algebras. We saw in Propostion 4.1 that the functor B^A (?) induces a map of K-theory spectra KA -! KB. This also restricts to a continuous functor of topological categories WA -! WB that induces a map of the plus contruction spectra above. We conclude that these two maps represent the same map in the stable category, since this functor commutes up to natural isomorphism with the functors used in comparing K+ with K. 8. COMPARISON WITH WALDHAUSEN'S K-THEORY OF SPACES 141 8. Comparison with Waldhausen's K-theory of spaces Now we compare the new algebraic K-theory with Waldhausen's algebraic K- theory of spaces. For this, let X be a connected pointed topological space, and* * let G = |GSX |, the geometric realization of the Kan loop group of the based singu* *lar complex of X. This is a topological group with non-degenerate identity. We let R = 1 (G+ ) (where the plus subscript is union with a disjoint basepoint) and we note that R is an S-algebra (IV.7.8) with k = ss0R = Z[ss0G]. Definition 8.1. Let HmndenoteWthe topological monoid of pointed G-equivari- ant homotopy equivalences of nm G+ with itself, and let BHmndenote its clas- sifying space. We have monoid maps Hmn- ! Hm+1n, and Hmn- ! Hmn+1which are induced by suspension and by addition of an identity map on the last wedge sum- mand and which are cofibrations. The algebraic K-theory of the space X is defin* *ed to be the space A(X) = Kf0Z[ss0G]x (colimm;nBHmn)+ . This is obviously equiva- lent to Waldhausen's definition [68, 2.2.1]. We shall also use the symbol A(X) * *to denote the spectrum associated to its delooping, and under this interpretation * *we will prove the following result. Theorem 8.2. The spectra K1 G+ and A(X) are homotopy equivalent, nat- urally in X. Observe that the functors 1m give maps of topological monoids W 1 W 1 Hmn- ! S ( n G+ ; n G+ ) which are easily seen to be compatible with suspension and addition of an ident* *ity map. Composing with the functors L and S ^L (?), we obtain maps of topological monoids W 1 W 1 Hmn- ! MS( nS ^L L G+ ; nS ^L L G+ ): We denote this composite functor by Lmn. The observation that the functor G+ ^(* *?) is naturally isomorphic to the functor R ^S (?) immediately implies that Lmnsen* *ds G-equivariant maps to R-module maps; therefore, we can interpret Lmnas a map of topological monoids Hmn- ! M"nR. Since forWm 2, HmnconsistsWof the subspace of those connected components of Map G ( nm G+ ; nm G+ ) which ss0 maps to GLn(R), we see that Lmn restricts to a map of monoids Hmn- ! gGLnR. We will show that in the colimit this map is a homotopy equivalence. Proposition 8.3. The map of topological monoids Ln :colimm Hmn- ! gGLnR is a homotopy equivalence. 142 VI. ALGEBRAIC K-THEORY OF S-ALGEBRAS Proof. We have defined Lmnvia a composition of functors so that it would be easy to see that it is a map of monoids; we rewrite this composition to make it easier to analyze homotopically. Consider the map of spaces W m W m W W fm :T ( nS ; n G+ ) -! MS( nSS; nSR ) (for fixed n) induced by the composite of the functors 1m, L, and S ^S (?). The colimit of the fm is the composite of the maps W m W m W W 1 colimm T ( nS ; n G+ ) -! S ( nS; n G+ ) W W 1 -! S [L]( nLS; nL G+ ) W W -! MS( nSS; nSR ); each of which is a homotopy equivalence. Via the obvious isomorphisms, the map Ln agrees with the restriction of this map to the connected components that con* *sist of weak equivalences, and so it is also a homotopy equivalence. __|_ | Since the inclusion of the identity in G is a cofibration, we see that induce* *d map colimm BHmn- ! BGgLnR is a homotopy equivalence, and hence the induced map on the plus constructions of the telescopes is a homotopy equivalence. Proof of Theorem 8.1. We need to show that we have a map of spectra. But the infinite loop space structure on A(X) comes from the operation wedge on the colimit of the topological categories whose objects are finite wedges of m G+ (* *for each m) and whose maps are the Hmn. The functors Lmnassemble to a continuous functor from this colimit to the category W which commutes with wedges and which coincides with the above homotopy equivalence on the plus constructions. We conclude that the map constructed above Kf0Z[ss0G] x (colimm;nBHmn)+ -! Kf0Z[ss0G] x BGgLR+ is a map of E1 spaces. Since G is a CW space, 1 G+ is a CW spectrum, so M"nR, gGLnR, BGgLnR, BGgLR, and BGgLR+ have the homotopy type of CW spaces; therefore, the plus construction of the previous section produces a spectrum homotopy equivalent to KR. We conclude that the spectrum A(X) is homotopy equivalent to KR. __|_ | Remark 8.4. (Linearization) The map R -! HZ ^S R is a map of S-algebras and a rational equivalence. The map HZ ^S (?): WR (_nSR ; _m SR ) -! WHZ^SR (_nSHZ^SR ; _m SHZ^SR ) induces an equivalence on rational homology. We conclude that the induced map KR -! K(HZ ^S R) is a rational equivalence. A comparison of the categories of modules for the S-algebra HZ ^S R and the simplicial ring Z[GSX] would then 8. COMPARISON WITH WALDHAUSEN'S K-THEORY OF SPACES 143 give a linearization result. We save this and other observations along these l* *ines for a future paper. 144 VI. ALGEBRAIC K-THEORY OF S-ALGEBRAS CHAPTER VII R-algebras and topological model categories In Chapter II, we set up the ground category of S-modules, and we developed the theory of S-algebras and their modules by exploiting the good formal properties of that category. In Chapter III, we set up a ground category of modules over a commutative S-algebra R that enjoys the same formal properties as the category * *of S-modules, and the previous three chapters gave applications of that theory. As* * we discuss in Section 1, we can go on to define R-algebras and their modules simply by changing ground categories from MS to MR . At this point, we face a homotopical problem. We want to use point-set level * *con- structions, such as bar constructions and constructions of topological Hochschi* *ld homology, that involve taking smash powers of a commutative R-algebra A. To make homotopical use of these constructions, we need to know that the underly- ing R-modules of these smash powers represent their smash powers in the derived category of R-modules. However, A need not have the homotopy type of a cell R-module, so we must approximate it by a weakly equivalent R-algebra with bet- ter properties. We first attacked this problem by use of the bar construction * *of Chapter XII, but we shall here deal with it by use of Quillen model categories. Thus we shall prove that all of our various categories of A1 and E1 ring sp* *ectra, R-algebras, commutative R-algebras, and modules over any of these are complete and cocomplete, tensored and cotensored, topologically enriched categories that admit canonical (closed) model structures in the sense of Quillen [54]. Since c* *ofi- brations and fibrations in the classical sense are important in our theory, we * *shall use the terms q-cofibration and q-fibration for the model category concepts. The proofs that our categories are so richly structured are almost entirely f* *ormal, and these formal structures do not solve or even address the motivating homotop* *ical problem since forgetful functors need not preserve q-cofibrant homotopy types. However, we shall see that the problem can be solved by combining the formal theory with the homotopical analysis of the linear isometries operad. 145 146 VII. R-ALGEBRAS AND TOPOLOGICAL MODEL CATEGORIES Much of the formal theory in this chapter is based on ideas and results origi* *nally due to Hopkins and McClure (in part in [31], but we have also benefited from ma* *ny profitable conversations) or to McClure, Schw"anzl and Vogt [52]. 1. R-algebras and their modules We fix a commutative S-algebra R and work in the symmetric monoidal category MR of R-modules. Definition 1.1. An R-algebra is a monoid in MR . A commutative R-algebra is a commutative monoid in MR . As in algebra, we obtain free R-algebras by "extension of scalars" from S to R. To show this, we use an alternative description of R-algebras and commutative R-algebras, which again is the same as in algebra. Say that a map j : R -! A of R-algebras is central if the following diagram commutes: R ^S A _____o_____//_A ^S R j^id|| |id^j| fflffl| fflffl| A ^S A OE__//_AooOEA_^S A We learned the following interpretation of this definition from McClure. Remark 1.2. The center of an associative k-algebra A with product OE can be written as the equalizer displayed in the diagram O"E C(A) _____//A____//_//_Homk(A; A); eOEo here "OE(a)(b) = ab and fOEo(a)(b) = ba. This suggests that the center C(A) of* * an S-algebra A should be defined as the equalizer displayed in the diagram "OE C(A) ____//_A___//_//_FS(A; A): eOEo The definition of a central map j : R - ! A then says precisely that j factors through C(A). Lemma 1.3. An R-algebra A is an S-algebra with a central map R -! A of S-algebras. A commutative R-algebra A is a commutative S-algebra with a map R -! A of S-algebras. 1. R-ALGEBRAS AND THEIR MODULES 147 Proof. Trivially, if A is an R-algebra, then its unit j : R -! A is a central map of R-algebras. Conversely, if A is an S-algebra and j : R -! A is a map of S-algebras, then A is a left R-module via the composite j^id OE R ^S A ____//_A ^S A___//_A: There is a symmetrically defined right action of R on A that makes A an (R; R)- bimodule. Centrality ensures that the left and right actions agree under the co* *mmu- tativity isomorphism of their domains. The product of A therefore factors throu* *gh A ^R A to give the required R-algebra structure. __|_ | We leave the proofs of the next few results as exercises; as in the proofs ab* *ove, one first writes down the proof of the algebraic analogue and then replaces ten* *sor products with smash products. Proposition 1.4. If Q is an S-algebra, then R ^S Q is the free R-algebra gen- erated by Q, hence R ^S TM is the free R-algebra generated by an S-module M. If Q is a commutative S-algebra, then R ^S Q is the free commutative R-algebra generated by Q, hence R ^S PM is the free commutative R-algebra generated by M. Remark 1.5. We may think of R ^S (S ^L BX) as the "free" R-algebra gener- ated by a spectrum X and R ^S (S ^L C X) as the "free" commutative R-algebra generated by a spectrum X. However, in view of II.1.3 (see also IIIx1), this is* * a misnomer since the right adjoints of these functors from the category of spectr* *a to the category of R-algebras or commutative R-algebras are weakly equivalent rath* *er than equal to the obvious forgetful functors. Proposition 1.6. Let f : R -! R0 and g : R -! R00be maps of commutative S-algebras. Then R0^R R00is both the coproduct of R0 and R00in the category of commutative R-algebras and the pushout of f and g in the category of commutative S-algebras. More generally, let f : A - ! A0 and g : A - ! A00be maps of commutative R-algebras. Then A0^A A00is the pushout of f and g in the category of commutative R-algebras. As in algebra, we can define the notion of a module over an R-algebra A, but * *it turns out to be equivalent to the notion of a module over A regarded just as an S-algebra. Recall III.3.1. Definition 1.7. Let A be an R-algebra. A left or right A-module is a left or right A-object in MR . 148 VII. R-ALGEBRAS AND TOPOLOGICAL MODEL CATEGORIES The free A-module generated by an S-module M is A ^R (R ^S M) ~=A ^S M: This gives an isomorphism of monads that implies the following result. Lemma 1.8. Let A be an R-algebra. A module over A regarded as an S-algebra is the same thing as a module over A regarded as an R-algebra. That is, an acti* *on A ^S M -! M necessarily factors through an action A ^R M -! M. Similarly, if M and N are A-modules, then M ^A N is the same whether defined using a coequalizer diagram in the category of R-modules or in the category of S-modules. Lemma 1.9. Let A be an R-algebra, and let M be a right and N a left A-module. Then M ^A N can be identified with the coequalizer M ^(A;R)N displayed in the diagram __^Rid____ M ^R A ^R N __________////M ^R N___//_M ^(A;R)N; id^R The analogous result holds for function A-modules. Proof. The proof is a formal categorical chase of the following schematic di* *a- gram: M ^S R ^S N l llll || llll || uullll ffflffl|flffl| M ^S A ^S N _______//_//_M ^S_N______//M6^A6N m m OOO | | m m O| | | mm O| fflffl| fflffl|m fflffl| M ^R A ^R N _______////_M ^R_N_____//M ^(A;R)N: Here the left vertical arrow is an epimorphism, and this implies that the diago* *nal dotted arrow factors through the dotted right vertical arrow. __|_ | Although we have only one notion of an A-module, it is helpful to think of its study as divided into an "absolute theory", in which we take the ground ring to* * be S, and a "relative theory", in which we take the ground ring to be R. The absol* *ute theory is a special case of the study of modules over algebras that we develope* *d in Chapter III. In particular, III.1.4 shows that FAX is weakly equivalent to A ^ X for a CW spectrum X. Here the free functor FA is isomorphic to the composite functor A ^R (R ^S FS) from spectra to A-modules. Again, the term free is a misnomer since the right adjoint of FS is only weakly equivalent to the forgetf* *ul functor. The theory of cell and CW A-modules and the definition of the derived category of A-modules are part of the absolute theory. 2. TENSORED AND COTENSORED CATEGORIES 149 The previous lemma shows that the absolute smash product ^A and function module functors FA are isomorphic to the relative functors, so that M ^A N and FA(M; N) are R-modules. Of course, if A is a commutative R-algebra, then these are A-modules and duality theory applies. In the relative theory, if we replace (R; S) by (A; R), with concomitant changes of notations for various functors, t* *hen all of the statements in Chapter III which make sense remain true. Note, for example, that we have relative versions of III.3.10 and of the pairings discuss* *ed in IIIx5. The results on pairings give the following generalization of III.6.12. Proposition 1.10. Let R be a commutative S-algebra, A be an R-algebra, and M and N be A-modules. Then FA(M; M) is an R-algebra and FA(M; N) is an (FA(N; N); FA(M; M))-bimodule. Of course, the case R = A is of particular interest. 2. Tensored and cotensored categories of structured spectra As in Ix1, consider the categories P and S of prespectra and spectra indexed * *on a universe U. It was proven in [37, p.17-18] that these categories are topologi* *cally enriched, in the sense that their Hom sets are based topological spaces such th* *at composition is continuous. For prespectra D and D0, P(D; D0) is topologized as a subspace of the product over indexing spaces V of the function spaces F (DV; D0* *V ). Since maps between spectra are just maps between their underlying prespectra, t* *his fixes the topology on S (E; E0). It was also observed in [37, p.18] that all of* * the functors introduced in that volume are continuous and all of the adjunctions pr* *oven in it are given by homeomorphisms of Hom sets. For example, by [37, I.3.3], there are natural homeomorphisms (2.1) S (E ^ X; E0) ~=T (X; S (E; E0)) ~=S (E; F (X; E0)) for spaces X and spectra E and E0, where T denotes the category of based spaces. In categorical language [32, x3.7], (2.1) states that S is tensored with tensor* *s E^X and cotensored with cotensors F (X; E). Adjoining disjoint basepoints to unbased spaces X, we obtain similar homeomorphisms involving the category U of unbased spaces. We give a formal definition in the unbased context. Definition 2.2. Let E be a category enriched over the category U of unbased spaces. Then E is tensored if there is a functor E : E x U - ! E , continuous in both variables, together with a natural homeomorphism E (E E X; E0) ~=U (X; E (E; E0)) for spaces X and objects E and E0 of E . We write for E when E is clear from the context. Dually, E is cotensored if there is a functor FE : U opx E -! E* * , 150 VII. R-ALGEBRAS AND TOPOLOGICAL MODEL CATEGORIES continuous in both variables, together with a natural homeomorphism U (X; E (E; E0)) ~=E (E; FE (X; E0)): As in the motivating example (2.1), FE will always admit an explicit descript* *ion. The tensors are more interesting and less familiar. We will give a way of descr* *ibing them for many spaces X in the next section. Again, by the argument illustrated in [37, p.18-19], colimits and limits of s* *pectra are continuous in the sense that the isomorphisms (2.3) S (colim Ei; F ) ~=limS (Ei; F ) and (2.4) S (F; limEi) ~=limS (F; Ei) are homeomorphisms. The continuity can also be deduced categorically. There are valuable general notions of indexed colimits and limits in enriched categories, which are defined and discussed in Kelly [32, x3.1]. Indexed colimits include tensors with spaces* * and continuous colimits as special cases, and dually for limits. We shall not repe* *at the general definition, since we shall not have occasion to use it, and we shal* *l rely on the following result of Kelly [32, 3.69-3.73] to deduce the existence of ind* *exed colimits and limits. Definition 2.5. A category E enriched over U is topologically cocomplete if it has all indexed colimits and topologically complete if it has all indexed li* *mits. Theorem 2.6 (Kelly). Let E be a category enriched over the category of bas* *ed or unbased spaces. Then E is topologically cocomplete if it is cocomplete and a* *dmits tensor products and is topologically complete if it is complete and admits cote* *nsor products. In particular, the given colimits and limits are continuous. Our various categories of structured ring, module, and algebra spectra inherit subspace topologies on their Hom sets. Thus they are all topologically enriched* *. All of the functors and adjunctions that we have constructed in this paper are cont* *inu- ous, by the cited arguments of [37, p.18-19]. We claim that our various categor* *ies of rings, modules, and algebras are topologically cocomplete and complete. For modules, this is immediate from II.1.4, III.1.1, and inspection. If R is* * an S-algebra, M is an R-module, and X is a based space, then (2.7) MR (M ^ X; M0) ~=T (X; MR (M; M0)) ~=MR (M; S ^L F (X; M0)): We deduce the first isomorphism from the first isomorphism of (2.1) by first wr* *it- ing MS(M; M0) as the equalizer of a pair of maps S (M; M0) - ! S (LM; M0) and then writing MR (M; M0) as the equalizer of a pair of maps MS(M; M0) -! 2. TENSORED AND COTENSORED CATEGORIES 151 MS(R ^S M; M0). We deduce the second isomorphism from the first by use of the isomorphisms M ^ X ~=M ^S 1 X and S ^L F (X; M) ~=FS(1 X; M): Proposition 2.8. For any S-algebra R, MR is topologically cocomplete and complete. Its tensors M ^ X and all other indexed colimits are created in MS or, equivalently, in S . Its cotensors FS(1 X; M) and all other indexed limits are created in MS or, equivalently, by applying the functor S ^L (?) to indexed lim* *its created in S . Now consider the categories of R-algebras and of commutative R-algebras. We agree to denote these categories by AR and C AR , respectively. We must enrich these categories over U , since there are no "trivial maps" to take as basepoin* *ts of Hom sets. We have already observed in IIx7 that the categories AR and C AR are complete and cocomplete. Continuing that discussion, we obtain the following result. The proof works equally well in the categories of A1 and E1 ring spec* *tra, where the result is due to Hopkins and McClure [31] and, in the E1 case, is the main technical result of McClure, Schw"anzl, and Vogt [52, Thm A]. Theorem 2.9. For any commutative S-algebra R, the categories AR of R- algebras and C AR of commutative R-algebras are topologically cocomplete and complete. Their cotensors and all other indexed limits are created in MS or, equivalently, by applying the functor S ^L (?) to indexed limits created in S . Proof. By II.7.1 (compare II.4.5), we have monads T and P in the category of R-modules whose algebras are the R-algebras and commutative R-algebras, and these monads are continuous (e.g., by inspection when R = S and use of Proposi- tion 1.4). Now II.7.2 and II.7.4 apply to show that AR and C AR are cocomplete.