MODERN FOUNDATIONS FOR STABLE HOMOTOPY THEORY A. D. ELMENDORF, I. KRIZ, AND J. P. MAY Contents Introduction 1 1. Spectra and the stable homotopy category 6 2. Smash products and twisted half-smash products 11 3. The category of S-modules and its derived category 13 4. The smash product of S-modules 15 5. A1 and E1 ring spectra and their categories of modules 19 6. The smash product of R-modules and function R-modules 22 7. Tor and Ext in topology and algebra 26 8. Universal coefficient and K"unneth spectral sequences 29 9. Algebraic constructions in the derived category of R-modules 31 10. Algebra structures on localizations and on quotients by ideals 34 11. The specialization to MU-modules and algebras 37 References 39 Introduction It is a truism that algebraic topology is a very young subject. In some of it* *s most fundamental branches, the foundations have not yet reached a state of shared co* *nsen- sus. Our theme will be stable homotopy theory and an emerging consensus on what its foundations should be. The consensus is different than would have been the * *case as recently as a decade ago. We shall illustrate the force of the change of par* *adigm with new constructions of some of the most basic objects in modern algebraic to* *pol- ogy, namely the various spectra and cohomology theories that can be derived from complex cobordism. The two following articles will give introductions to comple* *tions 1 2 A. D. ELMENDORF, I. KRIZ, AND J. P. MAY in stable homotopy theory and to equivariant stable homotopy theory. The three papers have a common theme: the relationship between commutative algebra and stable homotopy theory, both relations of analogy and relations of application. Stable homotopy theory began around 1937 with the Freudenthal suspension the- orem. In simplest terms, it states that, if q is small relative to n, then ssn+* *q(Sn) is independent of n. Stable phenomena had of course appeared earlier, at least imp* *lic- itly: reduced homology and cohomology are examples of functors that are invaria* *nt under suspension without limitation on dimension. Stable homotopy theory emerged as a distinct branch of algebraic topology with Adams' introduction of his epon* *ymous spectral sequence and his spectacular conceptual use of the notion of stable ph* *enom- ena in his solution to the Hopf invariant one problem. Its centrality was reinf* *orced by two related developments that occurred at very nearly the same time, in the * *late 1950's. One was the introduction of generalized homology and cohomology theories and especially K-theory, by Atiyah and Hirzebruch. The other was the work of Th* *om which showed how to reduce the problem of classifying manifolds up to cobordism to a problem, more importantly, a solvable problem, in stable homotopy theory. * *The reduction of geometric phenomena to solvable problems in stable homotopy theory has remained an important mathematical theme, the most recent major success bei* *ng Stolz's use of Spin cobordism to study the classification of manifolds with pos* *itive scalar curvature. In an entirely different direction, the early 1970's saw Qui* *llen's introduction of higher algebraic K-theory and the recognition by Segal and othe* *rs that it could be viewed as a construction in stable homotopy theory. With algeb* *raic K-theory as an intermediary, there has been a growing volume of work that relat* *es algebraic geometry to stable homotopy theory. With Waldhausen's introduction of the algebraic K-theory of spaces in the late 1970's, stable homotopy became a b* *ridge between algebraic K-theory and the study of diffeomorphisms of manifolds. Within algebraic topology, the study of stable homotopy theory has been and remains the focus of much of the best work in the subject. The study of nilpotence and peri* *odic phenomena by Hopkins, Mahowald, Ravenel, and many others has been especially successful. We shall focus on the study of structured ring, module, and algebra spectra. * *This study plays a significant role in all of the directions of work that we have ju* *st men- tioned and would have been technically impossible within the foundational conse* *nsus that existed a decade ago. Stable homotopy theory demands a category in which to work. One could set up the ordinary Adams spectral sequence ad hoc, as Adams did, but it would be ugly* * at best to set up the Adams spectral sequence based on a generalized homology theo* *ry that way. One wants objects - called spectra - that play the role of spaces in * *unstable MODERN FOUNDATIONS FOR STABLE HOMOTOPY THEORY 3 homotopy theory, and one wants a category in which all of the usual constructio* *ns on spaces are present and, up to homotopy, the suspension functor is an equival* *ence. At this point, we introduce a sharp distinction: there is a category of point-s* *et level objects, and there is an associated homotopy category. There has been consensus on what the latter should be, up to equivalence of categories, since the fundam* *ental work of Boardman in the 1960's. The change in paradigm concerns the point-set l* *evel category that underlies the stable homotopy category. There is an analogy with algebra that is fundamental to an understanding of t* *his area of mathematics. Suppose given a (discrete) commutative ring R. It has an associated category MR of (Z -graded) chain complexes, there is a notion of hom* *otopy between maps of chain complexes, and there is a resulting homotopy category hMR. However, this is not the category that algebraists are interested in. For exam* *ple, if R-modules M and N are regarded as chain complexes concentrated in degree zero, then, in the derived category, the homology of their tensor product shoul* *d be their torsion product TorR*(M; N). Formally, the fundamental invariants of cha* *in complexes are their homology groups, and one constructs a category that reflects this. A map of chain complexes is said to be a quasi-isomorphism if it induces* * an isomorphism of homology groups. The derived category DR is obtained by adjoining formal inverses to the quasi-isomorphisms. The best way to make this rigorous * *is to introduce a notion of cell R-module such that every quasi-isomorphism between cell R-modules is a chain homotopy equivalence (Whitehead theorem) and every chain complex is quasi-isomorphic to a cell R-module. Then DR is equivalent to * *the ordinary homotopy category of cell R-modules. See [15, 21]. This is a topolog* *ist's way of thinking about the appropriate generalization to chain complexes of proj* *ective resolutions of modules. We think of the sphere spectrum S as the analog of R. We think of spectra as analogs of chain complexes, or rather as a first approximation to the defini* *tive analogs, which will be S-modules. We let S denote the category of spectra. There is a notion of homotopy of maps between spectra, and there is a resulting homot* *opy category hS . The fundamental invariants of spectra are their homotopy groups, * *and a map of spectra is a weak equivalence if it induces an isomorphism of homotopy groups. The stable homotopy category, which we denote by hS , is obtained by formally inverting the weak equivalences. This is made rigorous by introducing * *CW spectra. A weak equivalence between CW spectra is a homotopy equivalence and every spectrum is weakly equivalent to a CW spectrum. Then hS is equivalent to the ordinary homotopy category of CW spectra. Now the category MR has an associative and commutative tensor product. If we regard R as a chain complex concentrated in degree zero, then R is a unit for t* *he 4 A. D. ELMENDORF, I. KRIZ, AND J. P. MAY tensor product. A differential R-algebra A is a chain complex with a unit R -! A and product A R A -! A such that the evident associativity and unity diagrams commute. It is commutative if the evident commutativity diagram also commutes. These are, obviously enough, point-set level structures. Algebraists would have* * trou- ble taking seriously the idea of an algebra defined in DR, with unit and produc* *t only defined in that category. The category S has a smash product but, in contrast with the tensor product, it is not associative, commutative, or unital. The induced smash product on the stable homotopy category hS is associative and commutative, and it has S as uni* *t. Topologists routinely study ring spectra, which are objects E of hS with a unit j : S - ! E and product OE : E ^ E - ! E such that the evident unit diagrams commute; that is, OE O (j ^ id) = id= OE O (id^j) in hS . Similarly, E is assoc* *iative or commutative if the appropriate diagrams commute in hS . Given that the point-set level smash product is not associative or commutative, it would seem at first s* *ight that these up to homotopy notions are the only ones possible. It is a recent discovery that there is a category MS of S-modules that has an associative and commutative smash product ^S [11]. Its objects are spectra with additional structure, and we say that a map of S-modules is a weak equivalence * *if it is a weak equivalence as a map of spectra. The derived category DS is obtain* *ed from MS by formally inverting the weak equivalences, and DS is equivalent to the stable homotopy category hS . Again, this is made rigorous by a theory of CW S-modules that is just like the theory of CW spectra. There is a natural unit m* *ap : S^SM -! M; it is not an isomorphism but it induces an isomorphism on passage to DS. There is a modification ?S of ^S that is defined on S-modules equipped w* *ith a map S -! M and that has S as a strict unit; we will not go into that in the pre* *sent account, but it is vital to some of the more sophisticated applications. In the category MS, we have a point-set level notion of a ring spectrum R tha* *t is defined in terms of maps j : S -! R and OE : R^SR -! R in MS. We require that t* *he unit diagrams commute in MS, in the sense that OE O (j ^S id) = = OE O (id^Sj)* *, and we require that the associativity diagram commutes. Such an R is called an A1 * *ring spectrum; if the commutativity diagram also commutes, then R is called an E1 r* *ing spectrum. Here "A1 " stands historically for "associative up to an infinite seq* *uence of higher homotopies"; similarly, "E1 " stands for "homotopy everything", meaning * *that the product is associative and commutative up to all higher coherence homotopie* *s. With the definitions just given the higher homotopies are hidden in the definit* *ion of the associative and commutative smash product in MS, but the definitions are* * in fact precisely equivalent to earlier definitions in which the higher homotopies* * were exhibited in terms of an "operad action". It is tempting to simply call these o* *bjects MODERN FOUNDATIONS FOR STABLE HOMOTOPY THEORY 5 associative and commutative ring spectra, but that would be a mistake. These te* *rms have long established meanings, as associative and commutative rings in the sta* *ble homotopy category, and the more precise point-set level notions do not make the older notions obsolete: there are plenty of examples of associative or commutat* *ive ring spectra that do not admit structures of A1 or E1 ring spectra. It is par* *t of the new paradigm that one must always be aware of when one is working in the derived category and when one is working on the point-set level. Now fix an E1 ring spectrum R. An R-module M is an S-module together with a map : R ^S M -! M such that the evident unit and transitivity diagrams commute, the former stating that O (j ^S id) = . Let MR be the category of R-modules. Again we have a homotopy category hMR and a derived category DR that is obtained from it by inverting the weak equivalences, by which we mean m* *aps of R-modules that are weak equivalences of underlying spectra. The construction* * of DR is made rigorous by a theory of cell R-modules, the one slight catch being t* *hat, unless R is connective, in the sense that its homotopy groups are zero in negat* *ive degrees, we cannot insist that cells be attached only to cells of lower dimensi* *on, so that our cell R-modules cannot be restricted to be CW R-modules. These categori* *es enjoy all of the good properties that we have described in the special case R =* * S. There is an associative and commutative smash product over R, and it has a unit* * map : R ^R M -! M that becomes an isomorphism on passage to the derived category. We can therefore go on to define A1 and E1 R-algebras A in terms of point-set level associative and commutative multiplications A ^R A -! A, and we can also define derived category level associative and commutative R-algebras A exactly * *like the classical associative and commutative ring spectra. It is the derived category DR that we wish to focus on in describing the curr* *ent state of the art in stable homotopy theory. We can mimic classical commutative algebra in this category. In particular, for an ideal I and multiplicatively c* *losed subset Y in the coefficient ring R* = ss*(R), we will show how to construct quo* *tients M=IM and localizations M[Y -1]. When applied with R taken to be the representing spectrum MU for complex cobordism, these constructions specialize to give simple constructions of various spectra that are central to modern stable homotopy the* *ory, such as the Morava K-theory spectra. Moreover, we shall see that these spectra * *are MU-algebra spectra. This account is largely a summary of the more complete and technical paper [1* *1], to which the reader is referred for further background and detailed proofs. 6 A. D. ELMENDORF, I. KRIZ, AND J. P. MAY 1. Spectra and the stable homotopy category We here give a bare bones summary of the construction of the stable homotopy category, referring to [16] and [11] for details and to [22] for a more leisure* *ly exposi- tion. We aim to give just enough of the basic definitional framework that the r* *eader can feel comfortable with the ideas. By Brown's representability theorem [6], if E* is a reduced cohomology theory on based spaces, then there are CW complexes En such that, for CW complexes X, En(X) is naturally isomorphic to the set [X; En] of homotopy classes of based maps X -! En. The suspension isomorphism En(X) ~=En+1(X) gives rise to a homotopy equivalence "oen: En -! En+1. The object E = {En; "oen}is called an - spectrum. A map f : E -! E0of -spectra is a sequence of homotopy classes of maps fn : En -! E0nthat are compatible up to homotopy with the equivalences "oenand "oe0n. The category of -spectra is equivalent to the category of cohomology the* *ories on based spaces and can be thought of as an intuitive first approximation to the stable homotopy category. However, this category does not have a usable theory * *of cofibration sequences and is not suitable for either point-set level or homotop* *ical work. For that, one needs more precise objects and morphisms that are defined without* * use of homotopies but that still represent cohomology theories and their maps. More subtly, one needs a coordinate-free setting in order to define smash products s* *ensibly. The nth space En relates to the n-sphere and thus to R n. Restricting to spaces* * En is very much like restricting to the standard basis of R1 when doing linear al* *gebra. A coordinate-free spectrum is indexed on the set of finite dimensional subspa* *ces V of a "universe" U, namely a real inner product space isomorphic to the sum R1 * *of countably many copies of R. In detail, writing W -V for the orthogonal compleme* *nt of V in W , a spectrum E assigns a based space EV to each finite dimensional su* *bspace V of U, with structure maps ~=W-V "oeV;W: EV -! EW when V W ; these maps are required to be homeomorphisms. Here W X is the function space F (SW ; X) of based maps SW -! X, where SW is the one-point co* *m- pactification of W . The structure maps are required to satisfy an evident tran* *sitivity relation when V W Z. 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 "oeV;W|| |"oe0V;W| fflffl|W-V fW fflffl| W-V EW _________//_W-V E0W: MODERN FOUNDATIONS FOR STABLE HOMOTOPY THEORY 7 We obtain the category S = S U of spectra indexed on U. We obtain an equiv- alent category if we restrict to any cofinal family of indexing spaces. If we * *drop the requirement that the maps "oeV;Wbe homeomorphisms, we obtain the notion of a prespectrum and the category P = PU of prespectra indexed on U. The for- getful functor ` : S - ! P has a left adjoint L. When the structure maps "oeare inclusions, (LE)(V ) is just the union of the spaces W-V EW for V W . We write oe : W-V EV - ! EW for the adjoint structure maps, where V X = X ^ SV . Examples 1.1. Let X be a based space. The suspension prespectrum 1 X is the prespectrum whose V th space is V X; the structure maps oe are the evident iden- tifications W-V V X ~= W X. The suspension spectrum of X is 1 X = L1 X. Let QX = [V V X, where the union is taken over the inclusions obtained from the adjoints of the cited identifications. Then (1 X)(V ) = Q(V X). The functor 1 from based spaces to spectra is left adjoint to the functor that assigns the * *zeroth space E0 = E({0}) to a spectrum E. More generally, for a fixed subspace Z U, define 1ZX to be the analogous prespectrum whose V th space is V -ZX if Z V and a point otherwise and define 1ZX = L1ZX. Then 1Z is left adjoint to the functor that sends a spectrum to its Zth space EZ; these functors are generally* * called "shift desuspensions". 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). Function spectra are easier. We set F (X; E)(V ) = F (X; EV ) and fi* *nd that this functor on prespectra preserves spectra. If we topologize the set S (* *E; E0) as a subspace of the product over V of the function spaces F (EV; E0V ) and let* * T be the category of based spaces with sets of maps topologized as function spaces, * *then there result homeomorphisms S (E ^ X; E0) ~=T (X; S (E; E0)) ~=S (E; F (X; E0)): Recall that a category is said to be cocomplete if it has all colimits and comp* *lete if it has all limits. Proposition 1.2. The category S is complete and cocomplete. Proof.Limits and colimits are defined on prespectra spacewise. Limits preserve * *spec- tra, and colimits of spectra are obtained by use of the left adjoint L. __|_ | 8 A. D. ELMENDORF, I. KRIZ, AND J. P. MAY We write Y+ for the union of a space Y and a disjoint basepoint. A homotopy in the category of spectra is a map E ^ I+ - ! E0. We have cofibration and fibrati* *on sequences that are defined exactly as on the space level (e.g. [29]) and enjoy* * the same homotopical properties. Let [E,E'] denote the set of homotopy classes of m* *aps E -! E0; we shall later understand that, when using this notation, E must be of the homotopy type of a CW spectrum. For based spaces X and Y with X compact, we have [1 X; 1 Y ] ~=colimn[nX; nY ]: Fix a copy of R 1 in U. In the equivariant generalization of the present the* *ory, it is essential not to insist that R 1 be all of U, but the reader may take U =* * R 1 here. We write 1n= 1Rn. For n 0, the sphere spectrum Sn is 1 Sn. For n > 0, the sphere spectrum S-n is 1nS0. We write S for the zero sphere spectrum. The nth homotopy group of a spectrum E is the set [Sn; E] of homotopy classes of ma* *ps Sn -! E, and this fixes the notion of a weak equivalence of spectra. The adjunc* *tions of Examples 1.1 make it clear that a map f of spectra is a weak equivalence if * *and only if each of its component maps fZ is a weak equivalence of spaces. The sta* *ble homotopy category hS is constructed from the homotopy category of spectra by adjoining formal inverses to the weak equivalences, a process that is made rigo* *rous by CW approximation. The theory of CW spectra is developed by taking sphere spectra as the domains of attaching maps of cells CSn = Sn ^ I [16, Ix5]. The one major difference fr* *om the space level theory of CW complexes is that we have to construct CW spectra * *as unions E = [En, where E0 is the trivial spectrum and where we are allowed to at* *tach cells of arbitrary dimension when constructing En+1 from En. There results a no* *tion of a cell spectrum. We define a CW spectrum to be a cell spectrum whose cells a* *re attached only to cells of lower dimension. Thus CW spectra have two filtrations* *, the