RATIONAL S1-EQUIVARIANT STABLE HOMOTOPY THEORY. J.P.C.Greenlees The author is grateful to the Nuffield Foundation for its support. Author addresses: School of Mathematics and Statistics, Hicks Building, Sheffield S3 7RH. UK. E-mail address: j.greenlees@sheffield.ac.uk Abstract.We make a systematic study of rational S1-equivariant cohomology t* *heories, or rather of their representing objects, rational S1-spectra. In Part I we construct a complete algebraic model for the homotopy catego* *ry of S1- spectra, reminiscent of the localization theorem. The model is of homologic* *al dimension one, and simple enough to allow practical calculations; in particular we ob* *tain a classifi- cation of rational S1-equivariant cohomology theories. In Part II we identify the algebraic counterparts of all the usual S1-spe* *ctra and con- structions on S1-spectra. This enables us in Part III to give a rational a* *nalysis of a number of interesting phenomena, such as the Atiyah-Hirzebruch spectral seq* *uence, the Segal conjecture, K-theory and topological cyclic cohomology. ii Contents Chapter 0. General Introduction. 1 0.1. Motivation 1 0.2. Overview 2 Part I The algebraic model of rational T-spectra. Chapter 1. Introduction to Part I. 13 1.1. Outline of the algebraic models. 13 1.2. Reading Guide. 17 1.3. Haeberly's example. 18 1.4. McClure's Chern character isomorphism for F-spaces. 18 Chapter 2. First steps. 23 2.1. Natural cells and basic cells 23 2.2. Separating isotropy types. 26 2.3. The single strand spectra E. 29 2.4. Operations. 32 Chapter 3. The Adams spectral sequence. 35 3.1. The Adams short exact sequence. 35 3.2. Construction of the Adams resolution. 36 3.3. Convergence of the Adams resolution. 38 3.4. Torsion Q[cH ]-modules and geometric implications. 39 Chapter 4. Categorical reprocessing. 43 4.1. Recollections about derived categories. 43 4.2. Split linear triangulated categories. 46 iii iv CONTENTS 4.3. The algebraicization of the category of T-spectra over H. 51 Chapter 5. Assembly and the standard model. 55 5.1. Assembly. 55 5.2. Global assembly. 57 5.3. The standard model category. 59 5.4. The algebraicization of rational T-spectra. 63 5.5. Notation. 66 Chapter 6. The torsion model. 71 6.1. Practical calculations. 71 6.2. The torsion model 72 6.3. Equivalence of derived categories of standard and torsion models. 75 Part II Algebraic counterparts to standard constructions. Chapter 7. Introduction to Part II 81 7.1. General outline. 81 7.2. Modelling of the smash product and the function spectrum. 84 7.3. Modelling functors changing equivariance. 86 7.4. Modelling Eilenberg-MacLane spectra and related objects. 88 7.5. Functors between split triangulated categories. 88 Chapter 8. Basic algebra for models and their derived categories. 91 8.1. Euler classes and F-finite torsion OF-modules. 91 8.2. Products in the torsion model. 94 8.3. The tensor hom adjunction. 97 8.4. Hom, tensor and torsion functors in standard models. 101 8.5. Hom, tensor and torsion functors on derived categories. 105 8.6. Products in the standard model. 107 Chapter 9. Products, smash products and function T-spectra. 109 9.1. Maps between injective spectra. 109 9.2. Models of smash products. 110 9.3. Models of function spectra. 112 Chapter 10. Induction, coinduction and geometric fixed points. 117 10.1. Forgetful, induction and coinduction functors. 117 CONTENTS v 10.2. Geometric fixed points. 119 Chapter 11. Algebraic inflation and deflation. 125 11.1. Algebraic inflation and Hausdorff OF-modules. 125 11.2. Algebraic inflation and deflation of OF-modules. 128 11.3. Inflation and its right adjoint on the torsion model category. 130 Chapter 12. Inflation, Lewis-May fixed points and quotients. 135 12.1. The topological inflation and Lewis-May fixed point functors. 135 12.2. Correspondence of Algebraic and geometric inflation maps. 139 12.3. A direct approach to the Lewis-May fixed point functor. 141 12.4. Lewis-May fixed points on objects in the standard model. 143 12.5. Quotient functors. 146 Chapter 13. Homotopy Mackey functors and related constructions. 149 13.1. The homotopy Mackey functor on A. 149 13.2. Eilenberg-MacLane spectra. 152 13.3. coMackey functors and spectra representing ordinary homology. 155 13.4. Brown-Comenetz spectra. 157 Part III Applications. Chapter 14. Introduction to Part III. 163 14.1. General Outline 163 14.2. Prospects and problems. 165 Chapter 15. Classical miscellany. 167 15.1. The collapse of the Atiyah-Hirzebruch spectral sequence. 167 15.2. Orbit category resolutions. 169 15.3. Suspension spectra. 171 15.4. K-theory revisited. 172 15.5. The geometric equivariant rational Segal conjecture for T. 175 Chapter 16. Cyclic and Tate cohomology. 179 16.1. Cyclic cohomology. 179 16.2. Rational Tate spectra. 180 16.3. The integral T-equivariant Tate spectrum for complex K-theory. 181 Chapter 17. Cyclotomic spectra and topological cyclic cohomology. 185 vi CONTENTS 17.1. Cyclotomic spectra. 185 17.2. Free loop spaces. 188 17.3. Topological cyclic cohomology of cyclotomic spectra. 190 Appendix A. Mackey functors. 195 Appendix B. Closed model categories. 201 Appendix C. Conventions. 207 C.1. Conventions for spaces and spectra. 207 C.2. Standing conventions. 209 Appendix D. Indices. 211 D.1. Index of definitions and terminology. 211 D.2. Index of notation. 215 Bibliography 225 CHAPTER 0 General Introduction. 0.1.Motivation Spaces with actions of the circle group T are of particular interest. Loops o* *ccur in many constructions, and it is often appropriate to take into account the action of t* *he circle by rotation; in particular the free loop space has been the object of much study. * *This in turn leads towards the use of the circle group in cyclic cohomology; the refinements* * of topological Hochschild homology and topological cyclic constructions are also important in * *algebraic K-theory. More prosaically the circle is simply the first infinite compact Lie * *group, and it plays a fundamental role in the understanding of all positive dimensional gr* *oups. For any one of these reasons it is important to understand equivariant cohomology t* *heories for spaces with circle action. To obtain a reasonably broad and simple picture, we consider the case of rati* *onal co- homology theories; these have been considered before for special classes of spa* *ces (see for example [5]), but this appears to be the first attempt to obtain a complete* * algebraic picture. In any case, the understanding of the rational case is a necessary fi* *rst step to- wards a general understanding of T-equivariant cohomology theories. It is well * *known [14] that, for finite groups, all cohomology theories are products of ordinary cohom* *ology the- ories, but this is false for the circle group. A test case of particular inter* *est is rational topological K-theory. The example of J.-P.Haeberly [16] shows that, by contrast* * with the case of finite groups of equivariance, there is no Chern character isomorphism.* * It follows that T-equivariantly some topological structure remains, even after rationaliza* *tion. The author began the present work to understand the T-equivariant Chern character, * *the T- equivariant Segal conjecture, the Tate construction on T-equivariant K-theory a* *nd several other T-equivariant rational objects that had come to light. The list of conten* *ts contains a list of examples treated here. From now on we let T denote the circle group. We only consider closed subgrou* *ps, and the letters H; K and L will denote finite subgroups. The family of all finite s* *ubgroups will be denoted by F. We work rationally throughout, without displaying this in the * *notation; for example Sn denotes the rationalized n-sphere. 1 2 0. GENERAL INTRODUCTION. 0.2. Overview Equivariant cohomology theories are represented by equivariant spectra, and w* *e shall conduct most of the investigation at the represented level. This gives more pre* *cise infor- mation both about individual theories and about natural transformations between* * them; indeed, the only loss is any geometric interpretation of the cohomology theory * *concerned, which is inevitable in any general study. It is important to be explicit that w* *e only consider cohomology theories which admit suspension isomorphisms for arbitrary represent* *ations; these are sometimes known as `genuine' or `RO(G)-graded' cohomology theories. * *The corresponding representing objects are G-spectra. For these too there are adje* *ctives to emphasize the type of spectra concerned: they are `genuine' G-spectra or G-spec* *tra `in- dexed on a complete G-universe'. Since these cohomology theories and these G-s* *pectra form the most natural classes to consider, we shall not use these adjectives un* *less required for emphasis. As made clear by the title, we consider the circle group G = T. Before summarizing our results we begin by putting the circle group into cont* *ext. In fact the circle stands at a watershed: for finite groups of equivariance ration* *al cohomology theories may be analysed completely, and any group more complicated than the ci* *rcle is substantially harder to understand. The main problem in analyzing spectra is to choose basic objects which are ea* *sy to work with and which give theorems of practical use. It is natural to be guided by on* *e's favourite algebraic invariant, and this suggests analysis in terms of Moore spectra or Ei* *lenberg- MacLane spectra. For finite groups of equivariance both approaches work well, a* *nd one may analyse rational spectra completely. There are two reasons for this: firstl* *y the group has no topology, and secondly the classifying space has no rational cohomology.* * The first fact means the category of Mackey functors is very simple, and the second means* * that the classes of Eilenberg-MacLane spectra, of Moore spectra and of Brown-Comenetz sp* *ectra coincide, so that all their characteristic properties can be used at once. Both* * simplifying factors fail for infinite groups, and the three basic classes are distinct. Th* *is means that different methods must be used: in essence we base our analysis for the circle * *group on a slightly embellished version of equivariant homotopy with its primary operati* *ons. The reason such a simple invariant suffices is that the rank of the circle group is* * one. In general the injective dimension of the category of rational Mackey functors and the Kru* *ll dimension of the cohomology of its classifying space are both equal to the rank of the gr* *oup. When the rank is one there is no room for extension problems, and some hope of a sim* *ple answer. However, even for the group O(2), it is necessary to take into account a topolo* *gy on the space of subgroups, and to work with sheaves: it is no longer possible to treat* * different conjugacy classes of subgroups entirely separately. This explains why it is wor* *thwhile to treat the single case of the circle in such detail. The work is broken into three parts: Part I in which we construct the algebra* *ic models for various classes of T-spectra, Part II in which we identify the algebraic co* *unterparts of various general constructions, and Part III in which we consider several classe* *s of examples of particular interest. Each part has a detailed introduction of its own, but * *we give a 0.2. OVERVIEW 3 general outline here. Part I begins by discussing K-theory. On the one hand, we give Haeberly's ex* *ample showing that K-theory cannot be described simply using ordinary cohomology. On * *the other hand, we give a generalization of McClure's result that the K-theory Atiyah-Hir* *zebruch spectral sequence collapses for F-free spaces. This suggests the necessity of * *the present work and that it is practical. We then turn to the main business of constructin* *g a model: in this introduction we describe the model in an aesthetically satisfying way, * *but do not attempt to explain the proof that it is a model. The introduction to Part I giv* *es a different approach to the model which does suggest the proof. We would prefer to achieve * *these two ideals simultaneously. To motivate the form of the model, one should recall the classical localizati* *on theorem for semifree T-spaces. This states that if X is a finite space which only has i* *sotropy groups T and 1, then the inclusion of the fixed point space XT -! X induces an isomorp* *hism in Borel cohomology once the Euler classes E = {1; c1; c21; : :}:are inverted: ~= -1* T -1 * * T E-1H*(ET+ ^T X) -! E H (ET+ ^T X ) = E H (BT+) H (X ): We conclude that N = H*(ET+ ^T X), regarded as a module over Q[c1] = H*(BT+) is very nearly enough to identify the homology of the fixed point space XT, but we* * need to pick out a vector subspace V = H*(XT) of E-1N which is a basis in the sense* * that E-1N ~=E-1H*(BT+) V . In particular, if X is free then N is E-torsion. Now T-equivariant cohomology theories are represented by T-spectra, and the l* *ocaliza- tion theorem suggests a model which turns out to be a complete invariant. To de* *scribe it, we first note that there is a natural homotopy-level analogue of the set of iso* *tropy groups which occur. This uses the geometric K-fixed point functor X 7-! K X, which is * *the func- tor extending the K-fixed point functor on spaces, in the sense that K (1 Y ) =* * 1 (Y K); it also enjoys similar formal properties to the space-level functor. We then de* *fine the set of isotropy groups of a spectrum X to be the set of subgroups K for which the g* *eometric fixed point spectra K X are non-equivariantly essential. This gives the notion * *of a free T-spectrum (alternatively characterized as a T-spectrum X for which ET+ ^ X -! * *X is an equivalence). We therefore suppose given a collection H of finite subgroups * *of T, and we may consider the class of H-free spectra (i.e. those with isotropy in H), an* *d the class of H-semifree spectra (i.e. those with isotropy in H [ {T}). The reader should * *concentrate on the case H = {1}, which gives the usual classes of free and semifree spectra* *, and on the case H = F: the class of F-semifree spectra is the class of all T-spectra. * *However the additional generality makes the picture clearer, and the two special cases are * *representative of the two classes of examples: those with H finite, and those with H infinite.* * Analagous to the ring H*(BT+) we have the ring of operations OH = C(H; Q)[c]; where C(H; Q) denotes the Q-valued functions on the discrete set H, and c is of* * degree -2. The notation is chosen to suggest that OH is a ring of functions on H. This* * ring is Noetherian if H is finite and not otherwise. We let eH 2 C(H; Q) = (OH)0 denot* *e the idempotent with support H 2 H, and we let cH = eH c. Next we need the set E = * *EH of Euler classes. If H = {1} this is simply the multiplicative subset {1; c1; c* *21; : :}:of OH 4 0. GENERAL INTRODUCTION. used for the localization theorem above, but in general it needs a little more * *explanation. For any finite subset OE H we have an associated idempotent eOE2 OH, and we ha* *ve an Euler class cOE= eOEc + (1 - eOE), which is not a homogeneous element of OH. Th* *e effect of cOEon an OH-module M = eOEM (1 - eOE)M is to multiply by c on the first fac* *tor and do nothing to the second: thus the result of inverting cOEon M is again a grade* *d module: eOEM[c-1] (1 - eOE)M. Thus our second ingredient is the set EH = {ckOE| OE H finite,k 0} of Euler classes. The category modelling semifree H-spectra is then the categor* *y AH of OH-modules N with a specified graded vector space V to act as a basis of E-1N. * * It is convenient to package this as saying that we are given a basing map fi : N -! (E-1OH) V which becomes an isomorphism when E is inverted. This makes clear that a morphi* *sm in AH is a diagram M - ! N ff # # fi 1OE -1 (E-1OH) U - ! (E OH) V: We refer to N as the nub and V as the vertex. We also refer to an OH-module wit* *h specified basing map as a based OH-module, and to a morphism : M -! N for which there is a compatible map OE as a based map. Note that if H is a singleton the existenc* *e of a basing isomorphism E-1N ~=E-1OH V for some V is automatic, but in general it p* *uts a restriction on the modules N. The connection with topology arises since OF = [EF+; EF+]T*, and hence this a* *cts on both ssT*(EF+ ^ X) and ssT*(DEF+ ^ "EF ^ X) for any X; if X is H-semifree this * *action factors through the projection OF -! OH. Furthermore, since c is of negative d* *egree and any element of ssT*(EF+ ^ X) is supported on a finite subspectrum, one sees* * that E -1ssT*(EF+ ^ X) = 0. Next, we have a map DEF+ ^ X -! DEF+ ^ "EF ^ X ' DEF+ ^ "EF ^ T X with cofibre DEF+ ^ EF+ ^ X ' EF+ ^ X. Since the homotopy of the cofibre is Euler-torsion, its homotopy i * * j ssA*(X) := ssT*(DEF+ ^ X) -! ssT*(DEF+ ^ "EF ^ T X) = ssT*(DEF+ ^ "EF) ss*(T * * X) is therefore an object of AH . Now we may state our main classification theorem. Classification Theorem: For any collection H of finite subgroups of the circle * *T, the above invariant induces bijections (i) {H-free rational spectra}= ' ! {Euler-torsion OH-modules}= ~= 0.2. OVERVIEW 5 where ' denotes homotopy equivalence, and ~=denotes isomorphism, and (ii) {H-semifree rational spectra}= ' ! {based OH-modules}= ~= where ' denotes homotopy equivalence, and ~=denotes isomorphism. In particular* *, ra- tional T-equivariant cohomology theories are in bijective correspondence to iso* *morphism classes of based OF-modules. In practice this is derived as a corollary of a theorem identifying the categ* *ories of spectra in algebraic terms. More precisely, recall that the derived category of a grade* *d abelian cat- egory is the category of differential graded objects with homology isomorphisms* * inverted, although for practical purposes a more concrete construction is essential. The* * theorem identifies the categories of spectra as the derived category of the associated * *algebraic cate- gory: H-free T-spectra' D(Euler torsion OH-modules) and H-semifree T-spectra' D(based OH-modules): Furthermore, cofibre sequences of spectra correspond to triangles under these e* *quivalences. The point here is that both algebraic categories turn out to be abelian and one* * dimensional, so that morphisms in the derived category can be calculated from a short exact * *sequence involving Hom and Ext in the abelian category. It is sometimes more practical to identify the place of a spectrum X in the c* *lassification by a different route. This amounts to identifying first EF+ ^ X and T X, and th* *en the map qX : "EF ^ T X = "EF ^ X -! EF+ ^ X of which X is the fibre. It is not enough to identify the effect of qX in homo* *topy: one must also take into account the twisting given by representations, and in gener* *al this requires both primary and secondary information. Nonetheless, there is a secon* *d model for semifree H-spectra based on this approach, which we call the torsion model.