* * In the commutative case, the construction of colimits is quite simple since Propos* *ition 1.6 gives coproducts and pushouts, and it is trivial to construct coequalizers * *from them. Moreover, by an easy bootstrap argument from the continuity of colimits in the ground category of spectra, coequalizers in AR and C AR are continuous. Now the following categorical result completes the proof. __|_ | Proposition 2.10. Let T : C -! C be a continuous monad defined on a topologically enriched category C and let C [T] be the category of algebras ove* *r T. Assume that C is topologically cocomplete and complete. (i)The forgetful functor C [T] -! C creates all indexed limits. (ii)If C [T] has continuous coequalizers, then C [T] has all indexed colimit* *s. Proof. The first part is the enriched version of [42, VI.2, Ex 2]. Our proof* * of the second part is due to Hopkins [31]. (The later argument of [52, 2.7] is sli* *ghtly flawed.) Let (C; ) be a T-algebra and X be a space. We must construct their 152 VII. R-ALGEBRAS AND TOPOLOGICAL MODEL CATEGORIES tensor T-algebra. Let C X denote their tensor in C . Define : TC X - ! T(C X) to be the adjoint of the composite map of spaces _T___ X ____//_C (C; C X) //C (TC; T(C X)); where the first arrow is adjoint to the identity map C X -! C X. Let T also denote the free functor from C to C [T]. Define C C[T]X to be the coequalizer in C [T] of the pair of maps T(id) T(TC X) ____//_//_T(C X): OT We easily check the required adjunction C [T](C C[T]X; C0) ~=U (X; C [T](C; C0)) by using the fact that C [T](C; C0) is the equalizer of C(;id) 0 C (C; C0)____//_C (TC; C ) and C(id;0) 0 C (C; C0)__T_//_C (TC; TC0)___//_C (TC; C ): By Linton's theorem (II.7.5), C [T] is cocomplete since it has coequalizers. By Kelly's theorem (Theorem 2.4), C [T] has all indexed colimits. __|_ | In particular, FS(1 X+ ; A) is the cotensor of a space X and an R-algebra or commutative R-algebra A. The diagonal on X and the product on A induce the product on FS(1 X+ ; A). The following instance of a general categorical observation explains the relationship between the smash product A ^ X+ in the category of R-modules and the tensor A X in the category of R-algebras or commutative R-algebras. Proposition 2.11. For R-algebras A and spaces X there is a natural map of R-modules ! : A ^ X+ -! A X such that ! = idif X = {*} and the following transitivity diagrams commute: (A ^ X+ ) ^ Y+ !^id//_(A X) ^ Y+__!__//(A X) Y ~=|| |~=| fflffl| fflffl| A ^ (X x Y )+___________!___________//A (X x Y ): 3. GEOMETRIC REALIZATION AND CALCULATIONS OF TENSORS 153 For x 2 X, let ix : A -! A^X+ be the map induced by the inclusion {x}+ -! X+ . A map f : A ^ X+ -! B of spectra into an R-algebra B such that each composite f O ix : A -! B is a map of R-algebras uniquely determines a map of R-algebras "f: A X -! B such that f = "fO !. The same statement holds for commutative R-algebras. Proof. We have a natural map AR (A X; B) ~=U (X; AR (A; B)) -! U (X; MR (A; B)) ~=MR (A ^ X+ ; B); and ! is the image of the identity map of A X. The rest is easy diagram chasin* *g, using the natural map MR (A^X+ ; B) -! S (A^X+ ; B) for the last statement. __* *|_ | Remark 2.12. For R-algebras A and B, the previous result says that a map A X -! B of R-algebras determines and is determined by a map A ^ X+ -! B of spectra that is pointwise a map of R-algebras. A similar construction and result apply whenever one has a tensored category E with a continuous forgetful functor to spectra. For objects A and B in E , we define a homotopy to be a map h : A I -! B. Then h is induced by a homotopy A ^ I+ - ! B through maps in E . 3. Geometric realization and calculations of tensors To prepare for our construction of model structures and our study of thh, we explain how to calculate tensors E X for certain spaces X, and we use this calculation to study pushouts and cofibrations in the context of R-algebras. Our main tool is geometric realization, and the reader is urged to read the first t* *wo sections of Chapter X, which give a down to earth study of the geometric realiz* *ation of simplicial spectra, before reading this section. Fix a topologically complete and cocomplete category E with a continuous for- getful functor to spectra. We have the notion of a simplicial object E* in E . * *There are two notions of the geometric realization of such an object. We can first fo* *rget down to the category of simplicial spectra and take the geometric realization |* *E*| there, or we can rework the definition and carry out the construction entirely * *in E , obtaining the internal geometric realization |E*|E . Explicitly, |E*|E is the c* *oend Z (3.1) |E*|E = Eq E q: The following relationships between these two kinds of geometric realization generalize and clarify observations of McClure, Schw"anzl, and Vogt [52, 4.3, 4* *.4] about the category of E1 ring spectra. We defer the proofs to the end of the section. 154 VII. R-ALGEBRAS AND TOPOLOGICAL MODEL CATEGORIES Proposition 3.2. Let X* be a simplicial space and let A 2 E . Then there is a natural isomorphism A E |X*| ~=|A E X*|E ; of objects of E . The realization of underlying simplicial spectra is more amenable to homotopi* *cal analysis than the internal realization. In favorable cases, the realization |E** *| will again be an object of E , but this is not formal. We shall prove in Xx1 that t* *his holds for all of the categories of interest to us. In such cases, the two geom* *etric realizations are isomorphic. In particular, the following result holds. Proposition 3.3. Let R be any commutative S-algebra, such as R = S. For simplicial R-algebras A*, there is a natural isomorphism of R-algebras |A*| ~=|A*|AR ; and similarly for simplicial commutative R-algebras. Corollary 3.4. For R-algebras A and simplicial spaces X*, there is a natural isomorphism of R-algebras A AR |X*| ~=|A AR X*|; and similarly for commutative R-algebras. In the following discussion, we let E denote either AR or C AR and write for E . We use the term R-algebra in either case. The computation of A |X*| just given applies particularly effectively to simplicial sets X*, regarded as * *discrete simplicial spaces. We have a categorical coproduct q in E . This is ^R in the commutative case, but it is the "free product" in the non-commutative case. In * *the commutative case, the codiagonal map O : A q A -! A is the product on A. In both cases, the unit j : R -! A is the unique map from the initial object. Sinc* *e a discrete set n_with n points is the coproduct of its elements and the functor A* * (?) preserves coproducts, An_is the coproduct of n copies of A. To calculate A|X*|, we need only identify the induced face and degeneracy operators on coproducts of copies of A in terms of the structure maps O and j. In order to understand homotopy theory in E , we need to understand A I. We shall describe it in terms of a bar construction that is defined on R-algebr* *as. Recall that we defined the bar construction B(M; R; N) for a commutative S- algebra R and R-modules M and N in IV.7.2. We shall later use the evident generalization in which we replace R and its modules by a commutative R-algebra A and its modules. We here introduce a variant that applies equally well to eit* *her commutative or non-commutative R-algebras. In the commutative case, it is just 3. GEOMETRIC REALIZATION AND CALCULATIONS OF TENSORS 155 the specialization of the cited generalization in which the given A-modules are restricted to be commutative A-algebras. Definition 3.5. Let A be an R-algebra, and let f : A -! A0and g : A -! A00 be maps of R-algebras. These maps and the identity maps of A0and A00determine maps of R-algebras : A0q A -! A0 and : A q A00-! A00 Define a simplicial R-algebra fi*(A0; A; A00) by replacing ^S and OE by q and O* * in IV.7.2. Then define an R-algebra fiR (A0; A; A00) by fiR (A0; A; A00) = |fiR*(A0; A; A00)|: There is an evident natural map of R-algebras : fiR (A0; A; A00) -! A0qA A00 from the bar construction to the displayed pushout. Define the double mapping cylinder R-algebra M(A0; A; A00) by (3.6) M(A0; A; A00) = A0qA (A I) qA A00 and observe that the map I -! {pt} induces a collapse map : M(A0; A; A00) -! A0qA A00: We have the following identification of these two constructions. Proposition 3.7. Let A be an R-algebra with given maps to R-algebras A0and A00. Then there is a natural isomorphism fiR (A; A; A) ~=A I of R-algebras over A and under A q A, and there is a natural isomorphism fiR (A0; A; A00) ~=M(A0; A; A00) of R-algebras over A0qA A00and under A0q A00. Proof. Let I* be the standard simplicial 1-simplex with realization I. It h* *as p + 2 p-simplices, and a simple comparison of its face and degeneracy operations (e.g., [43, p.14]) with those of the bar construction shows that we have a natu* *ral identification of simplicial R-algebras fiR*(A; A; A) ~=A I*: The rest follows. __|_ | 156 VII. R-ALGEBRAS AND TOPOLOGICAL MODEL CATEGORIES In fact, one can see this quite directly, since the only non-degenerate simpl* *ices of I* are a 1-simplex 1 and its faces, and similarly for fiR*(A; A; A). We use this to obtain a result about cofibrations that will be at the heart of our construction of model structures on E . Let T : MR - ! E be the free R- algebra functor; thus T must be interpreted as P in the commutative case. Since T preserves tensors and pushouts and since R = T(*), we have TCM ~=R qTM (TM I): Proposition 3.8. For any R-module M, and any map of R-algebras TM -! A, the natural map of R-algebras : M(TCM; TM; A) -! TCM qTM A is homotopic rel A to an isomorphism. Proof. For a based space X, it is trivial to see that the map CX [X (X ^ I+ ) -! CX that retracts the cylinder onto the base of the cone is homotopic to a homeomor- phism. Working in the category of R-modules, the same argument works with X replaced by M. Applying the functor T, the cited map then becomes the map ae : R qTM (TM I) qTM (TM I) -! R qTM (TM I) that retracts the second copy of TM I onto the base of the first. We have M(TCM; TM; A) ~=R qTM (TM I) qTM (TM I) qTM A; and is obtained by applying the functor (?) qTM A to ae. The conclusion fol- lows. __|_ | We shall prove in XII.2.3 that the functor T preserves cofibrations of R-modu* *les, and this makes the following result plausible. We shall have more to say about * *this in Section 6. Proposition 3.9. For any pushout diagram of R-algebras TM ______//A | | | |i fflffl| fflffl| TCM ____//_B; the map i is a cofibration of R-modules and therefore of spectra. 3. GEOMETRIC REALIZATION AND CALCULATIONS OF TENSORS 157 Proof. The essential point is just that the unit map j : R -! TM is the in- clusion of a wedge summand of R-modules and a retract of R-algebras. From this, we find that the induced map A -! TM q A of R-algebras is also the inclusion of a wedge summand of R-modules and a retract of R-algebras. By the previous lemma and proposition, the pushout is isomorphic under A to the bar construc- tion fiR (TCM; TM; A). All of the degeneracy operators of fiR*(TCM; TM; A) are inclusion of wedge summands of R-modules, and it follows that fiR*(TCM; TM; A) is proper in the sense of X.1.2. This implies that the map from the zero skelet* *on TCM q A into fiR (TCM; TM; A) is a cofibration, and the conclusion follows. __* *|_ | We shall also need the following elementary complement. Lemma 3.10. Let {Ai} be a sequence of maps of R-algebras that are cofibrati* *ons of spectra. Then the underlying spectrum of the colimit of the sequence computed in the category of R-algebras is the colimit of the sequence computed in the ca* *tegory of spectra. Proof. The colimit in the category of spectra computes the colimit in the ca* *t- egory of R-modules and satisfies (colim Ai) ^R (colim Ai) ~=colim(Ai^R Ai): Therefore the spectrum level colimit inherits an R-algebra structure from the A* *i, and the universal property in the category of R-algebras follows from the unive* *rsal property in the category of R-modules. __|_ | We must still prove Propositions 3.2 and 3.3. Let sC denote the category of simplicial objects in a category C . Proof of Proposition 3.2. For a space Y , let U (*; Y ) be the evident si* *m- plicial space with q-simplices U (q; Y ). This functor of Y is right adjoint * *to geometric realization, (3.11) U (|X*|; Y ) ~=sU (X*; U (*; Y )): Similarly, for an object F of E , let FE (*; F ) be the evident simplicial obje* *ct of E with q-simplices FE (q; F ). This functor of F is right adjoint to the inter* *nal geometric realization, (3.12) E (|E*|E ; F ) ~=sE (E*; FE (*; F )): 158 VII. R-ALGEBRAS AND TOPOLOGICAL MODEL CATEGORIES These adjunctions, together with tensor and cotensor adjunctions, give the chain of natural isomorphisms E (E E |X*|; F )~=U (|X*|; E (E; F )) ~=sU (X*; U (*; E (E; F )) ~=sU (X*; E (E; FE (*; F )) ~=sE (E E X*; FE (*; F )) ~=E (|E E X*|E ; F ): The conclusion follows. __|_ | Proof of Proposition 3.3. Our interest is in the examples E = AR and E = C AR , but the argument is general. In all cases where realizations |E*| in* *herit structure present in E , the induced structure "arises pointwise". To explain w* *hat this means, note that we have an adjunction like those of (3.11) and (3.12) for simplicial spectra K* and spectra L, namely (3.13) S (|K*|; L) ~=sS (K*; F ((*)+ ; L)); where F ((*)+ ; L) has q-simplices F ((q)+ ; L). Now let E* be a simplicial obj* *ect of E and F be an object of E . When |E*| is again an object of E , we have the subspace E (|E*|; F ) S (|E*|; F ) of maps in E . This subspace coincides under the adjunction (3.13) with the sub- space of sS (E*; F ((*)+ ; F )) consisting of those points f = {fq} such that t* *he adjoint f"q: Eq ^ (q)+ - ! F of fq : Eq -! F ((q)+ ; F )) restricts to a map Eq -! F in E on the copy of Eq in Eq ^ (q)+ determined by each point of q. By Proposition 2.11 and Remark 2.12, such a map "fqextends uniquely to a map "gq: Eq E q -! F in E . In turn, under the tensor-cotensor adjunction, "gq corresponds to a map gq : Eq -! FE (q; F ) in E . The function {fq} -! {gq} determines an adjunction (3.14) E (|E*|; F ) ~=sE (E*; FE (*; F )): Comparison of (3.12) and (3.14) gives the conclusion. An alternative argument based on the properties of the monads T and P is also possible. The adjunctions above can be used to check that T|A*| ~=|T(A*)|AR : The functor T commutes with |?| on simplicial R-modules, the functor |?|AR pre- serves coequalizers, and a comparison of coequalizer diagrams gives the result.* * __|_ | 4. MODEL CATEGORIES OF RING, MODULE, AND ALGEBRA SPECTRA 159 4. Model categories of ring, module, and algebra spectra We shall prove that our various categories of structured spectra admit model structures. A more general, axiomatic, framework is possible; compare Blanc [5]. We assume familiarity with the language of model categories, by which we under- stand closed model categories in Quillen's original sense [54]. A good expositi* *on is given in [17]. We explain our results in this section and prove them in the * *next. In this paper, cofibrations and fibrations in any of our categories mean maps* * that satisfy the homotopy extension property (HEP) or covering homotopy property (CHP) in that category. Cofibrations in this sense will play a central role in* * the work of the next section. It is a pity that the language of model categories ha* *s, in the literature, been superimposed on the classical language, with resulting ambigui* *ty. We shall use q-cofibrations and q-fibrations for the model theoretic terms. In all of our model categories, the weak equivalences in the model sense will* * be those maps in the category which are weak equivalences of underlying spectra. We say that the weak equivalences are created in S . Observe that a retract of a w* *eak equivalence is a weak equivalence. Recall that a q-fibration or q-cofibration * *in a model category is said to be acyclic if it is a weak equivalence. Implicitly or explicitly, we must constantly think in terms of diagrams __ff_ E //X?? g "" i||" |p| fflffl|fflffl|" F __fi_//Y; where the square is given to be commutative and we seek a lift g that makes both triangles commute. We say that i has the left lifting property (LLP) with respe* *ct to a class of morphisms P if there exists such a lift g for any square in which p 2 P. We say that p satisfies the right lifting property (RLP) with respect to* * a class of morphisms I if there exists such a lift g for any square in which i 2 * *I . For example, a Serre fibration of spectra is a map that satisfies the CHP with repect to the set of "cone spectra" n fifi o 1qCSn fi q 0 and n 0 : This means that it is a map that satisfies the RLP with respect to the set of inclusions i0 : 1qCSn -! 1qCSn ^ I+ : Again, a retract of a Serre fibration is a Serre fibration. The q-fibrations in* * S will be the Serre fibrations. The following definition will allow us to give succinct statements of our res* *ults. 160 VII. R-ALGEBRAS AND TOPOLOGICAL MODEL CATEGORIES Definition 4.1. Let C be a model category with a forgetful functor to S that creates weak equivalences and let E be a category with a forgetful functor to C* * . We say that C creates a model structure in E if E is a model category whose weak equivalences are created in S and whose q-fibrations are created in C . That is* *, a map in E is a q-fibration if it is a q-fibration when regarded as a map in C . * *The q-cofibrations in E must then be those maps which satisfy the left lifting prop* *erty with respect to the acyclic q-fibrations. Our categories are enriched, and our model structures will reflect this. Quil* *len defined the notion of a simplicial model category in [54, IIx2], and the approp* *riate topological analogue of his definition reads as follows. Definition 4.2. A model category E is topological if it is topologically com* *plete and cocomplete and if, for any q-cofibration i : E -! F and q-fibration p : X -! Y , the induced map (4.3) (i*; p*) : E (F; X) -! E (E; X) xE (E;YE)(F; Y ) is a Serre fibration of spaces which is acyclic if either i or p is acyclic. Theorem 4.4. The category S is a topological model category with respect to the weak equivalences and Serre fibrations. If T : S -! S is a continuous monad such that the category S [T] of T-algebras has continuous coequalizers and satisfies the "Cofibration Hypothesis", then S creates a topological model stru* *cture in S [T]. We think of the first statement as the specialization to the identity monad o* *f the second. We shall specify the "Cofibration Hypothesis" shortly. It will obviou* *sly be satisfied by the identity monad and by the monad L, and arguments like those of the previous section verify it for the monads TL and PL that define A1 and E1 ring spectra. Corollary 4.5. The categories of L-spectra and of A1 and E1 ring spectra are topological model categories. Of course, we are far more interested in our categories of modules and algebr* *as. The crux of the proof of Theorem 4.4 is the adjunction S [T](TX; A) ~=S (X; A) for spectra X and T-algebras A. For S-modules, this must be replaced by the adjunction MS(S ^L LX; M) ~=S (X; FL (S; M)) for S-modules M that we obtain by composition of the first adjunction of II.2.2 with the freeness adjunction for the monad L. Thus we must change our forgetful 4. MODEL CATEGORIES OF RING, MODULE, AND ALGEBRA SPECTRA 161 functor from the obvious one to the functor FL (S; ?). Since FL (S; ?) is natur* *ally weakly equivalent to M, by I.8.7, the weak equivalences are unchanged. However, the q-fibrations are changed. Although the functor TX = S ^L LX from spectra to S-modules is not a monad, the proof of Theorem 4.4 will apply verbatim to it to give the following result. Theorem 4.6. The category MS is a topological model category with weak equi* *v- alences created in S . Its q-fibrations are those maps f : M -! N of S-modules such that F (id; f) : FL (S; M) -! FL (S; N) is a Serre fibration of spectra. To understand this, it is useful to think in terms of the "mirror image categ* *ory" M Sof counital L-spectra specified in II.2.1. By II.2.7 and composition (see II* *.6.1), we have a continuous monad FL (S; L(?)) on S whose algebras are the counital L- spectra. We have a topological equivalence of categories M S- ! MS that carries N to S ^L N. By II.2.5, S ^L FL (S; M) is naturally isomorphic to S ^L M for any L-spectrum M. Thus the monad that defines counital L-spectra is transported under the equivalence to the functor T relevant to the construction of the model structure on MS. The equivalence has the effect of changing the forgetful funct* *or. The proof of Theorem 4.4 will apply equally well if we change our ground cate* *gory to MS. Theorem 4.7. If T : MS -! MS is a continuous monad such that the cate- gory MS[T] of T-algebras has continuous coequalizers and satisfies the "Cofibra* *tion Hypothesis", then MS creates a topological model structure in MS[T]. Of course, the description of the q-fibrations as maps f such that FL (S; f) * *is a Serre fibration persists. Again, the Cofibration Hypothesis will be specified s* *hortly and holds in our examples. Corollary 4.8. The categories of S-algebras, commutative S-algebras, and modules over an S-algebra R are topological model categories. Now that we have a model structure on MR , we can generalize Theorem 4.7 by changing its ground category to MR . Theorem 4.9. Let R be a commutative S-algebra. If T : MR - ! MR is a continuous monad such that the category MR [T] of T-algebras has continuous co- equalizers and satisfies the "Cofibration Hypothesis", then MR creates a topolo* *gical model structure in MR [T]. Corollary 4.10. The categories of algebras and commutative algebras over a commutative S-algebra R are topological model categories. 