sequential filtration {En} that gives the order in which cells are attached, a* *nd the skeletal filtration {Eq} , where Eq is the union of the cells of dimension at m* *ost q. We say that a map between CW spectra is cellular if it preserves both filtratio* *ns. In fact, by redefining the sequential filtration appropriately, we can always arra* *nge that the sequential filtration is preserved. We have three basic results, whose proo* *fs are very little different from their space level counterparts. Theorem 1.3 (Whitehead). If E is a CW spectrum and f : F - ! F 0is a weak equivalence of spectra, then f* : [E; F ] -! [E; F 0] is an isomorphism. There* *fore a weak equivalence between CW spectra is a homotopy equivalence. Theorem 1.4 (Cellular approximation). Let A be a subcomplex of a CW spec- trum E, let F be a CW spectrum, and let f : E -! F be a map whose restriction MODERN FOUNDATIONS FOR STABLE HOMOTOPY THEORY 9 to A is cellular. Then f is homotopic relative to A to a cellular map. Therefor* *e any map E -! F is homotopic to a cellular map, and any two homotopic cellular maps are cellularly homotopic. Theorem 1.5 (Approximation by CW spectra). For a spectrum E, there is a CW spectrum E and a weak equivalence fl : E -! E. On the homotopy category hS , is a functor such that fl is natural. It follows that the stable category hS is equivalent to the homotopy category* * of CW spectra. Homotopy-preserving functors on spectra that do not preserve weak equivalences are transported to the stable category by first replacing their va* *riables by weakly equivalent CW spectra. Observe that there has been no mention of space level CW complexes in our de- velopment so far. The total lack of hypotheses on the spaces and structural map* *s of our prespectra allows considerable point-set level pathology, even if, as usual* * in mod- ern algebraic topology, we restrict attention to compactly generated weak Hausd* *orff spaces. Recall that a space X is weak Hausdorff if the diagonal subspace is clo* *sed in the compactly generated product X x X. More restrictively, a space X is said to* * be LEC (locally equiconnected) if the inclusion of the diagonal subspace is a cofi* *bration. We record the following list of special kinds of prespectra both to prepare for* * our discussion of smash products and to compare our definitions with those adopted * *in the original treatments of the stable homotopy category. Definition 1.6.A prespectrum D is said to be (i)-cofibrant if each oe : W-V DV ! DW is a based cofibration (that is, satisfies the based homotopy extension property). (ii)CW if each DV is LEC and has the homotopy type of a CW complex. (iii)strictly CW if each DV is a based CW complex and the structure maps oe a* *re the inclusions of subcomplexes. A spectrum E is said to be -cofibrant if it is isomorphic to LD for some -cofib* *rant prespectrum D; E is said to be tame if it is of the homotopy type of a -cofibra* *nt spectrum. If E is a spectrum, then the maps "oeare homeomorphisms. Therefore the underl* *ying prespectrum `E is not -cofibrant unless it is trivial. However, it is a very w* *eak condition on a spectrum that it be tame. We shall see that this weak condition is enough to avoid serious point-set topological problems. If D is a -cofibrant prespectrum, then the maps "oeare inclusions and therefore LD(V ) is just the u* *nion of the spaces W-V DW . We have the following relations between CW prespectra 10 A. D. ELMENDORF, I. KRIZ, AND J. P. MAY and CW spectra. Remember that CW spectra are defined in terms of spectrum level attaching maps. Theorem 1.7. 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. In particular, spectra of the homotopy types of CW spectra are tame. Implicitly or explicitly, early constructions of the stable homotopy category* * re- stricted attention to the spectra arising from strict CW prespectra. This is fa* *r too restrictive for serious point-set level work, and it is also too restrictive to* * admit a sensible equivariant analogue. Note that such a category cannot possibly be com* *plete or have well-behaved point-set level function spectra. One reason for focusing on -cofibrant spectra is that they are built up out o* *f their component spaces in a simple fashion. Proposition 1.8. If E = LD, where D is a -cofibrant prespectrum, then E ~=colimV 1VDV; 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 E. * *The maps of the colimit system are shift desuspensions of based cofibrations. Another reason is that general spectra can be replaced functorially by weakly equivalent -cofibrant spectra. Proposition 1.9. There is a functor K : PU -! PU, called the cylinder functor, such that KD is -cofibrant for any prespectrum D, and there is a natural spacew* *ise weak equivalence of prespectra KD - ! D. On spectra E, define KE = LK`E. Then there is a natural weak equivalence of spectra KE -! E. In practice, if one is given a prespectrum D, perhaps indexed only on integer* *s, and one wishes to construct a spectrum from it that retains homotopical information* *, one forms E = LKD. Then ssn(E) = colimqssn+qDq: If D is an -spectrum that represents a given cohomology theory on spaces, then E = LKD is a genuine spectrum that represents the same theory. MODERN FOUNDATIONS FOR STABLE HOMOTOPY THEORY 11 2. Smash products and twisted half-smash products The construction of the smash product of spectra proceeds by internalization * *of an "external smash product". The latter is an associative and commutative pairi* *ng S U x S U0 ! S (U U0) for any pair of universes U and U0. It is constructed by starting with the pres* *pectrum 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. In order to obtain smash products internal to a single universe U, we exploit* * the "twisted half-smash product". The input data for this functor consist of two un* *iverses U and U0, an unbased space A with a given 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 mu* *st 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 cate* *gories. When A is a point, ff is a choice of a linear isometry f : U -! U0 and we write* * f* for the twisted half-smash product. For a prespectrum D, 0-f(V ) -1 0 (f*D)(V 0) = D(V ) ^ SV ; where V = f (V \ imf): For a spectrum E, f*E is obtained by application of L to the prespectrum level construction. The functor f* is left adjoint to the more elementary functor f* * *specified by (f*E0)(V ) = E0(f(V )). For general A and ff, the intuition is that A n E is obtained by suitably topologizing the union of the ff(a)*(E). Another intuitio* *n is that the twisted half-smash product is a generalization to spectra of the "untw* *isted" functor A+ ^ X on based spaces X. This intuition is made precise by the followi* *ng "untwisting formula" relating twisted half-smash products and shift desuspensio* *ns. Proposition 2.1. For a map A -! I (U; U0) and an isomorphism V ~= V 0, where V U and V 0 U0, there is an isomorphism of spectra A n 1VX ~=A+ ^ 1VX0 that is natural in spaces A over I (U; U0) and based spaces X. The twisted-half smash product functor enjoys essentially the same formal pro* *per- ties as the space level functor A+ ^ X. The functor A n E is homotopy-preservin* *g 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 pres* *erves 12 A. D. ELMENDORF, I. KRIZ, AND J. P. MAY homotopy equivalences in the variable A. The following central technical result* * is an easy consequence of Propositions 1.8 and 2.1. Theorem 2.2. 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. Since A n E is a CW spectrum if A is a CW complex and E is a CW spectrum, this has the following consequence. Corollary 2.3. 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. Now, as before, restrict attention to a particular universe U and write S = S* * U; again, the reader may think of U as R1 . We are especially interested in twist* *ed half-smash products defined in terms of the following spaces of linear isometri* *es. Notations 2.4. 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 ident* *ity 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 [18, p.1] with structural maps fl, called the* * linear isometries operad. Points f 2 L (j) give functors f* that send spectra indexed * *on Uj to spectra indexed on U. Applied to a j-fold external smash product E1 ^ . .^.E* *j, there results an internal smash product f*(E1^. .^.Ej). All of these smash prod* *ucts become equivalent in the stable category hS , but none of them are associative * *or commutative on the point set level. More precisely, the following result holds. Theorem 2.5. Let St S be the full subcategory of tame spectra and let hSt be its homotopy category. On St, the internal smash products f*(E ^ E0) determined by varying f 2 L (2) are canonically homotopy equivalent, and hSt is symmetric monoidal under the internal smash product. For based spaces X and tame spectra E, there is a natural homotopy equivalence E ^ X ' f*(E ^ 1 X). MODERN FOUNDATIONS FOR STABLE HOMOTOPY THEORY 13 This implies formally that we have arrived at a stable situation. As for spac* *es, the suspension functor is given by E = E ^ S1 and is left adjoint to the loop func* *tor given by E = F (S1; E). The cofibre Cf of a map f : E -! E0 of spectra is the pushout E0[f CE. Theorem 2.6. The suspension functor : hSt -! hSt is an equivalence of cate- f 0 gories. A cofibre sequence E- ! E - ! Cf in St gives rise to a long exact seque* *nce of homotopy groups . .-.! ssq(E) -! ssq(E0) -! ssq(Cf) -! ssq-1(E) -! . .:. Proof.For based spaces X, 1 X is naturally isomorphic to (11X) ^ S1 because both functors are left adjoint to the zeroth space functor. Thus, for tame spec* *tra E, the previous theorem gives a natural homotopy equivalence E = E ^ S0 ' f*(E ^ 1 S0) ~=f*(E ^ 11S0) ^ S1: Therefore is an equivalence of categories with inverse obtained by smashing wi* *th the (-1)-sphere spectrum S-1 = 11S0. It follows categorically that E ' f*(E ^ S-1) and that the unit and counit j : E -! E and " : E -! E of the adjunction are homotopy equivalences. The last statement is a standard c* *on- sequence of the fact that maps can now be desuspended. __|_ | Note that only actual homotopy equivalences, not weak ones, are relevant to t* *he last two results. For this reason among others, hStis a technically convenient * *halfway house between the homotopy category of spectra and the stable homotopy category. 