* * We show it is equivalent to the standard model described above, and it is often the eas* *iest route to placing a spectrum in the classification. There are really three stages to the proof of these theorems. Firstly one sho* *ws, using idempotents in the Burnside rings of finite subgroups, that for F-free spectra * *it is essentially enough to deal with the case of free spectra. Next, one constructs an Adams sp* *ectral sequence for free spectra, which collapses to a short exact sequence and gives * *a means of calculation. Because of the particularly simple algebraic behaviour of O1 = Q[c* *1] this is enough to identify the entire triangulated category. The final stage is to take* * this work and process it: this stage is essentially formal. Once we have algebraic models for various categories of spectra we naturally * *want to understand familiar topological constructions in algebraic terms. This is the b* *usiness of Part II. We have followed the order suggested by logic, and therefore begin by * *studying the smash product and function spectrum constructions, and then go on to functors c* *hanging equivariance. Unfortunately the smash product and function spectrum are by far* * the 6 0. GENERAL INTRODUCTION. most complicated examples, and require more algebraic machinery than any of the* * other examples we consider. Furthermore, their complexity means that we are not able * *to show that our description is functorial, and our approach is necessarily indirect. T* *his highlights a shortcoming of our method: the correct proof of our results would follow tha* *t used by Quillen in modelling rational homotopy of simply connected spaces. The func* *torial identification of smash products and function spectra would then be automatic. * *At present, such a proof is not accessible, but the present results strongly suggest that s* *uch a proof exists. In any case, the model of the smash product is essentially the left der* *ived tensor product, and the model of function spectra is its right adjoint. There are two * *warnings here: in the categories of H-free spectra, there are not enough flat objects, so the * *left derived tensor product must be calculated in a larger category; it results in an Euler-* *torsion object since it coincides with the suspension of the right derived torsion product. Wi* *th this caveat, if the spectra X and Y are modelled by M and N respectively then X ^ Y is modelled byM L N: There is also a caveat for function objects, which we now explain. It is conv* *enient in both cases to consider the larger algebraic category in which no condition is p* *laced on the behaviour of Euler classes. For H-free spectra this is the category of all OH-m* *odules, and for H-semifree spectra it is the category of all maps N -! E-1OH V . It turns * *out that the internal Hom functor in the abelian category is the composite functor Hom * *(M; N), where Hom (M; N) is an object in the category with no condition on behaviour u* *nder inversion of Euler classes, and where is the right adjoint to the inclusion of* * the smaller category. For example, in the case of H-free spectra Hom (M; N) is simply the O* *H-module of OH-morphisms, and for an arbitrary OH-module M0, the Euler-torsion module M * *is defined to be the kernel of M -! E-1M. In the semifree case both functors are h* *arder to describe, and we refer the reader to Chapter 8. It turns out that the right * *adjoint of M 7-! M L N is not the right derived functor of P 7-! Hom (N; P ), but rather * *it is P 7-! R RHom (N; P ). Thus if the spectra Y and Z are modelled by N and P , th* *en The internal function spectrum of maps from Y to Z is modelledRby RHom (N; P * *): An essential step in identifying the function spectrum on objects is to give a * *functorial identification of the product. In these terms we may say that if Xiis modelled * *by Mithen Y The internal product of the spectra Xiis modelledRby Mi; i and this model is functorial. The other topological functors we consider can be modelled functorially, and * *we shall discuss only the full category of T-spectra. The forgetful functor and its lef* *t and right adjoints, induction and coinduction, are straightforward. Similarly the geomet* *ric fixed point functor X 7-! T X is the passage-to-vertex functor given as part of the s* *tructure. The first interesting functor is the geometric fixed point functor K : T - sp* *ectra -! T=K - spectra for a finite subgroup K. This turns out to be easy to describe: w* *e simply let e 2 C(F; Q) denote the idempotent supported on the set [ K] of subgroups co* *ntaining K. The algebraic model of_K_ is multiplication by_e; this make sense since eOF * *is naturally identified with the ring O __Fof operations for T = T=K. As usual, the Lewis-M* *ay fixed 0.2. OVERVIEW 7 point functor K : T - spectra -! T=K - spectra (the spectrum K X is written XK * *in [18]) is much harder to understand, and we only describe its behaviour here for* * F-free and F-contractible spectra, referring the reader to Chapters 11 and 12 for details_* *of how these are spliced. On F-contractible spectra X ' "EF ^ T X, we have K (X) = "EF^ T X, so this is easy. We have seen that an F-spectrum X is modelled by an Euler-tors* *ion OF- module N; from the form of Euler classes it follows that this is equivalent to * *specifying the function [N] : F-! torsionQ[c] - modules H 7-! eH N: The Lewis-May fixed point functor groups these modules together according to th* *e be- haviour of the subgroup on passage_to quotient. More precisely,_we observe that* * passage to quotient q : T -! T=K = T defines a map q* : F -! F on finite subgroups. If * *the function [N] models the F-free spectrum X then the function [K N] modelling K X* * is the map __ F_ -! torsionQ[c]L- modules H 7-! q*(H)=__H[N](H): A little thought shows that it is not a trivial matter to see how the F-free an* *d F- contractible parts should be spliced together. Because the Lewis-May fixed poin* *t functor is so complicated, we actually approach it via its left adjoint, the inflation * *map infTT=K: T=K - spectra -! T - spectra. This is the functor given by regarding a T=K spec* *trum as a T-spectrum by pullback along the quotient, and then building in representa* *tions (it is written q# in [18], but more commonly i* by abuse of notation; we shall stick t* *o the more descriptive notation)._From our description_of Lewis-May_fixed points it is eas* *y_to_deduce inflation on F-contractible_and F-free specta. On F-contractible_spectra Y ' "E* *F^ TY we have infTT=KY = "EF ^ TY . If [P ] is the model of the F-spectrum Y then the* * model [infTT=KP ] of infTT=KY is the composite q* __[P] F -! F -! torsionQ[c] - modules: In cases where N is Euler-torsion, the right adjoint of the inflation map is al* *so its left adjoint; it therefore also gives a model for the topological quotient when X is* * K-free. The final chapter of Part II turns to ordinary cohomology and its variants. A* *fter Eilen- berg and Steenrod we define a cohomology theory to be ordinary if its coefficie* *nts are non- zero only in degree 0, and similarly in homology. For each integer q, an equiva* *riant coho- mology theory FG*(.) specifies a contravariant additive functor G=H+ 7-! FGq(G=* *H+) = FHq on the stable category of orbits; such a functor is called a Mackey functor. As* * in the clas- sical case, ordinary cohomology theories are classified by their non-zero Macke* *y functor M in degree 0, and we write H*G(.; M) for this theory and HM for its representing* * spectrum. Similarly, for each integer q a homology theory F*G(.) defines a covariant addi* *tive functor G=H+ 7-! FqG(G=H+) on the stable category of orbits; such a functor is called a* * coMackey functor. Ordinary homology theories are classified by their associated coMackey* * functors N, and we write HG*(.; N) for this functor and JN for the representing spectrum* *. For finite groups G the stable orbit category is self-dual, so that a coMackey func* *tor can also be viewed as a Mackey functor; in this case the ordinary homology theory classi* *fied by a 8 0. GENERAL INTRODUCTION. Mackey functor M is also represented by HM. However, for positive dimensional g* *roups such as the circle, the functor given by a homology theory cannot usually be vi* *ewed as a Mackey functor. Our first task is to identify objects of the form HM and JN in our model; we * *find that they are well behaved but by no means trivial. Finally, whenever one has an in* *jective Mackey functor I one may consider the cohomology theory defined by Brown-Comene* *ntz I-duality hIqG(X) = Hom(ss_Gq(X); I); and its representing spectrum hI. Again, in the case of a finite group all rati* *onal Mackey functors are injective, and HM = JM = hM. Indeed, this is the basis of a simple* * proof that all rational cohomology theories are ordinary for finite groups. However, * *for the circle group the spectrum hI is rather complicated, and in particular it is unbounded;* * we identify it exactly in our model. In Part III we apply the general theory of Parts I and II to several examples* * of particular interest. First we answer a number of obvious general questions. To begin with,* * we relate the model we have used to the use of Postnikov towers and the use of cells. In * *fact, we can understand the Atiyah-Hirzebruch spectral sequence H*T(X; K_*T) =) K*T(X) f* *or F-free spectra X completely, in terms of our model. It collapses at the E2 page if an* *d only if KT*(EF+) is injective over OF. The latter condition holds for complex K-theory* *, so we recover McClure's theorem that the Atiyah-Hirzebruch spectral sequence for the * *rational K-theory of an F-space collapses at E2. However, in general there are arbitrar* *ily long differentials. The contrast with the simplicity of the one dimensional nature o* *f the category of Euler-torsion OF-modules suggests that the Postnikov tower is a poor way to * *study T- spectra. On the other hand, because of the simplicity of the graded maps betwee* *n cells, we can contemplate homological algebra over it, and it is easy to construct a c* *onvergent spectral sequence based on cellular resolutions with a calculable E2 term. Unfo* *rtunately the spectral sequence does not appear to be useful in general. We do not have the means to detect purely unstable phenomena, but the splitti* *ng the- orem of Segal and tom Dieck shows that suspension spectra of T-spaces are very * *special, and we briefly comment on the implications of this for their algebraic model. Finally we return to complex K-theory and identify its algebraic model. It i* *s simple to describe in terms of representation theory, and is well behaved algebraicall* *y (`formal' in the torsion model). However there remain many interesting questions that we* * have not treated. Firstly, a qualitative comparison of the F-spectrum Euler classes * *and the K- theory Euler classes is sufficient for our purpose, but an exact comparison usi* *ng the Chern character, along the lines of Crabb's work [5], would be illuminating. Secondly* *, it would be interesting to compare our model with that of Brylinski [3]. Presumably these q* *uestions would be useful preparation for the more substantial project of modelling T-equ* *ivariant elliptic cohomology as constructed by Grojnowski [8] and Ginzburg-Kapranov-Vase* *rrot [6]. The other motivating problem was that of understanding the T-equivariant anal* *ogue of the Segal conjecture. We had the ironic situation that we understood the harder* * profinite part by virtue of work on the Segal conjecture for finite groups, whilst we cou* *ld not un- derstand the rational part. Using the model described here, it is now an easy e* *xercise to 0.2. OVERVIEW 9 identify DET+ in the torsion model as the composite E-1OF Q E-1OF -! E-1OF -! E-1OF=OF -! Q[c1; c-11]=Q[c1] where the first map is the product. It is quite instructive to view this as a s* *pecial case of the identification of the function spectrum. Turning to more specialised examples, we reach Tate cohomology theories in th* *e sense of [14]. This construction on T-spectra corresponds precisely to Tate cohomology i* *n commu- tative algebra in the sense of [10]. Perhaps more interesting is our study of t* *he integral Tate spectrum of complex equivariant K-theory. We are able to identify the exact ho* *motopy types of both t(KZ) ^ EF+ and t(KZ) ^ "EF and the map q of which t(KZ) is the f* *ibre: the first is rational, and identified using our general theory, and the second * *is formed from K-theory with suitable coefficients by inflating and smashing with "EF. Finally we turn to examples gaining their importance from algebraic K-theory.* * The mo- tivation for the notion of a cyclotomic spectrum comes from the free loop space* * X = map(T; X) on a T-fixed space X. This has the property that if we take K-fixed p* *oints we obtain the T=K-space map(T=K; X), and if we identify the circle T with the circ* *le T=K by the |K|th root isomorphism we recover X. For spectra one also needs to worry* * about the indexing universe, but a cyclotomic spectrum is basically one whose geometr* *ic fixed point spectrum K X, regarded as a T-spectrum, is the original T-spectrum X. Aft* *er the suspension spectrum of a free loop space, the principal example comes from the * *topological Hochschild homology of T HH(F ) of a functor F with smash products. Given such * *a cyclo- tomic spectrum X one may construct the topological cyclic spectrum T C(X) of B"* *okstedt- Hsiang-Madsen [2], which is a non-equivariant spectrum. An intermediate constru* *ction of some interest is the T-spectrum T R(X). Although these constructions are princi* *pally of interest profinitely, it is instructive to identify the cyclotomic spectra in o* *ur model and fol- low the constructions through. In fact we show that cyclotomic spectra, are tho* *se spectra X so that the function [N] : F -! torsionQ[c] - modules modelling EF+ ^ X is co* *nstant, and so that the structure map E-1OF V -! N commutes with any translation of the finite subgroups. It therefore factors through E-1OF V -! (E-1OF)=OF V , and * *the map (E-1OF)=OF V -! N is a direct sum of copies of Q[c; c-1]=Q[c] V -! [N](1). Furthermore, we may recover Goodwillie's theorem that for any cyclotomic spectr* *um X we have T C(X) = XhT: topological cyclic cohomology coincides with cyclic cohom* *ology in the rational setting. This summarises the contents of the body. There are also a number of appendic* *es. Ap- pendix A gives the structure of rational Mackey functors, and is of independent* * interest: in particular the category is of projective and injective dimension 1. Appendi* *x B gives Quillen closed model category structure on the algebraic categories. Finally we* * suggest the reader glance at Appendix C summarising our conventions. There are also a numb* *er of indices. It is appropriate to comment briefly on reading this document. Formally, Part* * I is the basis of all that follows, and is cumulative. Part II consists of an introduct* *ory chapter, followed by the treatment of four classes of examples. Since it gives algebraic* * models of 10 0. GENERAL INTRODUCTION. topological constructions it must therefore develop the relevant algebra before* * comparing it to topology. Thus Chapters 8 and 11 are purely algebraic, and are prerequis* *ites for Chapters 9 and 12 respectively. Otherwise the chapters are independent of each* * other, but the geometric results depend on Part I. Finally, the chapters of Part III * *are again independent, and depend only on Part I and the appropriate results from Part II* *. We have made some effort to ensure it is possible for the trusting reader to read a par* *t without previously reading its predecessors. We expect there will be those only interested in Chapters 1 to 3. There may * *also be those wanting to gain a feel for the behaviour of certain functors, who may fin* *d Part II worthwhile, even without reading Part I. Finally, there may be those who want t* *o begin with Part III and read earlier chapters as necessary. The author is grateful to the Nuffield Foundation for its support, to the Uni* *versities of Georgia (Athens) and Chicago for their hospitality, and to the towns of Karl* *sruhe and Worms. The author also thanks L.Hesselholt, J.P.May and N.P.Strickland for usef* *ul com- ments and conversations. Part I The algebraic model of rational T-spectra. 