162 VII. R-ALGEBRAS AND TOPOLOGICAL MODEL CATEGORIES In fact, Theorems 4.7 and 4.9 both apply, and they give the same model struct* *ures since they give the same q-fibrations and weak equivalences. We prefer to think* * of the model structure as created in MR , since that makes visible more information about the q-cofibrations. While the model category theory dictates what the q- cofibrations must be, the proofs of the theorems will lead to explicit descript* *ions. Definition 4.11. Let T be a monad in S as in Theorem 4.7. A relative cell T-algebra Y under a T-algebra X is a T-algebra Y = colimYn, where Y0 = X and Yn+1 is obtained from Yn as the pushout of a sum of attaching maps TSq -! Yn along the coproduct of the natural maps TSq -! TCSq. When X is an initial T-algebra, we say that Y is a cell T-algebra. Relative and absolute cell T-alge* *bras are defined in precisely the same way for a monad T in MR as in Theorem 4.9, except that the sphere spectra Sq are replaced by the sphere R-modules SqR. Remark 4.12. The functor T : S -! S [T], being a left adjoint, preserves coproducts. Thus, when attaching a coproduct of cells TCSq to Yn to obtain Yn+1, we are considering a pushout in S [T] of the general form TE _____//_A | | (4.13) | i| fflffl| fflffl| TCE ____//_B; where E is a wedge of spheres, and similarly when the ground category is MS or MR . The Cofibration Hypothesis is just the minimum condition necessary to obtain homotopical control over these pushout diagrams and their colimits. It holds in our examples by Proposition 3.9 and Lemma 3.10. Cofibration Hypothesis. The map i in any pushout of the form (4.13) is a cofibration of spectra (for Theorem 4.4) or of S-modules (for Theorem 4.7) or of R-modules (for Theorem 4.9). The underlying spectrum of the T-algebra colimit of a sequence of cofibrations of T-algebras is their colimit as a sequence of m* *aps of spectra. Actually, for the model structure in Theorems 4.7 and 4.9, we only need the maps i to be cofibrations of spectra. However, the stronger R-module cofibration condition holds in practice and is important in the applications. Theorem 4.14. Under the hypotheses of Theorems 4.4, 4.7, and 4.9, a map of T-algebras is a q-cofibration if and only if it is a retract of a relative cell* * T-algebra. Moreover, any q-cofibration is a cofibration of underlying spectra (in Theorem 4.4) or of underlying S-modules (in Theorem 4.7) or of underlying R-modules (in Theorem 4.9). 5. THE PROOFS OF THE MODEL STRUCTURE THEOREMS 163 By the Cofibration Hypothesis, the second statement will follow from the firs* *t. In all of our categories of T-algebras, the trivial spectrum is a terminal obje* *ct and every T-algebra is q-fibrant. By the previous result, a T-algebra is q-cofibra* *nt if and only if it is a retract of a cell T-algebra. Note in particular that the u* *nit R -! A of a q-cofibrant R-algebra or commutative R-algebra is a cofibration of R-modules. As in our discussion of Theorem 4.6, the proof of the previous theorem will a* *pply to give the following expected conclusion. Theorem 4.15. For an S-algebra R, such as R = S, a map of R-modules is a q-cofibration if and only if it is a retract of a relative cell R-module. Thus, in the case of R-modules, model category theory just brings us back to the cell theory that we took as our starting point. We can turn this around. We certainly want the weak equivalences and q-cofibrations in MR to be the weak equivalences of underlying spectra and the retracts of relative cell R-modules.* * Since the weak equivalences and q-cofibrations determine the q-fibrations, we see tha* *t the q-fibrations specified in Theorem 4.6 are in fact forced on us by the cell theo* *ry that we began with. 5. The proofs of the model structure theorems We must prove Theorems 4.4, 4.6, 4.7, 4.9, 4.14, and 4.15. For uniformity of treatment, let C be either S or MR for a commutative S-algebra R. Logically, of course, we should treat the case R = S before going on to the general case. Let T be a continuous monad in C such that C [T] has continuous coequalizers. By Proposition 2.10, we already know that C [T] is complete, cocomplete, tensored * *and cotensored, and indeed has all indexed limits and colimits. It is clear that if* * g O f is defined and two of f, g, and g O f are weak equivalences, then so is the third.* * It is also clear that the collections of q-fibrations, q-cofibrations, and weak equiv* *alences are closed under composition and retracts and contain all isomorphisms. It rema* *ins to prove that arbitrary maps factor appropriately and that the q-fibrations sat* *isfy the right lifting property (RLP) with respect to the acyclic q-cofibrations. T* *he essential point is that Quillen's "small object argument" applies to construct * *the required factorizations. A general version of Quillen's original argument is gi* *ven in [17, x6], and we shall give a modified version of that argument. Definition 5.1. For the purposes of this section, define a finite pair of sp* *ectra to be a pair of the form (1qB; 1qA), where B is a finite based CW complex, A is a subcomplex, and q 0. Define a finite pair of L-spectra to be a pair obtai* *ned by applying L to a finite pair of spectra. Define a finite pair of R-modules to* * be a pair obtained by applying FR to a finite pair of spectra. 164 VII. R-ALGEBRAS AND TOPOLOGICAL MODEL CATEGORIES As a matter of esoterica, we actually only need A, not B, to be finite in our arguments. Lemma 5.2. Let F be a set of maps in C [T], each of which is of the form TE -! TF for some finite pair (F; E) in C . Then any map f : X -! Y in C [T] factors as a composite __i__ 0_p__// X //X Y; where p satisfies the RLP with respect to each map in F and i satisfies the LLP with respect to any map that satisfies the RLP with respect to each map in F . Proof. Let X = X0. We construct a commutative diagram i0 in X0 _____//X1____//_._._.//_Xn____//Xn+1____//_. . . (5.3) f=p0|| p1|| |pn| |pn+1| fflffl| fflffl| fflffl| |fflffl Y __id_//_Y____//. ._.__//Y__id__//_Y_____//. . . as follows. Suppose inductively that we have constructed pn. Consider all maps from a map in F to pn. Each such map is a commutative diagram of the form __ff_ TE //Xn | |p (5.4) | | n fflffl| |fflffl TF __fi_//Y: Summing over such diagrams, we construct a pushout diagram of the form P ` ff TE ______//Xn | | | |in ` fflffl| fflffl| TF _____//Xn+1: The maps fi induce a map pn+1 : Xn+1 - ! Y such that pn+1 O in = pn. Let X0 = colim Xn, let i : X - ! X0 be the canonical map, and let p : X0 -! Y be obtained by passage to colimits from the pn. Constructing lifts by passage to coproducts, pushouts, and colimits in C [T], we see that each in and therefore * *also i satisfies the LLP with respect to maps that satisfy the RLP with respect to m* *aps 5. THE PROOFS OF THE MODEL STRUCTURE THEOREMS 165 in F . Assume given a commutative square _ff0_ 0 TE //X== g - - i|| - |p| fflffl|- fflffl| TF _fi_//_Y; where i is in F . To verify that p satisfies the RLP with respect to i, we must construct a map g that makes the diagram commute. The Cofibration Hypothesis implies that X0is constructed as the colimit of a sequence of cofibrations of s* *pectra. By [37, App.3.9], a cofibration of spectra is a spacewise closed inclusion. The* *refore, using that T is the free functor from C to C [T], we see by III.1.7 that the na* *tural map (5.5) colimC [T](TE; Xn) -! C [T](TE; X0) is a bijection. This ensures that ff0: TE -! X0 factors through some Xn, giving us one of the commutative squares (5.4) used in the construction of Xn+1. By construction, there is a map TF -! Xn+1 whose composite with the natural map to X0 gives a map g as required. __|_ | Lemma 5.6. Any map f : X - ! Y in C [T] factors as p O i, where i is an acyclic q-cofibration that satisfies the LLP with respect to any q-fibration an* *d p is a q-fibration. Proof. Let F be the set of pairs obtained by letting (B; A) in Definition 5.1 run through all pairs of spaces (CSn ^ I+ ; CSn ^ {0}+), n 0. By chasing throu* *gh adjunctions and using the definition of q-fibrations and q-cofibrations, we see* * that a map is a q-fibration if and only if it satisfies the RLP with respect to every * *map in F and that every map in F is a q-cofibration, of course an acyclic one. Note the relevance of the first adjunction of II.2.2 when C = MS: this is where the defi* *nition of q-fibrations in Theorem 4.6 is forced on us. Now use Lemma 5.2 to factor f. Then that lemma says that p is a q-fibration and that i satisfies the LLP with respect to all q-fibrations. In particular, i is a q-cofibration. We use the cy* *linders (?) I to define homotopies in the category C [T], as discussed in Remark 2.12. Then the free functor T and the adjoint forgetful functor preserve homotopies. A formal argument shows that each in is the inclusion of a deformation retraction* * of T-algebras, and it follows that i is also a deformation retraction. Therefore i* * is an acyclic q-cofibration. __|_ | Lemma 5.7. q-fibrations satisfy the RLP with respect to acyclic q-cofibratio* *ns. 166 VII. R-ALGEBRAS AND TOPOLOGICAL MODEL CATEGORIES Proof. This is formal. Let f : E -! F be any acyclic q-cofibration. We must show that f satisfies the LLP with respect to q-fibrations. By the previous lem* *ma, we may factor f as f = p O i, where i : E -! E0 is an acyclic q-cofibration that does satisfy the LLP with respect to q-fibrations and p : E0- ! F is a q-fibrat* *ion. Since f and i are weak equivalences, so is p. Since f satisfies the LLP with re* *spect to acyclic q-fibrations, there exists g : F -! E0 such that g O f = i and p O g* * = idF. Clearly p and g, together with the identity map on E, express f as a retract of* * i. Since i satisfies the LLP with respect to q-fibrations, so does f. __|_ | Lemma 5.8. Any map f : X -! Y in S [T] factors as p O i, where i is a q-cofibration and p is an acyclic q-fibration. Proof. This is another application of Lemma 5.2. Let A F be the set of pairs obtained by letting (B; A) in Definition 5.1 run through all pairs of spa* *ces (CSn; Sn), n 0. By tracing through adjunctions again, we see that a map of T-algebras is an acyclic q-fibration if and only if it satisfies the RLP with r* *espect to all maps in A F and that each map in A F is thus a q-cofibration. In the factorization f = p O i that we now obtain from Lemma 5.2, that lemma says that p is an acyclic q-fibration and i is a q-cofibration. __|_ | This completes the proofs of Theorems 4.4, 4.6, 4.7, and 4.9, and we turn to Theorem 4.14. As in the proof of Lemmas 5.2 and 5.8, a relative cell T-algebra E - ! E0 satisfies the LLP with respect to the acyclic q-fibrations and is thus* * a q-cofibration. Let f : E -! F be a q-cofibration. The proof of Lemma 5.8 gives a factorization of f as the composite of a relative cell T-algebra i : E -! E0 * *and an acyclic q-fibration p : E0 -! F . As in the proof of Lemma 5.7, there exists g : F -! E0 such that g O f = i and p O g = idF, and p and g express f as a ret* *ract of i. We must still prove that C [T] is topological, in the sense of Definition 4.2* *. As in [54, SM7(a), p.II.2.3], the description of q-cofibrations as retracts of rel* *ative cell T-algebras implies that we need only check that the map (4.3) is a Serre fibrat* *ion when i : E -! F is in the set F defined in the proof of Lemma 5.6 and an acyclic Serre fibration when i : E -! F is in the set A F defined in the proof of Lemma 5.8. The freeness adjunction for the monad T reduces this to the case of spectra or of R-modules, and further adjunctions then reduce it to its known space level analog. The reader should be convinced that the construction of model structures is a nearly formal consequence of the monadic descriptions of our various notions of structured ring, module, and algebra spectra. Remark 5.9. In [34], categories of A1 and E1 k-algebras and their modules were defined, and derived categories of modules were constructed, using a cell 6. THE UNDERLYING R-MODULES OF q-COFIBRANT R-ALGEBRAS 167 theory based on "sphere and cone modules". Replacing the ground category S with the ground category Mk of differential graded k-modules, the arguments of this section apply to give model structures to the analogous algebraic categori* *es. Unlike the treatments in [54, 5, 17], with this approach there is not the sligh* *test reason to restrict attention to bounded below k-modules. 6. The underlying R-modules of q-cofibrant R-algebras Let R be a fixed q-cofibrant commutative S-algebra. We study the underlying R- modules of q-cofibrant R-algebras and commutative R-algebras. The main point is to prove that the point-set level iterated smash products of q-cofibrant R-alge* *bras represent their smash product in the derived category DR , but we also prove a * *key technical lemma on cofibrations. We begin with the simpler non-commutative case, and we do not need R to be q-cofibrant in the following two results. Recall from III.7.3 that smash produc* *ts of cell R-modules are cell R-modules. Proposition 6.1. Let A and B be R-algebras that are cell R-modules relative to R. Then their coproduct A q B is a cell R-module relative to R. In more deta* *il, A q B is the colimit of an expanding sequence of relative cell R-modules {Cn} s* *uch that C0 = R and, for n 1, Cn=Cn-1 is the wedge of the two monomial word modules of length n in A=R and B=R. Proof. The monomial word modules in R-modules M and N are the smash products M ^R N ^R M ^R . . . and N ^R M ^R N ^R . .:. By II.7.4, we see that A q B is constructed via a coequalizer diagram in MR T(TA _ TB) ____//_//_T(A __B)_//A q B: Writing out the source and target of the pair of parallel arrows as wedges of s* *mash products and restricting to those wedge summands with at most n smash factors, we define Cn to be the coequalizer of the resulting restricted parallel pair of* * arrows. Clearly there result compatible maps Cn -! Cn+1 and Cn -! A q B such that A q B is the colimit of the Cn. In view of the use of the action maps TA -! A and TB -! B in II.7.4, we see that the wedge of the monomial words in A and B of length at most n maps onto Cn. That is, elements of word monomials involving A^R A or B ^R B are identified in the coequalizer with elements of word monomia* *ls of lower length. Let Un be the coproduct in the category of R-modules under R of the two mono- mial words in A and B of length n, so that the copies of R in these R-modules under R are identified. Then Un is a relative cell R-module. Let Vn be the union of the subcomplexes of Un that are obtained by replacing any one A or B in eith* *er 168 VII. R-ALGEBRAS AND TOPOLOGICAL MODEL CATEGORIES of the monomial words by its submodule R. The isomorphisms R ^R A - ! A and R ^R B -! B induce a map Vn -! Cn. Inspection of the restricted coequal- izer diagrams (and comparison with algebra for intuition) shows that there resu* *lt pushout diagrams of R-modules Vn _______//Cn | | | | fflffl| fflffl| Un+1 ____//_Cn+1: Inductively, the Cn and A q B are cell R-modules relative to R. __|_ | Theorem 6.2. If A is a q-cofibrant R-algebra, then A is a retract of a cell R-module relative to R. Thus the unit R -! A is a q-cofibration of R-modules. Proof. If M is a cell R-module, then Mj is a cell R-module for j 1 and (TM; R) is a relative cell R-module. Moreover, since M -! CM is cellular, TM -! TCM is the inclusion of a subcomplex in a relative cell R-module. Now suppose that (A; R) is a relative cell R-module and that we have a pushout diag* *ram of R-algebras TM ______//A | | | | fflffl| fflffl| TCM ____//_B: As in the proof of Proposition 3.9, B is isomorphic to the geometric realizatio* *n of a simplicial R-module that is proper because its degeneracies are given by inclus* *ions of wedge summands. The previous proposition implies that its R-module of p- simplices is a cell R-module relative to R. Moreover, the face and degeneracy m* *aps are sequentially cellular. Therefore, by X.2.7, (B; R) is isomorphic to a rela* *tive cell R-module, and A is a subcomplex. By passage to colimits, any cell R-algebra is a relative cell R-module. The conclusion follows from Theorem 4.14. __|_ | In the commutative case, the argument fails because we must pass to orbits ov* *er actions of symmetric groups. Tracing the proof of III.7.3 back to that of I.6.* *1, we see that it depends on the homeomorphism L (j) ~=L (1) induced by a linear isomorphism f : Uj - ! U. Since this homeomorphism is not j-equivariant, we cannot deduce that symmetric powers of CW L-spectra are, or even have the homotopy types of, CW L-spectra, although they do have the homotopy types of CW spectra. For this reason, we cannot conclude that the symmetric power Mj=j of a cell R-module M has the homotopy type of a cell R-module; we refer the reader ahead to VIII.2.7 for an analysis of the homotopy type of its underl* *ying spectrum. We get around this problem by use of the following result, which gives 6. THE UNDERLYING R-MODULES OF q-COFIBRANT R-ALGEBRAS 169 canonical CW S-module approximations of smash products of "extended powers" of CW spectra. Here the jth extended power of X is defined to be DjX = (LX)j=j ~=L (j) nj Xj: We adopt the convention that D0X = S. Theorem 6.3. Let {X1; : :;:Xn} be CW spectra, let ji 0, and consider the following commutative diagram of L-spectra: ^ ^ ^ __L__// LD X L(S ^L LDjiXi) L ji i ^L (id^L )|| |^L| ^ fflffl| ^ |fflffl ______// D X : L (S ^L DjiXi) ^L L ji i All spectra in the diagram have the homotopy types of CW-spectra, all maps in the diagram are homotopy equivalences of spectra, and ^S(S ^L LDjiXi) has the homotopy type of a CW S-module. Proof. By [37, VI.5.2 or VIII.2.4], Xj has the homotopy type of a j-CW spectrum indexed on Uj. By XI.1.7, L (j) has the homotopy type of a j-CW complex. Therefore, by the equivariant form of I.2.6, L (j) n Xj has the homoto* *py type of a j-CW spectrum indexed on U. Thus, by [37, I.5.6], DjX has the homotopy type of a CW-spectrum. By I.4.7 and II.1.9, LDjX has the homotopy type of a CW L-spectrum and S ^L LDjX has the homotopy type of a CW S- module. These conclusions pass to smash products by I.6.1 and III.7.3. The top horizontal arrow is a homotopy equivalence of L-spectra by I.4.7 and I.8.5, and* * the bottom horizontal arrow is a homotopy equivalence of spectra by XI.2.4. We claim that the right vertical arrow and therefore the left vertical arrow are also ho* *motopy equivalences of spectra. Indeed, if all ji 1, then use of I.5.4 and I.5.6 shows* * that the right vertical arrow is isomorphic to (L (n) x L (j1) x . .x.L (jn)) nj1x...xjn (Xj11^ . .^.Xjnn) flnid|| fflffl| L (j1 + . .+.jn) nj1x...xjn (Xj11^ . .^.Xjnn): Since fl is a (j1x . .x.jn)-equivariant homotopy equivalence, the map before passage to orbits is an equivariant homotopy equivalence by the equivariant ver* *sion of I.2.5. If any ji= 0, then use of I.6.1 reduces us to the case when a single * *ji= 0, and in this case the conclusion follows from I.8.6. __|_ | Now return to consideration of our given q-cofibrant commutative S-algebra R. 170 VII. R-ALGEBRAS AND TOPOLOGICAL MODEL CATEGORIES Definition 6.4. Define ER to be the collection of R-modules of the form R ^S (S ^L DjX); where X is any spectrum of the homotopy type of a CW-spectrum and j 0. Define ER to be the closure of ER under finite ^R -products, wedges, pushouts along cofibrations, colimits of countable sequences of cofibrations, and homoto* *py equivalences, where all of these operations are taken in the category of R-modu* *les. That is, if {M1; : :;:Mn} ER, then M1 ^R . .^.RMn 2 ER, and so forth. Observe that ER contains all R-modules of the homotopy types of cell R-module* *s, that being the collection that would be obtained if we only allowed j = 1 in our initial class. One point of the definition is the following observation. Its pr* *oof is just like that of Theorem 6.2, and we shall say more about the commutative case shortly. Theorem 6.5. The underlying R-module of a q-cofibrant R-algebra or commu- tative R-algebra A is in ER. Another point is the following reassuring consequence of Theorem 6.3 and the definition. Proposition 6.6. The underlying spectrum of an R-module in ER has the ho- motopy type of a CW-spectrum. These lead to the main point, which is that we have control of the behavior of derived smash product of R-modules that are in ER. Theorem 6.7. Let R be a q-cofibrant commutative S-algebra. Choose a cell R-module M and a weak equivalence of cell R-modules fl : M -! M for each M 2 ER. Then, for any finite subset {M1; : :;:Mn} of ER, fl ^R . .^.Rfl : M1 ^R . .^.RMn -! M1 ^R . .