3. The category of S-modules and its derived category For f 2 L (j) and Ei 2 St, Theorem 2.2 implies that the inclusion {f} L (j) induces a homotopy equivalence f*(E1 ^ . .^.Ej) -! L (j) n (E1 ^ . .^.Ej): The proof of Theorem 2.5 above is entirely based on the use of such equivalence* *s. It therefore seems natural to think of L (j) n (E1 ^ . .^.Ej) as a canonical j-fold smash product. It is still not associative, but it seems * *closer to being so. However, to take this idea seriously, we must take note of the diffe* *rence between E and its "1-fold smash product" L (1) n E. The space L (1) is a monoid 14 A. D. ELMENDORF, I. KRIZ, AND J. P. MAY under composition, and the formal properties of twisted half-smash products imp* *ly a natural isomorphism L (1) n (L (1) n E) ~=(L (1) x L (1)) n E; where, on the right, L (1)xL (1) is regarded as a space over L (1) via the comp* *osition product. This product induces a map : (L (1) x L (1)) n E -! L (1) n E, and the inclusion {1} -! L (1) induces a map j : E -! L (1) n E. Thus it makes sense to consider spectra E with an action : L (1) n E -! E of the monoid L (1). It * *is required that the following diagrams commute: j (L (1) x L (1)) n E_____//L (1) n E and E I____//L (1) n E III L (1)n|| || =IIIII || fflffl| fflffl| I$$Ifflffl| L (1) n E ____________//_E E: Definition 3.1.An S-module is a spectrum E together with an action of L (1). A map f : E ! E0 of S-modules is a map of spectra such that the following diagr* *am commutes: L (1)nf 0 L (1) n E __________//L (1) n E E || |E0| fflffl| fflffl| E ________f________//_E0: We let MS denote the category of S-modules. A number of basic properties of the category of spectra are directly inherite* *d by the category of S-modules. Theorem 3.2. The category of S-modules is complete and cocomplete, with both limits and colimits created in the underlying category S . If X is a based spac* *e and M is an S-module, then M ^ X and F (X; M) are S-modules, and the spectrum level fibre and cofibre of a map of S-modules are S-modules. A homotopy in the category of S-modules is a map M ^ I+ - ! M0. A map of S-modules is a weak equivalence if it is a weak equivalence as a map of spectra. The derived category DS is constructed from the homotopy category of S-modules by adjoining formal inverses to the weak equivalences. There is a theory of CW * *S- modules that is exactly like the theory of CW spectra, and, again, the construc* *tion of DS is made rigorous by CW approximation. We have a free functor L from spect* *ra to S-modules specified by LE = L (1) n E. The "sphere S-modules" that we take as MODERN FOUNDATIONS FOR STABLE HOMOTOPY THEORY 15 the domains of attaching maps when defining CW S-modules are the free S-modules LSn. Using the freeness adjunction MS(LE; M) ~=S (E; M); it is easy to prove Whitehead, cellular approximation, and approximation by CW S-modules theorems exactly like those stated for spectra in Section 1, and DS is equivalent to the homotopy category of CW S-modules. There is one catch: al- though S and all other suspension spectra are S-modules in a natural way, using* * the untwisting isomorphism of Proposition 2.1 and the projection L (1) -! {*}, S do* *es not have the homotopy type of a CW S-module. However, it is not hard to see that the categories hS and DS are equivalent. Theorem 3.3. The following conclusions hold. (i)The free functor L : S - ! MS carries CW spectra to CW S-modules. (ii)The forgetful functor MS -! S carries S-modules of the homotopy types of CW S-modules to spectra of the homotopy types of CW spectra. (iii)Every CW S-module M is homotopy equivalent as an S-module to LE for some CW spectrum E. (iv)If E 2 St, for example if E is a CW spectrum, then j : E - ! LE is a homotopy equivalence of spectra. (v) If M has the homotopy type of a CW S-module, then : LM - ! M is a homotopy equivalence of S-modules. Therefore the free and forgetful functors establish an adjoint equivalence betw* *een the stable homotopy category hS and the derived category DS. 4.The smash product of S-modules One of the most surprising developments of recent years is the discovery of a* *n as- sociative and commutative smash product in the category of S-modules. We proceed to define it. To begin with, observe that the monoid L (1) x L (1) acts from t* *he right on L (2) and acts from the left on L (i) x L (j), via instances of the st* *ructural maps fl of the linear isometries operad. Another instance of fl gives rise to a* * map (4.1) fl: L (2) xL (1)xL (1)L (i) x L (j) -! L (i + j): The space on the left is the balanced product (formally a coequalizer) of the t* *wo specified actions by L (1) x L (1). The essential, elementary, point is that t* *his map is a homeomorphism if i 1 and j 1. To see this, choose linear isometric isomorphisms s : U - ! Ui and t : U - ! Uj. Composition on the right with s t 16 A. D. ELMENDORF, I. KRIZ, AND J. P. MAY gives vertical homeomorphisms in the commutative diagram fl L (2) xL (1)xL (1)L (i) x L (j)__//_L (i + j) | | | | fflffl| fl fflffl| L (2) xL (1)xL (1)L (1) x L (1)____//L (2); and the lower map flis clearly a homeomorphism. Note also that L (1) acts from * *the left on L (2) and that this action commutes with the right action of L (1) x L * *(1). Regard L (1) x L (1) as a space over I (U2; U2) via the direct sum of isometr* *ies map. If M and N are S-modules, then L (1)xL (1) acts from the left on the exter* *nal 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: The smash product over S of M and N is simply the balanced product (again, formally a coequalizer) (4.2) M ^S N = L (2) nL (1)xL (1)(M ^ N): The left action of L (1) on L (2) induces a left action of L (1) on M ^S N that* * gives it a structure of S-module. Use of the transposition oe 2 2 and the commutativi* *ty of the external smash product easily gives a commutativity isomorphism o : M ^S N -! N ^S M: More substantially, there is a natural associativity isomorphism (M ^S N) ^S P ~=M ^S (N ^S P ): In fact, using the case i = 2 and j = 1 of the homeomorphism fl, we obtain isom* *or- phisms (M ^S N) ^S 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 ^S (N ^S P ). In view of the generality of the homeomorphisms (4.1), the argument iterates to gi* *ve (4.3) M1 ^S . .^.SMj ~=L (j) nL (1)j(M1 ^ . .^.Mj); where the iterated smash product on the left is associated in any fashion. On passage to the derived category DS, the smash product of S-modules just constructed can be used interchangeably with the internal smash product on the MODERN FOUNDATIONS FOR STABLE HOMOTOPY THEORY 17 stable category hS . To see this, one defines the latter by use of a linear iso* *metric isomorphism f : U2 -! U (not just an isometry). With this choice, it is not har* *d to check the following result. Proposition 4.4. For spectra E and F , there are isomorphisms of S-modules LE ^S LF ~=L (2) n E ^ F ~=Lf*(E ^ F ): For CW S-modules M and N, M ^S N is a CW S-module with one (p + q)-cell for each p-cell of M and q-cell of N. Together with Theorem 2.6, this gives the starting point towards the proof of* * the following fundamental theorem. Theorem 4.5. Let E be a CW spectrum and M be a CW S-module. (i)For spectra F in St, there is a natural spectral sequence Torss*S(ss*E; ss*F ) =) ss*((f*(E ^ F )): (ii)If OE : F -! F 0is a weak equivalence between spectra in St, then f*(id^OE) : f*(E ^ F ) -! f*(E ^ F 0) is a weak equivalence of spectra. (iii)For S-modules N in St, there is a natural spectral sequence Torss*S(ss*M; ss*N) =) ss*(M ^S N): (iv)If OE : N -! N0 is a weak equivalence of S-modules in St, then id^SOE : M ^S N -! M ^S N0 is a weak equivalence of S-modules. (v) The composite natural map of spectra f*(M ^ N) -! L (2) n M ^ N -! M ^S N is a weak equivalence when the S-module N is in St. Therefore the equivalence of categories DS -! hS carries the smash product over S to the internal smash product of spectra. We must still address the question of units. Proposition 4.6. There is a natural map of S-modules : S ^S N -! N, and is a homotopy equivalence of spectra if N is a CW S-module. Therefore induces a natural isomorphism of functors on DS. 18 A. D. ELMENDORF, I. KRIZ, AND J. P. MAY Proof.When N is the free S-module LE = L (1) n E generated by a spectrum E, is given by the map S ^S LE = L (2) nL (1)xL (1)(L (0) n S0) ^ (L (1) n E) ~=(L (2) xL (1)xL (1)L (0) x L (1)) n (S0 ^ E) flnid --! L (1) n E = LE: Here the map fl, although not a homeomorphism, is a homotopy equivalence. There- fore, by Theorem 2.2, is a homotopy equivalence when E 2 St. For general N, the map just constructed for LN induces the required map for N by a comparison * *of coequalizer diagrams. By Theorem 3.3(iii), we conclude directly that is a homo* *topy equivalence when N is a CW S-module. __|_ | There is one case when is an isomorphism. It turns out that the map flof (4.