12 CHAPTER 1 Introduction to Part I. This chapter motivates Part I and provides a map for it. In Section 1.1 we exp* *lain the strategy used in Part I to analyse the category of rational T-spectra, and in S* *ection 1.2 give a brief guide to help readers with particular interests. This is followed * *in Sections 1.3 and 1.4 by accounts of Haeberly's example and a generalization of McClure's the* *orem: this is designed to show there is a need for analysis and some hope of achieving it. 1.1.Outline of the algebraic models. The main business of Part I is to construct a complete algebraic model of the* * category of rational T-spectra. Since spectra represent cohomology theories, this gives * *a complete algebraic classification of rational T-equivariant cohomology theories. Having* * given the overview in the General Introduction, we concentrate here on the practical appr* *oach. In fact, we lead the reader through the investigative process to the algebraic mod* *el of T- spectra. This should help explain the how geometric information is packaged in * *the model, and how the algebraic model can be used. The main problem in analyzing T-spectra is to choose basic objects which are * *easy to work with and which give theorems of practical use. We explained in the introdu* *ction that the building blocks familiar from finite groups of equivariance are not suitabl* *e: Eilenberg- MacLane spectra, Moore spectra and Brown-Comenetz spectra form distinct classes* *. This means that different methods must be used. The redeeming feature is that there is no complication at all from representa* *tion theory since the Weyl groups are all connected. This means we can return to geometric * *intuition and concentrate on isotropy groups. It is appropriate for our present purpose t* *o think of T-spectra as generalized stable spaces. It is standard practice in transformat* *ion groups to consider various fixed point spaces XH of a space X. In particular, spaces w* *ith a free action are especially approachable. One reason for this is that only one subgro* *up occurs as an isotropy group. In the rational case the behaviour at each finite subgroup i* *s reasonably similar and reasonably simple. Therefore it is common to consider spaces X all * *of whose isotropy groups are finite. These are variously called F-spaces, F-free spaces,* * almost free spaces, or spaces without fixed points. We shall call them F-spaces, and concen* *trate on the fact that they are equivalent to spaces constructed from cells G=H x En wit* *h H finite. 13 14 1. INTRODUCTION TO PART I. In any case, our analysis follows this time-honoured pattern, by breaking any* * object X into into F-free and F-contractible parts by the isotropy separation cofibering qX X -! X ^ "EF -! X ^ EF+: We thus consider X in two parts: the F-contractible object X(T) = X ^ "EF and t* *he F- free object X(F) = X ^ EF+. The object X(T) is determined by its T-homotopy gro* *ups as rational vector spaces. The main content of the analysis is therefore in und* *erstanding F-objects such as X(F), and how they may be stuck to F-contractible objects X(T* *). By use of idempotents in Burnside rings it is easy to see that X(F) splits as a we* *dge of objects X(H), one for each finite subgroup H, where only the isotropy group H is releva* *nt to X(H). The category of these will be called the category of T-spectra over H and* * denoted T-Spec=H ; the mathematical core of the whole enterprise is the analysis of thi* *s category of objects X(H). It turns out that ssT*(X(H)) is a torsion module over the ring OH* * = Q[cH,] in which cH is an Euler class, and of degree -2, and that the category T-Spec=H* * of objects X(H) is equivalent to the derived category of differential graded torsion Q[cHQ* *]-modules. The object X(F) is thus determined by the torsion module ssT*(X(F)) over OF = * *H Q[cH.] Because we are working rationally it is not difficult to calculate homotopy gro* *ups of any precisely described spectrum, so this description is of practical use. Finally we must determine the assembly map qX : X(T) -! X(F). Note first that ssT*(X(T)) is not naturally a module over OF, and also that ssT*(qX ) may be ze* *ro without qX being zero. The answer is to take into account the twisting available from repr* *esentations of T. This twisting is measured by Euler classes, and since there are Thom isom* *orphisms for arbitrary F-spectra we may consider the ring E-1OF formed from OF by invert* *ing all Euler classes. We denote this ring tF*, sinceLit is in fact the F-Tate cohomolo* *gyQof S0 in the sense of [14]. It turns out that tF*is HQ in positive even degrees and H* * Q in even degrees 0. By construction, tF*is a OF-module, and qX determines a map ^qX: tF* ssT*(X(T)) -! ssT*(X(F)) in the derived category of differential graded OF-modules. It transpires that ^* *qXis a com- plete invariant of qX , so that X is determined by the rational vector space ss* *T*(X(T)), the torsion Q[cH-]modules ssT*(X(H)), and the derived OF-map ^qX. Continuing from t* *his stage, it is not hard to identify which triples (ssT*(X(T)); ssT*(X(F)); ^qX) occur, a* *nd to identify the relevant algebraic triangulated category. In fact we may consider the torsionLmodel category Atwhose objects are maps t* *F*V -! T of OF-modules, T being a sum HT (H) with T (H) a torsion Q[cH ]-module. It * *turns out that this category is abelian and of injective dimension 2. One may therefo* *re consider differential graded objects in At, and invert homology isomorphisms to form the* * derived category DAt . This category is equivalent to the category of rational T-spec* *tra, and provides the complete algebraic model we seek. However we prefer not to emphasi* *ze this model: the analysis is only possible by introducing a second model, which we c* *all the standard model. This proves to be more convenient for most purposes. The real d* *ifficulty is that, since At is of dimension 2, it is rather hard to get a precise hold on* * morphisms in the derived category. On the other hand the standard model is of dimension * *1. The identification of the standard model is the most important result of the analys* *is. 1.1. OUTLINE OF THE ALGEBRAIC MODELS. 15 It will help to explain the construction of algebraic models for four triangu* *lated categories of T-spectra in increasing order of complexity. They are (i) the category of fr* *ee T-spectra, or more generally the category T-Spec=H of T-spectra in which only the isotrop* *y group H is important, (ii) the category of T-Spec=F of F-spectra, (iii) the category* * T-Specsf of semifree T-spectra and (iv) the category of all rational T-spectra. For eac* *h of these categories C, we find an abelian category A = A C of dimension 1, and a linear* *ization functor ssA*: C -! AC. Because the abelian category AC is so simple in each cas* *e, it is possible to reconstruct the original triangulated category C from it. Recall th* *at the derived category of an abelian category A is the category formed from the category of d* *ifferential graded objects by inverting homology isomorphisms; if A is finite dimensional, * *the derived category may be constructed explicitly. Theorem 1.1.1. If C is one of the above four categories of rational T-spectra* *, there is a category A = AC which is abelian and one dimensional so that there is an equi* *valence of triangulated categories C ' DA ; where DA is the derived category of A. Hence in particular, for any objects X * *and Y of C, there is a natural short exact sequence 0 -! ExtA(ssA*(X); ssA*(Y )) -! [X; Y ]T*-! Hom A(ssA*(X); ssA*(Y )) -! 0; which splits unnaturally. Before making the theorem explicit for the four categories we make some gener* *al remarks about the levels at which the theorem is useful. Firstly, every geometric obje* *ct X of C has an algebraic model ssA*(X) and there is a bijection between isomorphism cla* *sses in C and isomorphism classes in A . Next, if we know the algebraic models of two * *objects X and Y , the short exact sequence allows us to use the algebra of the abelian * *category to calculate the group [X; Y ]T*of maps between them. Finally, we may model all* * primary constructions (such as formation of cofibres, smash products, function spectra,* * composition of functions and calculation of Toda brackets) in the algebraic category. This* * much is internal to the category, but in addition, all homotopy functors of T-spectra h* *ave their algebraic counterparts. It is very illuminating to identify the algebraic behav* *iour of various well known functors. We now make Theorem 1.1.1 explicit in the four cases. Theorem 1.1.2. If C = T-Spec=H is the category of T-spectra over H, then A i* *s the category of torsion Q[cH ]-modules. The functor ssA*is simply T-equivariant hom* *otopy ssT*. This category A is abelian and one dimensional. Accordingly, for two T-spectra * *X and Y over H there is a split short exact sequence 0 -! ExtQ[cH](ssT*(X); ssT*(Y )) -! [X; Y ]T*-! Hom Q[cH](ssT*(X); ssT*(Y ))* * -! 0: __|_| The proof of this will be completed in Section 4.3. The short exact sequence * *is Theorem 3.1.1, and it is the central result of the analysis of Part I. 16 1. INTRODUCTION TO PART I. Theorem 1.1.3. If C = T-Spec=F is the categoryLof F-spectra, then A is the * *full subcategory of OF-modules M of the form M = H M(H) for torsion Q[cH ]-modules M(H). We refer to these as F-finite torsion modules, and they may also be desc* *ribed as the OF-modules annihilated by inverting all Euler classes. The functor ssA** *is simply T-equivariant homotopy ssT*. The category of F-finite torsion modules is abelia* *n and one dimensional. Accordingly, for two F-spectra X and Y there is a split short exac* *t sequence 0 -! ExtOF(ssT*(X); ssT*(Y )) -! [X; Y ]T*-! Hom OF(ssT*(X); ssT*(Y )) -! 0* *: __|_| The proof of this will also be completed in Section 4.3. Theorem 1.1.4. If C = T-Specsf is the category of semi-free spectra, then A i* *s the category whose objects are morphisms M - ! Q[c; c-1] V of Q[c]-modules (for s* *ome graded vector space V ) which become isomorphisms when c is inverted. This cate* *gory A is abelian and one dimensional. The functor ssA*is defined by i j ssA*(X) := ssT*(X ^ DET+) -! ssT*(X ^ DET+ ^ "EF): Accordingly, for two semifree T-spectra there is a split short exact sequence 0 -! ExtA(ssA*(X); ssA*(Y )) -! [X; Y ]T*-! Hom A(ssA*(X); ssA*(Y )) -! 0:* * __|_| Finally the model of all rational T-spectra is as follows. Theorem 1.1.5. If C = T-Spec then A is the category whose objects are morphis* *ms M -! tF* V of OF-modules (for some graded vector space V ) which become isomor- phisms when all Euler classes are inverted (i.e. the kernel and cokernel are F-* *finite torsion modules). This category A is abelian and one dimensional. The functor ssA*is de* *fined by i j ssA*(X) := ssT*(X ^ DEF+) -! ssT*(X ^ DEF+ ^ "EF): Accordingly, for two T-spectra there is a split short exact sequence 0 -! ExtA(ssA*(X); ssA*(Y )) -! [X; Y ]T*-! Hom A(ssA*(X); ssA*(Y )) -! 0:* * __|_| The proof of this is given in Section 5.4. It should be emphasized that Hom A* *(M; N) and ExtA (M; N) are routinely computable, and that, because we are working rational* *ly, there is usually no serious trouble in calculating ssA*(X). Part I begins with the concrete and moves towards the abstract in two steps. * *Thus we begin with the cohomology theories, move on to homotopy theory, pass to algebra* * by an Adams spectral sequence, and finally package this in categorical terms. Here i* *s a more detailed outline of contents. We begin with two sections which can be expressed in classical terms. These g* *ive evidence that there is some complexity in rational T-equivariant cohomology theories, bu* *t not too much. In particular they give some evidence for the simplicity of F-objects. 1.2. READING GUIDE. 17 After this, the discussion is conducted in the Lewis-May [18] stable category* * of T-spectra. The first step is to introduce the basic building blocks and the methods for br* *eaking general objects up. This gives us the setting to construct an Adams spectral sequence,* * which provides the connection between topology and algebra. Once the Adams spectral s* *equence for T-Spec=H has been constructed we need only do some algebra and certain for* *mal manipulations to obtain and exploit all the algebraic models. We have taken the* * view that an abstract machine should only be introduced when there is a particular case o* *n which its operation can be illustrated. Accordingly we have not described the transit* *ion from an Adams spectral sequence to an algebraic model (in Section 4.2) until we have co* *nstructed the simplest instance to which it applies. On the other hand Section 4.2 may be* * relevant in quite different settings, and it is written axiomatically so that it can be rea* *d and applied independently of the preceding sections. Once the general analysis is completed we consider standard T-spectra and con* *structions on T-spectra in Part II. In Part III we consider in more detail certain example* *s of estab- lished interest. More detailed accounts of the contents of Parts II and III may* * be found in their introductions. 1.2. Reading Guide. Some readers may not wish to read all of the material in Part I, so we provid* *e further guidance here. Those only interested in the Atiyah-Hirzebruch spectral sequence for the K-th* *eory of an F-space will only need to read Sections 1.3, 1.4, 2.1, referring to Appendix* * A for the necessary facts about Mackey functors. Sections 1.3 and 1.4 are not used elsewh* *ere in Part I. We shall return to the Atiyah-Hirzebruch spectral sequence in Section 15.1 o* *f Part III, where we give more complete results. Those interested in Mackey functors should read Section 2.1 and then refer to* * Appendix A. Mackey functors are not used until we consider ordinary cohomology theories * *in Chapter 13 from Part II. The central material constructing the main Adams spectral sequence for the ca* *tegories of F-spectra and T-spectra over H is to be found in Chapters 2 and 3. Maps fro* *m F- contractible spectra to F-free spectra are deduced in Sections 5.1 and 5.2. Thi* *s is sufficient to answer most direct questions about particular T-spectra, and may satisfy som* *e readers. On the other hand readers wishing to understand the shape of the algebraic mode* *ls without reading these chapters. In Chapter 4, we explain the abstract process of reaching an algebraic model * *from an Adams spectral sequence and we illustrate it for T-spectra over H. However the * *goal of a full algebraic model is fulfilled in Chapter 5. We deduce the remaining topolog* *ical input from the Adams spectral sequence in Sections 5.1 and 5.2, and construct the alg* *ebraic model in Section 5.3. It is then a simple matter to show in Section 5.4 that th* *e algebra does indeed model the topology. Chapter 6 completes the circle by introducing t* *he torsion model, closely following geometric intuition, and by showing that it gives a mo* *del equivalent to the standard model. 18 1. INTRODUCTION TO PART I. 1.3. Haeberly's example. We give Haeberly's example [16] showing there is no Chern character isomorphi* *sm, for T- equivariant K-theory. This simply involves constructing a T-space X whose equiv* *ariant K- theory is concentrated in even degrees, but whose ordinary cohomology with coef* *ficients in the rationalized representation ring functor is nonzero in odd degrees. Since t* *he homotopy functors of the K-theory spectrum are in even degrees the K-theory cannot be a * *product of copies of ordinary cohomology. In the next section we give a proof of a gene* *ralization of McClure's result that there is a Chern isomorphism for T-spaces X with XT trivi* *al. To explain Haeberly's example it is convenient to consider the group = T x T* *0where both T and T0are copies of the circle group. The group has a 3-dimensional com* *plex representation V = (1 t t2) t0, where t is the natural representation of T o* *n C, and similarly for T0. We may consider the unit sphere S(V ) as a -space, give it a* * disjoint basepoint and then form the T-space X = S(V )+=T0. We could equally well descri* *be X as a copy of CP+2on which T acts via s(z0 : z1 : z2) = (z0 : sz1 : s2z2). From* * the first description it is easy to calculate the K-theory since we have K*T(X) = K*(S(V * *)+), because S(V ) is free as a T0-space. Indeed, the cofibre sequence S(V )+ -! S0 -! SV of* * -spaces gives an exact sequence (V )i 0 i i+1 V . .-.! Ki(SV ) -! K (S ) -! K (S(V )+) -! K (S ) -! . .:. Now by Bott periodicity Ki(SV ) is R() if i is even and 0 if i is odd, and beca* *use the degree 0 Euler class (V ) = (1-t0)(1-tt0)(1-t2t0) is not a zero divisor in R() = Z[t; * *t-1; t0; (t0)-1] we find K0T(X) = R()=(V ) andK1T(X) = 0: In particular the K-theory of X is entirely in even degrees. On the other hand from the second description2it is not hard to see that X ha* *s isotropy groups T; C2 and 1. Furthermore XC2 = (St _S0)+ and X may be given a T-CW struc* *ture with two free 1-cells, one free 2-cell and one free 3-cell. Hence for any Macke* *y functor M we see that H*T(X; M) is the cohomology of a complex 0 d1 d2 3M(T) -d!M(C2) 2M(1) -! M(1) -! M(1); and it is easy to see that d1 is surjective. Thus H3(X; M) = M(1), and in parti* *cular if M is the rationalized representation ring Mackey functor this is the non-zero gro* *up Q. 1.4. McClure's Chern character isomorphism for F-spaces. McClure has observed that the if X is an F-space then the Atiyah-Hirzebruch s* *pectral sequence for the K-cohomology of X does collapse at E2. His proof involves appe* *aling to unstable results and the work of Slominska. We shall give a proof of the corre* *sponding statement for any cohomology theory whose homotopy functors are concentrated en* *tirely in even degrees, and of the corresponding statement for homology theories. Of c* *ourse this applies in particular to K theory, by the Bott periodicity theorem. In Section * *15.1 of Part 3 we shall give a necessary and sufficient condition for the collapse of the Atiy* *ah-Hirzebruch spectral sequence for F-spaces, which will give an alternative to the proof of * *this section. 