^.RMn is a weak equivalence of R-modules. That is, the derived smash product of the Mi in the category DR is represented by their point-set level smash product. Proof. When R = S and each Mi is in ES, Theorem 6.3 gives the conclusion. The conclusion for general Mi follows by standard commutation formulas relating smash products to the chosen operations. For general R and Mi= R ^S Ni, where Ni2 ES has CW S-approximation Ni, R ^S Niis a CW R-approximation of Mi. Here we have the identification (R ^S N1) ^R . .^.R(R ^S Nn) ~=R ^S (N1 ^S . .^.SNn); 6. THE UNDERLYING R-MODULES OF q-COFIBRANT R-ALGEBRAS 171 and similarly and compatibly for the Ni. By Theorem 6.5, R is in ES, hence the result for S implies the result for these Mi. The result for general Mi follows* * as in the case R = S. __|_ | Observe that the fl and their smash products are necessarily homotopy equiva- lences of underlying spectra, since these are CW homotopy types. We conclude with the following lemma on cofibrations. It will imply that the simplicial R-modules used in the following chapters are proper, in the sense of X.2.2. As explained at the start of Xx2, we abuse language by writing about unions, images, and inclusions when we should be writing more precisely about maps from a suitable coend to Aq. The abuse is justified by the conclusion, sin* *ce a cofibration of spectra is a spacewise closed inclusion [37, I.8.1]. Lemma 6.8. Let A be a q-cofibrant R-algebra or a q-cofibrant commutative R- algebra. Let sAq Aq be the "union of the images" of the maps si= (id)i^ j ^ (id)q-i: Aq-1 -! Aq: Then the "inclusion" sAq Aq is a cofibration of R-modules. In particular, the unit j : R -! A is a cofibration of R-modules. Proof. In the non-commutative case, (A; R) is a relative cell R-module. Its qth smash power inherits such a structure, by III.7.3, and sAq is a subcomplex. In the commutative case, we can apply the same brief argument, once we observe that (A; R) is a suitably general kind of relative cell R-module. Thus we consi* *der generalized relative cell R-modules that are constructed with (cell, sphere) pa* *irs replaced by pairs of the form (N ^ Bq+; N ^ Sq-1+), where N runs through all finite smash products over R of R-modules of the form (SnR)j=j or (CSnR)j=j for integers n and for j 1. Here the (Bq; Sq-1) are ordinary space level (cell, sp* *here) pairs. Observe that these R-modules are finite colimits of compact R-modules, so that III.1.7 applies to them. Let P be the monad in the category of R-modules that defines commutative R-algebras. Obviously PM and PCM are relative cell R-modules in this generalized sense when M is a wedge of sphere R-modules. Equally obviously, the smash product over R of two such generalized relative ce* *ll R-modules is another such. Suppose that A is such a commutative R-algebra and consider a pushout diagram PM ______//A | | | | fflffl| fflffl| PCM ____//_B: As explained in 3.5-3.9, B is isomorphic to the geometric realization of the pr* *oper simplicial R-module fiR*(PCM; PM; A). Remember that, here in the commutative case, the coproduct used in VIIx3 is the smash product over R. We may construct 172 VII. R-ALGEBRAS AND TOPOLOGICAL MODEL CATEGORIES the geometric realization by first using degeneracy identifications, which serv* *e sim- ply to eliminate redundant wedge summands from the relevant coend, and then face identifications; compare X.2.6. The latter identifications can be expresse* *d by pushout diagrams of R-modules g fiRq(PCM; PM; A) ^ @q+ ____//_Fq-1fiR (PCM; PM; A) | | | | fflffl| fflffl| fiRq(PCM; PM; A) ^ q+ ______//FqfiR (PCM; PM; A): Of course, we think of (q; @q) as a model for (Bq; Sq-1). Proceeding inductively and using III.1.7 and the proof of III.2.2, we can make g a sequentially cellul* *ar map and deduce that the qth filtration is a generalized relative cell R-module.* * By passage to colimits, so is any commutative cell R-algebra. The same holds for smash powers. The inclusion of a subcomplex is a cofibration, by reduction to t* *he obvious case of cell pairs, and the conclusion follows. __|_ | Remark 6.9. By their characterization in terms of the LLP, coproducts of q- cofibrations are q-cofibrations In any model category. This observation may help clarify the previous result, since it ensures that the smash product over R of * *two q-cofibrant commutative R-algebras is again a q-cofibrant commutative R-algebra. CHAPTER VIII Bousfield localizations of R-modules and algebras We study Bousfield localizations in this chapter. For any S-algebra R and cell R-module E, we show that MR admits a new model category structure in which the weak equivalences are the E-equivalences. With this model structure, a fact* *or- ization of the trivial map M -! * as the composite of an E-acyclic E-cofibration and an E-fibration constructs the localization of M at E. Restricting to a q-cofibrant commutative S-algebra R, we combine formal con- structions with the homotopical analysis of the previous chapter to prove that localizations at E of cell R-algebras can be constructed as cell R-algebras, and similarly for commutative cell R-algebras. Of course, this applies quite genera* *lly since any R-algebra is weakly equivalent to a cell R-algebra. That is, we can c* *on- clude that Bousfield localizations of R-algebras and commutative R-algebras are again such. In the case R = S, Hopkins and McClure had an unpublished argu- ment, sketched in e-mails to us, that Bousfield localizations of E1 ring spect* *ra are E1 ring spectra. We deduce that the derived category of E-local R-modules is equivalent to the full subcategory of the derived category of RE -modules whose objects are those* * RE - modules that are E-local as R-modules. In particular our new derived categories* * of SE -modules are intrinsically important to a complete understanding of the clas* *sical Bousfield localizations of spectra. As a simple direct application, we deduce that KO and KU are commutative ko and ku-algebras since they are Bousfield localizations of ko and ku obtained* * by inverting the Bott elements. By neglect of structure, they are therefore commut* *ative S-algebras. This solves a problem that was first studied in McClure's 1978 PhD thesis. We refer the reader to [25] for a discussion of the special cases of Bousfield localization that give localizations and completions of R-modules at ideals of * *the coefficient ring R*. 173 174 VIII. BOUSFIELD LOCALIZATIONS OF R-MODULES AND ALGEBRAS 1. Bousfield localizations of R-modules Let R be an S-algebra, such as S itself, and suppose given a cell R-module E. We shall construct Bousfield localizations of R-modules at E. The treatment is based on Bousfield's papers [11, 12], but in the latter he was handicapped by working in a primitive category of spectra that did not admit a model category structure. A map f : M -! N of R-modules is said to be an E-equivalence if the induced map id^R f : E ^R M -! E ^R N is a weak equivalence. Homologically, we should call such maps ER*-equivalences, and we shall often refer to them as E-acyclic maps. An R-module W is said to be E-acyclic if E ^R W ' *, and a map f is E-acyclic if and only if its cofiber is* * E- acyclic. We say that an R-module L is E-local if f* : DR (N; L) -! DR (M; L) is* * an isomorphism for any E-equivalence f or, equivalently, if DR (W; L) = 0 for any * *E- acyclic R-module W . Since this is a derived category criterion, it suffices to* * test it when W is a cell R-module. A localization of M at E is a map : M -! ME such that is an E-equivalence and ME is E-local. Of course, the formal properties of such localizations discussed in [11, 12] carry over verbatim to the present con* *text. We shall construct a model structure on MR that implies the existence of E- localizations of R-modules. Theorem 1.1. The category MR admits a new structure as a topological model category in which the weak equivalences are the E-equivalences and the cofibrat* *ions are the q-cofibrations in the model structure already constructed. The fibratio* *ns in the new model structure are the maps that satisfy the right lifting property wi* *th respect to the E-acyclic q-cofibrations. Although the theorem gives the best way to think about the new model structur* *e, it will be convenient to construct it in a way that parallels the proofs in VII* *x5. To that end, we give apparently different definitions of E-fibrations and E-cofibr* *ations. Definition 1.2. A map f : M -! N is an E-fibration if it has the right lifting property with respect to the E-acyclic inclusions of subcomplexes in ce* *ll R-modules. A map f : M - ! N is an E-cofibration if it satisfies the left lifti* *ng property with respect to the E-acyclic E-fibrations. The following comparisons will emerge during our proof of Theorem 1.1. Lemma 1.3. A map is an E-cofibration if and only if it is a q-cofibration. Lemma 1.4. A map is an E-fibration if and only if it satisfies the RLP with respect to the E-acyclic q-cofibrations. 1. BOUSFIELD LOCALIZATIONS OF R-MODULES 175 An R-module L is said to be E-fibrant if the unique map L - ! * is an E- fibration. Proposition 1.5. An R-module is E-fibrant if and only if it is E-local. Proof. The argument is the same as that of [11, 3.5]. As in [11, 3.6], one c* *hecks that the class of E-equivalences in the derived category DR admits a calculus of left fractions. This implies [11, 2.5] that an R-module L is E-local if and onl* *y if f* : DR (N; L) -! DR (M; L) is a surjection for any E-equivalence f : M -! N. Since we are working in derived categories, there is no loss of generality to a* *ssume that f is the inclusion of a subcomplex in a cell R-module. If L is E-fibrant, the RLP already gives surjectivity on the point-set level, hence on the level of homotopy classes. If L is E-local, we have surjectivity on the level of homoto* *py classes and deduce it on the point-set level by use of HEP. __|_ | Theorem 1.6. Every R-module M admits a localization : M -! ME . Proof. We may factor the trivial map M - ! * as the composite of an E- acyclic E-cofibration : M -! ME and an E-fibration ME -! *. __|_ | Localizations of the underlying spectra of R-modules at spectra can be recove* *red as special cases of our new localizations of R-modules at R-modules. Therefore,* * up to equivalence, the localization of an R-module at a spectrum can be constructed as a map of R-modules. Proposition 1.7. Let K be a CW-spectrum and let E be the R-module FR K. Regarded as a map of spectra, a localization : M -! ME of an R-module M at E is a localization of M at K. Proof. By IV.1.9, we have K*(M) ~=ER*(M) for R-modules M. Therefore an E-equivalence of R-modules is a K-equivalence of spectra. If W is a K-acyclic spectrum, then FR W is an E-acyclic R-module since E ^R FR W is equivalent to FR (K ^W ). Therefore, if N is an E-local R-module, then [W; N] ~=[FR W; N]R = 0 and N is a K-local spectrum. The conclusion follows. __|_ | The argument generalizes to show that, for an R-algebra A, the localization of an A-module at an R-module E can be constructed as a map of A-modules. Proposition 1.8. Let A be a q-cofibrant commutative R-algebra, let E be a ce* *ll R-module and let F be the A-module A ^R E. Regarded as a map of R-modules, a localization : M -! MF of an A-module M at F is a localization of M at E. 176 VIII. BOUSFIELD LOCALIZATIONS OF R-MODULES AND ALGEBRAS We prove Theorem 1.1 in the rest of the section. Of course, MR is topological* *ly complete and cocomplete. It is clear that retracts of E-equivalences, E-cofibra* *tions, and E-fibrations are again such (and similarly with cofibrations and fibrations* * as in the statement of the theorem). The following result motivates our definition of* * E- fibrations in terms of inclusions of subcomplexes rather than general q-cofibra* *tions. Let #X denote the cardinality of the set of cells of a cell R-module X and let c be a fixed infinite cardinal that is at least the cardinality of ER*(SR ). Long* * exact sequences and the commutation of homology with colimits imply that if X -! Y is the inclusion of a subcomplex in a cell R-module Y such that #Y c, then the cardinality of ER*(Y=X) is at most c. Let T be the set of E-acyclic inclusions* * of subcomplexes in cell R-modules Y such that #Y c. Then T is a test set for E-fibrations. Lemma 1.9. A map f : M -! N is an E-fibration if and only p has the RLP with respect to maps in T . Proof. Arguing as in [11, 11.2,11.3], we see that for any proper inclusion X* * -! Y of a subcomplex in a cell R-module Y , there is a subcomplex X0 Y such that #X0 c, X0 is not contained in X, and X0\ X -! X0 is E-acyclic. We construct X0 as the union of a sequence of subcomplexes Xn of Y such that #Xn c, Xn is not contained in X, and the map ER*(Xn=Xn \ X) -! ER*(Xn+1=Xn+1 \ X) is zero. The fact that homology commutes with colimits and that ER*(Y=X) = 0 allows us to perform the inductive step by adjoining finite subcomplexes of Y * *to Xn to kill elements of ER*(Xn=Xn \ X). We conclude that if f has the RLP with respect to maps in T , then it has the RLP with respect to X -! X0[ X since it has the RLP with respect to X0\ X -! X0. By transfinite induction, it follows that f has the RLP with respect to X -! Y . __|_ | Lemma 1.10. Any map f : M -! N factors as a composite p M- i!M0-! N; where p is an E-fibration and i is an E-acyclic q-cofibration that satisfies th* *e LLP with respect to E-fibrations. Proof. The construction is exactly like that in the proof of VII.5.2, with T here playing the role of F there. However, since we do not have compactness, we must perform the construction transfinitely. We carry the construction through * *to the least ordinal of cardinality greater than c. We can then use set theory rat* *her than (VII.5.5) to ensure the requisite factorization ff0 in the cited proof. Th* *e re- sulting map p is certainly an E-fibration. The construction by successive pusho* *uts 1. BOUSFIELD LOCALIZATIONS OF R-MODULES 177 of wedges of maps in T , for ordinals n with successors, and passage to colimit* *s, for limit ordinals, shows that the constructed map i is E-acyclic and satisfies the* * LLP with respect to the E-fibrations. To see that i, despite its transfinite constr* *uction, is a q-cofibration, we must specify a sequential filtration inductively. Given* * the sequential filtration on Mn, where n has a successor, we obtain a sequential fi* *ltra- tion on Mn+1 by using III.2.3 to arrange that the pushout that constructs Mn+1 is a diagram of sequentially cellular maps. If n is a limit ordinal and we have compatible sequential filtrations on the Mm for m < n, then each cell of Mn has* * a preassigned filtration and we take the qth filtration of Mn to be the union of * *the qth filtrations of the Mm for m < n. __|_ | Remark 1.11. The previous proof uses that if i : X - ! Y is an E-acyclic q- cofibration and f : X -! M is any map, then the pushout j : M -! M [X Y is E- acyclic. This holds because i and j are cofibrations of R-modules with isomorph* *ic quotients M=X ~=(M [X Y )=M. Lemma 1.12. The following conditions on a map f : M -! N are equivalent. (i)f is an E-acyclic E-fibration. (ii)f is an E-acyclic map that satisfies the RLP with respect to all q-cofib* *rations. (iii)f is an acyclic q-fibration. Proof. Obviously (ii) implies (i) and (iii) implies (ii). Assume (i). Clearl* *y f is a q-fibration, and we must prove that it is a weak equivalence. By factoring a * *weak equivalence from a cell R-module to M as the composite of an acyclic q-cofibrat* *ion and a q-fibration, we can construct an acyclic q-fibration p : M0 -! M, where M0 is a cell R-module. By VII.5.8, we may factor f O p as the composite of a q-cofibration i : M0 -! N0 and an acyclic q-fibration p0 : N0 -! N. By III.2.3 and the proof of VII.5.2, we can arrange that N0 is a cell R-module that contai* *ns M0 as a subcomplex. Since f O p is E-acyclic, so is i. Summarizing, we have the diagram p M0 ______M0=____//_M= z i|| rzz fOp|| f|| fflffl|z fflffl| fflffl| N0 __p0_//N______N: Since f O p is an E-fibration and i is an E-acyclic inclusion of a subcomplex i* *n a cell R-module, there exists a lift r. This expresses f O p as a retract of the * *weak equivalence p0. Therefore f O p and f are weak equivalences. __|_ | Observe that Lemma 1.3 is an immediate consequence of Lemma 1.12. 178 VIII. BOUSFIELD LOCALIZATIONS OF R-MODULES AND ALGEBRAS Proof of Lemma 1.4. It suffices to show that an E-fibration satisfies the* * RLP with respect to all E-acyclic q-cofibrations i : X -! Y . Factor i as in Lemma * *1.8 and consider the resulting diagram j X _____//X0>> " i ||" " |p| fflffl|fflffl|" Y ______Y; where p is an E-fibration and j is an E-acyclic q-cofibration that satisfies th* *e LLP with respect to the E-fibrations. Clearly p is E-acyclic, hence Lemma 1.10 impl* *ies that it satisfies the RLP with respect to the cofibration i. There results a l* *ift , and and p show that i is a retract of j. Since j satisfies the LLP with respec* *t to all E-fibrations, so does i. __|_ | Proof of Theorem 1.1. We have proven one of the factorization axioms in Lemma 1.10, and the remaining axioms for a model structure are now direct con- sequences of the corresponding axioms for the original model structure on MR . * * __|_ | 2. Bousfield localizations of R-algebras In this section, we restrict R to be a q-cofibrant commutative S-algebra and * *let E be a cell R-module. We shall prove the following pair of theorems. Theorem 2.1. For a cell R-algebra A, the localization : A - ! AE can be constructed as the inclusion of a subcomplex in a cell R-algebra AE . Moreover,* * if f : A -! B is a map of R-algebras into an E-local R-algebra B, then f lifts to a map of R-algebras f": AE - ! B such that f"O = f; if f is an E-equivalence, then "fis a weak equivalence. Theorem 2.2. For a commutative cell R-algebra A, the localization : A -! AE can be constructed as the inclusion of a subcomplex in a commutative cell R- algebra AE . Moreover, if f : A - ! B is a map of R-algebras into an E-local commutative R-algebra, then f lifts to a map of R-algebras f": AE - ! B such that "fO = f; if f is an E-equivalence, then "fis a weak equivalence. Proofs. The idea is to replace the category MR by either the category AR or C AR in the work of the previous section. Most arguments go through with little change, the crucial exception being the part of the proof of Lemma 1.10 that is singled out in Remark 1.11. The problem there is that, to prove the analogue of the cited lemma in full generality, we would have to allow A to be an arbitrary R-algebra or commutative R-algebra. However, to keep homotopical control, we need A to be a cell R-algebra. This is enough to prove our theorems, although we don't actually obtain new model structures on AR and C AR . 2. BOUSFIELD LOCALIZATIONS OF R-ALGEBRAS 179 For definiteness, consider the non-commutative case. Proceeding as in the pro* *ofs of VII.5.2 and Lemma 1.10, we let A0 = A and construct a transfinite sequence i0 in (2.3) A0 ____//_A1___//_._._._//An____//_An+1___//_. . . as follows. Suppose inductively that we have constructed An and that n has a successor. Consider all diagrams of R-modules __i__ __ff_// (2.4) Y oo X An; where i is in T . Using the free R-algebra functor T on R-modules, we take the sum over such diagrams of the adjoint maps "ff: TX - ! An and construct the pushout diagram of R-algebras P ` "ff TX _____//_An (2.5) | | | |in ` fflffl| fflffl| TY _____//An+1: If n is a limit ordinal and Am has been constructed for m < n, we let An be the colimit of the Am . We take AE to be the colimit up to the least ordinal of cardinality greater than c. Then any map of R-modules from a cell R-module X with #X c into AE factors through some An, and we let : A -! AE be the canonical map. Regarded as an R-module, AE is E-fibrant and therefore E-local. To see this, consider a diagram of R-modules __ff0_ X //AE=;= g - i|| - - fflffl|- Y where i is in T . We must construct a map g that makes the diagram commute. Since ff0 factors through some An, we have one of the diagrams (2.4) used in the construction of An+1. By construction, there is a map TY - ! An+1 the adjoint of whose composite with the natural map to AE is a map g as required. By arguments like those in Lemmas 1.10 and 1.12, i : A -! AE is the inclusion of a subcomplex in a cell R-algebra. We must prove that i is an E-equivalence. By the commutation of homology with directed`colimits, only the pushout maps An -! An+1 are at issue. Observe that TX ~=T(_X). Lemma 2.6 below shows that the left vertical arrow in (2.5) is an E-equivalence. In the commutative case, we must replace T by P, and here Lemma 2.7 shows that the left vertical arrow in the commutative analogue of (2.