* *1) is a homeomorphism when i = j = 0; that is, non-obviously since L (1) is a mono* *id but not a group, the domain of (4.1) is then a point. This implies that S ^S S * *~=S. More generally, it implies that the smash product over S precisely generalizes * *the smash product of based spaces. Proposition 4.7. For based spaces X and Y , 1 (X ^ Y ) ~=1 X ^S 1 Y: We shall not display the constructions here, but the twisted half-smash produ* *ct functor AnE has a right adjoint twisted function spectrum functor F [A; E0) and* * the external smash product has a right adjoint function spectrum functor. Using the* *se functors and appropriate equalizer diagrams, dual to the coequalizer diagrams t* *hat were implicit in the definition of ^S, we can construct function S-modules. Theorem 4.8. There is a function S-module functor FS(M; N) such that MS(L ^S M; N) ~=MS(L; FS(M; N)): Theorem 4.9. Let M be a CW S-module. (i)If OE : N ! N0 is a weak equivalence of S-modules, then FS(id; OE) : FS(M; N) -! FS(M; N0) is a weak equivalence of S-modules. (ii)There is an induced adjunction isomorphism DS(L ^S M; N) ~=DS(L; FS(M; N)): (iii)There is a natural weak equivalence of spectra FS(M; N) -! F (M; f*N). MODERN FOUNDATIONS FOR STABLE HOMOTOPY THEORY 19 Therefore the equivalence of categories DS -! hS induced by the forgetful funct* *or from S-modules to spectra carries the function S-module functor FS to the inter* *nal function spectrum functor F . Proposition 4.10. The adjoint N - ! FS(S; N) of the unit map N ^S S - ! N becomes a natural equivalence on passage to the derived category DS. 5. A1 and E1 ring spectra and their categories of modules Intuitively, A1 ring spectra are as near to associative rings with unit as o* *ne can get in stable homotopy theory, and E1 ring spectra are as near as one can get * *to commutative rings. Definition 5.1.An A1 ring spectrum is an S-module R together with maps of S- modules j : S -! R and OE : R ^S R ! R such that the following diagrams commute: j^Sid id^Sj id^SOE S ^S R ____//_R ^S Roo__R ^S S and R ^S R ^S R ____//_R ^S R LLL rrr LLL |OE rrr OE^Sid| OE|| LLL | rrro | fflffl| LL%%fflffl|yyrr fflffl|OE R R ^S R _________//_R; R is an E1 ring spectrum if the following diagram also commutes: R ^S RH______o_____//R ^S R HHH vvvv HHH vvv OE H$$H zzvOEv R: Definition 5.2.Let R be an A1 ring spectrum. A (left) R-module M is an S- module together with a map : R ^S M ! M of S-modules such that the following diagrams commute: j^Sid id^S S ^S M ____//_R ^S M and R ^S R ^S M ____//_R ^S M MMM MMM | OE^Sid| | MMM | | | MM&&fflffl| fflffl| fflffl| M R ^S M _________//_M: A map f : M - ! M0 of R-modules is a map of S-modules such that the following diagram commutes: id^f R ^S M ____//_R ^S M0 || |0| fflffl| fflffl| M ____f____//_M0: 20 A. D. ELMENDORF, I. KRIZ, AND J. P. MAY We let MR denote the category of R-modules. If R is an E1 ring spectrum, then an R-module is the same thing as a left mo* *dule over R regarded as an A1 ring spectrum, exactly as in algebra. From here, we c* *an mimic vast areas of algebra, one particularly striking direction being the deve* *lopment of topological Hochschild homology. However, we shall concentrate on the genera* *lized analog of stable homotopy theory that we obtain by studying the homotopy theory of R-modules for a fixed E1 ring spectrum R. Everything that makes sense is al* *so true for A1 ring spectra and their left and right modules. The following obser* *vation is exactly the same as for S-modules in Theorem 3.2. Theorem 5.3. The category of R-modules is complete and cocomplete, with both limits and colimits created in the underlying category S . If X is a based spac* *e and M is an R-module, then M ^ X and F (X; M) are R-modules, and the spectrum level fibre and cofibre of a map of R-modules are R-modules. Here we have the following complement. Proposition 5.4. Let M and N be R-modules and let K be an S-module. Then M ^S K and FS(K; N) are R-modules such that MR(M ^S K; N) ~=MR(M; FS(K; N)); and FS(M; K) is also an R-module. A homotopy in the category of R-modules is a map M ^ I+ - ! M0. A map of R-modules is a weak equivalence if it is a weak equivalence as a map of spectra* *. The derived category DR is constructed from the homotopy category hMR by adjoining formal inverses to the weak equivalences; again, the process is made rigorous b* *y the approximation of general R-modules by cell R-modules. Cell theory is based on the free R-module functor F from spectra to R-modules. It is the composite of the free S-module functor L and a free R-module functor MS -! MR, and, as usual, it is characterized by an adjunction MR(FE; M) ~=S (E; M): This formal property is vital, but, for calculational utility, it is also essen* *tial that F enjoys the following homotopical property. To ensure this, we must assume that * *R is tame and the unit j : S ! R is a cofibration of S-modules. Fortunately, the cyl* *inder functor K of Proposition 1.9 carries E1 ring spectra to weakly equivalent -cof* *ibrant E1 ring spectra, and we can also arrange the cofibration hypothesis without los* *s of generality. We do not assume that R has the homotopy type of a CW spectrum. MODERN FOUNDATIONS FOR STABLE HOMOTOPY THEORY 21 Proposition 5.5. If E is a CW spectrum, there is a natural weak homotopy equiva- lence from the internal smash product R^E to FE. If E is a wedge of sphere spec* *tra, then ss*(FE) is the free ss*(R)-module with one generator of degree n for each * *wedge summand Sn. Part of the point is that FS is only weakly equivalent to R, not equal to it.* * We think of the free R-modules FSn as "sphere R-modules". For cells, we note that * *the cone functor CE = E ^ I commutes with F, so that CFSn ~=FCSn. Thus, via the adjunction, maps out of sphere R-modules and their cones are induced by maps on the spectrum level. Using this, we can simply parrot the theory of cell spectr* *a in the context of R-modules, reducing proofs to the spectrum level via adjunction.* * We easily obtain the analogs of the Whitehead theorem and of the approximation by cell R-modules theorem, and DR is equivalent to the homotopy category of cell R- modules. If R is connective, but not otherwise, we obtain the cellular approxim* *ation theorem when we restrict attention to CW R-modules, namely cell R-modules such that cells are only attached to cells of lower dimension. The category DR has all homotopy limits and colimits; they are created as the corresponding constructions on the underlying diagrams of spectra. Thus we have enough information to quote the categorical form of Brown's representability th* *eorem given in [6]. Adams' analog [3] for functors defined only on finite CW spectra * *also applies in our context, with the same proof. Theorem 5.6 (Brown). A contravariant functor k : DR ! Setsis representable 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 5.7 (Adams). A contravariant group-valued functor k 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. In fact, Brown's theorem is the kind of formal result that can be derived in * *any (closed) model category in the sense of Quillen (see [8] for a good exposition)* *, and we have the following result. Serre fibrations of spectra are maps that satisf* *y the covering homotopy property with respect to the set of cone spectra n fifi o 1qCSn fi q 0 and n 0 : Relative cell R-modules M -! N are constructed exactly like cell R-modules, exc* *ept that one starts the inductive construction of N = [Nn with N0 = M. 22 A. D. ELMENDORF, I. KRIZ, AND J. P. MAY We write Cofibrations and Fibrations here to avoid confusion with cofibrations (HEP) and fibrations (CHP); the ambiguous use of the same term for both the classical and the model theoretic concepts is one of the banes of the literatur* *e. Theorem 5.8. The category of R-modules is a model category. Its weak equivalen* *ces and Fibrations are those maps of R-modules that are weak equivalences or Serre fibrations of spectra. Its Cofibrations are the retracts of relative cell R-mod* *ules. 6. The smash product of R-modules and function R-modules Continuing to work with our fixed E1 ring spectrum R, we mimic the definition of tensor products of modules over algebras. Definition 6.1.For R-modules M and N, 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. Then M ^R N has a canonical R-module structure induced from the R-module structure of M or, equivalently, N. Of course, S is an E1 ring spectrum and our new M ^S N coincides with our old M ^S N. The functor ^R preserves colimits in each of its variables, and sma* *sh products with spaces commute with ^R, in the sense that X ^ (M ^R N) ~=(X ^ M) ^R N and (M ^R N) ^ X ~=M ^R (N ^ X): Therefore the functor ^R commutes with cofibre sequences in each of its variabl* *es. We have analogous relations with smash products over S and an adjunction that c* *an be thought of as completing Proposition 5.4. Proposition 6.2. For an S-module K, K ^S (M ^R N) ~=(K ^S M) ^R N and (M ^R N) ^S K ~=M ^R (N ^S K): Moreover, MS(M ^R N; K) ~=MR(N; FS(M; K)): The associativity and commutativity of the smash product over S is inherited * *by the smash product over R. MODERN FOUNDATIONS FOR STABLE HOMOTOPY THEORY 23 Theorem 6.3. There is a natural commutativity isomorphism of R-modules o : M ^R N -! N ^R M: There is also a natural associativity isomorphism of R-modules (M ^R N) ^R P ~=M ^R (N ^R P ): The action : R ^S N -! N of an R-module N factors through a natural unit map of R-modules : R ^R N -! N. If N is a cell R-module, then : R ^R N ! N is a homotopy equivalence of spectra and thus a weak equivalence of R-modules. We can deduce not only formal but also homotopical properties of ^R from corr* *e- sponding properties of ^S. For example, the proof of the unit equivalence just * *stated reduces to the case of sphere R-modules, where the conclusion is a consequence * *of the following result. As in Section 4, we use an isomorphism of universes f : U U * *! U to define the internal smash product f*(E ^ F ). Proposition 6.4. Let E and F be spectra and let N be an R-module. There is a natural isomorphism of R-modules FE ^R N ~=LE ^S N: There is also a natural isomorphism of R-modules FE ^R FF ~=Ff*(E ^ F ): If M and N are cell R-modules, then M ^R N is a cell R-module with one (p + q)-* *cell for each p-cell of M and q-cell of N. Similarly, this proposition and Theorem 4.5 imply the following analog of the* * latter. Theorem 6.5. If M is a cell R-module and OE : N -! N0 is a weak equivalence of R-modules in St, then id^ROE : M ^R N -! M ^R N0 is a weak equivalence of R-modules. We construct ^R as a functor DR x DR ! DR by approximating the variables by cell R-modules, and we need only approximate one of the variables provided that* * the other is tame. That is, if N 2 St, then the derived smash product of M and N is represented by M ^R N. We have a function spectrum functor FR to go with our smash product. It is defined as the equalizer of a certain pair of maps FS(M; N) - ! FS(R ^S M; N). The details are dictated by the expected adjunction. Again, FR(M; N) inherits a structure of R-module from M or, equivalently, N. 24 A. D. ELMENDORF, I. KRIZ, AND J. P. MAY Proposition 6.6. For R-modules N and P and an S-module K, MR(K ^S N; P ) ~=MS(K; FR(N; P )): If M is an R-module, then MR(M ^R N; P ) ~=MR(M; FR(N; P )): Therefore FR(M ^R N; P ) ~=FR(M; FR(N; P )): Formal arguments from the adjunction, as in algebra, give a natural associati* *ve and unital composition pairing (6.7) ss : FR(N; P ) ^R FR(M; N) -! FR(M; P ): Parenthetically, we note that this gives rise to a host of examples of A1 ri* *ng spectra; in fact, R itself need only be an A1 ring spectrum in the following r* *esult. Proposition 6.8. For R-modules M and N, FR(M; M) is an A1 ring spectrum and FR(M; N) is an (FR(N; N); FR(M; M))-bimodule. Proposition 6.9. Let E be a spectrum and M be an R-module. There is a natural isomorphism of R-modules FR(FE; M) ~=FS(LE; M): The functor FR(M; N) converts colimits and cofibre sequences in M to limits a* *nd fibre sequences. It preserves fibre sequences in N. Using the previous result t* *o deal with sphere R-modules, we obtain the analog of Theorem 6.5. Proposition 6.10. 