1.4. MCCLURE'S CHERN CHARACTER ISOMORPHISM FOR F-SPACES. 19 Before stating the theorem, we recall that for each integer k it is appropria* *te to consider the entire system of homotopy groups ssHk(X) = [G=H+ ^ Sk; X]T as H runs through all subgroups of T. It is appropriate to regard this as a functor ss_Tk(X) : G* *=H+ 7-! [G=H+ ^Sk; X]T, on the category of stable orbits. An additive functor of this f* *orm is called a Mackey functor; we examine the algebraic structure of the category of rationa* *l Mackey functors in Appendix A, but for the present we only need the basic terminology.* * In line with the usual abbreviation we write the coefficient functor ss_Tk(K) as K_Tk. Since the orbits are the equivariant analogues of points, an ordinary cohomol* *ogy theory is one for which the cohomology of each orbit is concentrated in degree zero. T* *hus ordi- nary cohomology theories correspend to Mackey functors M, and they are represen* *ted by Eilenberg-MacLane spectra HM. Theorem 1.4.1. If K is any rational T-spectrum with homotopy functors K_Tm= 0* * for all odd integers m then for any F-space X there are isomorphisms (a) Y T K*T(X) ~= H*T(2nX; K_-2n) n2Z and (b) M T KT*(X) ~= HT*(2nX; K_2n): n2Z This follows from a geometric statement. Theorem 1.4.2. If K is any rational T-spectrum with homotopy functors K_Tm= 0* * for all odd integers m then (a) Y T F (EF+; K) ' F (EF+; 2nH(K_2n)) n2Z and (b) _ T K ^ EF+ ' EF+ ^ 2nH(K_2n): n2Z To see how Theorem 1.4.1 follows from 1.4.2 we use a lemma which is immediate* * from the definition of EF+ and its unreduced suspension "EF. Lemma 1.4.3.For any F-spectrum X, (a) X ^ "EF ' * and hence X ' EF+ ^ X; also (b) for any T-spectrum Y we have F (X; Y ^ "EF) ' * and hence F (X; Y ^ EF+) ' F (X; Y ). __|_| By 1.4.3 (a), Theorem 1.4.1 follows by applying F (X; ) to Part (a) of 1.4.2 * *and X^ to Part (b) of 1.4.2 and taking homotopy groups. 20 1. INTRODUCTION TO PART I. Proof: We turn to the proof of 1.4.2. Note first that it is enough to prove Par* *t (b); indeed, by 1.4.3 (b), Part (a) follows by applying F (EF+; .) toWthe equivalence of Par* *t (b). It is enough to construct a T-map : K ^ EF+ -! EF+ ^ n2Z2nH(K_2n) which is an H-equivalence for all finite subgroups H. By the Whitehead theorem it is su* *fficient that induces an isomorphism of ssH*for all finite subgroups H. By 1.4.3 (b) ag* *ain, it is equivalent to give the composite _ 0: K ^ EF+ -! 2nH(K_2n); n2Z and since this wedge is equivalent to the product we may specify 0by giving its* * components. These are elements of the cohomology groups [K ^ EF+; HM]*T= H*T(K ^ EF+; M) for various Mackey functors M. Accordingly we set about calculating the cohomology* * of K ^ EF+. The idea is to filter EF+ so that the subquotients are analogues of cells, bu* *t with all elements of finite order as isotropy groups. This extends the idea of [9]. Thus* * we note that if H L we have a projection T=H -! T=L, and that the subgroups of finite order* * form a directed set. We may therefore let T==F+:= holim!T=H+ where the limit is over a* *ll finite H subgroups H (or over a cofinal sequence if that appears more comfortable). Anal* *ogously, if H is a finite subgroup of order n we may let V (H) denote the representation* * tn with kernel H, and there are maps mV (H) -! mV (L) (of degree |L=H|m ) for all m. We* * let S(mV (F))+ := holim!S(mV (H))+ for 0 m 1. The usefulness of these constructio* *ns H is summarized in a lemma. Lemma 1.4.4. The infinite sphere S(1V (F))+ is a model for EF+. We thus have* * a filtration * = S(0V (F))+ S(1V (F))+ S(2V (F))+ . . .S(1V (F))+= EF+ and the subquotients are generalized cells S(mV (F))+=S((m - 1)V (F))+' S2m-2^ T==F+ for 1 m < 1. Proof: Since (S(mV (H)))L = ; if L 6 H or S(mV (H)) if L H the fact that S(1V * *(F))+ is a universal space is clear. To identify the quotients we use the fact that * *the cofibre sequences S((m - 1)V (H))+ -! S(mV (H))+ -! S2m-2^ T=H+ fit into a direct system. __|_| In other words we have EF+ = T==F+ [ T==F+ ^ e2 [ T==F+ ^ e4 [ T==F+ ^ e6 [ . .:. Thus, for any spectrum K, we may form the spectral sequence of the filtered s* *pectrum K ^ EF+ which will have the form Es;t1= Hs+tT(K ^ (EF(s)+=EF(s-1)+); M) ) Hs+tT(K ^ EF+; M): 1.4. MCCLURE'S CHERN CHARACTER ISOMORPHISM FOR F-SPACES. 21 Indeed, from the form of the filtration, we find the spectral sequence is conce* *ntrated in the first quadrant in terms with even s where we have E2m;t1= HtT(K ^ T==F+; M): Of course, using the change of groups isomorphism H*T(K ^ T=H+; M) = H*H(K; M),* * we have a Milnor exact sequence 0 -! lim1Ht-1H(K; M) -! HtT(K ^ T==F+; M) -! lim HtH(K; M) -! 0: H H It is in the analysis of this exact sequence that it is essential we are workin* *g rationally. Indeed, because H is finite, every rational H-spectrum is a product of Eilenber* *g-MacLane spectra and these are necessarily also Moore spectra. It now follows that, prov* *ided K has its homotopy functors in even degrees, the groups HtH(K; M) are only nonzero fo* *r even t. The collapse of the spectral sequence is thus ensured once we show the lim1term* *s vanish. In fact the restriction maps HtL(K; M) -! HtH(K; M) are surjective. Perhaps the quickest way to see this is to note that H*H(HM0; * *M) = [HM0; HM]*H= HomH (M0; M), for any Mackey functors M0and M. We may then use the corresponding fact for Mackey functors, that HomL(M0; M) -! HomH (M0; M) is surjective. This surjectivity is due to the fact that all Weyl groups are co* *nnected, and it is easily deduced from Appendix A. We conclude that if K has all its homotopy functors in even degrees then Y H*T(K ^ EF+; M) = lim H*H(2mK; M); m2Z H and in particular we can find a map 02m: K ^ EF+ -! 2mH(K_T2m) inducing the identity in ssH2m(o) for all finite subgroups H. The map _ T 0: K ^ EF+ -! 2nH(K_2n) n2Z is thus an F-equivalence and hence is a homotopy equivalence as required. __|* *_| In Section 15.1 of Part III we shall complete the picture of Atitiyah-Hirzebr* *uch spectral sequences for F-spaces by giving an analysis without hypothesis on the rational* * cohomol- ogy theory. We characterize those theories K*T(.) for which the spectral sequen* *ce always collapses at E2, show that arbitrarily high differentials occur, and give a geo* *metric expla- nation of them in terms of universal examples. The behaviour of the spectral se* *quence for arbitrary spaces X is much more complicated. 22 1. INTRODUCTION TO PART I. CHAPTER 2 First steps. This chapter introduces all the basic ingredients for our analysis. In Section * *2.1 we sum- marise what is known about the homotopy groups of the cells T=H+, and use idemp* *otents of the Burnside rings to break them up; in the following section we show this e* *xtends to a complete splitting of the category of F-spectra as a product of categories, o* *ne for each finite subgroup. In Section 2.3 we analyse the summands E of EF+; their simp* *licity is fundamental to the success of our programme. Finally, in Section 2.4 we identif* *y the ring of self-maps of E, which acts as operations on the homotopy groups of all T-* *spectra over H, as the principal ideal domain Q[cH.] 2.1.Natural cells and basic cells The idea is that in any context one should understand complicated objects by * *first understanding the building blocks and the way they can be stuck together. In p* *ractice this involves a complete description of the full subcategory on the basic build* *ing blocks. There are inevitably many different choices for the set of basic objects, and f* *or practical purposes it is useful to choose those for which the basic subcategory is as sim* *ple as possible. The ideal is typified by representation theory over C where where every morphis* *m between simple modules is a scalar multiplication. The natural building blocks for locally nice G-spaces are the G-cells G=H+ ^S* *n. The full category on these is complicated because of the group theory (which enters via * *the fixed point spaces (G=H)K ) and because of the topology (since we need to know the ho* *motopy groups of spheres). Working over the rationals brings the topology under reason* *able control, but the group theoretic complication is still considerable. However, if we work* * stably, we obtain a reasonable supply of idempotents with which to chop up the cells into * *simpler pieces, for which the residual group theory and topology is much simpler. It is time to be quite definite, and we return to the case G = T. This case i* *s particularly favourable for several reasons. Firstly T is abelian, secondly there is only o* *ne infinite subgroup, and finally all Weyl groups are connected so that there is hardly any* * group theoretic complication at all. There are two types of natural cells: the T-fixe* *d cells Sn and the cells T=H+ ^ Sn with finite isotropy group. The stable rational maps betwee* *n them are not hard to calculate, using the standard change of groups isomorphisms and tom* * Dieck 23 24 2. FIRST STEPS. splitting. Here and elsewhere we shall refer always to homology grading unless * *otherwise specified, and we use the homology suspension (tM)n = Mn-t. Lemma 2.1.1. (a) The graded self maps of the fixed cell S0 are [S0; S0]T*= Q (QF [c-1]) where QF denotes a rational vector space with basis F, and c-1 is an element of* * degree 2. (b) The other maps involving the fixed cell are [S0; T=H+]T*= A(H) and [T=H+; S0]T*= A(H) where A(H) is the rational Burnside ring of H. (c) Maps between two cells with finite isotropy groups are [T=H+; T=K+]T*= A(H \ K) A(H \ K) For the present c-1 is simply a notational device, but we shall eventually in* *troduce an operation c of degree -2 which will act as the notation suggests. Proof: The tom Dieck splitting theorem [18, V.9.1] states that for any T-space X M ssT*(X) = ss*(X) ss*(E(T=H)+ ^T=HXH ): H Part (a) follows, as does Part (b). For Parts (b) and (c) we use the change of groups isomorphism [T ^H X; Y ]T*~=[X; Y ]H* and the Wirthm"uller isomorphism [18, II.6.5] [Y; T ^H X]T*~=[Y; S1 ^ X]H*; where X is an H-spectrum and Y is a T-spectrum. To complete the proof of (c) we* * note that as an H-spectrum T=K+ is S(U)+ where U is a representation with kernel L =* * H \K. Now there is a cofibre sequence of H-spaces H=L+ -! H=L+ -! S(U)+ where the first map is 1 - g for some generator g of the cyclic group H=L. Sinc* *e the group is abelian g induces the identity in ssH0, and Part (c) follows. __|_| The point to notice is that there is vertical complication (i.e. there are no* *ntrivial maps of positive degree) and there is horizontal complication (i.e. the maps of degr* *ee zero do not form a simple ring). The vertical complexity is topological in origin, coming f* *rom the facts that T is of dimension 1 and that BT has positive dimensional rational cohomolo* *gy. This is unavoidable, and indeed it is the source of the interesting mathematics. On * *the other hand we may eliminate the horizontal complexity simply by using simpler buildin* *g blocks. For this we need idempotents from the Burnside rings. Recall that for any fin* *ite subgroup ~=Q H we have A(H) = [S0; S0]H and the ring isomorphism OE : A(H) -! KH Q whose * *Kth 2.1. NATURAL CELLS AND BASIC CELLS 25 coordinate takes f : S0 -! S0 to the degree of the fixed point map H (f) : S0 -* *! S0, where H is the geometric H-fixed point functor. We let eK denote the idempotent* * which is 1 on the Kth factor and 0 on the others. This notation is consistent with re* *striction in the sense that if K H0 H then resHH0eK = eK . Notice that for any H-spectrum X* * we have eK X = (eK S0) ^ X and hence we have non-equivariant equivalences L(eK X) * *' * if L 6= K, whilst K eK X = K X. Definition 2.1.2.The basic cells are the T-spectra oenT= Sn oenH= T+ ^H eH Sn for various integers n and finite subgroups H of T. We shall need a couple of easy observations about basic cells. Lemma 2.1.3.(i) The duals of the basic cells are as follows: Doe0T= oe0Tand D* *oe0H' -1oe0H. (ii) If K H there is an equivalence oe0K' T+ ^H eK S0: (iii) The natural cells can be obtained from the basic cells by the formula _ T=H+ = oe0K: KH W 0 Note in particular that this means that T==F+= H oeH, which completes the co* *nnection with Section 1.4 on McClure's theorem. Proof: (i) The duality result follows from the fact that any map f : S0 -! S0 i* *s self-dual. (ii) The projection H=K+ -! S0 is an equivalence on K fixed points, so that it * *induces an equivalence eK H=K+ -'!eK S0. Now by consistency of eK under restriction, eK H=K+ ' eK (H+ ^K S0) ' H+ ^K eK S0; so that H+ ^K eK S0 ' eK S0. Inducing up to T, we find oe0K' T+ ^K eK S0, as re* *quired. W 0 (iii) For the splitting we choose an H-equivalence S0 -'! KeK S correspondin* *g to ~= Q 0 ' theWsplitting A(H) -! K Q and extend it to a T-equivalence T=H+ = T+ ^H S -! __ KT+ ^H eK S0. Now use Part (ii). |_| Obviously the analogue of 2.1.1 (a) for basic cells is identical, but Parts (* *b) and (c) are much simpler. Lemma 2.1.4.(a) The graded self maps of the fixed cell oe0Tare [oe0T; oe0T]T*= Q (QF [c-1]) where QF denotes a rational vector space with basis F, and c-1 is an element of* * degree 2. (b) The other maps involving the fixed cell are [oe0T; oe0H]T*= Q 26 2. FIRST STEPS. and [oe0H; oe0T]T*= Q (c) Maps between two cells with finite isotropy groups are [oe0H; oe0H]T*= Q Q and [oe0H; oe0K]T*= 0 if H 6= K. Proof: The result for [oe0H; oe0T]T*is clear from the definitions, and the resu* *lt for [oe0T; oe0H]T*fol- lows by duality. Part (c) follows by applying the idempotent eH to the proof of* * 2.1.1 (c). __|_| There are certain features worth special attention, beyond the fact that ther* *e are no non-trivial maps oe0H-! oe0Kif H 6= K. Firstly notice that except for degree z* *ero the graded ring [oe0T; oe0T]T*is entirely in odd degrees, and hence its product str* *ucture is trivial. The second feature is the fact that every non-zero map out of oe0Tis an isomorp* *hism: this is responsible for the finite dimensionality of the category of T -Mackey funct* *ors. The curious reader may wish to read Appendix A; this will complete the justif* *ication of Section 1.4, but will otherwise not be needed until Part 2. We also pursue the * *analysis of the graded category of stable maps between cells a little further in Section 15* *.2. 2.2.Separating isotropy types. Building on the observations on cells in the Section 2.1, we pull the whole c* *ategory of rational T-spectra apart on similar lines. Obviously the fixed cell oe0Tbeh* *aves quite differently from the others, so it is appropriate to separate T -fixed phenomen* *a from F- phenomena. The success of the present method depends on the fact that, in addit* *ion, the effects of different finite subgroups are completely decoupled. This idea is fi* *rmly based in geometric intuition since we are separating a spectrum X into parts X(H) which * *only have geometric fixed points at H (i.e. K X(H) is non-equivariantly contractible if K* * 6= H). Let T-Spec be the stable homotopy category of rational T -spectra, and consid* *er the full subcategory T-Spec=T of F-contractible T -spectra X (i.e. with H X non-equivari* *antly contractible for all finite subgroups H), and the full subcategory T-Spec=F of * *F-spectra. We view the category T-Spec as being generated by these two subcategories. More* * formally we consider the isotropy separation cofibering EF+ ^ X -! X -! "EF ^ X : the last term "EF ^ X lies in T-Spec=T since H (E"F ^ X) ' (H "EF) ^ (H X), whi* *ch is non-equivariantly contractible for all finite subgroups H, and the first ter* *m EF+ ^ X is visibly an F-spectrum. We thus let X(T) = E"F ^ X and X(F) = EF+ ^ X. Our approach is to analyze the categories T-Spec=T and T-Spec=F rather completely* * and then study how to assemble them to make T-Spec. The analysis of T-Spec=F forms * *the main contentQof the work, and it begins by observing that the decoupling implie* *s that T-Spec=F ' HT-Spec=H where T-Spec=H is the full subcategory on T -spectraWX with K X non-equivariantly contractible if K 6= H. In particular X(F) ' H X(H)* * in 2.2. SEPARATING ISOTROPY TYPES. 27 T-Spec=F , for suitable spectra X(H) in T-Spec=H . In a sense to be made precis* *e below T-Spec=H has global dimension 1 and is thus easily controlled. The analysis of T-Spec=T is straightforward obstruction theory, together with* * the clas- sical analysis of the nonequivariant category of rational spectra. The obstruct* *ion theory is easily done integrally. Proposition 2.2.1.