* *5) 180 VIII. BOUSFIELD LOCALIZATIONS OF R-MODULES AND ALGEBRAS is an E-equivalence. Finally, in both cases, Lemma 2.9 shows that the right ver* *tical arrow An -! An+1 is an E-equivalence. We prove the lifting statement for a map f : A -! B by inductively lifting f * *to maps fn : An -! B. We proceed by passage to colimits when n is a limit ordinal. For the inductive step when fn : An -! B is given and n has a successor, we app* *ly the fact that B is E-fibrant to lift the evident composites X -! TX -! An -! B to maps of R-modules Y -! B and then apply freeness to obtain maps of R- algebras TY -! B that lift the maps of R-algebras TX -! B. Passage to pushouts then gives the required map fn+1 : An+1 -! B. It is clear that "fmust be an E-equivalence and therefore a weak equivalence if f is an E-equivalence. * * __|_ | Lemma 2.6. If f : M - ! N is an E-equivalence of cell R-modules, then Tf : TM -! TN is an E-equivalence of R-modules. Proof. If f : M -! N and f0 : M0 -! N0 are E-equivalences, then, factoring id^R f ^R f0 : E ^R M ^R M0 -! E ^R N ^R N0 as the composite (id^R f ^R id)(id^R id^R f0) and using the commutativity and associativity of ^R and the fact that all R-modules in sight are cell R-modules* *, we see that id^R f ^R f0 is an equivalence and thus that f ^R f0 is an E-equivalen* *ce. Inductively, fj : Mj -! Nj is an E-equivalence for all j 0. __|_ | Lemma 2.7. If f : M - ! N is an E-equivalence of cell R-modules, then Pf : PM -! PN is an E-equivalence of R-modules. Proof. We must show that fj=j : Mj=j -! Nj=j is an E-equivalence for all j 0. By III.6.1, this will hold if id ^fj : (Ej)+ ^j Mj -! (Ej)+ ^j Nj is an E-equivalence. By the previous proof, we have an E-equivalence before passage to orbits. Using the skeletal filtration of Ej, we may set up a natural spectral sequence H*(j; ER*(Mj)) =) ER*((Ej)+ ^j Mj) and so deduce the conclusion. __|_ | 3. CATEGORIES OF LOCAL MODULES 181 Lemma 2.8. Suppose given a pushout diagram of R-algebras f A _____//C i || |j| fflffl|fflffl| B __g__//D; where i is an E-acyclic inclusion of a subcomplex in a cell R-algebra and C is a cell R-algebra. Then j is E-acyclic. The same conclusion holds in the case of commutative R-algebras. Proof. Recall the bar construction fiR (B; A; C) of VII.3.5. By definition o* *r by VII.3.7, we may interpret fiR (B; A; C) as the homotopy pushout of i and f. Sin* *ce i is a cofibration of R-algebras, the natural map fiR (B; A; C) -! D is a homot* *opy equivalence of R-algebras under C. Moreover, the map C -! fiR (B; A; C) factors as the composite of a map j : C -! fiR (A; A; C) and the map fiR (i; id; id) : fiR (A; A; C) -! fiR (B; A; C): Here j is a homotopy equivalence of R-modules by XII.1.2 and X.1.2. The map fiR (i; id; id) is the geometric realization of a map of proper simplicial R-mo* *dules, where properness is defined in X.2.2. Properness holds in the case of R-algebras since, by VII.6.2, the inclusions of degeneracy sub R-modules are inclusions of subcomplexes in relative cell R-modules. It holds in the case of commutative R- algebras by VII.6.8. The smash product over R with E commutes with geometric realization by X.1.4. Since i is an E-equivalence, we find in the R-algebra ca* *se that the map E ^R fiRq(i; id; id) on q-simplices is a homotopy equivalence for * *each q because it is a weak equivalence between relative cell R-modules. In the com- mutative case, this map is the smash product over R of the weak equivalence E ^R A -! E ^R B with the identity map on Aq ^R C and is therefore a weak equivalence by VII.6.7. In either case, we conclude from X.2.4 that fiR (i; id;* * id) is an E-equivalence. __|_ | 3. Categories of local modules Again, let R be a q-cofibrant commutative S-algebra and E be a cell R-module. let RE be a q-cofibrant commutative R-algebra whose unit is a localization of R at E. The fact that Bousfield localization preserves R-algebras and commutative R-algebras gives a powerful tool for the construction of new R-algebras and ope* *ns up a new approach to the study of Bousfield localizations. To see the latter, let us compare the derived category DRE to the stable homo* *topy category DR [E-1] associated to the model structure on MR determined by E. Here DR [E-1] is obtained from DR by inverting the E-equivalences and is equivalent * *to 182 VIII. BOUSFIELD LOCALIZATIONS OF R-MODULES AND ALGEBRAS the full subcategory of DR whose objects are the E-local R-modules. Observe tha* *t, for a cell R-module M, we have the canonical E-equivalence = j ^ id: M ~=R ^R M -! RE ^R M: The following observation is the same as in the classical case. Lemma 3.1. If M is a finite cell R-module, then RE ^R M is E-local and there- fore is the localization of M at E. Proof. If W is an E-acyclic R-module, then DR (W; RE ^R M) ~=DR (W ^R DR M; RE ) = 0 since W ^R DR M is E-acyclic and RE is E-local. __|_ | We say that localization at E is smashing if, for all cell R-modules M, RE ^R* * M is E-local and therefore i is the localization of M at E. The following observa* *tion is due to Wolbert [69]. Proposition 3.2 (Wolbert). If localization at E is smashing, then the cat* *e- gories DR [E-1] and DRE are equivalent. These categories are closely related even when localization at E is not smash* *ing, as the following elaboration of Wolbert's result shows. Theorem 3.3. The following three categories are equivalent. (i)The category DR [E-1] of E-local R-modules. (ii)The full subcategory DRE [E-1] of DRE whose objects are the RE -modules that are E-local as R-modules. (iii)The category DRE [(RE ^R E)-1] of (RE ^R E)-local RE -modules. This implies that the question of whether or not localization at E is smashing is a question about the category of RE -modules, and it leads to the following factorization of the localization functor. In the classical case R = S, this sh* *ows that our new commutative S-algebras SE and their categories of modules are intrinsic to the theory of Bousfield localization. Theorem 3.4. The E-localization functor DR -! DR [E-1] is equivalent to the composite of the extension of scalars functor RE ^R (?) : DR -! DRE and the (RE ^R E)-localization functor (?)RE^RE : DRE - ! DRE [(RE ^R E)-1]: 3. CATEGORIES OF LOCAL MODULES 183 Corollary 3.5. Localization at E is smashing if and only if all RE -modules are E-local as R-modules, so that DR [E-1] DRE DRE [(RE ^R E)-1]: The proofs of the results above are based on the following generalization of * *III.4.4 and a special case of III.4.1 from the ground category S to the ground category* * R. The proofs are the same as those of the cited results. Proposition 3.6. Let A be an R-algebra with unit j : R -! A. The forgetful functor j* : DA -! DR has the functor A ^R (?) : DR -! DA as left adjoint, and it also has a right adjoint j# : DR -! DA. Proof of Theorem 3.3. We apply the previous result to j : R - ! RE , obtaining DRE (N; j# M) ~=DR (j*N; M) and DRE (RE ^R M; N) ~=DR (M; j*N) for R-modules M and RE -modules N. We claim that the functors j* and j# of the first adjunction restrict to give inverse adjoint equivalences between DR [* *E-1] and DRE [E-1], and we also claim that an RE -module N is (RE ^R E)-acyclic if and only if j*N is E-acyclic. These claims will give the conclusion. If W is an E-acyclic R-module, then RE ^R W is an (RE ^R E)-acyclic RE -module since (RE ^R W ) ^RE (RE ^R E) ~=RE ^R (W ^R E) ' *: Therefore, by the second adjunction, if N is an (RE ^R E)-local RE -module, then j*N is an E-local R-module. If V is an (RE ^R E)-acyclic RE -module, then j*V is an E-acyclic R-module since (j*V ) ^R E ~=j*(V ^RE (RE ^R E)) ' *: Therefore, by the first adjunction, if M is an E-local R-module, then j# M is an (RE ^R E)-local RE -module and thus j*j# M is again an E-local R-module. We claim that if if M is E-local, then the counit " : j*j# M -! M of the first adjunction is a weak equivalence of R-modules. To see this, consider the 184 VIII. BOUSFIELD LOCALIZATIONS OF R-MODULES AND ALGEBRAS commutative diagrams: DRE (RE ^R Sq; j#TM) j*hhhhhhhh TTTTT~=TT hhhh TTTTT tthhhhhh * T))T DR (j*(RE ^R Sq); j*j# M) ______________________________//_DR (Sq; j*j# M) "* || "*|| fflffl| * fflffl| DR (j*(RE ^R Sq); M) ___________________________________//DR (Sq; M): The left vertical composite "* O j* is an instance of the first adjunction. The* * right diagonal is an instance of the second adjunction. The horizontal arrows are ind* *uced by : Sq -! RE ^R Sq and are isomorphisms since is localization at E and M and j*j# M are E-local. Therefore the maps "* in the diagram are isomorphisms and " is a weak equivaleance. If N is an RE -module such that j*N is E-local, then the unit i : N -! j# j*N of the first adjunction is a weak equivalence since the composite j*i * # * " * j*N- ! j j j N- ! j N is the identity and " is a weak equivalence. Since j# j*N is (RE ^R E)-local, t* *his also implies that N is (RE ^R E)-local and so completes the proof. __|_ | Proof of Theorem 3.4. Any E-local R-module is isomorphic in DR to one of the form j*N, where N is an RE -module that is E-local as an R-module. Therefore the E-localization of any R-module M is given by a map : M - ! ME , where ME is an RE -module. Such a map factors uniquely through a map " : RE ^R M -! ME in DRE . Clearly " is an E-equivalence of R-modules and therefore an (RE ^R E)-equivalence of RE -modules. This proves the claimed factorization of localization at E. __|_ | Proof of Corollary 3.5. Localization at E is smashing if and only if all R-modules of the form RE ^R M for an R-module M are E-local. In this case, if M is an RE -module, then, as an R-module, M is a retract of RE ^R M and is therefore also E-local. __|_ | 4. Periodicity and K-theory We illustrate the constructive power of our results on R-algebras by showing * *that the algebraic localizations of R considered in Chapter V take R to commutative R-algebras on the point set level and not just on the homotopical level studied* * in V.2.3. Thus let X be a set of elements of R* and consider R[X-1]. We saw in V.2* *.3 4. PERIODICITY AND K-THEORY 185 that R[X-1] is a commutative and associative R-ring spectrum whose product is an equivalence under R. For an R-module M, we have the canonical map = ^R id: M ~=R ^R M -! R[X-1] ^R M: Proposition 4.1. For any R-module M, is the Bousfield localization of M at R[X-1]. Proof. Upon smashing with R[X-1], becomes an equivalence with inverse given by the product on R[X-1]. Thus is an R[X-1]-equivalence. By a standard argument, R[X-1]^R M is R[X-1]-local since it is an R[X-1]-module spectrum. __* *|_ | So far we have been working homotopically, in the derived category DR . Theo- rem 2.1 allows us to translate to the point-set level: R is a cell R-algebra, s* *o its localization at E = R[X-1] can be constructed as a map of R-algebras. We are entitled to the following conclusion. Theorem 4.2. The localization R -! R[X-1] can be constructed as the unit of a cell R-algebra. By multiplicative infinite loop space theory [49] and our model category stru* *cture on the category of S-algebras, the spectra ko and ku that represent real and co* *mplex connective K-theory can be taken to be q-cofibrant commutative S-algebras. It is standard (see e.g. [47, IIx3]) that the spectra that represent periodic K-theor* *y can be reconstructed up to homotopy by inverting the Bott element fiO 2 ss8(ko) or fiU 2 ss2(ku). That is, KO ' ko[fi-1O] and KU ' ku[fi-1U]: We are entitled to the following result as a special case of the previous one. Theorem 4.3. The spectra KO and KU can be constructed as commutative ko and ku-algebras. Restricting the unit maps ko - ! KO and ku - ! KU along the unit maps S - ! ko and S - ! ku, we see that KO and KU are commutative S-algebras. McClure studied the problem of obtaining such a structure in his thesis. He pro* *ved that KO and KU are H1 ring spectra, this being a weakened up-to-homotopy analogue of the notion of an E1 ring spectrum, with some additional structure; see [14, VIIx7]. More recently, in unpublished work, he returned to the problem and proved that the completion of KU at a prime p is an E1 ring spectrum. Of course, this also follows from our work since completion at p is another example of a Bousfield localization. Wolbert [69] is studying the algebraic structure of the derived categories of* * mod- ules over the connective and periodic versions of the real and complex K-theory S-algebras. 186 VIII. BOUSFIELD LOCALIZATIONS OF R-MODULES AND ALGEBRAS Remark 4.4. It is often important in algebraic K-theory to invert a "Bott e* *le- ment" fi 2 K2(R), where R is a suitable discrete ring. Since multiplicative inf* *inite loop space theory implies that the algebraic K-theory spectrum KR can be taken to be a q-cofibrant commutative S-algebra, our arguments construct KR[fi-1] as a commutative KR-algebra. Remark 4.5. For finite groups G, Theorem 4.3 applies with the same proof to construct the periodic spectra KOG and KUG of equivariant K-theory as commu- tative koG and kuG -algebras. As explained in [26], this leads to an elegant pr* *oof of the Atiyah-Segal completion theorem in equivariant K-cohomology and of its analogue for equivariant K-homology. CHAPTER IX Topological Hochschild homology and cohomology As another application of our theory, we construct the topological Hochschild h* *o- mology of R-algebras with coefficients in bimodules. The relevance to THH of a theory such as ours has long been known. McClure and Staffeldt gave a good intr* *o- duction of ideas in [53, x3], and in fact our paper provides the foundations th* *at were optimistically assumed in theirs (with reference to a four author paper in prep* *a- ration that will never exist). Our analogue of B"okstedt's topological Hochsch* *ild homology [8] is under active investigation by a number of people, and we shall * *just lay the foundations. Actually, in Sections 1 and 2, we give two different constructions. For a q- cofibrant commutative S-algebra R, a q-cofibrant R-algebra A, and an (A; A)- bimodule M, we first define T HHR (A; M) to be the derived smash product M ^Ae A, exactly as in algebra (for flat algebras over rings). With this definition, * *we prove that algebraic Hochschild homology can be realized as the homotopy groups of the topological Hochschild homology of suitable Eilenberg-Mac Lane spectra, and we construct spectral sequences for the calculation of the homotopy and homology groups of T HHR (A; M) in general. We then define thhR (A; M) by mimicking the standard complex for the com- putation of algebraic Hochschild homology and we prove that, when M is a cell Ae-module, thhR (A; M) and T HHR (A; M) are equivalent. When M = A, the resulting construction thhR (A) has exceptionally nice formal properties. For * *ex- ample, it is immediate from the construction that thhR (A) is a commutative R- algebra when A is. While A is not equivalent to a cell Ae-module, we shall use our standing hypotheses that R and A are q-cofibrant to prove that T HHR (A) is equivalent to thhR (A). Further formal properties of thhR (A) are explained in the brief Section 3, w* *hich contains the results of the recent paper [52] of McClure, Schw"anzl, and Vogt. * *They exploit the tensor structure of the category of commutative S-algebras to give a 187 188 IX. TOPOLOGICAL HOCHSCHILD HOMOLOGY AND COHOMOLOGY conceptual description of thhR (A) as A S1 when A is a commutative R-algebra. Their paper was based on the now obsolete definitions of a preliminary draft of our paper, it had some errors of detail, and we have since found simpler ways to carry out their intriguing application of our theory. 1. Topological Hochschild homology: first definition We assume given an algebra A over a commutative S-algebra R and an (A; A)- bimodule M. We here define the topological Hochschild homology and cohomol- ogy spectra T HHR (A; M) and T HHR (A; M). The former presumably generalizes B"okstedt's original definition [8] (see also [10]), although a precise compari* *son has not been established. There is no precursor of the latter in the literature. We* * mimic the conceptual definition of Hochschild homology and cohomology given by Cartan- Eilenberg [15, IXxx3-4]. In the next section, we give an alternative construct* *ion that mimics Hochschild's original definition in terms of the standard complex [* *30] and compare definitions. We are only interested in relative (A; A)-bimodules, that is, those for which* * the induced left and right actions of R agree under transposition of M and R, and we define the enveloping R-algebra of A by Ae = A ^R Aop: Then an (A; A)-bimodule M can be viewed as either a left or a right Ae-module. We usually view A itself as a left Ae-module and our general bimodule M as a right or a left Ae-module, whichever is convenient. If A is commutative, then Ae = A ^R A, the product Ae -! A is a map of R-algebras, and A can be viewed as an (Ae; A)-bimodule. We assume once and for all that our given commutative S-algebra R is q-cofibr* *ant in the model category of commutative S-algebras and that A is q-cofibrant in the model category of R-algebras or of commutative R-algebras. There is no loss of generality in these assumptions since we could replace any given pair (A; R) by a weakly equivalent pair that satisfies our assumptions. By VII.6.7, these assumptions ensure that if fl : A - ! A is a weak equivalence of R-modules, where A is a cell R-module, then fl ^ fl : A ^R A -! A ^R A is a weak equivalence of R-modules. Thus the underlying R-module of Ae repre- sents A ^R A in the derived category DR . Definition 1.1. Working in derived categories, define topological Hochschild homology and cohomology with values in DR by T HHR (A; M) = M ^Ae A and T HHR (A; M) = FAe(A; M): 1. TOPOLOGICAL HOCHSCHILD HOMOLOGY: FIRST DEFINITION 189 If A is commutative, then these functors take values in the derived category DA* *e. On passage to homotopy groups, define e * * T HHR*(A; M) = TorA*(M; A) and T HHR (A; M) = ExtAe(A; M): When M = A, we delete it from the notations. Since we are working in derived categories, we are implicitly taking M to be a cell Ae-module in the definition of T HHR (A; M) and approximating A by a weakly equivalent cell Ae-module in the definition of T HHR (A; M). When A is commutative, we have the following observation, which will be amplified in the next section. Proposition 1.2. If A is a commutative R-algebra, then T HHR (A) is isomor- phic in DAe to a commutative Ae-algebra. Proof. By VII.6.9, Ae is a q-cofibrant commutative R-algebra since A is as- sumed to be one, and A is clearly a commutative Ae-algebra. Let A -! A be a weak equivalence of Ae-algebras from a q-cofibrant commutative Ae-algebra A to A. Then, by VII.6.5 and VII.6.7, the commutative Ae-algebra A ^Ae A is isomorphic in DAe to T HHR (A). __|_ | The module structures on T HHR (A; M) have the following standard implication. Proposition 1.3. If either R or A is the Eilenberg-Mac Lane spectrum of a commutative ring, then T HHR (A; M) is a product of Eilenberg-Mac Lane spectra. There is no analogue in the literature of our T HHR (A; M) except in the abso* *lute case R = S, and there is no analogue of our T HHR (A; M) even then. However, the relationship between algebraic and topological Hochschild homology becomes far more transparent when one works in full generality. To describe this relationsh* *ip, we must first fix notations for algebraic Hochschild homology and cohomology. F* *or a commutative graded ring R*, a graded R*-algebra A* that is flat as an R*-modu* *le, and a graded (A*; A*)-bimodule M*, we write e p;q * * p;q * * HHR*p;q(A*; M*) = Tor(A*)p;q(M*; A*) and HHR*(A ; M ) = Ext(A*)e(A ; M ); where p is the homological degree and q is the internal degree. When M* = A*, we delete it from the notation. Observe that there is an evident epimorphism (1.4) : M* -! HHR*0;*(A*; M*) and that is an isomorphism if the left and right action of A on M are related * *by ` = r O o. If A* is commutative, then HHR**;*(A*) is a graded A*-algebra and is a ring homomorphism; see [15, XIx6]. 