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 of R-modules. In the derived category DR, FR(M; N) means FR(M; N), where M is a cell approximation of M. The unit equivalence for the smash product implies its func* *tion module analog. Corollary 6.11. The adjoint N ! FR(R; N) of the unit map R ^R N ! N induces a natural isomorphism of functors on the derived category DR. Summarizing, we obtain the following derived category level conclusion. MODERN FOUNDATIONS FOR STABLE HOMOTOPY THEORY 25 Theorem 6.12. The derived category DR is symmetric monoidal under the product derived from ^R, and DR(M ^R N; P ) ~=DR(M; FR(N; P )): There is a formal theory of duality (explained in [16, Ch. III]) that now app* *lies to DR. We define the dual of an R-module M to be DRM = FR(M; R). We have an evaluation map " : DRM ^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 (6.13) : FR(L; M) ^R N -! FR(L; M ^R N): By composition with FR(id; ), specializes to a map (6.14) : DRM ^R M -! FR(M; M): We say that M is "finite", or "strongly dualizable", if it has a coevaluation m* *ap j : R -! M ^R DRM such that the following diagram commutes in DR: j R ________//_M ^R DRM (6.15) j|| |o| fflffl| fflffl| FR(M; M) oo___DRM ^R M: The definition has many purely formal implications. The map of (6.14) is an isomorphism in DR if either L or N is finite. The map of (6.15) is an isomorph* *ism if and only if M is finite, and the coevaluation map j is then the composite o-* *1j in (6.16). The natural map ae : M -! DRDRM is an isomorphism if M is finite. The natural map ^ : FR(M; N) ^R FR(M0; N0) -! FR(M ^R M0; N ^R N0) is an isomorphism if M and M0 are finite or if M is finite 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, when M is a finite cell R-module it does not follow that N has the homotopy type of a finite cell R-module. Theorem 6.16. A cell R-module is finite in the sense just defined if and only* * if it is a wedge summand up to homotopy of a finite cell R-module. The analogy with finitely generated projective modules in algebra should be c* *lear. 26 A. D. ELMENDORF, I. KRIZ, AND J. P. MAY 7. Tor and Ext in topology and algebra Still restricting for definiteness to an E1 ring spectrum R and its modules,* * we define Tor and Ext groups as the homotopy groups of derived smash product and function modules. Definition 7.1.For R-modules M and N, define TorRn(M; N) = ssn(M ^R N) and ExtnR(M; N) = ss-n(FR(M; N)): Note that TorR*(M; N) and Ext*R(M; N) are ss*(R)-modules. We emphasize that the smash product and function spectra are understood to be taken in the derived category DR. At this point in our exposition, we act as tr* *aditional topologists, taking it for granted that all spectra and modules are to be appro* *ximated as cell modules, without change of notation, whenever necessary. Various proper* *ties reminiscent of those of the classical Tor and Ext functors follow directly from* * the definition and the results of the previous sections. The intuition is that the * *definition gives an analogue of the differential Tor and Ext functors (alias hyperhomology* * and cohomology functors) in the context of differential graded modules over differe* *ntial graded algebras. In particular, the grading should not be thought of as the res* *olution grading of the classical torsion product, but rather as a total grading that su* *ms a resolution degree and an internal degree; this idea will be made precise by the* * grading of the spectral sequences that we shall describe for the calculation of these f* *unctors. Proposition 7.2. TorR*(M; N) satisfies the following properties. (i)If R, M, and N are connective, then TorRn(M; N) = 0 for n < 0. (ii)A cofibre 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. The commutativity and associativity relations for the smash product imply var* *ious further properties. We content ourselves with the following examples: TorR*(M; N) ~=TorR*(N; M) and TorR*(M ^R N; P ) ~=TorR*(M; N ^R P ): MODERN FOUNDATIONS FOR STABLE HOMOTOPY THEORY 27 Say that a spectrum N is coconnective if Nn = 0 for n > 0. Proposition 7.3. 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)Cofibre sequences N0 ! N ! N00and 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)): (iv)The functor Ext*R(?; N) carries wedges to products. Passing to homotopy groups from the pairings (6.7), we obtain the following f* *urther property. As usual, for a spectrum E, abbreviate En = ssn(E) = E-n : Proposition 7.4. There is a natural, associative, and unital system of pairings* * of R*-modules ss* : Ext*R(M; N) R* Ext*R(L; M) -! Ext*R(L; N): The formal duality theory of the previous section implies the following resul* *t, together with various other such isomorphisms. Proposition 7.5. For a finite cell R-module M and any R-module N, T orRn(DRM; N) ~=Ext-nR(M; N): Thinking of the derived category DR as a stable homotopy category, we may cha* *nge notations and reinterpret the functors Tor and Ext as prescribing homology and cohomology theories in this category. Definition 7.6.Let M and E be R-modules. Define ERn(M) = ssn(E ^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. 28 A. D. ELMENDORF, I. KRIZ, AND J. P. MAY Corollary 7.7. For a finite cell R-module M and any R-module E, ERn(DRM) ~=E-nR(M): Since the equivalence between the classical stable homotopy category and the * *de- rived category of S-modules preserves smash products and function spectra, we o* *btain all of the usual homology and cohomology theories by taking R = S. We also obtain the classical algebraic Tor and Ext groups as special cases, b* *y spe- cialization to Eilenberg-MacLane spectra. Thus let R be a discrete commutative * *ring for a moment. Recall that HR denotes a spectrum whose zeroth homotopy group is R and whose remaining homotopy groups are zero. It follows from multiplicative in* *finite loop space theory [20] that the Eilenberg-Mac Lane spectrum HR = K(R; 0) is an E1 ring spectrum. Analogously, if M is an R-module, then HM can be constructed as an HR-module. We shall see a quick and easy construction shortly. Granting t* *his, we have the following result. Theorem 7.8. For a discrete commutative ring R and R-modules M and N, TorR*(M; N) ~=TorHR*(HM; HN) and Ext*R(M; N) ~=Ext*HR(HM; HN): Under the second isomorphism, the topologically defined pairing Ext*HR(HM; HN) R Ext*HR(HL; HM) -! Ext*HR(HL; HN) coincides with the algebraic Yoneda product. The proof is clear enough: we just check the axioms for Tor and Ext. We can elaborate this result to an equivalence of derived categories. Recall * *from [28] or [15, Ch.III] that the derived category DR is obtained from the homotopy cate* *gory of chain complexes over R by formally inverting the quasi-isomorphisms, exactly* * as we obtained the category DHR from the homotopy category of HR-modules by inverting the weak equivalences. The algebraic theory of cell and CW chain complexes over R in the latter source makes the analogy precise and gives a treatment of tensor products and Hom functors in DR that exactly parallels our treatment of ^HR and FHR . The proof of the equivalence is quite easy. The category DHR is equivale* *nt to the homotopy category of CW HR-modules and cellular maps. It is a simple matter to see that CW HR-modules have associated chain complexes. This gives a functor DHR - ! DR. An inverse functor is obtained by applying Brown's representability theorem. Indeed, for a given chain complex X, the functor k on DHR specified * *by k(M) = DR(C*(M); X) satisfies the hypotheses of that result, and we let (X) represent this functor. Specialization to R-modules regarded as chain complexes MODERN FOUNDATIONS FOR STABLE HOMOTOPY THEORY 29 concentrated in degree zero gives the promised construction of Eilenberg-MacLane HR-modules from R-modules. Theorem 7.9. The cellular chain functor C* on HR-modules induces an equivalence of categories DHR - ! DR. The functor C* satisfies H*(C*(M)) ~=ss*(M) and carri* *es the functors ^HR and FHR to the functors R and Hom R. The inverse equivalence satisfies ss*((X)) ~=H*(X) and carries the functors R and Hom R to the functors ^HR and FHR . Proof.By construction, we have an adjunction DR(C*(M); X) ~=DHR (M; (X)); and one checks that its unit and counit are isomorphisms. The statements relati* *ng ^HR and FHR to R and Hom R are all consequences of the fact that 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): __|_ | 8. Universal coefficient and K"unneth spectral sequences Returning to our general E1 ring spectrum R, we find spectral sequences for * *the calculation of our Tor and Ext groups that are analogous to the Eilenberg-Moore (or hyperhomology) spectral sequences in differential homological algebra. Comp* *are [9, 13, 15]. They may be viewed as giving universal coefficient and K"unneth sp* *ectral sequences for homology and cohomology theories on R-modules, and they specialize to give such spectral sequences for homology and cohomology theories on spectra. Theorem 8.1. For R-modules M and N, there are natural spectral sequences of differential R*-modules E2p;q= TorR*p;q(M*; N*) =) TorRp+q(M; N) and Ep;q2= Extp;qR*(M*; N*) =) Extp+qR(M; N): Moreover, the pairing FR(M; N) ^R FR(L; M) ! FR(L; N) induces a pairing of spec- tral sequences that coincides with the algebraic Yoneda pairing Ext *;*R*(M*; N*) R* Ext*;*R*(L*; M*) -! Ext*;*R*(L*; N*) on the E2-level and that converges to the induced pairing of Ext groups. 