(i) H X is non-equivariantly contractible for all finite su* *bgroups H if and only if the inclusion X = X ^ S0 -! X ^ "EF is a T-homotopy equivalenc* *e. (ii) Maps between objects of T-Spec=T are calculated by [E"F ^ X; "EF ^ Y ]T*= [TX; TY ]*: __|_| For a vector space W of dimension d we let X[W ]Wbe a d-fold wedge of copies * *on X, and if W is a graded vector space we let X[W ] = nnX[Wn]. Since the nonequivar* *iant rational spectrum TX is equivalent to a wedge S0[ss*(TX)] of spheres, we find X* *(T) ' E"F ^ S0[ss*(TX)]. Corollary 2.2.2.The functor T induces an equivalence of categories T-Spec=T ' Graded vector spaces. __|_| If the equivalence is also to preserve cofibre sequences, one must use the de* *rived category of differential graded vector spaces on the right hand side. Now return to the analysis of F-spectra. The idea is that any F-spectrum may * *be formed from natural cells with finite isotropy groups and hence from basic cells with * *finite isotropy groups. Since basic cells with one isotropy type cannot be non-trivially attach* *ed to those of another the splitting follows. Theorem 2.2.3. (Decoupling of finite isotropy groups.) (i) For any F-spectrum X and each finite subgroup H of T there is a spectrum X(* *H) in T-Spec=H formed from basic cells with isotropy H so that there is an equivalen* *ce _ X -'! X(H): H (ii) Given two F-spectra X and Y the natural map Y ~= [X(H); Y (H)]T -! [X; Y ]T H is an isomorphism. To obtain a functorial construction we use the universal example EF+. Here w* *e let E = EF+(H), and we may describe it more geometrically. Indeed we may let Fk denote the family of finite subgroups of order k, and hence obtain the filtrat* *ion [ ET+ = EF1+ EF2+ EF3+ . . . EFk+= EF+: k 28 2. FIRST STEPS. The cofibre of EFk-1+-! EFk+is the space E for H of order k. Note in particu* *lar that E<1> = ET+, but that in general E is not of the form X+ for any space X. The* * content of 2.2.3 is that the filtration splits rationally (see [12] for the generalizat* *ion to arbitrary compact Lie groups). W Now since anyWF-spectrum X is equivalent to EF+ ^ X and EF+ ' H E, one then finds X ' H E ^ X. Since K commutes with smash products it follows that X(H) ' E ^ X. In any case we obtain the categorical consequence we need. Corollary 2.2.4. There is an equivalence of triangulated categories Y T-Spec=F ' T-Spec=H H W __ induced by the functors X 7-! (X(H))H and (X(H))H 7-! H X(H). |_| We turn to the proof of 2.2.3. The basic tool is the following. Lemma 2.2.5. If H is a finite subgroups of T and if X is formed from basic ce* *lls oenHwith isotropy H and if Y is a F-spectrum formed using basic cells oenKwith isotropy * *K 6= H then [X; Y ]T = 0 andX ^ Y ' *: Proof: The fact that [oe0H; Y ]T = 0 follows by induction on the number of cell* *s of Y and passage to direct limits. The fact that [X; Y ]T = 0 now follows by wedges cof* *ibres and limits in the X variable. To see that X ^ Y ' * note that D(oe0H) ' -1oe0Hso that oe0H^ oe0K' * and arg* *ue by induction and limits. __|_| Proof of 2.2.3: If 2.2.3 (i) holds for X, we shall say X splits, and we note fi* *rst that Part (ii) of the 2.2.3 holds for any pair of F-spectra X and Y which both split. Ind* *eed we have _ _ Y _ [ X(H); Y (K)]T = [X(H); Y (K)]T: H K H K W T Since [X(H); K6=HY (K)] = 0 by 2.2.5 we conclude _ _ Y [ X(H); Y (K)]T = [X(H); Y (H)]T H K H as required. We now turn to the proof of Part (i), in which the basic ingredients are the * *splitting of cells as in 2.1.3, and an inductive step as follows. Lemma 2.2.6. If X -! Y -! Z is a cofibre sequence of F-spectra and both X and* * Y split, so too does Z. 2.3. THE SINGLE STRAND SPECTRA E. 29 Before proving 2.2.6, we note how the theorem follows. Indeed any F-spectrum* * has an inductive filtration so that the subquotients are natural F-cells. In other * *words X ' holim!Xn where X0 = * and where there are cofibre sequences Cn -! Xn -! Xn+1whe* *re n Cn is a wedge of natural cells with finite isotropy. The result therefore follo* *ws for Xn by induction on n, and since there is a cofibre sequence _ _ Xn -! Xn -! holim Xn n n ! n it also follows for X. It thus remains only to prove 2.2.6. Proof of 2.2.6: By hypothesis X and Y both split, so we may form the diagram X -! Y - ! Z '# '# W f W HX(H) -! KY (K) : W By Part (ii) of the theoremWf ' H f(H) for suitable maps f(H) : X(H) -! Y (H). It follows that Z ' H Z(H) where Z(H) is the cofibre of f(H). This cofibre is* * of the required form by the cellular approximation theorem for basic cells. __|_| __ |_| 2.3.The single strand spectra E. It turns out that E and its skeleta are very basic objects of T-Spec=H . F* *urthermore since Y ' Y ^ E for any object Y of T-Spec=H , it follows that [E; E]T* **acts as an algebra of operations on objects of T-Spec=H . It is therefore fitting to sp* *end a little time considering maps between the skeleta of E. Topologists may like to thin* *k of E as a version of CP+1 massively simplified by the facts that j2 = 0, and James p* *eriodicity is trivial. From the construction of E as EF+(H) we find its basic structure. Lemma 2.3.1.E may be constructed with a single basic cell oe2mHin each eve* *n di- mension: E = oe0H[ oe0H^ e2 [ oe0H^ e4 [ : ::: __|_| Using this cell structure we may consider the skeleta E(2m). These are the* * fundamen- tal objects of T-Spec=H . Since we only have contributions from T-isotropy and * *H-isotropy we should identify the H-homotopy types. Proposition 2.3.2.(a) E is H-equivariantly equivalent to eH S0. (b) E(2m)is H-equivariantly equivalent to eH (S0 _ S2m+1) W Proof: For Part (a) we apply eH to the H-equivalences S0 ' EF+|H ' K E|H . Now for Part (b) we argue by induction on m. Indeed the case m = 0 states th* *at oe0His H-equivariantly eH (S0 _ S1), which follows by applying eH to the H-equi* *valence G=H+|H ' S0 _ S1 (which is true at the unstable level). The fact that the attac* *hing map 30 2. FIRST STEPS. is an H-equivariant equivalence on the common cell follows from (a). __|_| For objects X of T-Spec=H the non-trivial part of the Mackey functor ssTn_(X* *) comes from ssTn(X) = [S0; X]Tnand ssHn(X). Indeed by 2.1.3 _ ssKn(X) = [G=K+; X]Tn= [ oe0K; X]Tn LK and this is [oe0H; X]Tnif H K and 0 otherwise. A form of the Whitehead theorem* * follows. Lemma 2.3.3. (Naive Whitehead theorem in T-Spec=H .) If X and Y are F-spectra* * over H, a map f : X -! Y is an equivalence provided it is a ssH*isomorphism. Proof: Since X and Y are F-spectra it is enough to verify that f induces an iso* *morphism in ssK*for all finite subgroups K; by the above discussion this follows from th* *e particular case H = K. __|_| We record the homotopy groups of the skeleta of E. Lemma 2.3.4. (a) For any natural number k M ssT*(E(2k)) = 2i+1Q andssH*(E(2k)) ~=Q 2k+1Q: 0ik (b) M ssT*(E) = 2i+1Q andssH*(E) ~=Q: i0 Proof: The values of ssH*are immediate from 2.3.2. The T-homotopy groups are e* *asily calculated from the cell structure by induction, using the fact that ssT*(oe0H)* * = Q. __|_| It follows that each of the attaching maps oe2m+1H-! E(2m)-! E(2m)=E(2m-2)= oe2mH is nontrivial in [oe2m+1H; oe2mH]T = Q. The following uniqueness result will have fundamental implications. Proposition 2.3.5.Let k be a natural number or 1, and let X be a T-spectrum o* *f the form X = oe0H[ oe0H^ e2 [ . .[.oe0H^ e2k. If the attaching maps oe2n+1H-! X2n -! X2n=X2n-2= oe2nHare nontrivial for all n then X is equivalent to E(2k). Three particular cases are of special interest. Corollary 2.3.6. (a)(Thom): If V (H) denotes a one dimensional representation* * with kernel H, there is an equivalence E ^ SV (H)' E ^ S2: (b) (James): For any natural number m, and for 0 k 1 there is an equivalence E(2m+2k)=E(2m-2)' 2mE(2k): 2.3. THE SINGLE STRAND SPECTRA E. 31 (c) (Atiyah): For any natural number m there is an equivalence D(E(2m)) ' -2m-1E(2m): __|_| Part (a) is the statement that there are universal Thom isomorphisms for T-sp* *ectra over H, Part (b) is the strong analogue of James periodicity, and Part (c) is the st* *raightforward analogue of Atiyah duality. We warn that DE is not an object of T-Spec=H , a* *nd in particular its form is quite different from the pattern suggested in Part (c). Proof: By the naive Whitehead theorem 2.3.3, it suffices to construct a map f :* * X -! E(2k)which is an isomorphism in [oe0H; .]T*. We do this by obstruction theor* *y, defining f on successive skeleta. In fact define f on X(0)= oe0Hto be the inclusion of t* *he 0-cell. By 2.3.2 [oerH; E(2k)]T = 0 for 0 < r < 2k + 1 and so this extends uniquely to * *a map f. Now it is easy to calculate ssH*(X2n) by induction on n and see that ssH*(X) * *~=ssH*(E(2k)). The map f is certainly an isomorphism in ssH0, and to see it is an isomorphism * *in ssH2k+1we follow through the construction, supposing by induction on n that fn gives an e* *quivalence X(2n)' E(2n). One extension to fn+1 may be obtained by completing the map of cofibre sequences oe2n+1H-! X(2n) -! X2n+2 # '# oe2n+1H-! E(2n) -! E(2n+2); in which the left hand vertical is an equivalence by hypothesis on X; by unique* *ness this gives the only possibility. __|_| The main use of the Thom isomorphism is to give Euler classes. In fact we sha* *ll refer 1^e(V ) V (H) 2 to the composite E = E ^ S0 -! E ^ S ' E ^ S as the Euler cla* *ss cH . It induces a map cH : X -! X ^ S2 for any object X of T-Spec=H . More exac* *tly the operation cH is the part of the Euler class appearing in T-Spec=H ; the Eul* *er class for V (H) makes contributions to T-Spec=K for all subgroups K H. If K is of o* *rder d it is perhaps best to view cKQas analogous to the dth cyclotomic polynomial d* *(z). In fact by analogy with 1 - zn = d|nd(z), if H is of order n we have c(V (H)) act* *ing on the category of spectra over K as cK if K H, and as 1 otherwise. We shall disc* *uss this global behaviour of Euler classes futher in Section 8.1, but for the present we* * need only worry about the behaviour of Euler classes over one subgroup, and we now may ma* *ke this more precise. Indeed, if we define "Eby the cofibre sequence E -! S0 -! "* *E it behaves quite analogously to "EF. Lemma 2.3.7.There are equivalences (i) S1V (H)^ E ' * (ii) S(1V (H))+ ^ E ' E (iii) "E^ S1V (H)' S1V (H). Proof: Part (i) is clear by considering the various -fixed points, and the othe* *r parts follow from the defining cofibre sequences. __|_| 32 2. FIRST STEPS. The other useful fact is that the skeleton "E(2k-1)behaves on T-Spec=H ra* *ther like the spheres SkV (H); we therefore use the notation oekV (H)= "E(2k-1). Corollary 2.3.8. (i) For any object Y of T-Spec=H there are equivalences Y ^o* *ekV (H)' Y ^ S2k. (ii) The inclusions oekV (H)-! oe(k+1)V (H)induce multiplication by cH in the h* *omotopy of any object of T-Spec=H . Proof: Both parts follow from the corresponding facts for SkV (H). IndeedWwe ha* *ve a cofi- bre sequence S(kV (H))+ -! S0 -! SkV (H)and the first term splits as KH E(* *2k-1). Only the term E(2k-1)survives smashing with an object of T-Spec=H . __|_| 2.4. Operations. As remarked above, for any object X of T-Spec=H , we have an equivalence X ' * *X ^ E and hence [E; E]T*acts on X, and on all homology and cohomology theo* *ries on T-Spec=H . The best thing of all is that this ring of operations is the prin* *cipal ideal domain Q[cH,]and is thus of global dimension 1. The aim of the next chapter is * *to construct an Adams spectral sequence based on these operations, which has the uncharacter* *istically simple form of a short exact sequence. We begin by calculating the ring of operations. Theorem 2.4.1. (Homotopy operations in T-Spec=H .) The ring of self-maps of E* * is [E; E]T*= Q[cH ]: where cH (of degree -2) is the Euler class of V (H) in the sense that it is the* * composite 1^e(V ) V (H) 2 E = E ^ S0 -! E ^ S ' E ^ S . Proof: We calculate the additive structure of [E; E]T*by passage to limit* *s from the case of finite skeleta. Of course we know that [oe0H; E]T*= Q so that by ind* *uction on k we see that [E(2k); E]T*is nonzero only in degrees 0; -2; -4; : :;:-2k,* * in each of which it is Q. The additive result follows using the Milnor exact sequence. Choose a generator cH : E -! 2E of [E; E]T-2; for the multiplicat* *ive statement it is sufficient to show that all composites cnHfor n 0 are essentia* *l. For this it is enough to show that cH induces an isomorphism in ssT2kfor k 1. This fol* *lows by the argument used in the proof of 2.3.5, since cH is the unique extension of th* *e composite E(2) -! E(2)=E(0)' oe2H-! 2E, in which the first map is the collapse map and the last is the inclusion of the bottom cell. To see that cH may be chosen to be the Euler class it is enough to observe th* *at the Euler class is nontrivial in ssT2. But the cofibre of the Euler class map is E ^ G* *=H+ ' oe1H, which has ssT*concentrated in degree 2. __|_| For us the fundamental object will be the Q[cH-]module ssT*(X); notice that c* *H decreases the degree of a homotopy element by 2. 2.4. OPERATIONS. 33 Theorem 2.4.2. For any object X of T-Spec=H , the Q[cH-]module ssT*(X) is tor* *sion in the sense that for any element x 2 ssT*(X) there is a natural number n so that * *(cH )nx = 0. Proof: In fact x is supported on a finite T-Spec=H -subcomplex X0 X in the sens* *e that x = i*x0for some x0 2 ssT*(X0) where i denotes the inclusion. Since ssT*(X0) i* *s bounded below, it is immediate that cnHx0= 0 for sufficiently large n. __|_| Note in particular that this implies that Q[cH ]itself does not occur as ssT** *(X) for any object X of T-Spec=H . Accordingly, the function spectrum F (E; E) is no* *t an object of T-Spec=H , so that T-Spec=H does not have arbitrary products. We should record the Q[cH-]modules for the spectra we have already studied. Lemma 2.4.3.There are Q[cH-]module isomorpisms ssT*(E) = I(H); where I(H) = -2Q[cH ; c-1H]=Q[cH ]and ssT*(E(2k)) = (Q[cH ]=ck+1H): Proof: The first calculation was part of the proof of 2.4.1, and the second fol* *lows since the inclusion E(2k)-! E induces an isomorphism of ssTifor i 2k + 1. __|_| The fundamental fact here is that ssT*(E) is Q[cH ]-injective, and this ex* *plains the notation. Indeed the Q[cH ]-module 2I(H) = Q[cH ; c-1H]=Q[cH ]is cH -divisible * *and hence injective. We have chosen to name I(H) rather than its double suspension becaus* *e I(H) is concentrated in degrees 0; 2; 4; 6; : :.: 34 2. FIRST STEPS. CHAPTER 3 The Adams spectral sequence. This chapter forms the core of Part I. In it we construct an Adams spectral seq* *uence for the category of T-spectra over H, and observe that it collapses to a short exact se* *quence: this gives us control over morphisms between F-spectra. In Section 3.1 we state the * *theorem and outline the strategy. In Section 3.2 we construct an Adams resolution. In* * Section 3.3 we prove the Hurewicz and Whitehead theorems, and deduce convergence from t* *hem, provided the codomain is bounded below. Finally, in Section 3.4 we are able to * *deduce the general case, because of the simplicity of the algebra. 3.1.The Adams short exact sequence. Our aim now is to prove the following theorem. Theorem 3.1.1. (The Adams short exact sequence.) For any spectra X and Y in T-Spec=H there is a short exact sequence 0 -! ExtQ[cH](ssT*(X); ssT*(Y )) -! [X; Y ]T*-! Hom Q[cH](ssT*(X); ssT*(Y )* *) -! 0: We note that one immediate consequence is a Whitehead theorem based on ssT*for T-Spec=H -spectra. Unlike the naive Whitehead theorem 2.3.3 based on ssH*, thi* *s has real content, and will be a major step in the proof (see 3.3.2 below). On the o* *ther hand, the resulting theorem allows us to be more quantitative and recover ssH*from ss* *T*. Indeed we know ssT*(oe0H) = Q so that a little homological algebra gives the result. Corollary 3.1.2.For any spectrum Y in T-Spec=H there is a short exact sequen* *ce 0 -! ssT*(Y )=cH -! ssH*(Y ) -! -1ann(cH ; ssT*(Y )) -! 0: __|_| The strategy is the usual one. The result is rather straightforward when ssT** *(Y ) is injective as a Q[cH ]-module, and the general case follows by resolving Y using spectra w* *ith Q[cH ]- injective ssT*. The procedure is somewhat complicated in the present case by th* *e fact that S0 is not an object of T-Spec=H and by the fact that T-Spec=H does not admit ar* *bitrary products. The pattern of proof is to establish the theorem in the following cases in tu* *rn. 3.1.3.Y = E and arbitrary X. 35 36 3. THE ADAMS SPECTRAL SEQUENCE. 3.1.4.Y bounded below of finite type and arbitrary X. 3.1.5.Arbitrary Y and finite X. 3.1.6.Any Y with ssT*(Y ) injective and arbitrary X. 3.1.7.The general case. Of course the case when ssT*(Y ) is injective is particularly easy. Lemma 3.1.8. If the Q[cH-]module ssT*(Y ) is injective and the functor ssT*: [X; Y ]T*-! Hom Q[cH](ssT*(X); ssT*(Y )) is an isomorphism for X = oe0Hthen it is an isomorphism for all objects X of T-* *Spec=H . Proof: Since ssT*(Y ) is Q[cH ]-injective, the codomain is a cohomology theory * *of X, and ssT*is a natural transformation of cohomology theories on T-Spec=H . Since any * *object of T-Spec=H can be constructed from cells oenH, the result follows. __|_| This shows that 3.1.5 implies 3.1.6. It also shows that 3.1.3 follows from a * *lemma. Lemma 3.1.9. Passage to homotopy induces an isomorphism ~= T 0 T [oe0H; E]T*-! Hom Q[cH](ss*(oeH); ss*(E)): Proof: Both sides are simply Q in degree 0, and the calculation of ssT*(E) i* *n 2.4.3 showed that the inclusion oe0H-! E induces an isomorphism in ssT0. __|_| The other easy step is that 3.1.4 implies 3.1.5, which follows because any ma* *p from a finite spectrum X factors through a finite subspectrum, and similarly in algebr* *a. More precisely if X is finite, ssT*(X) is finitely generated and hence [X; Y ]T*= li* *m![X; Yff]T*, and * *ff Hom Q[cH](ssT*(X); ssT*(Y )) = lim!Hom Q[cH](ssT*(X); ssT*(Yff)). Since Q[cH ]i* *s Noetherian the ff result for Hom implies the corresponding result for Ext, and by naturality of t* *he short exact sequence we obtain the result for [X; Y ]T*by passage to limits from thos* *e of [X; Yff]T*. The remaining steps are that 3.1.3 implies 3.1.4 and that 3.1.6 implies 3.1.7* *. Each of these require a substantial amount of work, and we explain them in the three fo* *llowing sections. 