190 IX. TOPOLOGICAL HOCHSCHILD HOMOLOGY AND COHOMOLOGY There is also a map (1.5) oe : A* -! HHR*1;*(A*) that sends an element a to the 1-cycle 1 a in the standard complex, and oe is a derivation if A is commutative. Specialization of IV.4.1 gives the following result. Observe that (Aop)* = (A* **)op. Theorem 1.6. There are spectral sequences of the form E2p;q= TorR*p;q(A*; Aop*) =) (Ae)p+q; e)* R E2p;q= Tor(Ap;q(M*; A*) =) T HHp+q(A; M); and Ep;q2= Extp;q(Ae)*(A*; M*) =) T HHp+qR(A; M): If A* is a flat R*-module, so that the first spectral sequence collapses, then * *the initial terms of the second and third spectral sequences are, respectively, HHR**;*(A*; M*) and HH*;*R*(A*; M*): This is of negligible use in the absolute case R = S, where the flatness hypo* *thesis is unrealistic. However, in the relative case, it implies that algebraic Hochs* *child homology and cohomology are special cases of topological Hochschild homology and cohomology. Theorem 1.7. Let R be a (discrete, ungraded) commutative ring, let A be an R-flat R-algebra, and let M be an (A; A)-bimodule. Then HHR*(A; M) ~=T HHHR*(HA; HM) and HH*R(A; M) ~=T HH*HR(HA; HM): If A is commutative, then HHR*(A) ~=T HHHR*(HA) as A-algebras. Proof. By VII.1.3 and the naturality of multiplicative infinite loop space t* *heory [49], we can construct HA as an HR-algebra, commutative if A is so. The results* * of IVx2 construct HM as an (HA; HA)-bimodule. Thus the statement makes sense. The spectral sequences collapse since their internal gradings are concentrated * *in degree zero. We will prove the last statement below. __|_ | We concentrate on homology henceforward. In the absolute case R = S, it is natural to approach T HHS*(A; M) by first determining the ordinary homology of T HHS (A; M), using the case E = HFp of the following spectral sequence, and then using the Adams spectral sequence. A spectral sequence like the following one was first obtained by B"okstedt [9]. An interesting case, essentially T HH(* *ku), was worked out by McClure and Staffeldt [53], who assumed without proof that 1. TOPOLOGICAL HOCHSCHILD HOMOLOGY: FIRST DEFINITION 191 the second spectral sequence in the following theorem could be constructed. The flatness hypotheses required when E is only a commutative ring spectrum are sti* *ll unrealistic in the absolute case, but the situation is saved by the lack of nee* *d for such hypotheses when E is a commutative S-algebra, such as HFp. Observe that there is a natural map (1.8) i = id^OE : M ~=M ^Ae Ae -! M ^Ae A = T HHR (A; M): Theorem 1.9. Let E be a commutative ring spectrum. If E*(R) is a flat R*- module, or if E is a commutative S-algebra, there is a spectral sequence of dif* *fer- ential E*(R)-modules of the form E2p;q= TorE*Rp;q(E*A; E*(Aop)) =) Ep+q(Ae): If E*(Ae) is a flat (Ae)*-module, or if E is a commutative S-algebra, there is a spectral sequence of differential E*(R)-modules of the form e) R E2p;q= TorE*(Ap;q(E*(M); E*(A)) =) Ep+q(T HH (A; M)): In either case, if E*(A) is E*(R)-flat, so that E2p;q= HHE*(R)*;*(E*(A); E*(M)) in the second spectral sequence, then the composite R E*(M) ____//_E20;*_//_E10;*__//E*(T HH (A; M)) coincides with i* : E*(M) -! E*(T HHR (A; M)). Proof. When E is just a commutative ring spectrum, both spectral sequences are immediate from IV.6.2. When E is a commutative S-algebra, both spectral sequences are immediate from IV.6.4 (see also IV.6.6). The statement about i is clear from the discussion of the edge homomorphism in IVx5. __|_ | Applied to Eilenberg-Mac Lane spectra, the following complement implies the last statement of Theorem 1.7. Clearly Proposition 1.2 implies that if A is a c* *om- mutative R-algebra, then T HHR (A) is a commutative R-ring spectrum. Moreover, by Corollary 3.8 below, there is then a natural map (in DR ) (1.10) ! : A ^ S1+-! T HHR (A): Proposition 1.11. Let A be a commutative R-algebra and assume sufficient hypotheses that Theorem 1.9 gives a spectral sequence E2p;q= HHE*(R)p;q(E*(A)) =) Ep+q(T HHR (A)): 192 IX. TOPOLOGICAL HOCHSCHILD HOMOLOGY AND COHOMOLOGY Then this is a spectral sequence of differential E*(A)-algebras such that E2 ha* *s the standard product in Hochshild homology. Moreover, the composite R E*(A) _oe_//_E21;*__//E11;*__//_E*(T HH (A))= imi* coincides with the restriction of (!)* : E*(A ^ S1+) -! E*(T HHR (A))= imi* to the wedge summand A. Proof. We may use the standard complex for the computation of algebraic Hochschild homology in the construction of the spectral sequences in IVx5. We can then construct a pairing of the resolutions constructed there that realizes* * the standard product and so deduce a pairing of spectral sequences. We omit details, since the result will be more transparent with the alternative construction of * *the spectral sequence (albeit under different hypotheses) in the next section. For * *the last statement, under the usual stable splitting of S1+as S0 _ S1, the restrict* *ion of ! to the wedge summand A coincides with i. Under the equivalence with the standard complex description of T HH in the next section, ! lands in simplicial filtration one. Thinking of A as (FS ^Ae A), where F is the free Ae-module functor, we find that the restriction of ! to A provides a factorization through A^AeA of the first stage of the inductive construction of the spectral sequence* * given in IVx5. Again, this will be more transparent with the alternative construction* * of the spectral sequence. __|_ | Remark 1.12. We have given Theorem 1.9 in the form appropriate to classical stable homotopy theory. It is perhaps more natural to give a version that makes sense from the point of view of the multiplicative homology theories ER* on R- modules of IV.1.7, where E is a commutative R-algebra. The ground ring in this context is E* = ER*(R). We leave the details to the reader. The essential point* * is the relative case of III.3.10. 2. Topological Hochschild homology: second definition We again assume given a q-cofibrant commutative S-algebra R, a q-cofibrant R-algebra or q-cofibrant commutative R-algebra A, and an (A; A)-bimodule M. Write Ap for the p-fold ^R -power of A, and let OE : A ^R A -! A and j : R -! A be the product and unit of A. Let ` : A ^R M -! M and r : M ^R A -! M 2. TOPOLOGICAL HOCHSCHILD HOMOLOGY: SECOND DEFINITION 193 be the left and right action of A on M. We have canonical cyclic permutation isomorphisms o : M ^R Ap ^R A -! A ^R M ^R Ap: The following definition precisely mimics the definition of the standard comple* *x for the computation of Hochschild homology, as given in [15, p. 175]. The topologic* *al analogue of passage from a simplicial k-module to a chain complex of k-modules is passage from a simplicial spectrum E* to its geometric realization |E*|. Definition 2.1. Define a simplicial R-module thhR (A; M)* as follows. Its R- module of p-simplices is M ^R Ap. Its face and degeneracy operators are 8 >>id ^(id)i-1^ OE ^ (id)p-i-1if 1 i < p >: p-1 (`^ (id) ) O o if i = p and si= id^(id)i^ j ^ (id)p-i. Define thhR (A; M) = |thhR (A; M)*|: When M = A, we delete it from the notation, writing thhR (A)* and |thhR (A)*|. Since geometric realization converts simplicial R-modules to R-modules, by X.1.5, thhR (A; M) and thhR (A) are R-modules. Observe that the maps ip = id^jp : M ~=M ^R Rp -! M ^R Ap specify a map of simplicial R-modules from the constant simplicial R-module M__ to thhR (A; M)*; it induces a natural map of R-modules i = |i*| : M -! thhR (A; M): Inspection of the simplicial structure shows that, if A is commutative, there i* *s a natural map of R-modules ! : A ^ S1+-! thhR (A) with image in the simplicial 1-skeleton; see (3.2) and Corollary 3.8 below. Mor* *e- over, we then have the following observation. Proposition 2.2. Let A be a commutative R-algebra. Then thhR (A) is natu- rally a commutative A-algebra with unit i : A -! thhR (A), and thhR (A; M) is a thhR (A)-module. By neglect of structure, thhR (A) is a commutative R-algebra. Proof. Clearly thhR (A)* is a simplicial commutative R-algebra, thhR (A; M)* is a simplicial thhR (A)*-module, and i* : A_- ! thhR (A)* is a map of simplici* *al commutative R-algebras. Since all structure in sight is preserved by geometric realization, by X.1.5, this implies the result. __|_ | 194 IX. TOPOLOGICAL HOCHSCHILD HOMOLOGY AND COHOMOLOGY The definition of T HHR (A; M) was homotopical and led directly to spectral sequences for its calculational study. The definition of thhR (A; M) is formal * *and algebraic. We must establish a connection between these two definitions. The starting point is the relative two-sided bar construction BR (M; A; N), w* *hich is defined for a commutative S-algebra R, an R-algebra A, and right and left A- modules M and N. The definition is the same as that of B(M; R; N) = BS(M; R; N) in IV.7.2, except that smash products over S are replaced by smash products over R. By XII.1.2 and X.1.2, there is a natural map : BR (A; A; N) -! N of A-modules that is a homotopy equivalence of R-modules. More generally, by use of the product on A and its action on the given modules, we obtain a natural map of R-modules : BR (M; A; N) -! M ^A N: The proof of the following result is the same as that of its special case IV.7.* *5. Proposition 2.3. For a cell A-module M and an A-module N, : B(M; A; N) -! M ^A N is a weak equivalence of R-modules. The relevance of the bar construction to thh is shown by the following observ* *a- tion, which is the same as in algebra. We agree to write BR (A) = BR (A; A; A): Observe that BR (A) is an (A; A)-bimodule; on the simplicial level, BR0(A) = Ae. Lemma 2.4. For (A; A)-bimodules M, there is a natural isomorphism thhR (A; M) ~=M ^Ae BR (A): Proof. If M__is the constant simplicial (A; A)-bimodule at M, then M ^Ae BR (A) ~=|M__^Ae BR*(A; A; A)|: We have canonical isomorphisms M ^R Ap ~=M ^Ae (Ae ^R Ap) ~=M ^Ae (A ^R Ap ^R A) given by the permutation of Aop= A past Ap. Inspection shows that these commute with the face and degeneracy operations and so induce the stated isomorphism. * *__|_ | Since the natural map : BR (A) - ! A of (A; A)-bimodules is a homotopy equivalence of R-modules, this has the following immediate consequence, by III.* *3.8. 2. TOPOLOGICAL HOCHSCHILD HOMOLOGY: SECOND DEFINITION 195 Proposition 2.5. For cell Ae-modules M, the natural map id^ R thhR (A; M) ~=M ^Ae BR (A)- ! M ^Ae A = T HH (A; M) is a weak equivalences of R-modules, or of Ae-modules if A is commutative. While we are perfectly happy, indeed forced, to assume that M is a cell Ae- module in our derived category level definition of T HH, we are mainly interest* *ed in the case M = A of our point-set level construction thh, and A is not of the Ae-homotopy type of a cell Ae-module except in trivial cases. However, we have the following result. Theorem 2.6. Let fl : M -! A be a weak equivalence of Ae-modules, where M is a cell Ae-module. Then the map thhR (id; fl) : thhR (A; M) -! thhR (A; A) = thhR (A) is a weak equivalence of R-modules, or of Ae-modules if A is commutative. There- fore T HHR (A; M) is weakly equivalent to thhR (A). Proof. With the notation of VII.6.4, it is clear from VII.6.5 that M and A are both in ER, and it follows from VII.6.7 that the map thhRp(id; fl) of p-sim* *plices is a weak equivalence for each p. The following result gives that the simplici* *al R-modules BR*(A) and thhR*(A) are proper, in the sense of X.2.2, and X.2.4 gives the conclusion. __|_ | Proposition 2.7. For right and left A-modules M and N, BR*(M; A; N) is a proper simplicial R-module. For an (A; A)-bimodule M, thhR*(A; M) is a proper simplicial R-module. Proof. The condition of being proper involves only the degeneracy and not the face operators of a simplicial R-module. In our cases, the degeneracies are obtained from the unit j : R -! A and, since smashing over R with M and N in the first statement and with M in the second preserves cofibrations of R-module* *s, the result in both cases is an immediate consequence of VII.6.8. __|_ | Returning to the study of spectral sequences in the previous section, we find* * that use of the standard complex gives us spectral sequences under different flatness hypotheses, just as in IVx7. We consider the spectral sequences derived in X.2.9 from the simplicial filtration of thhR (A; M). For simplicity, we restrict atte* *ntion to the absolute case R = S. Theorem 2.8. Let E be a commutative ring spectrum, let A be an S-algebra, and let M be a cell Ae-module. If E*(A) is E*-flat, then there is a spectral se* *quence of the form E2p;q= HHE*p;q(E*(A); E*(M)) =) Ep+q(thhS(A; M)): 196 IX. TOPOLOGICAL HOCHSCHILD HOMOLOGY AND COHOMOLOGY The composite S E*(M) ____//_E20;*__//E10;*__//E*(thh (A; M)) coincides with i* : E*(M) -! E*(thhS(A; M)). If A is a commutative S-algebra then the spectral sequence E2p;q= HHE*p;q(E*(A)) =) Ep+q(thhS(A)) is a spectral sequence of differential E*(A)-algebras, and the composite S E*(A) _oe__//E21;*__//_E11;*_//_E*(thh (A))= imi* coincides with the restriction of (!)* : E*(A ^ S1+) -! E*(thhS(A))= imi* to the wedge summand A. Proof. Using VII.6.2 or VII.6.7 and our standing q-cofibrancy hypothesis to see that our point-set level constructions can be used to compute derived smash products, we see that the E1-terms are exactly the standard complexes for the c* *om- putation of the algebraic Hochschild homology groups in the E2-term. The standa* *rd product on the standard complex is realized on E1, and the rest is clear. __|_* * | 3. The isomorphism between thhR (A) and A S1 We here explain a reinterpretation of the definition of thhR (A) that was obs* *erved by McClure, Schw"anzl, and Vogt [52]. Recall that the category C AR of commuta- tive R-algebras C AR is tensored and cotensored over the category U of unbased topological spaces, so that we have adjunction homeomorphisms (3.1) C AR (A X; B) ~=U (X; C AR (A; B)) ~=C AR (A; F (X+ ; B)): As in VII.3.7, we easily obtain an identification of simplicial commutative R- algebras (3.2) thhR (A)* ~=A S1*: by writing out the standard simplicial set S1*whose realization is the circle a* *nd comparing faces and degeneracies. We give a slightly different proof of the fol* *lowing result. We think of S1 as the unit complex numbers. 3. THE ISOMORPHISM BETWEEN thhR(A) AND A S1 197 Theorem 3.3 (McClure, Schw"anzl, Vogt). For commutative R-algebras A, there is a natural isomorphism of commutative R-algebras thhR (A) ~=A S1: ` 1 1 The product of thhR (A) is induced by the codiagonal S1 S -! S . The unit i : A -! thhR (A) is induced by the inclusion {1} ! S1. Proof. We may identify S1 with the pushout in the diagram @I ______//_I | | | | fflffl| fflffl| {pt} ____//_S1: We arrange our identification to map {pt} to {1}. Applying the functor A (?), we obtain a pushout diagram A @I ______//_A I | | | | fflffl| |fflffl A {pt} ____//_A S1: By VII.3.7, we have BR (A) ~=A I. By Lemma 2.4, thhR (A) ~=A ^Ae BR (A). This gives the isomorphism thhR (A) ~=A S1 by a comparison of pushouts. The statement about the product follows from the isomorphism of coproducts ` 1 (A R S1) ^R (A R S1) ~=A (S1 S ): Since i is determined by {pt} -! S1, the last statement is clear. __|_ | Corollary 3.4. The pinch map S1 - ! S1 _ S1 and trivial map S1 - ! * induce a (homotopy) coassociative and counital coproduct and counit : thhR (A) -! thhR (A) ^A thhR (A) and " : thhR (A) -! A that make thhR (A) a homotopical Hopf A-algebra. Proof. Since S1_ S1 is the pushout of S1 - * -! S1 and the functor A (?) preserves pushouts, we see from VII.1.6 that thhR (A) ^A thhR (A) ~=thhR (A) (S1 _ S1): The rest is clear. __|_ | The next few corollaries are based on the case X = S1 of the adjunctions (3.1* *). Of course, left adjoints preserve colimits. Corollary 3.5. The functor thhR (A) preserves colimits in A. 198 IX. TOPOLOGICAL HOCHSCHILD HOMOLOGY AND COHOMOLOGY The word "naive" in the following corollary refers to the fact that we only c* *on- sider spectra indexed on universes with trivial S1-actions here; genuine S1-spe* *ctra must be indexed on universes that contain all representations of S1. In the nai* *ve sense, we define commutative S1-S-algebras, and so on, simply by requiring S1 to act compatibly with all structure in sight. We think of S1 as acting trivially * *on R and A. Corollary 3.6. thhR (A) is a naive commutative S1-R-algebra. If B is a naive commutative S1-R-algebra and f : A -! B is a map of commutative R-algebras, then there is a unique map of naive commutative S1-R-algebras "f: thhR (A) -! B such that "fO i = f. Proof. The product on S1 gives a map ff : (A S1) S1 ~=A (S1 x S1) -! A S1: Its adjoint S1 -! C AR (thhR (A); thhR (A)) gives actions by elements of S1 via maps of commutative R-algebras, with the requisite continuity, and the adjuncti* *on immediately implies the universal property. __|_ | For an integer r, define OEr : S1 -! S1 by OEr(e2ssit) = e2ssirt: It is immediate that these induce power operations r on thhR (A). Corollary 3.7. There are natural maps of commutative R-algebras r : thhR (A) -! thhR (A) such that 0 = i"; 1 = id; r O s = rs; and the following diagrams commute: ff R thhR (A) S1 ____//_thh (A) rOEs || r+s|| fflffl| fflffl| thhR (A) S1 ff__//_thhR (A): Corollary 3.8. There is a natural S1-equivariant map of R-modules ! : A ^ S1+-! thhR (A) such that if B is a commutative R-algebra and f : A ^ S1+- ! B is a map of spectra such that the composite f O (id^ix) : A -! B, ix : {x}+ S1+, is a map of R-algebras for each x 2 S1, then f uniquely determines a map of R-algebras 3. THE ISOMORPHISM BETWEEN thhR(A) AND A S1 199 "f: thhR (A) -! B such that f = f"O !. Moreover, ! is obtained by passage to geometric realization from the natural map of simplicial spectra !* : A ^ (S1*)+ -! A S1*: Proof. This is immediate from VII.2.11. Its transitivity diagram and a natu- rality diagram imply the S1-equivariance. __|_ | The image of ! lies in the simplicial 1-skeleton. The intuition is that the r* *est of thhR (A) freely builds up the R-algebra structure. Remark 3.9. When A is an R-algebra, inspection of the simplicial structure shows that thhR (A)* is a cyclic spectrum, and it follows exactly as for cyclic* * spaces [16] that thhR (A) is a naive S1-R-module. We believe that thhR (A) can in fact* * be constructed as a genuine S1-spectrum that is cyclotomic in the sense of Hesselh* *olt and Madsen [29, 1.2], and we intend to return to this elsewhere. 200 IX. TOPOLOGICAL HOCHSCHILD HOMOLOGY AND COHOMOLOGY CHAPTER X Some basic constructions on spectra We have used geometric realization of simplicial spectra in several places, and we shall make further use of it to prove some of our earlier claims. We study * *it in the first two sections, concentrating on formal properties in Section 1 and * *on homotopical properties in Section 2. We then use geometric realization to defi* *ne homotopy colimits in Section 3. All of the basic definitions and much of the wo* *rk in the first three sections carries over to any of our model categories of module,* * ring, and algebra categories, as we have already indicated in VIIx3. We prefer to be more concrete in this service chapter. It will be evident that geometric realiz* *ation in the category of L-spectra or the category of R-modules for an S-algebra R is given by geometric realization in the underlying category of spectra. After discussing various special kinds of prespectra in Section 4, we use ho- motopy colimits to construct the "cylinder functor" in Section 5. This functor converts spectra to weakly equivalent -cofibrant spectra while preserving ring, module, and algebra structures. Much of the material of this chapter has long been known to the authors, and * *to others, but little if any of it has appeared in the literature. 1. The geometric realization of simplicial spectra We first recall from [42] the definition of a coend, or tensor product of fun* *ctors. Let be any small category and let C be any category that has all (small) colim* *its. Write _ for the categorical coproduct in C . Suppose given a functor F : opx -! C : Define the coend of F , denoted Z F (n; n) 201 202 X. SOME BASIC CONSTRUCTIONS ON SPECTRA to be the coequalizer of the pair of maps W _e__//_W OE:m!nF (n; m)_f__//_nF (n; n) where the restriction of e to the OEth summand is F (OE; id) and the restrictio* *n of f to the OEth summandRis F (id; OE). Using equalizers, we obtain the dual notion * *of the end of F , denoted F (n; n). Now recall that a simplicial object in any category C is a contravariant func* *tor from the simplicial category to C . We have the classical geometric realization functor, denoted |X*|, from simplicial spaces to spaces, and we need to extend * *it to the level of spectra. We shall begin with a spectrum level definition, and * *we shall then show how to interpret it in terms of the space level definition. Thi* *s will allow us to deduce many properties of the spectrum level functor simply by quot* *ing standard properties of the space level functor. Recall that, using the usual fa* *ce and degeneracy maps, we obtain a covariant functor from to spaces that sends q to the standard topological q-simplex q. Definition 1.1. Let K* be a simplicial spectrum. Define its geometric realiz* *a- tion to be the coend Z |K*| = Kq ^ (q)+ : Of course, the functor opx -! S that is implicit in the definition sends (p; q) to Kp ^ (q)+ . The geometric realization of a simplicial space X* is def* *ined similarly: Z |X*| = Xq x q: If X* is a simplicial based space, so that all its face and degeneracy maps are basepoint-preserving, then all points of each subspace *x q are identified to t* *he point (*; 1) 2 X0 x 0 in the construction of |X*|, hence Z |X*| = Xq ^ (q)+ : This places the two definitions in the same form. Actually, as with any cate- gorical colimit, the geometric realization of a simplicial spectrum is obtained* * by applying the spectrification functor L to the spacewise geometric realization o* *f its underlying simplicial prespectrum. In more detail, for a simplicial prespectrum K*, we have simplicial based spaces K*(V ) for indexing spaces V . Their geomet- ric realizations form a prespectrum with structural maps induced from those of * *K* as the composites W-V |K*(V )| ~=|W-V K*(V )| -! |K*(W )|: 1. THE GEOMETRIC REALIZATION OF SIMPLICIAL SPECTRA 203 If K* is a simplicial spectrum, then |K*| is obtained by applying L to this pre* *spec- trum. As we explain in the proof, the following result has two different senses, an* *d it is correct in both senses. Proposition 1.2. The geometric realization functor from simplicial spectra to spectra preserves homotopies. Proof. As on the space level [44, x11], we have two kinds of homotopy be- tween simplicial maps, and geometric realization carries both to homotopies of * *the usual sort. One kind is just a simplicial map with domain of the form K* ^ I+ , and geometric realization preserves this kind of homotopy by part (ii) of the n* *ext proposition applied to the constant simplicial space at I+ . The other is the c* *om- binatorial kind of homotopy that makes sense for maps between simplicial objects in any category [44, 9.1]. It can be viewed as a simplicial map with domain of the form K* ^ *[1]+ , where *[1] is the standard simplicial 1-simplex viewed as a discrete simplicial space, hence part (ii) of the next proposition also appli* *es to show that geometric realization preserves this kind of homotopy. __|_ | On the level of based spaces, it is standard that geometric realization commu* *tes with wedges and products and therefore with smash products. This easily implies a direct proof of (ii) and (iv) of the following result, and (i) can be viewed * *as a special case of (ii). Recall that functors on C are extended termwise to simpli* *cial objects in C ; for example (J* ^ K*)q = Jq ^ Kq for simplicial spectra J* and K* **. Proposition 1.3. Geometric realization enjoys the following properties. (i)For simplicial based spaces X*, there is a natural isomorphism 1 |X*| ~=|1 X*|: (ii)For simplicial based spaces X* and simplicial spectra K*, there is a nat* *ural isomorphism |K* ^ X*| ~=|K*| ^ |X*|: (iii)For simplicial spectra K* indexed on U and spaces A over I (U; U0), there is a natural isomorphism |A n K*| ~=A n |K*|: (iv)For simplicial spectra |J*| and |K*|, there is a natural isomorphism |J* ^ K*| ~=|J*| ^ |K*|; where external smash products are understood. 