30 A. D. ELMENDORF, I. KRIZ, AND J. P. MAY 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 it converges strongly. The Ext spectral se* *quence is of standard cohomological type, with dr : Ep;qr! Ep+r;q-r+1r: It lies in the right half plane. In the language of Boardman [5] (see also [12,* * App B]), it is conditionally convergent. It therefore converges strongly if, for each fi* *xed (p; q), there are only finitely many r such that dr is non-zero on Ep;qr. Setting M = FX in the two spectral sequences of Theorem 8.1, we obtain a universal coefficient spectral sequence. We have written the stars to indicate * *the way the grading is usually thought of in cohomology. Theorem 8.2 (Universal coefficient). For an R-module N and any spectrum 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 theorems. * *Re- placing N by FY and by FR(FY; R) in the two universal coefficient spectral sequ* *ences, we arrive at K"unneth spectral sequences. Theorem 8.3 (K"unneth). For any spectra X and Y , there are spectral sequences of the form TorR**;*(R*(X); R*(Y )) =) R*(X ^ Y ) and Ext*;*R*(R-*(X); R*(Y )) =) R*(X ^ Y ): Adams [1] first observed that one can derive K"unneth spectral sequences from universal coefficient spectral sequences, and he observed that, by duality, the* * four spectral sequences of Theorems 8.2 and 8.3 imply two more universal coefficient* * and two more K"unneth spectral sequences. He derived spectral sequences of this so* *rt under the hypothesis that his given ring spectrum E is the colimit of finite su* *bspec- tra 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 calculational hypothesis that requires case-by-case verification. * *It cov- ers some cases that are not covered by the results above, and conversely. The c* *ited MODERN FOUNDATIONS FOR STABLE HOMOTOPY THEORY 31 paper of Adams, and his later book [2], are prime sources for the first floweri* *ng of stable homotopy theory. While some of their foundational parts may be obsolete, their applications and calculational parts certainly are not. The following generalized K"unneth theorem admits a number of variants; see [* *11]. Theorem 8.4. Let E and R be E1 ring spectra and M and N be R-modules. As- sume that E or R is of the homotopy type of a CW spectrum. Then there is a spectral sequence of differential E*(R)-modules of the form Tor E*(R)p;q(E*(M); E*(N)) =) Ep+q(M ^R N): 9.Algebraic constructions in the derived category of R-modules If we replace the pair (S; R) by a pair (R; A) in Definition 5.1, we arrive a* *t the notion of an A1 or E1 algebra A over an E1 ring spectrum R. For example, the A1 ring spectra FR(M; M) of Proposition 6.8 are actually A1 R-algebras. Again,* * if A is an algebra over a discrete commutative ring R, then HA is an A1 HR-algebr* *a. Proceeding in this line, we can, for instance, construct R-modules whose homoto* *py groups realize the Hochschild homology of A with coefficients in (A; A)-bimodul* *es. However, we now proceed in a more homotopical direction, thinking of the deri* *ved category of R-modules as an analog of the stable homotopy category. From this p* *oint of view, we have the notion of an R-algebra up to homotopy, which is just like * *the classical notion of a ring spectrum in the stable homotopy category. Definition 9.1.An R-algebra 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 diag* *ram commutes in DR. j^id id^j R ^R AL ____//_A ^R Aoo__A ^R R LLL'LL |OE 'rrrrr LLL | rrror LL&&fflffl|xxrr A A is associative or commutative if the appropriate diagram commutes in DR. Lemma 9.2. If A and B are R-algebras, then so is A^R B. If A and B are associ* *ative or commutative, then so is A ^R B. By neglect of structure, an R-algebra A is a ring spectrum in the sense of cl* *assical 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: 32 A. D. ELMENDORF, I. KRIZ, AND J. P. MAY Similarly, for an R-algebra A, we have the evident homotopical notion of an A- module. These structures play a role in the study of DR analogous to the role played by ring spectra and their modules in classical stable homotopy theory. W* *hen R = S, S-algebras and their modules are equivalent to classical ring spectra an* *d their modules. We show in this section how to construct quotients M=IM and localizations M[Y -1] of modules over an E1 ring spectrum R and indicate in the next section when these constructions inherit a structure of R-algebra from an R-algebra str* *uc- ture on M. When specialized to MU, these results give highly structured versio* *ns of spectra that in the past were constructed by means of the Baas-Sullivan theo* *ry of manifolds with singularities or the Landweber exact functor theorem. At leas* *t at odd primes, the results give an entirely satisfactory, and very simple, treatme* *nt of algebra structures on the resulting MU-modules. We are interested in homotopy groups, and we make use of the isomorphisms (9.3) Mn = hS (Sn; M) ~=hMS(LSn; M) ~=hMR(FSn; M) to represent elements as maps of R-modules. For x 2 Rn, the composite map of R-modules (9.4) FSn ^R M x^id//_R ^R M____//_M is a module theoretic version of the map x. : nM - ! M, and we agree to write nM for FSn ^R M in this section. By Proposition 6.4, FSn ^R M is isomorphic as an R-module to LSn ^S M and, by Theorem 4.5, LSn ^S M is weakly equivalent as a spectrum to Sn ^ M. Therefore, the R-module nM is a model for the spectrum level suspension of M. Definition 9.5.Define M=xM to be the cofibre of the map (9.4) and let : 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 . Clearly we have a long exact sequence (9.6) . .-.! ssq-n(M)-x.!ssq(M)-*! ssq(M=xM) -! ssq-n-1(M) -! . .:. MODERN FOUNDATIONS FOR STABLE HOMOTOPY THEORY 33 If x is not a zero divisor for ss*(M), then * induces an isomorphism of R*-modu* *les (9.7) ss*(M)=x . ss*(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 (9.8) ss*(M)=(x1; : :;:xn)ss*(M) ~=ss*(M=(x1; : :;:xn)M); and similarly for a possibly infinite regular sequence X = {xi}. The following * *result implies that M=XM is independent of the ordering of the elements of the set X. * *We write R=X instead of R=XR. Lemma 9.9. For a set X of elements of R*, there is a natural weak equivalence (R=X) ^R M -! M=XM: In particular, for a finite set X = {x1; : :;:xn}, R=(x1; : :;:xn) ' (R=x1) ^R . .^.R(R=xn): If I denotes the ideal generated by X, then it is reasonable to define (9.10) M=IM = M=XM: However, this notation must be used with caution since, if we fail to restrict * *attention 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 modules are ob* *tained if we repeat a generator xi of I in our construction. We next construct localizations of R-modules at countable multiplicatively cl* *osed subsets Y of R*. Let {yi} be any cofinal sequence of Y , with yi2 Rni, so that * *every y 2 Y divides some yi. We may represent yi by an R-map FS0 -! FS-ni, which we also denote by yi. Let q0 = 0 and, inductively, qi= qi-1+ ni. The R-map yi^id -ni FS0 ^R M -! FS ^R M represents yi.. Smashing over R with FS-qi-1, we obtain a sequence of R-maps (9.11) FS-qi-1^R M -! FS-qi^R M: Definition 9.12. Define the localization of M at Y , denoted M[Y -1], to be the telescope of the sequence of maps (9.11). Since M ~= FS0 ^R M in DR, we may regard the inclusion of the initial stage FS0 ^R M of the telescope as a natura* *l map : M -! M[Y -1]. 34 A. D. ELMENDORF, I. KRIZ, AND J. P. MAY Since homotopy groups commute with localization, we see immediately that induces an isomorphism of R*-modules (9.13) ss*(M[Y -1]) ~=ss*(M)[Y -1]: As in Lemma 9.9, the localization of M is the smash product of M with the localization of R. Lemma 9.14. For a multiplicatively closed set Y of elements of R*, there is a* * natural equivalence R[Y -1] ^R M -! M[Y -1]: Moreover, R[Y -1] is independent of the ordering of the elements of Y . For se* *ts X and Y , R[(X [ Y )-1] is equivalent to the composite localization R[X-1][Y -1]. 10. Algebra structures on localizations and on quotients by ideals The behavior of localizations with respect to algebra structures is immediate. Proposition 10.1. Let Y be a multiplicatively closed set of elements of R*. If * *A is an R-algebra, then A[Y -1] is an R-algebra such that : A -! A[Y -1] is a map of R-algebras. If A is associative or commutative, then so is A[Y -1]. Proof.By Lemmas 9.2 and 9.14, it suffices to observe that R[Y -1] is an associa* *tive and commutative R-algebra with unit and product the equivalence R[Y -1] ^R R[Y -1] ' R[Y -1][Y -1] ' R[Y -1]: __|_ | This doesn't work for quotients since (R=X)=X is not equivalent to R=X. Howev* *er, we can analyze the problem by analyzing the deviation, and, by Lemma 9.9, we may as well work one element at a time. We have a necessary condition for R=x to be* * an R-algebra that is familiar from classical stable homotopy theory. Lemma 10.2. Let A be an R-algebra. If A=xA admits an R-algebra structure such that : A - ! A=xA is a map of R-algebras, then x : A=xA - ! A=xA is null homotopic as a map of R-modules. 