3.2.Construction of the Adams resolution. In this section and the next we prove that 3.1.3 implies 3.1.4 which is the p* *rincipal step in the proof of Theorem 3.1.1. The method is the usual one for constructin* *g Adams spectral sequences: we show in this section that there are enough injectives fo* *r which the theorem is known, so that we may construct an Adams resolution of Y , and in th* *e next we show that the resulting Adams resolution is convergent. The implication then fo* *llows as usual by applying [X; .]T to the Adams resolution: the E2 term is identified by* * 3.1.3, and convergence follows from that of the resolution. 3.2. CONSTRUCTION OF THE ADAMS RESOLUTION. 37 Suppose then that Y is bounded below and of finite type. We begin by construc* *ting an Adams resolution Y -! I0 -! I1 -! I2 -! . . . in T-Spec=H , where the associated sequence 3.2.1. ssT*(Y ) -! ssT*(I0) -! ssT*(I1) -! ssT*(I2) -! . . . is a resolution of ssT*(Y ) by injective Q[cH-]modules, and where Theorem 3.1.1* * is known to hold for [X; Ij]T*. Of course once 3.1.1 is proved we shall know that there is * *a resolution stopping at I1, but that is not obvious at present. In fact we show how to construct a ssT*-monomorphism Y -! I0 with I0 a bounde* *d below locally finite wedge of suspensions of E. The next step in the resolution is* * obtained by applying this to Y1 = cofibre(Y -! I0), and the rest of the resolution is const* *ructed by iterating the procedure. We shall use some elementary facts about Q[cH-]modules, which we list for con* *venience. Firstly, any finitely generated Q[cH-]module is a direct sum of cyclic modules,* * each of which is a suspension of either Q[cH ]itself or a module of form Q[cH ]=ckHfor some k* * 0. Next recall that a Q[cH ]-module M is injective if and only if it is cH divisible. * *In particular all direct limits of injectives are injective, and the module I(H) = -2Q[cH ; c* *-1H]=Q[cH ] is injective. Anyway, for any Q[cH ]-module M and any x 2 M of degree n there is a* * Q[cH ]- morphism OE : M -! nI(H) with OE(x) 6= 0: indeed, since I(H) is injective we ma* *y extend a homomorphism from the cyclic submodule Q[cHx]over M. Proposition 3.2.2.If Y is a locally finite and bounded below spectrum in T-Sp* *ec=H , there is a sequence of integers nitending to infinity with i, and a map _ k : Y -! niE; i inducing a monomorphism in ssT*. Furthermore if ssT*(Y ) is (c+1)-connected we * *may arrange that ssTn(k) is an isomorphism for n c + 3. Consequently the fibre Y1 of k is * *also a locally finite and bounded below spectrum in T-Spec=H , and if ssT*(Y ) is (c+1)-connec* *ted ssT*(Y1) is (c + 2)-connected. The apparently strange indexing will be explained below by the fact that the * *connectivity of ssT*(Y ) is one more than that of Y . Proof: By suspending Y as necessary we may suppose ssT*(Y ) is 0-connected. For any nonzero x 2 ssTn(Y ), by 3.1.3 we can find a map k(x) : Y -! n-1E * *so that k(x)*(x) 6= 0. Accordingly we let xirun over a Q-basis of ssT*(Y )Wwith nibeing* *Qthe degree of xi. By hypothesis, the integers nitend to infinity, and therefore iniE ' i* *niE. We may thus form the map k by giving it components k(xi): Y _ k : Y -! niE ' niE: i i This certainly gives a monomorphism in ssT*. Since ssT*(E) = I(H) is Q in * *degree 1, and otherwise concentrated in degrees greater than 2, the construction also * *gives an 38 3. THE ADAMS SPECTRAL SEQUENCE. isomorphism in the bottom two degrees, ssT1and ssT2. __|_| 3.3.Convergence of the Adams resolution. The convergence statement says that all of Y is ultimately squeezed into the * *injective spectra Ij. It is convenient to convert the Adams resolution 3.2.1 into the Ada* *ms tower .. . # Y2 - ! -2I2 # Y1 - ! -1I1 # Y = Y0 - ! I0: By construction, each of the maps in the tower induces zero in ssT*so that ssT** *(holim Ys) = 0. * * s However we want a stronger statement. Theorem 3.3.1. (Convergence.) If Y is a bounded below object of T-Spec=H , t* *hen the Adams tower converges in the sense that holimYs' *. s As usual, the Whitehead and Hurewicz theorems are the basis for the proof. Theorem 3.3.2. (Whitehead theorem for T-Spec=H .) If X and Y are spectra in T* *-Spec=H and f : X -! Y induces an isomorphism in ssT*then f is an equivalence. Proof: The hypothesis is equivalent to saying the cofibre Z of f is ssT*-acycli* *c, and it is enough to show Z is contractible. Indeed, by the naive Whitehead theorem 2.3.3* *, it is enough to show that ssH*(Z) = 0. This follows from the Hurewicz theorem 3.3.3 b* *elow. __|_| As remarked in its proof, the Whitehead theorem is the statement that a ssT*-ac* *yclic object of T-Spec=H is in fact contractible. The Hurewicz theorem is a little more qua* *ntitative. The extra suspension in the statement is due to that in the Wirthm"uller isomor* *phism. If X is a c-connected object of T-Spec=H it may be constructed with cells only in d* *imensions c+1 and above; since ssT*(oec+1H) is (c+1)-connected it follows that ssT*(X) is* * (c+1)-connected. Theorem 3.3.3. (Hurewicz theorem for T-Spec=H .) If Z is an object of T-Spec* *=H and ssT*(Z) is (c + 1)-connected then Z is c-connected. If in addition ssTc+2(* *Z) 6= 0 then ssHc+1(Z) 6= 0, so the result is sharp. Proof: We use the cofibre sequence G=H+ -! S0 -! SV (H). By the universal Thom * *iso- morphism 2.3.6(a) [SV (H); Z]T*~=[S2; Z]T*and the cofibre sequence gives the Gy* *sin sequence 3.3.4. : :-: ssTn-1(Z) cH-ssTn+1(Z) - ssHn(Z) - ssTn(Z) cH-ssTn+2(Z) - : ::: It follows that ssH*(Z) is c-connected as required. __|_| 3.4. TORSION Q[cH]-MODULES AND GEOMETRIC IMPLICATIONS. 39 Proof of 3.3.1: First note that the Hurewicz theorem 3.3.3 allows us to upgrad* *e the conclusion of 3.2.2 and deduce that if Y is c-connected then Y1 is (c + 1)-conn* *ected.QW Thus the connectivity of the terms Ys tends to infinity, and therefore sYs * *' sYs. Since the geometric fixed point functor K commutes withQwedges,Qthis product is* * in T-Spec=H . Finally the cofibre sequence holim Ys -! sYs -! sYs shows that * *the s spectrum holimYs lies in T-Spec=H . s We may therefore apply the Whitehead theorem 3.3.2 to deduce convergence from* * the fact that ssT*(holim Ys) = 0. __|_| s We have now established the existence and convergence of an Adams resolution * *of Y by locally finite bounded below wedges of suspensions of the spectra E, and we * *obtain a spectral sequence by applying [X; .]T*. We find Es;t2= Exts;tQ[c(ssT*(X); ssT*(* *Y )) by 3.1.3, H ] and we find that it converges to [X; Y ]T*by the convergence theorem, 3.3.1. Si* *nce Q[cH ] is of global dimension the spectral sequence takes the form of the short exact * *sequence of 3.1.1. This completes the proof that 3.1.3 implies 3.1.4. 3.4.Torsion Q[cH ]-modules and geometric implications. In this section we prove that 3.1.6 implies 3.1.7 in the proof of Theorem 3.1* *.1. This completes the proof of Theorem 3.1.1. In fact it simply remains to exploit the fact that realizable modules are tor* *sion (2.4.2), by showing it implies that any topologically realized Q[cH ]-module can be embe* *dded in a realizable injective, and that all torsion injectives are topologically realiza* *ble. L Proposition 3.4.1.If M is a torsion Q[cH-]module then there is an embedding M* * -! iniI(H) for suitable indexing set and integers ni. Furthermore, if M is inje* *ctive the embedding e may be taken to be an isomorphism. Proof: We consider the filtration [ 0 = F0M F1M F2M . . . FjM = M j of M, where FjM = {x 2 M | cjHx = 0}; this exhausts M because M is a torsion mo* *dule. Now the associated graded module GrM has trivial cH -action, so it is natural t* *o view it as a graded vector space. We shall construct an embedding e : M -! GrM I(H). S* *ince Hom Q[cH](M; N) = limHom Q[cH](FjM; N) it is sufficient to give e one filtratio* *n at a time, j using the notation 0 1 M ej: FjM -! @ FiM=Fi-1MA I(H) 0ij and starting with the unique homomorphism when j = 0. Now if we suppose that ej* * has been defined, we form ej+1as follows. The codomain is injective so ej extends t* *o a map ^ej on Fj+1M with the same codomain, and we use the canonical embedding Fj+1M=FjM -! Fj+1M=FjM I(H) as the final coordinate. This final coordinate ensures that no * *element of Fj+1M \ FjM lies in the kernel of e. 40 3. THE ADAMS SPECTRAL SEQUENCE. Now, if M is injective, we show that e is also an epimorphism. Indeed, if e i* *s not surjec- tive we may choose an element y not in the image, with torsion order as small a* *s possible. More precisely we suppose ckHy = 0 and that the image of e includes all element* *s annihi- lated by ck-1H; note that k 2 by construction of e. By hypothesis, we may choo* *se x so that e(x) = cH y, and since M is injective there is an element "xso that cH "x=* * x. Now, let y"= e("x), so that cH (y - "y) = 0. By construction of e there is an x0with e(x* *0) = y - "y; hence e(x0+ "y) = y, contradicting the choice of k. __|_| Corollary 3.4.2. Any injective torsion Q[cH ]-module I is realizable in the s* *ense that there is a T-spectrum X(I) over H so that ssT*(X(I)) ~=I. __|_| We may now complete the proof of Theorem 3.1.1. Indeed for any spectrum Y in T-Spec=HW, the Q[cH ]-module ssT*(Y ) is torsion by 2.4.2, and hence by 3.4.1 e* *mbeds in ssT*( ini-1E) for suitable indexing set and integers ni. By the general inj* *ective case 3.1.6, this algebraic embedding is realized by a map k : Y -! I0 of spectra. Si* *nce Q[cH ] has global dimension 1, the cofibre of k has injective ssT*and it is thus reaso* *nable to call it I1. Therefore we have a cofibre sequence Y -! I0 -! I1 realizing the Q[cH ]-* *injective resolution of ssT*(Y ), and, by 3.1.6, Theorem 3.1.1 holds for calculating [X; * *I0]T*and [X; I1]T*: Theorem 3.1.1 follows for Y . __|_| Of course we could have constructed a long Adams resolution for arbitrary Y w* *ithout dealing with the bounded below case first, but the proof of the convergence the* *orem 3.3.1 would have been obscured by algebraic complications. Since only the zeroth and first terms I0 and I1 are required, we shall use th* *e letters I and J to denote them. Although the construction is neither canonical nor functo* *rial we shall use the notation Y -! I(Y ) -! J(Y ) for an Adams resolution of an object Y of T-Spec=H with I(Y ) and J(Y ) being * *wedges of suspensions of E and with the induced sequence 0 -! ssT*(Y ) -! ssT*(I(Y )) -! ssT*(J(Y )) -! 0 an injective resolution of ssT*(Y ). The Adams short exact sequence 3.1.1 and the realization of injectives 3.4.2 * *are enough to allow us to make algebraic models, both of T-Spec=H and of T-Spec=F . In th* *e next section we begin to establish a suitable abstract framework for this categorica* *l reprocessing, but we shall first state the result we are going to prove. This should also pro* *vide a particular case to bear in mind in the course of the coming abstraction. Consider the homotopy category of differential graded Q[cH ]-modules. Hencef* *orth we often abbreviate `differential graded' to `dg'. Next we form the derived categ* *ory of dg Q[cH ]-modules: its objects are dg Q[cH ]-modules, and its morphisms are obtain* *ed from 3.4. TORSION Q[cH]-MODULES AND GEOMETRIC IMPLICATIONS. 41 the homotopy category of dg Q[cH ]-modules by inverting homology isomorphisms. * *More practically, maps in the derived category from M to N are simply [M; N] = [M; N] where M is projective as a Q[cH ]-module and equipped with a homology isomorphi* *sm M -! M, and N is injective as a Q[cH-]module and equipped with a homology isomo* *r- phism N -! N. In fact we want to restrict attention to torsion modules, and for* *m the derived category of dg torsion Q[cH-]modules. Since there are no projective tor* *sion Q[cH-] modules we are forced to use the construction of the derived category in terms * *of injectives; we have seen that a realizable Q[cH-]module N admits a realizable dg Q[cH-]modu* *le N. More precisely, we shall prove the following theorem. Theorem 3.4.3. There is an equivalence of triangulated categories between T-S* *pec=H and the full subcategory D(torsQ[cH ])of the derived category of dg Q[cH-]modul* *es torsion Q[cH ]-modules: T-Spec=H ' D(torsQ[cH ]): The proof will be completed in Section 4.3 after we have developed the necess* *ary algebra. 42 3. THE ADAMS SPECTRAL SEQUENCE. CHAPTER 4 Categorical reprocessing. An Adams spectral sequence arises from a homology functor on a triangulated cat* *egory with values in an abelian category. The algebraic archetype of a triangulated c* *ategory is the derived category of an abelian category. In this chapter we show that the * *algebraic simplicity of the categories which concern us allows us to deduce the structure* * of the triangulated categories of spectra from their abelianizations. We begin in Section 4.1 by recalling the construction of the derived category* * in an appropriate form. In Section 4.2 we give abstract conditions (one dimensionalit* *y being the most important) under which an abelian category determines a triangulated categ* *ory of which it is the abelianization, and in Section 4.3 we show this applies to the * *two cases we have encountered so far: torsion Q[cH ]-modules and F-finite torsion OF-modu* *les. We show in the next chapter that it also applies to a suitable abelianization of t* *he category of all T-spectra. 4.1. Recollections about derived categories. For a differential graded algebra R the associated derived category D(R) is o* *btained from the category of dg R-modules by forming the category of fractions in which* * homology isomorphisms are inverted. For a category of fractions to be useful one must gi* *ve a fairly concrete construction: this has the additional merit of proving its existence. * *Since most references restrict themselves to derived categories of ungraded algebras, and * *since we need to discuss derived categories of other abelian categories, we shall spend * *some time outlining well-known constructions. In this section we concentrate on giving a * *very concrete approach, but the construction is a special case of Quillen's construction of t* *he homotopy category of a closed model category [21]. We therefore use terminology consist* *ent with Quillen's, and give details in Appendix B of how to regard our algebraic model * *categories as closed model categories. We shall repeatedly use the following formality in constructing relevant deri* *ved cat- egories. This is more familiar to topologists in its dual counterpart, where i* *t is used to construct a category in which weakly equivalent spaces are isomorphic. The dual* * properties are then the cellular approximation theorem, and two theorems of J.H.C.Whitehea* *d. Suppose given a category C, a collection S of morphisms which we intend to in* *vert, and a collection I of fibrant objects (which have the character of injectives) * *such that the 43 44 4. CATEGORICAL REPROCESSING. following conditions hold. Conditions 4.1.1.(i) For every object X of C there is an element s : X -! X o* *f S with X 2 I (ii) If s : I -! I0is an element of S with I; I02 I then s is an isomorphism, a* *nd (iii) For every object I of I and morphism s : X - ! Y in S the induced map s** * : C(Y; I) -! C(X; I) is a bijection. We may then extend to a functor : C -! C by filling in the diagram X -! X f # # f Y -! Y using (iii). Next define a category S-1C by giving it the same objects as C and* * morphisms S-1C(X; Y ) = C(X; Y ), and a functor l : C -! S-1C by using . Lemma 4.1.2. The morphism l : C -! S-1C constructed above is the localization* * of the category C away from S. __|_| Now suppose given a graded algebra R with zero differential, and let C be the* * homotopy category of differential graded R-modules. More generally, we may suppose given* * a graded abelian category A, and let C be the category of differential graded objects of* * A. For dg A- objects X and Y , we shall denote the dg abelian group of A-morphisms by Hom(X;* * Y ), and the group of homotopy classes by [X; Y ]; it is useful to note that [X; Y ] = H* *0(Hom(X; Y )). Of course, a short exact sequence of dg A-objects induces a long exact sequen* *ce in homology. It follows that a cofibre sequence of dg A-objects also induces a lo* *ng exact sequence in homology, but cofibre sequences have the advantage that they are pr* *eserved by application of Hom(.; Y ) and Hom(X; .), and hence they also induce long exact * *sequences on applying [.; Y ] and [X; .]. We can only expect short exact sequences of dg * *A-objects to induce long exact sequences of [.; Y ] if Y consists of injective A-objects and* * of [X; .] if X consists of projective A-objects. Now let S be the class of homology isomorphisms and I be the collection of di* *fferential graded A-objects which may be formed from injective A-objects with zero differe* *ntial by a finite number of cofibre sequences. Proposition 4.1.3.If A is of finite homological dimension, then with these ch* *oices of S and I, Conditions 4.1.1 hold and hence the derived category D(A) may be forme* *d from the homotopy category of dg A-objects by inverting homology isomorphisms. Proof: The finiteness of homological dimension is only used to verify Condition* * (i), so we begin without assumption on A. Lemma 4.1.4. If J is an injective A-object with zero differential and E is ex* *act then [E; J] = 0. 4.1. RECOLLECTIONS ABOUT DERIVED CATEGORIES. 