204 X. SOME BASIC CONSTRUCTIONS ON SPECTRA (v)For simplicial spectra K*, there are natural isomorphisms |BK*| ~=B|K*| and |C K*| ~=C |K*|; and similarly for the monads B[1] and C [1] and for the corresponding re- duced monads that were defined in IIxx4-5. Proof. Part (i-iii) hold since left adjoints commute with colimits. Parts (* *iv) and (v) follow from parts (i)-(iii) and a Fubini theorem for iterated coends. * *__|_ | Proposition 1.4. For simplicial L-spectra K* and L*, there is a natural iso- morphism |K*| ^L |L*| ~=|K* ^L L*|: For an S-algebra R, such as R = S, and simplicial R-modules M* and N*, there is a natural isomorphism |M*| ^R |N*| ~=|M* ^R N*|: Proof. The first isomorphism is immediate from the previous proposition, and it directly implies the second when R = S. In view of the coequalizer descripti* *on of smash products over R, the case of general R follows by the commutation of coequalizers with geometric realization. __|_ | Proposition 1.5. The geometric realization of a simplicial A1 or E1 ring spectrum is an A1 or E1 ring spectrum. For a commutative S-algebra R, such as R = S, the geometric realization of a simplicial (commutative) R-algebra is a (commutative) R-algebra. The analogous preservation properties hold for modules. 2. Homotopical and homological properties of realization To discuss the behavior of geometric realization with respect to equivalences* * and CW homotopy types, and to obtain useful spectral sequences from its canonical filtration, we need the following technical definition. Definition 2.1. Let K* be a simplicial spectrum and let sKq Kq be the "union" of the subspectra sjKn-1, 0 j < q. Say that K* is proper if the "inclusion" sKq -! Kq is a cofibration for each q 0. The term "union" must be interpreted in terms of appropriate pushout diagrams. The corresponding "inclusions" must be interpreted, a priori, in terms of assoc* *iated maps. However, a cofibration of spectra is a spacewise closed inclusion [37, I.* *8.1]. Rigorous notation would make this section unreadable, so we shall use notations as if we were dealing with simplicial spaces, leaving the translation to rigoro* *us categorical language to the reader. One way to be precise about the degeneracy 2. HOMOTOPICAL AND HOMOLOGICAL PROPERTIES OF REALIZATION 205 subspectrum sKq and its associated map to Kq is to interpret the latter as the following map of coends Z Dq-1 Z Dq Kp ^ D(q_; p_)+ -! Kp ^ D(q_; p_)+ ~=Kq: Here D is the subcategory of consisting of the monotonic surjections (which index the degeneracy and identity maps), and Dq is its full subcategory of obje* *cts i_with 0 i q. The isomorphism is an application of Yoneda's lemma. With this interpretation, we can generalize the context to L-spectra or to R-modules for a fixed S-algebra R. Recall that colimits in the categories of L-spectra an* *d of R-modules are created in the category of spectra. Definition 2.2. A simplicial L-spectrum is proper if the canonical map of L- spectra sKq -! Kq is a cofibration for each q 0. A simplicial R-module is proper if the canonical map of R-modules sKq -! Kq is a cofibration for each q 0. Since mapping cylinders of R-modules are created in S , a cofibration of L- spectra or of R-modules is a cofibration of spectra, but not conversely. Note that the inclusion M -! N of a relative cell R-module (N; M) is a cofibration, * *by HELP, and that VII.4.14 gives that q-cofibrations of R-algebras and of commutat* *ive R-algebras are cofibrations of R-modules. Lemma 2.3. Let i : A - ! X be a cofibration of spectra, L-spectra, or R- modules. Then j = i ^ id: (A ^ q+) [ (X ^ @q+) -! X ^ q+ is a cofibration of spectra, L-spectra, or R-modules. Therefore, if K* is a pro* *per simplicial spectrum, L-spectrum, or R-module, then the inclusion (sKq ^ q+) [ (Kq ^ @q+) -! Kq ^ q+ is a cofibration for each q 1. Proof. With the usual conventions on products and smash products of pairs, we are given that (X; A) ^ (I+ ; {0}+) is a retract pair, and we must deduce that (X; A) ^ (q+; @q+) ^ (I+ ; {0}+) is a retract pair. There is a homeomorphism of pairs (q; @q) x (I; {0}) ~=(q x I; q x {0}): 206 X. SOME BASIC CONSTRUCTIONS ON SPECTRA In based notation, this clearly implies a homeomorphism of pairs (q+; @q+) ^ (I+ ; {0}+) ~=(I+ ; {0}+) ^ q+: The conclusion follows upon smashing with (X; A) and using the given retrac- tion. __|_ | Similarly, the theorems to follow are valid with essentially identical proofs* * in the contexts of spectra, L-spectra, and R-modules. Theorem 2.4. Let f* : K* -! K0*be a map of proper simplicial spectra, L- spectra, or simplicial R-modules. (i)If each fq : Kq -! K0qis a homotopy equivalence, then so is |f*| : |K*| -! |K0*|. (ii)In the L-spectrum case, and therefore also in the R-module case, if each fq : Kq -! K0qis a weak equivalence, then so is |f*| : |K*| -! |K0*|; in * *the spectrum case, this holds if all given and constructed spectra are tame. Proof. The proof is precisely parallel to the proof of the space level analo* *gs [45, A.4]. The essential point is just the gluing theorem to the effect that a push* *out of (weak) equivalences is a (weak) equivalence when corresponding legs of the given pushout diagrams are cofibrations. For tame spectra, the weak version of this statement is a consequence of I.3.4. For L-spectra, the weak version is a consequence of I.6.5. For 0 k < q, let skKq be the union over 0 j k of the subspectra sjKq-1 of Kq. We claim first that the inclusion sk-1Kq -! skKq is a cofibration for 0 < k < q. This holds vacuously if q = 1. We assume it for q - 1 and deduce it for q. Since a composite of cofibrations is a cofibration and K* is proper, the* * left vertical inclusion in the following commutative diagram is a cofibration: sk-1Kq-1 ____//_sk-1Kq \ skKq-1___//_sk-1Kq | | | (2.5) | | | fflffl| fflffl| fflffl| Kq-1 ___________//skKq-1_________//skKq: The left horizontal arrows are induced by sk and are isomorphisms with inverses induced by dk+1, and the right square is a pushout. Therefore the middle and ri* *ght vertical arrows are also cofibrations. This proves our claim. Since s0 : Kq-1 -! s0Kq is an isomorphism, we find by induction on q and, for fixed q, by induction on k that fq : skKq -! skK0qis a (weak) equivalence for e* *ach 2. HOMOTOPICAL AND HOMOLOGICAL PROPERTIES OF REALIZATION 207 k and q. In particular, fq : sKq -! sK0qis a (weak) equivalence for each q. As usual, |K*| is filtered, and we have successive pushouts sKq ^ @q+ ____________//_Kq ^ @q+ | | | | fflffl| fflffl| g (2.6) sKq ^ q+ _____//(sKq ^ q+) [ (Kq ^ @q+) ____//_Fq-1|K*| | | | | fflffl| fflffl| Kq ^ q+ ______________//Fq|K*|: Here the restrictions of the map g to sKq^ q+ and Kq^ @q+ are dictated by the coequalizer description of |K*|. The vertical arrows are cofibrations, the bot* *tom middle one by Lemma 2.3. Therefore the restrictions |f*| : Fq|K*| -! Fq|K0*| are (weak) equivalences by successive applications of the gluing theorem, and |f*| * *is a (weak) equivalence by passage to colimits over q, using III.1.7 in the weak cas* *e. __|_ | Theorem 2.7. Let K* be a simplicial spectrum, L-spectrum, or R-module. (i)If each Kq is a cell object, each degeneracy operator is the inclusion of* * a subcomplex, and each face operator is sequentially cellular, then |K*| is* * a cell object, and similarly for CW objects. (ii)If K* is proper and if each Kq has the homotopy type of a cell object, t* *hen so does |K*|, and similarly for CW objects provided that, in the R-module case, R is connective. Proof. The proofs follow the same outline as in the previous theorem, and pa* *rt (i) is clear from the second pushout in (2.6). For the CW case of (ii), If each* * Kq has the homotopy type of a CW object, then, by induction on q and, for fixed q, by induction on k, (2.5) shows that each skKq has the homotopy type of a CW spectrum. By induction on q, (2.6) then shows that each Fq|K*| has the homotopy type of a CW spectrum. Therefore |K*| has the homotopy type of a CW spectrum. The essential point is that if J, K, and L are CW homotopy types, then a push* *out diagram obtained from a cofibration J ! K and a map J ! L is equivalent to a pushout diagram obtained from the inclusion of a subcomplex J0 in a CW object K0 and a cellular map from J0 to a CW object L0. Since we are applying the cellular approximation theorem, we must assume that R is connective in the R- module case. By the gluing theorem, the pushout K [J L is therefore homotopy equivalent to the CW object K0[J0L0. Since colimits are constructed from wedges and coequalizers and thus from wedges and pushouts, it follows that the colimit* * of a sequence of cofibrations of objects of the homotopy types of CW objects has t* *he 208 X. SOME BASIC CONSTRUCTIONS ON SPECTRA homotopy type of a CW object. The proof of the statement about cell objects is similar; here III.2.2 substitutes for the cellular approximation theorem. __|_* * | Remark 2.8. A result similar to Theorem 2.4(i) was proven for simplicial LEC spectra (see x4) in [20]; in that context the argument proceeds by an immediate reduction to the space level analog. However, that proof does not work to give Theorem 2.7(ii) and does not apply to the R-module setting. The following is the spectrum level analogue of a frequently used space level spectral sequence. We have used it in our discussion of bar constructions and of topological Hochschild homology. For a spectrum E, we can apply Eq to the simplicial spectrum K* to obtain a simplicial abelian group Eq(K*). Taking the homology of its normalized chain complex, we obtain groups Hp(Eq(K*)), and we obtain the same groups if we take the homology of its unnormalized chain complex [43, 22.3]. If E is an R-module and K* is a simplicial R-module, then, with the notation of IV.1.7, we obtain groups Hp(ERq(K*)) the same way. If R is commu- tative, then each Hp(ER*(K*)) is an R*-module. If, further, E is a commutative R-ring spectrum, then Hp(ER*(K*)) is an E*-module. Theorem 2.9. Let K* be a proper simplicial spectrum and let E be any spec- trum. There is a natural homological spectral sequence {Erp;qK*} such that E2p;qK* = Hp(Eq(K*)) and {Erp;qK*} converges strongly to E*(|K*|). With E* replaced by ER*, the same statement holds for a proper simplicial R-module K* and an R-module E. Here, if R is commutative, then the spectral sequence is one of differential R*-modul* *es, and if E is a commutative R-ring spectrum, then the spectral sequence is one of differential E*-modules. Proof. Let Fp|K*| be the image in |K*| of the wedge over 0 q p of the spectra Kq ^ (q+). Then the inclusions Fp-1|K*| Fp|K*| are cofibrations, and we have isomorphisms Fp|K*|=Fp-1|K*| ~=(Kp=sKp) ^ (p=@p) ~=p(Kp=sKp): We apply E* to obtain an exact couple, and thus a spectral sequence, with Ep+q(Fp|K*|=Fp-1|K*|) = E1p;qK*: We now see that E1p;qK* is isomorphic to the p-chain group of the normalized ch* *ain complex of Eq(K*), and a diagram chase just like that of [44, p. 111] shows that d1 agrees with the differential of this chain complex. This identifies E2, and* * the convergence is standard. __|_ | 3. HOMOTOPY COLIMITS AND LIMITS 209 3. Homotopy colimits and limits Homotopy colimits and limits of spectra do not appear explicitly in the liter* *a- ture. However, in view of our results on geometric realization, these construct* *ions present no more difficulty in the category S of spectra than they do in the cat- egory T of based spaces. Of course, we are concerned with precise point-set level versions rather than with the cruder homotopical versions that are presen* *t in any Quillen model category. We record the definitions in this section. The same definitions apply in the category of R-modules for any S-algebra R. Let D be any small category. Let Bq(D) be the set of q-tuples f_= (f1; : :;:f* *q) of composable arrows of D, depicted f1 f2 fq (3.1) d0oo___d1 oo___. .o.o__dq; and let S(f_) = dq and T (f_) = d0 be the source and target of f_. We understand B0(D) to be the set O of objects of D. With the usual faces and degeneracies, B*(D) is a simplicial set whose geometric realization is the classifying space * *B(D). We first specify homotopy colimits. A D-shaped diagram of spectra is a covari* *ant functor D : D -! S . For any such D, there is a simplicial spectrum B*(*; D; D), the space level analogue of which was specified in [46, x12]. (The left variabl* *e * is a place holder.) The spectrum of q-simplices is the wedge over all f_2 Bq(D) of the spectra D(S(f_)). The faces and degeneracies are the standard ones of the t* *wo- sided bar construction [46, x7]. Applied to f_as in (3.1), the last face on Bq* *(D) forgets fq; the last face on the f_th wedge summand of Bq(*; D; D) is the map D(fq) : D(dq) -! D(dq-1). By definition, hocolim D is the geometric realization of this simplicial spectrum. Using the abbreviation B(?) = |B*(?)|, we may write this as (3.2) hocolim D = B(*; D; D): For example, if K_ : D - ! S is the constant functor at a spectrum K, then we see by inspection of definitions that (3.3) hocolimK_ ~=B(D)+ ^ K: A map f : D -! D0of diagrams is a natural transformation of functors, and, since B*(*; D; D) is clearly proper, Theorem 2.4 has the following immediate implicat* *ion. Proposition 3.4. If f : D -! D0 is a map of diagrams such that each f(d) is a homotopy equivalence, then hocolimf : hocolimD -! hocolimD0is a homotopy equivalence. If all given and constructed spectra are tame, the same holds for * *weak equivalences. 210 X. SOME BASIC CONSTRUCTIONS ON SPECTRA We shall need the following lemma on cofibrations. While we shall only use it in the context of based spaces, it works equally well in the context of spectra. Lemma 3.5. Let D0 be a subcategory of D such that any morphism of D with target in D0 is a morphism of D0. Let D be a functor from D to the category of spaces, based spaces, or spectra and let D0 be the restriction of D to D0. Then* * the induced map hocolim D0- ! hocolimD is a cofibration. Proof. We work with based spaces for definiteness, but the argument is the same in the other two cases. It suffices to construct a retraction r : (hocolim D) ^ I+ -! (hocolim D) [ ((hocolim D0) ^ I+ ) from the reduced cylinder to the reduced mapping cylinder of the inclusion. A point z = |(f_; x); u| ^ s of the domain is given by a composable tuple f_of ma* *ps as in (3.1), a point x 2 D(dq), a point u 2 q, and a point s 2 I. There is an i 0 such that the maps f1; . .;.fi are in D - D0 and the maps fi+1; . .;.fq are in * *D0. If i = 0, define r(z) = z. If i = q, define r(z) by replacing s by 0; that is, * *r is here just the retraction to the base of the cylinder. If 0 < i < q, write u = (tv; (* *1-t)w), where v 2 i-1, w 2 q-i, and t 2 I. Then define 8 <|(f; x); (0; w)| ^ (1 - 2t)s if0 t 1=2 r(z) = : |(f; x); ((2t - 1)v; (2 - 2t)w)| ^i0f1=2 t 1: It is straightforward to check that r is a well-defined retraction. __|_ | Although we shall not need it here, we take the opportunity to record our pre- ferred definition of homotopy limits, which is precisely dual to the definition* * of homotopy colimits. We suppose given a contravariant functor E : D -! S . We obtain a cosimplicial spectrum C*(E; D; *), a two-sided cobar construction. Its spectrum of q-cosimplices is the product over all f_2 Bq(D) of the spectra E(T (f_)). The cofaces and codegeneracies with target Cq(E; D; *) have f_th co- ordinate obtained by projecting onto the coordinate of the source that is index* *ed by the corresponding face or degeneracy applied to f_, and, for the zeroth cofa* *ce, composing with the map E(f1) : E(d0) -! E(d1). We define the geometric realization or totalization "Tot K*" of a cosimplicial spectrum K* to be the end Z (3.6) Tot K* = F ((q)+ ; Kq): Here we are using the evident functor opx -! S that sends (p; q) to F ((p)+ ; K* *q). 4. -COFIBRANT, LEC, AND CW PRESPECTRA 211 We define the homotopy limit of a contravariant functor E : D -! S to be the totalization (3.7) holim E = Tot C*(E; D; *): For example, if K_ : D - ! S is the constant functor at a spectrum K, then we see by use of adjunctions and inspection of definitions that (3.8) holimK_ ~=F (B(D)+ ; K): The essential point is that the definition makes perfect sense with the precise* * point- set level definitions of product and function spectra given in [37, pp. 13, 17]. 4. -cofibrant, LEC, and CW prespectra We here discuss several special types of prespectra that play an important te* *ch- nical role in point-set level studies in stable homotopy theory. We first put * *the notion of a -cofibrant spectrum (from I.2.4) into perspective by recalling the * *fol- lowing definitions from [37, Ix8] and [20]. A space X is said to be LEC (local* *ly equiconnected) if the inclusion of its diagonal subspace is an unbased cofibrat* *ion; see e.g. Lewis [35] for a discussion of such spaces. Definition 4.1. A prespectrum D is said to be (i)-cofibrant if its structure maps oe : W-V DV ! DW are based cofibrations. (ii)an inclusion prespectrum if its adjoint structure maps "oe: DV ! W-V DW are inclusions. (iii)cofibrant if its adjoint structure maps "oeare based cofibrations. (iv)LEC if it is -cofibrant and each space DV is LEC. (v)CW if it is LEC and each DV has the homotopy type of a CW complex. A spectrum E is said to be -cofibrant or LEC if it is isomorphic to LD for some -cofibrant or LEC prespectrum D. If E is a spectrum, then the maps "oeare homeomorphisms. Therefore, as a prespectrum, E is cofibrant, but it is not -cofibrant (unless it is trivial). A* *lthough we have concentrated on -cofibrant prespectra and spectra, the following result of Lewis [35] gives one reason for interest in the LEC notion. Lemma 4.2. A -cofibrant prespectrum is an inclusion prespectrum. An LEC prespectrum is cofibrant. 212 X. SOME BASIC CONSTRUCTIONS ON SPECTRA Our CW prespectra must not be confused with CW spectra; the latter are defined in terms of spectrum-level spheres and attaching maps. Since we are interested * *in notions that are appropriate for serious point-set level work and that admit us* *able equivariant generalizations, we have no interest in the old-fashioned and, to o* *ur minds, obsolete, notion of a CW prespectrum that requires CW complexes Dn and cellular structure maps Dn -! Dn+1. We have the following relations between CW prespectra and CW spectra [37, I.8.12-14]. Theorem 4.3. If D is a CW prespectrum, then LD has the homotopy type of a CW spectrum. If E is a CW spectrum, then each space EV has the homotopy type of a CW complex and E is homotopy equivalent to LD for some CW prespectrum D. Thus a spectrum has the homotopy type of a CW spectrum if and only if it has the homotopy type of LD for some CW prespectrum D. The first statement is an immediate consequence of the following description * *of spectra in terms of shift desuspensions of spaces [37, I.4.7]. The second is ba* *sed on use of the cylinder construction defined in the next section. Proposition 4.4. If D is an inclusion prespectrum, then LD ~=colim1VDV; where the colimit is computed as the prespectrum level colimit of the maps 1Woe : 1VDV ~= 1WW-V DV - ! 