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. To give a criterion for when R=x does have an R-algebra structure, we first note an easy formal lemma. Lemma 10.3. Let : R -! M be any map of R-modules. Then ( ^ id) O ' (id^) O : R -! M ^R M: MODERN FOUNDATIONS FOR STABLE HOMOTOPY THEORY 35 Theorem 10.4. Let x 2 Rm and assume that ssm+1 (R=x) = 0 and ss2m+1(R=x) = 0. Then R=x admits a structure of R-algebra with unit : R -! R=x. Therefore A=XA admits a structure of R-algebra such that : A -! A=XA is a map of R-algebras f* *or every R-algebra A and every sequence X of elements of R* such that ssm+1 (R=x) * *= 0 and ss2m+1(R=x) = 0 if x 2 X has degree m. Proof.Consider the following diagram in the derived category DR: (10.5) 2m+1R |x| fflffl| x m+1 R ________//_R l l | ll l || | x ^id vvll ss fflffl|x |fflffl m (R=x) ____//_R=x____//(R=x)o^Ro(R=x)____//___m+1o(R=x)_//_o___(R=x) OE oe |ss0| fflffl| 2m+2R: The map x is that specified by (9.4). The bottom row is the cofibre sequence th* *at results from the equivalence (R=x) ^R (R=x) ' (R=x)=x of Lemma 9.9, and the column is also a cofibre sequence. The composite x O is * *null homotopic since O x is null homotopic and the square commutes. Therefore there* * is a map such that ss O = , and is unique since ssm+1 (R=x) = 0. Since ss O O * *x = O x = 0, O x factors through a map 2m+1R -! R=x. Since ss2m+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 = . Now ss O oe O = ss O = , hence (ss O oe - id) = 0. Theref* *ore ss O oe - idfactors through a map 2m+2R -! m+1 (R=x). Again, such maps are null homotopic. Therefore ss O oe = id. Thus the bottom cofibre sequence splits (pro* *ving in passing that x : n(R=x) -! R=x is null homotopic, as it must be). A choice O* *E of a splitting gives a product on R=x. The unit condition OE O ( ^ id) = idis auto* *matic. To see that OE O (id^) = id, we observe that, by the lemma, (OE O (id^) - id) O = OE O (id^ - ^ id) O = 0: Therefore OE O (id^) - idfactors through a map m+1 R -! R=x. Again, such maps are null homotopic, hence OE O (id^) = id. This completes the proof that R=x is* * an R-algebra with unit . The rest follows from Lemmas 9.9 and 9.2. __|_ | 36 A. D. ELMENDORF, I. KRIZ, AND J. P. MAY The product on R=x can be described a little more concretely. The wedge sum (10.6) ( ^ id) _ oe : (R=x) _ m+1 (R=x) -! (R=x) ^R (R=x) is an equivalence. The product OE restricts to the identity on the first wedge * *summand and to the trivial map on the second wedge summand. Thus the product is determi* *ned by the choice of oe, and two choices of oe differ by a composite 0 (10.7) m+1 (R=x) __ss//_2m+2R ____//_(R=x) ^R (R=x): By the splitting (10.6) and the assumption that ssm+1 (R=x) = 0, we can view the second map as an element of ss2m+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. For an R-algebra A and an element x as in the theorem, we give A=xA ' (R=x)^R* * A the product induced by one of our constructed products on R=x and the given pro* *duct on A. We refer to any such product as a "canonical" product on A=xA. Observe that, by first using the product on A, the product on A=xA can be factored thro* *ugh 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 di- agram so constructed is a diagram of right A-modules via the product action of A on itself. This smashing with A can kill obstructions. Clearly, a map of A-modu* *les 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). This le* *ads to the following result. Theorem 10.8. Let x 2 Rm and assume that ssm+1 (R=x) = 0 and ss2m+1(R=x) = 0. Let A be an R-algebra and assume that ss2m+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 ss3m+3(A=xA) = 0, then A=xA is associative. Proof.The second arrow of (10.7) becomes zero after smashing with A since it is* * then given by an element of ss2m+2(A=xA) = 0. This proves the uniqueness statement. * *The commutativity statement follows since if OE is a canonical product on A=xA, the* *n so is OEo. The associativity statement requires consideration of the restriction * *of the iterated product to the wedge summands of A=xA ^R A=xA ^R A=xA. The details are similar to, but simpler than, those in the proof of Theorem 10.4. __|_ | Iterating and observing that passage to telescopes can kill obstructions, we * *arrive at the following fundamental conclusion. MODERN FOUNDATIONS FOR STABLE HOMOTOPY THEORY 37 Theorem 10.9. Assume that Ri = 0 if i is odd. 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-algebra, and it is commutative and associative. 11. The specialization to MU-modules and algebras The classical Thom spectra arise in nature as E1 ring spectra. In fact, it * *was inspection of their prespectrum level definition in terms of Grassmannians that* * first led to the theory of E1 ring spectra [19]. Of course, the homotopy groups of * *MU are concentrated in even degrees, and every non-zero element is a non zero divi* *sor. Thus the results above have the following immediate corollary. Theorem 11.1. Let X be a regular sequence in MU*, let I be the ideal generate* *d by X, and let Y be any sequence in MU*. Then there is an MU-algebra (MU=X)[Y -1] and a natural map of MU-algebras (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 un* *ique canonical product on (MU=X)[Y -1], and this product is commutative and associa- tive. In comparison with earlier constructions of this sort based on the Baas-Sulli* *van the- ory of manifolds with singularities or on Landweber's exact functor theorem (wh* *ere it applies), we have obtained a simpler proof of a substantially stronger resul* *t. We emphasize that an MU-algebra is a much richer structure than just a ring spectr* *um and that commutativity and associativity in the MU-algebra sense are much more stringent conditions than mere commutativity and associativity of the underlying ring spectrum. 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 [24], and we have the splitti* *ng maps 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*: 38 A. D. ELMENDORF, I. KRIZ, AND J. P. MAY 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-algebras and thus of ring spectra, a trivial diagram ch* *ase 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-algebra, commutative and associative if p > 2. The situation for p = 2 is interesting. We conclude f* *rom 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 field* * 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 10.1. If * *p > 2, they all have unique canonical commutative and associative MU-algebra structure* *s. Further study is needed when p = 2. In any case, this theory makes it unnecessa* *ry to appeal to Baas-Sullivan theory or to Landweber's exact functor theorem for t* *he construction and analysis of spectra such as these. With more sophisticated techniques, the second author [14] has proven that BP can be constructed as an E1 ring spectrum, and in fact admits uncountably many distinct E1 ring structures. There is much other ongoing work on the construct* *ion and application of new A1 and E1 ring spectra, by Hopkins, Miller, McClure, a* *nd others. The enriched multiplicative structures on rings and modules that we ha* *ve MODERN FOUNDATIONS FOR STABLE HOMOTOPY THEORY 39 discussed are rapidly becoming a standard tool in the study of periodicity phen* *omena in stable homotopy theory. References 1. J.F. Adams. Lectures on generalised cohomology. Springer Lecture Notes in Ma* *thematics Vol. 99, 1969, 1-138. 2. J.F. Adams. Stable homotopy and generalized homology. University of Chicago.* * 1974, 1994. 3. J.F. Adams. A variant of E.H.Brown's representability theorem. Topology, 10 * *(1971), 185-198. 4. J.M. Boardman. Stable homotopy theory. Thesis, Warwick 1964; mimeographed no* *tes from Warwick and Johns Hopkins Universities, 1965-1970. 5. J.M. Boardman. Conditionally convergent spectral sequences. Preprint. 1981. 6. E.H. Brown, Jr. Abstract homotopy theory. Trans. Amer. Math. Soc. 119 (1965)* *, 79-85. 7. R.R. Bruner, J.P. May, J.E. McClure, and M. Steinberger. H1 ring spectra and* * their applica- tions. Springer Lecture Notes in Mathematics Vol. 1176. 1986. 8. W.G. Dwyer and J. Spalinski. Homotopy theories and model categories. This vo* *lume. 9. S. Eilenberg and J.C. Moore. Homology and fibrations, I. Comm. Math. Helv. 4* *0 (1966), 199- 236. 10.A.D. Elmendorf, J.P.C. Greenlees, I. Kriz, and J.P. May. Commutative algebra* * in stable homo- topy theory and a completion theorem. Mathematical Research Letters 1 (1994)* *, 225-239. 11.A.D. Elmendorf, I. Kriz, and J.P. May. Rings, modules, and algebras in stabl* *e homotopy theory. In preparation. 12.J.P.C. Greenlees and J.P. May. Generalized Tate cohomology. Memoirs Amer. Ma* *th. Soc. To appear. 13.V.K.A.M. Gugenheim and J.P. May. On the theory and applications of different* *ial torsion products. Memoirs Amer. Math. Soc. 142, 1974. 14.I. Kriz. Towers of E1 ring spectra with an application to BP. Preprint. 1993. 15.I. Kriz, and J.P. May. Operads, algebras, modules, and motives. Asterisque. * *To appear. 16.L.G. Lewis, Jr., J.P. May, and M. Steinberger (with contributions by J.E. Mc* *Clure). Equivariant stable homotopy theory. Springer Lecture Notes in Mathematics Vol. 1213. 198* *6. 17.S. Mac Lane. Categories for the Working Mathematician. Springer-Verlag. 1971. 18.J.P. May. The Geometry of Iterated Loop Spaces. Springer Lecture Notes in Ma* *thematics Vol. 271. 1972. 19.J.P. May (with contributions by N. Ray, F. Quinn and J. Tornehave). E1 ring * *spaces and E1 ring spectra. Springer Lecture Notes in Mathematics Vol. 577. 1977. 20.J.P. May. Multiplicative infinite loop space theory. J. Pure and Applied Alg* *ebra, 26 (1982), 1-69. 21.J.P. May. Derived categories in algebra and topology. In the Proceedings of * *the Eleventh Inter- national Conference on Topology, Rendiconti dell Istituto Matematico dell Un* *iversita di Trieste. To appear. 22.J.P. May. Equivariant homotopy and cohomology theory. CBMS Lectures. In prep* *aration. 23.D.G. Quillen. Homotopical algebra. Springer Lecture Notes in Mathematics Vol* *. 43. 1967. 24.D.G. Quillen. On the formal group laws of unoriented and complex cobordism t* *heory. Bull. Amer. Math. Soc. 75 (1969), 1293-1298. 25.A. Robinson. Derived tensor products in stable homotopy theory. Topology 22 * *(1983), 1-18. 26.A. Robinson. Spectra of derived module homomorphisms. Math. Proc. Camb. Phil* *. Soc. 101 (1987), 249-257. 27.A. Robinson. The extraordinary derived category. Math. Z. 196(1987), 231-238. 40 A. D. ELMENDORF, I. KRIZ, AND J. P. MAY 28.J.L. Verdier. Categories derivees. Springer Lecture Notes in Mathematics Vol* *. 569, 1977, 262- 311. 29.G.W. Whitehead. Elements of homotopy theory. Springer-Verlag. 1978. Purdue University Calumet, Hammond, IN 46323 USA E-mail address: aelmendo@math.purdue.edu The University of Michigan, Ann Arbor, MI 48109-1003 USA E-mail address: ikriz@math.lsa.umich.edu The University of Chicago, Chicago, IL 60637 USA E-mail address: may@math.uchicago.edu