45 Proof: Suppose f : E -! J is a map of dg A-objects. Since d = 0 on J it follows* * that f = 0 on im(d) = ker(d). Hence f factors through E=ker(d); by injectivity of J* * this map may be extended along the inclusion E=ker(d) E, providing a map h : E -! J* * of A-objects. Since d = 0 on J we find dh + hd = f, so that f is null-homotopic. * *__|_| Corollary 4.1.5.Conditions 4.1.1 (ii) and (iii) hold for these choices of C, * *I and S. Proof: We first show Condition (iii) holds. If f : X -! Y is a homology isomorp* *hism, its cofibre E is exact. Since a cofibre sequence induces a long exact sequence * *in the first variable it is enough to show [E; I] = 0 for all objects I in I. Since I may be* * formed from injective A-objects using finitely many cofibre sequences this follows from Lem* *ma 4.1.4 using exact sequences in the second variable. Now for Condition (ii) we must find an inverse t to s : I -! I0. By Condition* * (iii) we ~= 0 have a bijection s* : [I0; I] -! [I; I], and similarly with I in the second v* *ariable. Now choose t so that s*t = id: this ensures ts = id, and since s*(st) = s*(id) we c* *onclude that st = id as well. __|_| Let us now turn to Condition (i) and begin with a useful observation. Lemma 4.1.6.If X -! Y -! Z is a cofibre sequence and two of the three objects* * are admit homology isomorphisms to an object of I, then so does the third. Proof: Since the property is invariant under isomorphism and suspension we may * *as well suppose that there are homology isomorphisms e : X -! I and f : Y -! J; it is e* *nough by Verdier's axiom and the 5-lemma to show that we may complete the square up to h* *omotopy X -! Y e # # f I . .>.J: But this is immediate from Condition (iii). __|_| The analogue for short exact sequences now follows. q Corollary 4.1.7.If 0 -! X -i! Y -! Z -! 0 is a short exact sequence and two * *of the three objects admit homology isomorphisms to an object of I, then so does t* *he third. Proof: The main point is that by Condition (iii), if we have a homology isomorp* *hism A -! B and one of the two objects admits a homology isomorphism to an object of* * I, so too does the other. If X is the term not known to admit a homology isomorphism to an object of I,* * we note that there is a homology isomorphism X -! F (q), where F (q) denotes the mappin* *g fibre of q, and apply 4.1.6. Similarly for Z we use the homology isomorphism C(i) -! * *Z where C(i) denotes the mapping cone of i. Finally for Y we use the short exact sequence 0 -! Y - ! C(i) -! X -! 0, the homology isomorphism C(i) -! Z and the first case. __|_| 46 4. CATEGORICAL REPROCESSING. Lemma 4.1.8. If A is of finite homological dimension then Condition 4.1.1 (i)* * also holds. Proof: An arbitrary dg A-object X has dg sub-object dX of boundaries, and both * *dX and X=dX have trivial differential. Accordingly, by 4.1.7 and the above remarks, it* * is sufficient to deal with the case that X has zero differential. For this we argue by induction on the length of injective resolution. Indeed * *for an injec- tive object we may use the identity and any A-object X of positive injective di* *mension lies in an exact sequence 0 -! X -! I -! Q -! 0, with I injective and Q of lower inj* *ective dimension. __|_| This completes the proof of Proposition 4.1.3 __|_| Finally it will be convenient to begin the construction of an Adams spectral * *sequence for objects of D(A). Consider the natural transformation : [X; Y ]derived-! HomA(H*(X); H*(Y )): Lemma 4.1.9. If H*(Y ) is injective then is an isomorphism. Proof: First note that if I is an injective object of A, regarded as a dg A-obj* *ect with zero differential, then [X; I] = H0(Hom(X; I)) = Hom(H*(X); I): Since [X; I] = [X; I]derivedwhen I is injective by construction, this proves th* *e result if Y has zero differential. It also shows that if H*(Y ) is injective, then there is a m* *ap Y -! H*(Y ) inducing the identity in homology. Thus [X; Y ]derived= [X; H*(Y )]derived, and* * the general case follows. __|_| Following the abuse of notation standard in topology we henceforth omit the s* *ubscript derived. Thus [X; Y ] now denotes maps in the derived category. 4.2. Split linear triangulated categories. The aim of this section is to show that a sufficiently simple triangulated ca* *tegory is determined by its abelianization. We treat this abstractly for the usual reaso* *ns: firstly it becomes apparent exactly what properties are being used, and secondly there * *are three instances of the phenomenon that we want to treat simultaneously. Thus we suppose given a triangulated category D, which might be a derived cat* *egory of some abelian category or a category of spectra; we shall use X; Y; : :t:o de* *note objects of D, and [X; Y ] to denote morphisms. We also suppose given a graded abelian c* *ategory A, and a functor H* : D -! A, exact in the sense that triangles are taken to lo* *ng exact sequences; in the applications H* is some form of homology. The idea is that H* ** is a linearization, so that we require in particular the Inverse Function Theorem ho* *lds in the sense that homology isomorphisms are invertible (i.e. that H* creates isomorphi* *sms). 4.2. SPLIT LINEAR TRIANGULATED CATEGORIES. 47 The example to bear in mind at present is the abelian category A = torsQ[cH ]* *-mod, and the triangulated category D = T-Spec=H , with ssT*providing the linearization. One can hope to incorporate non-linear phenomena by using derived functors in* * A; this is the situation when there is an Adams spectral sequence based on H*. One coul* *d think of this as a formal power series description of D with coefficients in A. Howe* *ver, if two or more derived functors are involved, there will usually be differentials in t* *he Adams spectral sequence, and one cannot hope to recover D from A alone. Suppose then * *that A has injective dimension 1, so that the Adams spectral sequence takes the form o* *f a short exact sequence 4.2.1. 0 -! ExtA(H*(X); H*(Y )) -! [X; Y ] -! HomA(H*(X); H*(Y )) -! 0: This happens in case D = T-Spec=H by Theorem 3.1.1. The Adams short exact sequence shows that objects X are classified by their i* *mages H*(X) in A. Indeed, if H*(X) ~=H*(Y ) we may lift an algebraic isomorphism to a* * map f : X -! Y in D; since it is a homology isomorphism by construction, it is an e* *quivalence in D. If we are to recover D from A we should therefore require that H* is surj* *ective on isomorphism classes of objects. It is sufficient to show enough injectives are * *in the image. Lemma 4.2.2.If enough injectives are in the image of H* then all objects of A* * are in the image of H*. By Corollary 3.4.2 this applies when D = T-Spec=H . Proof: First note that if I is a realizable injective then any retract of I is * *also realizable. Indeed a retract of I is of the form eI for some idempotent e : I -! I. Now if * *H*(X(I)) = I then we may lift e uniquely to a map X(e) : X(I) -! X(I) by the Adams short exa* *ct sequence, and X(e) is also idempotent. Thus H*(X(e)X(I)) = eI as required. f Now an arbitrary object M of A has an injective resolution 0 -! M -! I -! J * *-! 0, and we may realize I by X(I), J by X(J) and the map f by a unique map X(f) : X(* *I) -! X(J) by the Adams short exact sequence. Thus the fibre of X(f) realizes M. __|* *_| However we still cannot expect to recover D from A without further hypothesis* *, because the extension 4.2.1 will not generally be split, In the applications, every obj* *ect M of A has a splitting M = M+ M- so that Hom(Mffi; Nffl) = 0 and Ext(Mffi; Nffl) = 0 * *if ffi 6= ffl, and so that (M)ffl= (Mffl+1). For example this applies when A = torsQ[cH ]-mods* *ince any module splits as a sum of its even graded and odd graded parts. We shall ca* *ll such a category A a split one dimensional graded abelian category. We say that a tri* *angulated category D is split linear if there is a functor H* : D -! A, creating isomorph* *isms to a split one dimensional category A, which is surjective on isomorphism classes of* * objects and so that there are natural Adams short exact sequences 4.2.1. We shall see that * *a split one dimensional abelian category A is the linearization of a split linear triangula* *ted category D unique up to equivalence of categories. Furthermore, amongst categories D ari* *sing as homotopy categories, this equivalence preserves triangles. 48 4. CATEGORICAL REPROCESSING. We have motivated the above definitions with the example D = T-Spec=H , A = torsQ[cH ]-modand H* = ssT*, but there is a general example. Lemma 4.2.3. If A is a split one dimensional abelian category then the derive* *d category D(A) is a split linear triangulated category with linearization A. Proof: Homology gives a functor H* : D(A) -! A. We see this is surjective on ob* *jects by using dg objects with zero differential. The Adams short exact sequence is e* *stablished in the usual way. First, note that, by 4.1.9, it holds when H*(Y ) is injective* *, and it only remains to prove that all objects have Adams resolutions by objects of this typ* *e. In fact, homology gives a natural map [X; Y ] -! HomA(H*(X); H*(Y )); where [X; Y ] denotes maps in the derived category, which is an isomorphism whe* *n H*(Y ) is injective. Now, for an arbitrary object Y in the derived category there is a* * triangle Y -! I -! J so that 0 -! H*(Y ) -! H*(I) -! H*(J) -! 0 is an injective resolution of H*(Y ) in A. To see this, find an embedding e : H* **(Y ) -! I with I injective. Regard I as a dg object with zero differential and use the in* *jective case to lift e to a map Y -! I in the derived category. The mapping cone of this map ne* *cessarily has injective homology, and hence qualifies as J. This completes the proof that* * D(A) is a split linear triangulated category with linearization A. __|_| Proposition 4.2.4.If A is a split one dimensional abelian category and D is a* * split linear triangulated category D with linearization A, then D is equivalent to D(A). Fur* *thermore, the Adams short exact sequence is split. Proof: Consider the two split linear triangulated categories, D and E = D(A). E* *ach is equipped with an exact functor H* to A, which is essentially surjective on obje* *cts, and which creates isomorphisms. Furthermore we have an exact sequence 4.2.1 for obj* *ects of D, and similarly for E. We shall construct a functor p : D -! E which is an equivalence of categories* * over A. First note that the splitting of objects of A lifts to splittings of objects of* * D and E. Indeed if H*(X) = H+ H- then, since H* is essentially surjective, there is an object * *X+ with H*(X+) ~=H+. Furthermore, it follows from the Adams short exact sequence 4.2.1 * *that we may lift the projection H*(X) -! H+ to a map X -! X+; arguing similarly with H-* * we obtain a map X -! X+ _ X- which is a homology isomorphism and hence an equivale* *nce. We refer to objects X with X = X+ or X = X- as pure parity objects; note that i* *f X and Y are both of pure parity then the Adams short exact sequence either says [* *X; Y ] = HomA(H*(X); H*(Y )) if they are of the same parity, or [X; Y ] = ExtA(H*(X); H** *(Y )) if they are of opposite parity. 4.2. SPLIT LINEAR TRIANGULATED CATEGORIES. 49 Following Adams [1], we may consider d and e invariants of a map f : X -! Y i* *n D. Any map f has a d invariant d(f) := H*(f) 2 HomA(H*(X); H*(Y )), and if d(f) = * *0 then the class of the extension 0 -! H*(Y ) -! H*(C(f)) -! H*(X) -! 0 gives the e invariant, e(f) 2 ExtA(H*(X); H*(Y )). This much applies to any tri* *angulated category, and in our case the e invariant is the place of a morphism in the Ada* *ms short exact sequence 4.2.1. Now choose such a splitting for each object of D and E, and a preferred repre* *sentative from each isomorphism class of pure parity objects of E. Define p on objects by* * p(X) = p(X+)_p(X-) where p(Xffi) is the preferred representative of objects with homol* *ogy H*(Xffi). On morphisms f : X -! Y between pure parity objects we take p(f) to be the map * *with the same d or e invariant as f. For arbitrary X and Y we view f as a 2 x 2 matr* *ix of maps between the pure parity summands of X and Y ; this gives a splitting of the Ada* *ms short exact sequence 4.2.1, so that any map f : X -! Y has a well defined e invariant* *. We define p(f) by letting p act on each of the four matrix entries. By matrix mult* *iplication, the functoriality of p on all morphisms will follow from that on morphisms betw* *een pure parity objects. There are four cases to consider. f g Consider maps X -! Y - ! Z between pure parity objects. If X; Y and Z have t* *he same parity d(gf) = d(g)d(f) since H* is a functor, and hence p(gf) = p(g)p(f).* * Similarly if either f or g is a map between objects of the same parity, then functorialit* *y comes from the naturality of the e invariant in the sense that e(gf) = e(g)d(f) or e(gf) =* * d(g)e(f) as the case may be. Finally if both f and g are maps between objects with opposite* * parities then f and g both have Adams filtration 1, so gf has Adams filtration 2 and so * *gf = 0 because A is one dimensional; p(g)p(f) = 0 for analogous reasons. This complet* *es the construction of a functor p : D -! E; a functor in the reverse direction can be* * constructed in the same way, and their composites are isomorphic to the relevant identity f* *unctors. Thus p is an equivalence as required. __|_| It remains to consider the behaviour of p on triangles, and we begin the disc* *ussion without f g h further assumptions on the categories. Suppose given a triangle X -! Y -! Z -* *! X in D. First note that, because H* is exact, p commutes with suspensions. The ideal* * outcome p(f) p(g) p(h) would be to show that p(X) -! p(Y ) -! p(Z) -! p(X) is a triangle in E. To do t* *his, we would have to show that a map determines the triangle it lies in, up to auto* *morphism, using only categorical constructions on its d and e invariants. Lemma 4.2.5.Let f : X -! Y be a morphism in D. The d and e invariants of f determine (a) its mapping cone Z up to isomorphism * *and (b) the d invariants of the maps g : Y -! Z and h : Z -! X. Proof: For (a) we note that it is only necessary to determine the isomorphism c* *lass of H*(Z). Indeed it lies in an extension 0 -! C -! H*(Z) -! K -! 0; 50 4. CATEGORICAL REPROCESSING. where C is the cokernel and K the kernel of H*(f). In fact the extension class * *is the image of e(f) under the composite ExtA(H*(X); H*(Y )) -! ExtA(H*(X); C) -! ExtA(K; C): The construction clearly identifies d(g) and d(h). __|_| To proceed further we wish to complete the diagram p(f) g0 0 h0 -p(f) p(X) -! p(Y )- ! Z - ! p(X) -! p(Y ) . =# =# L ff.. R =# =# p(f) p(g) p(h) -p(f) p(X) -! p(Y )- ! p(Z) - ! p(X) -! p(Y ) in which the first row is obtained by embedding p(f) into a triangle in E. Sin* *ce p is functorial, p(g)p(f) = p(gf) = 0 and so a map ff exists so that the square L co* *mmutes; the problem is to choose the map to make the square R commute. If this is possi* *ble, then a 5-lemma argument in homology shows that ff is an equivalence in E, and so the* * second row is isomorphic to the first, and hence is a triangle as required. The standard method of discussing such problems is that of Toda brackets, but* * these are not available in an arbitrary triangulated category. Since they are available i* *n the cases of interest to us we shall impose further conditions. Note in particular that D* *(A) is the homotopy category of the category of dg A-objects so that it does have Toda bra* *ckets. Theorem 4.2.6. If A is a split one-dimensional abelian category, and D is a s* *plit linear triangulated category of which A is the linearisation and which is the homotopy* * category of a model category then the equivalence D ' D(A) of 4.2.4 is an equivalence of triangulated categories. Proof: We established in 4.2.4 that the derived category E of differential grad* *ed objects of A is a split linear triangulated category with linearization A, and it was c* *onstructed as a homotopy category. We shall show that any other triangulated category D as* * in the statement is equivalent to E. Consider the diagram of differential graded objects of A and homomorphisms (i* *.e. not up to homotopy) A - ! B -i! C -ss! A --! B . =# =# L ff..R =# =# OE 0 - A - ! B -! C -! B -! B in which C is the mapping cone of , and i and ss are the canonical maps. A map ff exists so that L strictly commutes if and only if OE ' 0, and choice* *s of ff correspond exactly to null-homotopies H of OE. The condition that R commutes u* *p to homotopy is then that ss ' ff, which is equivalent to the existence of a null-* *homotopy K 4.3. THE ALGEBRAICIZATION OF THE CATEGORY OF T-SPECTRA OVER H. 51 of OE so that 1 ' H - K. In other words, there is a solution ff so that both * *L and R commute if and only if the Toda bracket < ; OE; > [A; A] exists and contains the identity. Exactly similar arguments apply to the category D (see [21, I.3]), so that th* *e result will follow once we show that the map p preserves Toda brackets. This follows from t* *he dis- cussion in [1, Section 5], and specifically from diagram (5.1). In fact we form* * the analogue of Adams' diagram (5.1) for calculating the Toda bracket in the model* * category whose homotopy category is D. Since f, g, h are successive maps in a triangle,* * 1 is an element of the Toda bracket and there are choices of null-homotopies so that th* *e composite HF ' 1 in Adams' notation. Now apply p to the resulting diagram in D, and concl* *ude that 1 = p(1) 2 . __|_| 4.3.The algebraicization of the category of T-spectra over H. In this section we quickly complete the proof of Theorem 3.4.3, by observing * *that T-Spec=H is a split linear triangulated category arising as the homotopy categ* *ory of a model category. We also prove the analogous theorem for F-spectra, once we have* * intro- duced the appropriate abelian category. Of course we take D = T-Spec=H , A = torsQ[cH ]-mod, and H*(X) = ssT*(X). It * *is easily verified that torsQ[cH ]-modis abelian and has injective dimension 1. The split* * condition arises since Q[cH ]is concentrated in even degrees, so that for any Q[cH-]modul* *e M we may take M+ to be the even-graded part of M and M- to be the odd graded part. The A* *dams short exact sequence 3.1.1 shows that A is the linearization of D. Finally, it follows from the construction of Lewis-May that T-Spec=H is the * *homotopy category of a model category. The model category we have in mind is the catego* *ry of F-spectra, with weak equivalences created by the functor X 7-! eH ssT*(X). Sinc* *e ssT*(X) = eH ssT*(X) for objects of T-Spec=H ,we see that homology is well defined. We co* *nclude that Theorem 4.2.6 applies to show there is an equivalence p : T-Spec=H -'! D(torsQ[cH ]) of triangulated categories; this completes the proof of Theorem 3.4.3. __|_| It may be instructive to give an example illustrating how maps are represente* *d in D(torsQ[cH ]). 