1WDW: That is, the prespectrum level colimit is a spectrum that is isomorphic to LD. In particular, if D is -cofibrant, then LD is the colimit of shift desuspensi* *ons of space level based cofibrations. This makes the point-set level analysis of s* *uch spectra particularly convenient. The condition of being -cofibrant is quite wea* *k. It is clear from Theorem 4.3 that tame spectra, that is, spectra of the homotopy types of -cofibrant spectra, are considerably more general than spectra of the homotopy types of CW spectra. The output spectra of the standard infinite loop space machines are -cofibrant no matter what their input. The following closure properties of the category of -cofibrant spectra are more directly relevant to * *us. Lemma 4.5. The suspension and shift desuspension spectra of based spaces are -cofibrant. Proof. The prespectrum level structure maps of shift desuspensions are iden- tity maps or the inclusions of basepoints (which are always based cofibrations)* * [37, I.4.1]. Explicitly, 1VX = L1VX for an indexing space V U, where (1VX)(W ) = W-V X ifW V and (1VX)(W ) = {*} otherwise: __|_ | 4. -COFIBRANT, LEC, AND CW PRESPECTRA 213 Proposition 4.6. Assume given compact spaces Ai, based cofibrations Ai -! Bi, and indexing spaces Vi, where i runs through any indexing set. If W 1 iViAi ____//_E | || | | W fflffl| fflffl| i1ViBi ____//_F is a pushout of spectra and E is -cofibrant, then F is -cofibrant. If W 1 iLViAi _____//L | || | | W fflffl| fflffl| iL1ViBi ____//_M: is a pushout of L-spectra and L is tame, then M is tame. Proof. Let E = LD, where D is a -cofibrant prespectrum. Since E is a pre- spectrum level colimit, Aiis compact, and 1Viis adjoint to the Vith space funct* *or, we find that the given map 1ViAi -! E is induced by a map Ai -! Wi-ViDWi for some Wi Vi. There results a map of prespectra 1WiWi-ViAi -! D that induces the given map of spectra under the isomorphism 1ViAi ~=1WiWi-ViAi. This allows us to construct the pushout on the prespectrum level, where an insp* *ec- tion from the fact that the structure maps of the 1WiWi-ViAi and 1WiWi-ViBi are wedges of identity maps or inclusions of basepoints shows that the pushout * *is -cofibrant. This proves the first statement. Since there is no claim about the * *ac- tion of L, the second statement is an easy consequence, by comparisons of pusho* *ut diagrams and use of the fact that j : 1VA -! L1VA is a homotopy equivalence for any A and V . __|_ | Proposition 4.7. The external smash product of two -cofibrant spectra is - cofibrant. The j-fold external smash power of a -cofibrant spectrum is -cofibra* *nt as a j-spectrum. Proof. The smash product f ^ g of based cofibrations is a based cofibration since it is the composite of based cofibrations f ^ idand id^g. Indexing smash products on inner product spaces V V 0, as we may, we see immediately that the smash product of -cofibrant prespectra is a -cofibrant prespectrum. Similarly, for j-fold smash powers, we may index on j-fold sums V jand use the fact that t* *he jth smash power of a based cofibration is a based j-cofibration to see that the j-fold smash power of a -cofibrant prespectrum is a -cofibrant j-prespectrum. By the following lemma, these prespectrum level observations imply the desired spectrum level conclusions. __|_ | 214 X. SOME BASIC CONSTRUCTIONS ON SPECTRA Lemma 4.8. For prespectra D and D0, the units D -! `LD and D0- ! `LD0 of the spectrification adjunction induce an isomorphism of spectra L(D ^ D0) -! L(`LD ^ `LD0): Proof. Factoring the map in question through L(D ^`LD0) in the evident way, we see that it suffices to prove that L(D ^ D0) -! L(D ^ `LD0) is an isomorphis* *m. We have adjoint function prespectra and spectra [37, II.3.3] such that if E is a spectrum and D0 is a prespectrum, then F (D0; `E) is a spectrum. Moreover, a glance at the cited definition shows that F (D0; `E) ~=`F (LD0; E): This isomorphism of right adjoints implies the desired isomorphism of left ad- joints. __|_ | Although the fact will not be used in our work, results like those above also apply to LEC spectra [20]: the suspension and shift desuspensions of LEC spaces are LEC spectra, and the smash product of LEC spectra is LEC. For twisted half-smash products, we only have an up to homotopy version of the previous proposition. Proposition 4.9. If E 2 S U is -cofibrant and A is a compact space over I (U; U0), then A n E is -cofibrant. If E 2 S U is tame and A is a space over I (U; U0) that has the homotopy type of a colimit of a sequence of cofibrations between compact spaces, then A n E is tame. Proof. The first statement is immediate from the prespectrum level construc- tion of twisted half-smash products [37, VI.2.7]. For the second statement, we * *may assume that E = LD, where D is a -cofibrant prespectrum and, by I.2.5, we may also assume that A = colimAi for a sequence of cofibrations Ai- ! Ai+1between compact spaces Ai. Then [37, VI.2.5 and VI.2.18] give a concrete description of A n E as L(A n D), where the prespectrum A n D is obviously -cofibrant. __|_ | For example, the second statement applies when A has the homotopy type of a CW complex with finite skeleta. 5. The cylinder construction We show here that we can functorially replace an A1 or E1 ring spectrum R by a weakly equivalent -cofibrant A1 or E1 ring spectrum KR, and similarly for modules. This replacement already works on the prespectrum level. We have used it in several technical proofs, and we shall use it again later. An itera* *ted mapping cylinder functor K that sends prespectra to weakly equivalent -cofibrant prespectra was constructed in [37, I.6.8]. We shall use the language of homotopy 5. THE CYLINDER CONSTRUCTION 215 colimits to give a more conceptual version of the construction that allows us to prove that it preserves structured ring and module spectra. Construction 5.1. Let D be a prespectrum indexed on U. Define KD as follows. For an indexing space V , let V_be the category of subspaces V 0 V and inclusions. Define0a functor DV from V_to the category of based spaces by letti* *ng DV (V 0) = V -VDV 0. For an inclusion V 00-! V 0, V - V 00= (V - V 0) (V 0- V 00); 0-V 0000 0 00 0 and oe : V DV - ! DV induces DV (V ) -! DV (V ). Define (KD)(V ) = hocolimDV : An inclusion i : V - ! W induces a functor i_: V_- ! W__, the functor W-V com- mutes with homotopy colimits, and there is an evident isomorphism W-V DV ~= DW i_of functors V_- ! W__. Therefore i_induces a map oe : W-V hocolimDV ~=hocolim W-V DV ~=hocolim DW i_-! hocolimDW ; and this map is a cofibration by Lemma 3.5. Thus, with these structural maps, KD is a -cofibrant prespectrum. The structural maps oe : DV V 0-! DV specify a natural transformation to the constant functor at DV and so induce a map r : (KD)(V ) -! DV , and these maps r specify a map of prespectra. Regarding the object V as a trivial subcategory of V_, we obtain j : DV -! (KD)(V ). Clearly rj = id, and jr ' idvia a canonical homotopy since V is a terminal obje* *ct of V_. The maps j do not specify a map of prespectra, but they do specify a weak map, in the sense that joe ' oeW-V j : W-V DV - ! (KD)(W ), via a canonical homotopy. Clearly K is functorial and homotopy-preserving, and r is natural. The following example may be illuminating. Example 5.2. Let X be a based space and let D be the suspension prespectrum of X, so that DV = V X and the structure maps oe : W-V V X -! W X are the evident identifications. Via these identifications, the functor DV is isomo* *rphic to the constant functor at V X, hence (KD)(V ) ~=B(V_)+ ^ V X: The structure maps of KD are induced by the cited identifications and the maps B(i_). In this case, we can use the initial objects {0} of the V_ rather than * *the terminal objects V to obtain maps V X - ! (KD)(V ). Because the functors i_ preserve initial points, this gives a map of prespectra : D - ! KD such that r = id. We have simply fattened up the V X via the compatible system of contractible spaces B(V_). 216 X. SOME BASIC CONSTRUCTIONS ON SPECTRA Construction 5.1 is a conceptual version of [37, I.6.8], and the discussion of "preternaturality" given in [37, I.7.5-I.7.7] applies to it. As usual, we exten* *d the construction to spectra by setting KE = LK`E, and we then have a natural weak equivalence r = Lr` : KE -! E. The following result implies the second statement of Theorem 4.3. Proposition 5.3. (i)If each space DV has the homotopy type of a CW complex, then LKD has the homotopy type of a CW spectrum. (ii)If E has the homotopy type of a CW spectrum, then KE has the homotopy type of a CW spectrum, hence r : KE -! E is a homotopy equivalence. Proof. By Proposition 4.4, LKD ~= colim1n(KD)n, (KD)n = (KD)(R n), where the colimit is taken over the cofibrations 1n(KD)n ~=1n+1(KD)n -! 1n+1(KD)n+1: The conclusion of (i) follows since the colimit of a sequence of cofibrations of spectra of the homotopy types of CW spectra has the homotopy type of a CW spectrum. By [37, I.8.14], each space EV of a CW spectrum E has the homotopy type of a CW complex. Thus (ii) follows from (i) and the Whitehead theorem. __* *|_ | We must still discuss the behavior of the functor K with respect to smash pro* *d- ucts and twisted half-smash products. Proposition 5.4. Let D and D0be prespectra indexed on U and U0. Then there is a natural unital, associative, and commutative system of isomorphisms ! : KD ^ KD0- ! K(D ^ D0) over D ^ D0, where external smash products are understood. Proof. Recall that the prespectrum level external smash product D ^ D0 is naturally indexed on direct sums V V 0of indexing spaces V in U and V 0in U0. Clearly, with our restricted set of indexing spaces, the product category V_x V* *_0_is isomorphic to V__V_0_. By definition, (D ^ D0)(V V 0) = DV ^ D0V 0; with the evident structural maps. Since homotopy colimits are two-sided bar con- structions and geometric realization and simplicial bar constructions commute s* *uit- ably with products, we obtain isomorphisms (hocolim DV ) ^ (hocolim D0V)0~=hocolim(DV ^ D0V)0~=hocolim(D ^ D0)V V 0 that are evidently compatible with the retractions to DV ^ D0V 0. The coherence statements are easily verified. For the unital condition, we allow U = {0} , * *in 5. THE CYLINDER CONSTRUCTION 217 which case K is the identity functor; the space S0 is the unit for the external smash product. __|_ | Clearly this extends to j-fold external smash products, with all possible ass* *o- ciativity and equivariance. We next consider changes of universe, preparatory to considering twisted half-smash products. Lemma 5.5. Let f : U - ! U0 be a linear isometry. For prespectra D0 indexed on U0, Kf*D0 is isomorphic over f*D0 to f*KD0. For spectra E indexed on U, there is a natural map ! : f*KE -! Kf*E over f*E. Proof. For an indexing space V in U, f induces an isomorphism of categories V_ -! f(V_)_. By definition, (f*D0)(V ) = D0f(V ), with the evident structural maps. By inspection, (Kf*D0)(V ) = hocolim(f*D0)V ~=hocolim D0f(V=)(f*KD0)(V ); and these isomorphisms are compatible with the retractions to (f*D0)(V ). The functor f* is left adjoint to f* [37, p.58]. For a prespectrum D indexed on U, * *the unit D -! f*f*D of the adjunction induces a natural map KD -! Kf*f*D ~=f*Kf*D: The adjoint of this map is a natural map OE : f*KD - ! Kf*D over f*D. The spectrum level left adjoint to f* is f*E = Lf*`E. The unit D - ! `LD of the (L; `) adjunction induces natural maps Lf*D -! Lf*`LD = f*LD and LKD -! LK`LD = KLD: By [37, pp. 19, 58], the first of these is an isomorphism of spectra since f*`E* * = `f*E. Therefore OE specializes to give the required map f*KE = Lf*`LK`E ~=Lf*K`E -! LKf*`E -! LK`Lf*`E = Kf*E: __|_ | Lemma 5.6. For based spaces X, there is a natural map ! : 1 X -! K1 X such that r O ! = id. Proof. We can obtain ! by applying the previous lemma to i : {0} - ! U since, as noted in the proof of I.3.2, i*X = 1 X. The map ! so obtained is the same as the map of spectra induced by the map of prespectra described in Example 5.2. __|_ | Proposition 5.7. Let ff : A - ! I (U; U0) be a space over I (U; U0). For spectra E 2 S U, there is a natural map ! : A n KE -! K(A n E) over A n E. 218 X. SOME BASIC CONSTRUCTIONS ON SPECTRA Proof. According to [37, II.2.17], a map of spectra A n E -! E0 determines and is determined by maps of spectra ff(a)*E -! E0for a 2 A that satisfy a cert* *ain continuity condition. In particular, the identity map of A n E is determined by* * the evident maps (a) : ff(a)*E -! A n E. Composing maps ! from Lemma 5.5 with maps K(a), we obtain maps ff(a)*KE -! K(ff(a)*E) -! K(A n E): It is not hard to trace through the definitions to check the required continuity condition, and it is clear by pointwise inspection that the resulting map ! cov* *ers the retractions to A n E, r O ! = idn!. __|_ | There are coherence diagrams that relate the maps ! of the proposition to the isomorphisms recorded in I.2.2. Putting these results together, using the defin* *itions of L-spectra and their smash product and its unit map (I.4.2, I.5.1, I.8.3) and* * the definition of the L-spectrum structure on 1 X (I.4.5), we arrive at the followi* *ng conclusions. Theorem 5.8. If N is an L-spectrum with action , then KN is an L-spectrum with action the composite K L (1) n KN _!__//_K(L (1) n N)____//KN: Moreover, r : KN - ! N is a map of L-spectra. If X a based space, then ! : 1 X - ! K1 X is a map of L-spectra over 1 X. For L-spectra M and N, there is a natural map of L-spectra ! : KM ^L KN -! K(M ^L N) over M ^L N such that the following unit, associativity, and commutativity dia- grams commute: S ^L KN _!^id//_KS ^L KN || |!| fflffl| fflffl| KN oo_K___K(S_^L N); KL ^L KM ^L KN !^id//_K(L ^L M) ^L KN id^!|| |!| fflffl| fflffl| KL ^L K(M ^L N) __!___//K(L ^L M ^L N); 5. THE CYLINDER CONSTRUCTION 219 and KM ^L KN __o__//KN ^L KM !|| !|| fflffl| Ko fflffl| K(M ^L N) _____//K(M ^L N): Theorem 5.9. Let R be an A1 ring spectrum with unit j and product OE. Then KR is an A1 ring spectrum with unit and product the composites __!__ _Kj_//_ _!__//_ KOE_//_ S //KS KR and KR ^L KR K(R ^L R) KR: Moreover, r : KR - ! R is a map of A1 ring spectra. If R is an E1 ring spectrum, then so is KR. If M is an R-module (in the sense of II.3.3), then KM is a KR-module such that r : KM -! M is a map of KR-modules. 220 X. SOME BASIC CONSTRUCTIONS ON SPECTRA CHAPTER XI Spaces of linear isometries and technical theorems This chapter contains a number of deferred proofs concerning the structure of the linear isometries operad and the behavior of twisted half-smash products wi* *th respect to equivalences and cofibrations. As we said in the introduction, these results are at the technical heart of our work. We emphasize that these results were used to build the foundations of Chapter I and were not referred to later * *until VIIx6. Logically, they precede the formal introduction of S-modules. 1. Spaces of linear isometries Many of our results depend on understanding the point-set topological and ho- motopical properties of spaces of linear isometries. We collect together the re* *sults that we need in this section and the next. However, we begin with a result on limits of cofibrations of unbased spaces. To prove it, we need the following ge* *ner- alization of the standard fact that a cofibration which is a homotopy equivalen* *ce is the inclusion of a strong deformation retract; it applies when the given map* * is also the map of total spaces of a pair of fibrations. Lemma 1.1. Assume given a commutative diagram of spaces __i__ A //X p || |q| fflffl|fflffl|j B _____//Y in which p and q are fibrations and i and j are cofibrations and homotopy equiv- alences. Assume given a map r : X - ! B such that r O i = p together with a homotopy h : q ' j O rrelA. Then there is a map r : X -! A such that r O i = id and p O r = r together with a homotopy h : id' ir relA such that q O h = h. 221 222 XI. SPACES OF LINEAR ISOMETRIES AND TECHNICAL THEOREMS Proof. Both (X; A) and (X; A) x (I; @I) = (X x I; X x @I [ A x I) are DR- pairs. The standard lifting property for fibrations and trivial cofibrations gi* *ves r and h via the diagrams: __id_ __k__// A //A?? and X x @I [ A x I pX88 " h p i||r"" |p| || p pp |q| fflffl|fflffl|" fflffl|p fflffl| X __r__//B X x I ___h_____//Y; where k(x; 0; t) = x, k(x; 1; t) = ir(x), and k(a; s) = i(a). __|_ | Proposition 1.2. For n 1, assume given a commutative diagram of spaces en An _______//Xn pn|| qn|| fflffl| fflffl| An-1 _en-1//_Xn-1 in which the pn and qn are fibrations and the en are cofibrations and homotopy equivalences. Then the induced map e : A limAi- ! limXi X is the inclusion of a strong deformation retract and a cofibration. Proof. Proceeding inductively, we use the lemma to construct retractions rn : Xn -! An and homotopies hn : id' en O rn relAn that are compatible with the given fibrations. The roles of r and h in the lemma are played by rn-1 O qn and hn-1 O (qn x id). We obtain the retraction r : X - ! A and homotopy h : id ' eOr relA by passage to limits. By the standard (N)DR-pair criterion, to show th* *at e is a cofibration, we need only construct a map u : X -! I such that u-1(0) = * *A; of course, this is given by Urysohn's lemma if X is normal (e.g., metric). Sin* *ce each (Xn; An) is a DR-pair, there are maps vn : Xn -! I such that v-1n(0) = An. Let un = vn O ssn, where ssn : X ! Xn is the projection, and define X1 1 u(x) = ___nun(x): n=12 Then u is continuous and u(x) = 0 if and only ssn(x) 2 An for each n. __|_ | Remark 1.3. The preceding may appear to be a model category result, but it depends on properties peculiar to the classical cofibrations of topological spa* *ces. 1. SPACES OF LINEAR ISOMETRIES 223 Now Let U and U0 be universes and write U and U0 as the unions of expanding sequences of finite dimensional subspaces {Vn} and {Vn0}, with the topology of * *the union. Thus a subset N of U is open if it intersects each Vn in an open subset. This topology is finer than the evident metric topology. If we identify U with * *R 1 and think of R 1 as a subset of the product of countably many copies of R , then the intersection of R 1 with the product of the intervals (-1=q; 1=q) is an open neighborhood of zero which is not an open set in the metric topology. For finite dimensional inner product spaces V and V 0, the space I (V; V 0) o* *f lin- ear isometries from V to V 0is a smooth compact manifold. For a finite dimensio* *nal V , I (V; U0) is the union of the I (V; Vn0). Since a regular topological space* * that is the union of a sequence of compact spaces is paracompact, I (V; U0) is paracom- pact. As a union of smooth compact manifolds, I (V; U0) can be triangulated as a CW complex (and this also implies paracompactness). The space I (U; U0) is the inverse limit of the I (Vn; U0). As such, it is a subspace of the product of the I (Vn; U0), and it is therefore compactly gen- erated. Each projection I (Vi+1; Vj0) - ! I (Vi; Vj0) is a bundle. By checking that the trivializations extend as j increases, one can deduce that each projec* *tion I (Vi+1; U0) -! I (Vi; U0) is also a bundle. Recall that a space X is LEC if the diagonal map X -! X x X is a cofibration. It is standard that the inclusion {x} -! X is then a cofibration for all x 2 X;* * that is, every point is a nondegenerate basepoint. In fact, more generally, the incl* *usion of a retract in an LEC space is a cofibration [35, 3.1]. Proposition 1.4. The space I (U; U0) is LEC. Proof. Any CW complex is LEC [35, 2.4], hence each I (Vi; U0) is LEC. Since I (Vi; U0) is also contractible [37, II.1.5] (or see the following lemma), its * *diagonal map is a cofibration and a homotopy equivalence. The diagonal map of I (U; U0) is the inverse limit of the diagonal maps of the I (Vi; U0). Now the conclusion* * is immediate from Proposition 1.2. __|_ | Breaking with our rule of ignoring equivariant considerations, we prove the f* *ol- lowing result in full equivariant generality. As we have already used, I (U; U* *0) is contractible. Thus, trivially, I (U; U0) has the homotopy type of a CW com- plex. We record equivariant generalizations of these facts. We assume that some compact Lie group G acts on U and U0. Then G acts on I (U; U0) by conjugation. Lemma 1.5. For a G-space X, any two G-maps f; g : X -! I (U; U0) are homotopic. Proof. Write U0 = U01U02, where U01and U02are G-universes isomorphic to U0. Deformations of the identity on U0 to isometries U0 -! U01and U0 -! U02show 224 XI. SPACES OF LINEAR ISOMETRIES AND TECHNICAL THEOREMS that f and g are homotopic to maps f0 : X -! I (U; U01) and g0: X -! I (U; U02). Orthogonalization of the linear homotopy, (1 - t)f0+ tg0 shows that f0 ' g0. _* *_|_ | Lemma 1.6. Assume that there is