52 4. CATEGORICAL REPROCESSING. Example 4.3.1.Consider a non-trivial map f : oe1H-! oe0H. First we note that * *ssT*(oe0H) = Q, and there is an exact sequence 0 -! Q -! I(H) cH-!3I(H) -! 0: Thus p(oe0H) = F (cH ), which we may illustrate: .. . . .. | # | Q degree 11 # . | Q | degree 10 | # | Q degree 9 # . | Q | degree 8 | # | Q degree 7 # . | Q | degree 6 | # | Q degree 5 # . | Q | degree 4 | # | Q degree 3 # . | Q | degree 2 # Q degree 1 Here degree is displayed vertically, the vertical arrows are multiplication by * *cH , and the differential is displayed diagonally. Now the map f : oe1H-! oe0Hhas zero d-inv* *ariant, and its e-invariant is a non-zero element of Q ~=Ext(3Q; Q). Thus p(f) is non-trivi* *al in all even degrees and zero in all odd degrees. The cofibre of f is equivalent to E(2), and it is natural to choose an Adams resolution corresponding to the minimal injective* * resolution 0 -! ssT*(E(2)) -! I(H) -! 5I(H) -! 0: The reader may find it instructive to show that C(p(f)) is quasi-isomorphic to p(E(2)). __|_| An alternative approach to T-spectra over H which some might find more natura* *l would be to prove first the analogue for the category T-Spec=F of all F-spectra, but* * we have preferred to avoid the relatively unimportant complications of this for as long* * as possible. Before we state the theorem, we must introduce some further conditions. If Y is any F-spectrum then Y ' EF+ ^ Y so ssT*(Y ) is a module over the r* *ing of operations OF := [EF+; EF+]T*. Using the Decoupling Theorem 2.2.3 we find that * *OF := 4.3. THE ALGEBRAICIZATION OF THE CATEGORY OF T-SPECTRA OVER H. 53 Q Q HQ[cH ]in the sense that in degree n we have (OF)n = HQ[cH ]n. Let eH 2 (O* *F)0 denote the idempotent which is projection onto the Hth factor, and let c 2 (OF)* *-2 denote the total cyclotomic class c with Hth coordinate cH (i.e. cH = eH c). Arbitrary modules over this ring may be quite unpleasant, but by the Decoupli* *ng Theo- rem 2.2.3, the geometrically occurring modules ssT*(Y ) for an F-spectrum Y are* * all rather well behaved. Indeed they all decompose as the sum M ssT*(Y ) = ssT*(Y (H)) H where ssT*(Y (H)) = eH ssT*(Y ) is actually a torsion module over Q[cH ]= eH OF. We therefore consider the following two conditions on an OF-module M. Condition 4.3.2.M is a torsion OF-module in the sense that for each x 2 M the* *re is a number N so that cN x = 0. and Condition 4.3.3.M is F-finite in the sense that it is the direct sum of its s* *ubmodules M(H): M M = M(H): H We refer to the category of full subcategory of OF-modules satisfying Condition* *s 4.3.2 and 4.3.3 as torsion F-finite modules, and denote it torsOfF-mod. Theorem 4.3.4. There is an equivalence p : T-Spec=F -'! D(torsOfF) of triangulated categories. Proof: We take D = T-Spec=F and A = torsOfF-mod. We have motivated our definiti* *ons by the fact that homotopy gives a functor ssT*: T-Spec=F -! torsOfF-mod: Furthermore it is clear from the Lewis-May construction that T-Spec=F is the ho* *motopy category of the model category of F-spectra with weak equivalences defined as u* *sual. By the Whitehead Theorem 3.3.2 for T-Spec=H , weak equivalences of F-spectra may e* *qually well be characterized as ssT*-equivalences. Since the idempotents eH show that the category torsOfF-modsplits to give an * *equiva- lence Y torsOfF-mod' torsQ[cH ]-mod H it follows that torsOfF-modis a split one dimensional graded abelian category.* * The De- coupling Theorem 2.2.3 showed that there was a similar splitting of T-Spec=F , * *and the map ssT*is compatible with these splittings. Hence we obtain Adams short exact * *sequences of the appropriate form to show that A is the linearization of D. __|_| 54 4. CATEGORICAL REPROCESSING. CHAPTER 5 Assembly and the standard model. So far, we have worked entirely locally: one finite subgroup at a time.* * By the decoupling theorem, that is sufficient to deal with arbitrary F-spectra. However,* * finite isotropy is related to T-isotropy, so, to take the T-fixed part into account, we mu* *st relate the parts lying over various finite subgroups. In this chapter, we show how to as* *semble the finite and infinite isotropy into a single algebraic object giving a complete inva* *riant. We begin in Sections 5.1 and 5.2 by showing how to deduce Adams spect* *ral sequences for the semifree and general case from the free and F-free cases. This * *then provides an obvious algebraic candidate for the abelianization, which we introduce * *formally in Section 5.3. We are then equipped to apply the criteria of Section 4.2, and we* * show in Section 5.3 that the triangulated category of T-spectra is equivalent to the de* *rived category of the standard model category. We conclude in Section 5.5 by introducing some* * explicit notation for particular cases that we need later. 5.1.Assembly. Now that we understand the categories T-Spec=T and T-Spec=F , it is t* *ime to return to the question of how to fit them together. Of course if X is an obje* *ct of T-Spec=T , then X ' E"F ^ X so that if TY ' * (as happens if Y is an object of T* *-Spec=F ) then [Y; X]T*= 0. It is therefore enough to understand [X; Y ]T*. Now, * *as remarked above X ' "EF ^X ' "EF ^TX, and, since the nonequivariant spectrum TX splits * *as a wedge of spheres, the essential case is [E"F; Y ]T*. The obstacle to using ou* *r available machinery is that ssT*(E"F) is not a Q[cH-]module. We begin with the simpler case when Y has only one relevant isotropy * *group i.e. Y is an object of T-Spec=H . Theorem 5.1.1. If X ' X ^ "EF and Y is in T-Spec=H there is a short e* *xact sequence 0 -! ExtQ[cH](Q[cH ; c-1H]ssT*(X); ssT*(Y )) -! [X; Y ]T*-! Hom Q[cH](Q[cH ; c-* *1H]ssT*(X); ssT*(Y )) -! 0: Proof: As in the construction of the Adams short exact sequence it is e* *nough to construct a natural transformation OE : [X; Y ]T*-! Hom Q[cH](Q[cH ; c-1H] ssT*(X); ssT*(Y* * )) 55 56 5. ASSEMBLY AND THE STANDARD MODEL. and show it is an isomorphism when ssT*(Y ) is an injective Q[cH-]module. Furth* *ermore, by the splitting in 4.2.4 we may also suppose ssT*(Y ) is concentrated in even deg* *rees. We first explain how to define OE. The problem is to get Q[cH ]acting natural* *ly on some functor of ssT*(X). Observe that for any object Y of T-Spec=H and any other spe* *ctrum Z, Q[cH ]does act on F (Z; Y ), even though it is not usually an F-spectrum. We shall need a variant on a well known observation. Lemma 5.1.2. There is an equivalence E ^ F (E; X) ' E ^ X: Proof: The cofibre of the natural map from right to left is E ^ F (E"; X)* *, which is contractible. This follows from the fact that E may be formed from cells oe0* *H, whilst oe0H^ F (E"; X) ' F (Doe0H^ "E; X), and Doe0H^ "Eis contractible. __|* *_| Now observe that, since Y ' E ^ Y , we have an equivalence Y ^ DE ' Y .* * We may therefore define the natural transformation by taking f : X -! Y , replacin* *g it with f ^ 1 : X ^ DE -! Y ^ DE ' Y and then applying homotopy. Lemma 5.1.3. There is a natural isomorphism of Q[cH-]modules ssT*(E"^ DE ^ X) ~=Q[cH ; c-1H] ss*(T X): Proof: Since the fibre of "E-! "EF is the wedge of spectra E with K 6= H,* * and E ^ DE ' *, it follows that ssT*(E"^ DE ^ X) ~=ssT*(T(DE ^ X)), * *and T commutes with smash products it is enough to deal with the case TX = S0. But * *then ssT*(E"^ DE) = ssT*(DE)[c-1H], by 2.3.8, and we have seen that ssT*(DE* *) ~= Q[cH ]. __|_| It is worth making the general remark that if Z = Z0^ S1V (H)and Y is a T-spe* *ctrum over H, then cH acts invertibly on F (Z; Y ) since cH is the Euler class of V (* *H) on spectra over H. Since Q[cH ]is a graded field, the Q[cH-]module structure on ssT*(F (Z;* * Y )) follows from its structure as a vector space. This is relevant to the lemma, but it is * *also relevant to the proof of the theorem, since we already have the machinery to calculateW[* *X; Y ]T*as a rational vectorQspace when ssT*(Y ) is injective. In fact we have seen X ' i* *ni"EF, and so [X; Y ]T ~= i[ni"EF; Y ]T*, it is thus enough to deal with the special case * *X = "EF. The cofibre sequence EF+ -! S0 -! "EF thus gives an exact sequence . .-. [S0; Y ]T*- [E"F; Y ]T*- [EF+; Y ]T*- . .;. and we can identify the first term as [S0; Y ]T*~=Hom Q[cH](Q[cH ]; ssT*(Y )) and the third as [EF+; Y ]T*~=Hom Q[cH](Q[cH ; c-1H]=Q[cH ]; ssT*(Y )): 5.2. GLOBAL ASSEMBLY. 57 Since both of these are concentrated in even degrees, the sequence is short exa* *ct and the centre term is Hom Q[cH](Q[cH ; c-1H]; ssT*(Y )), at least as a rational vector* * space. By the re- marks above this also respects the Q[cH-]module structure. __|_| This proof generalizes readily to F-spectra Y which are nontrivial at only a * *finite number of subgroups. 5.2.Global assembly. We now need to discuss the case where Y has nontrivial isotropy at an infinit* *e number of subgroups. As discussed inQSection 4.3 above, for an F-spectrum Y we view ss* *T*(Y ) as a module over the ring OF = H Q[cH ]. Recall that eH 2 (OF)0 denotes the idemp* *otent which is projection onto the Hth factor, and c 2 (OF)-2 denotes the total cyclo* *tomic class c with Hth coordinate cH . The element c is particularly useful. Indeed, the representation V (H) only g* *ives rise to an Euler class K-equivariantly for K H, but one can pass to limits and obtain * *a suitable substitute which works for all finite subgroups. As we have seen before, "EF ha* *s a filtration by skeleta "EF(2k-1)analogous to a putative sphere SkV (F). Furthermore we have* * universal Thom isomorphisms for all F-spectra for these generalized spheres, and conseque* *ntly Euler classes. By passage to limit one sees that c = e(V (F)). Lemma 5.2.1.(a) For any F-spectrum Y there is an equivalence Y ^ SV (F)' Y ^ S2: (b) There is an equivalence SV (F)^ SkV (F)' S(k+1)V (F): (c) For any F-spectrum Y the composite Y = Y ^ S0 -! Y ^ SV (F)' Y ^ S2 induces multiplication by c. Proof: All parts are obtained from 2.3.6 and the following discussion by using * *the splitting of any F-spectrum. __|_| Having emphasized the analogy with spheres we should draw attention to a sign* *ificant difference: there is no spectrum T so that T ^ SV (F)' S0. This non-invertibili* *ty appears to introduce wrinkles into all approaches. We continue by analogy with the previous section. The complication is that, s* *ince F is infinite, sums and products do not coincide. Once again the module tF*:= ss* *T*(E"F ^ DEF+ ^X) is relevant, but it is not simply OF[c-1]ssT*(X): In fact it is very i* *lluminating to view tF*as the result of inverting all Euler classes in OF, but we defer thi* *s point of view until Section 8.1. The notation tF*is chosen since it is the coefficient r* *ing of F-Tate homology for S0; this will play no part in what follows, but suggests a number * *of parallels worth further investigation. Indeed, we need only observe that since OF acts on* * DEF+, 58 5. ASSEMBLY AND THE STANDARD MODEL. tF*is a OF-module. To understand the theorem which follows we should be explici* *t about the OF-module structure. Q Lemma 5.2.2. As a gradedLvector space tF*= ssT*(E"F ^ DEF+ ^ X) is H Q in de* *grees 0; -2; -4; -6; : :a:nd H Q in degrees 2; 4; 6; : :.: Furthermore there is a s* *hort exact sequence 0 -! OF -! tF*ffl-!2I -! 0 of OF-modules, and the map M Y c : tF2= Q -! Q = tF0 H H describing the extension is the natural inclusion of the sum in the product. Proof: The additive structure of tF*follows by applying ssT*(o ^ DEF+) to the c* *ofibre sequence EF+ -! S0 -! "EF; indeed the homotopy groups of EF+ ^ DEF+ ' EF+Q follow from 2.3.4 and are in positive degrees, whilst those of DEF+ ' H DE * *follow from 2.4.1, and are in non-positive degrees. This also gives the short exact se* *quence. To determine the effect of c : tF2-! tF0it is enough to consider the square tF2 -c! tF0 " " ssT2(F (E; S0) ^ "EF)cH-!ssT0(F (E; S0) ^ "EF); where the verticals are induced by the projection EF+ -! E onto a direct fac* *tor. We saw in the previous section that the bottom horizontal was cH . __|_| Although the occurrence of both sums and products might appear unattractive, * *both seem to play an essential role. The ring OF is fairly complicated, but we only need to extend the modules M w* *e consider slightly beyond the F-finite torsion modules in the sense of 4.3.2 and 4.3.3 (o* *ccurring as ssT*(Y ) for an F-spectrum Y ) to include modules like tF*. Note that for an ar* *bitrary module L, Hom OF(L; N(H)) = Hom Q[cH](L(H); N(H)), so that if N(H) is an injective Q[c* *H ]- module it is also injective over OF. However since OF is not Noetherian it does* * not follow that infinite direct sums of modules of this form are injective. Lemma 5.2.3. If N is F-finite and M is either of the form tF* U for a graded * *vector space U, or F-finite, then ExtsOF(M; N) = 0 for s 1 if N(H) is injective for a* *ll H or for s 2 in general. Q Proof: Note that if M is F-finite, Hom OF(M; N) = H Hom Q[cH](M(H); N(H)), so * *that the result holds in this case. The result holds if M = tF* U by 5.2.2, and the * *fact that OF is projective. __|_| This is sufficient to give us the Adams short exact sequence for arbitrary T-* *spectra. 5.3. THE STANDARD MODEL CATEGORY. 59 Theorem 5.2.4. If X is F-contractible and Y is in T-Spec=F there is a sho* *rt exact sequence 0 -! ExtOF(tF* ssT*(X); ssT*(Y )) -! [X; Y ]T*-! Hom OF(tF* ssT*(X); ssT*(Y * *)) -! 0: Proof: As in the construction of the Adams short exact sequence it is enough* * to construct a natural transformation OE : [X; Y ]T*-! Hom OF(tF* ssT*(X); ssT*(Y )) and show it is an isomorphism when ssT*(Y ) is an OF-module injective in an * *appropriate sense. We define OE for arbitrary T-spectra X. Observe that since Y ' EF+ ^ Y and* * DEF+ ^ EF+ ' EF+, we have an equivalence Y ^ DEF+ ' Y . Now define the natural tran* *sfor- mation by taking f : X -! Y , replacing it with f ^ 1 : X ^ DEF+ -! Y ^ DEF+* * ' Y and then applying homotopy: explicitly this gives a natural transformation OE : [X; Y ]T*-! Hom OF(ssT*(X ^ DEF+); ssT*(Y )): W n T F T T If X ' X ^ "EF then X ' i i"EF, and ss*(X ^ DEF+) ~=t* ss*( X), so that the* * Q codomain of OE is of the required form. Note also that in this case [X; Y ]T* * ~= i[ni"EF; Y ]T*, so that it is enough to deal with the special case X = "EF; we may also supp* *ose that ssT*(Y ) is concentrated in even degrees. We saw in the proof of 5.2.2 that applying ssT*(o^DEF+) to the cofibre seq* *uence EF+ -! S0 -! "EF gives a short exact sequence of OF-modules. Hence applying OE we o* *btain a diagram 0 - [S0; Y ]T* - [E"F; Y ]T* - [EF+; Y ]T* - 0 ~=# # #~= 0 - Hom(OF; ssT*(Y ))- Hom(tF*; ssT*(Y-)) Hom(ssT*(EF+); ssT*(Y ))- 0: The bottom lefthand zero follows from 5.2.3; the lefthand vertical is tautol* *ogically iso- morphic, the righthand vertical is isomorphic by 3.1.1. The central vertica* *l is thus an isomorphism by the 5-lemma. __|_| 5.3.The standard model category. The purpose of this section is to describe an abelian category A = A Fwhos* *e derived category is an algebraic model for the category of T-spectra, and whose homo* *logical algebra provides Adams spectral sequences for both the categories. There is a somewh* *at simpler abelian category A1 whose derived category gives an algebraic model of semif* *ree T-spectra, and similarly for other collections of proper subgroups. 60 5. ASSEMBLY AND THE STANDARD MODEL. fi F Definition 5.3.1.The objects of the standard model A are maps N -! t* V of OF-modules whose kernel and cokernel are F-finite torsion modules. Morphisms ar* *e com- mutative squares M -! N fi # # fl : 1OE F tF* U -! t* V We shall refer to N as the nub and V as the vertex of the object N -! tF* V . T* *he map fi is called the basing map. Remark 5.3.2. (a) In due course we will show that F-finite torsion modules ar* *e precisely the modules annihilated by inverting the set E of Euler classes. Since tF*is E * *local, the condition that fi : N -! tF* V has F-finite torsion kernel and cokernel is equi* *valent to saying that it becomes an isomorphism when E is inverted. Thus the role of fi i* *s to give a specific isomorphism E-1N ~=tF* V . Thus we may view objects of A as given by* * OF modules together with with a particular vector subspace V of E-1N so the the ca* *nonical extension of the inclusion V -! E-1N is an isomorphism. This explains why fi is* * called a basing map. (b) The standard model category AH for a set H of finite subgroups is obtained * *by replacing OF by eHOF. The case in which H is finite is significantly simpler, but we shal* *l just make explicit the case H = {1}. Here the objects are Q[c]-module maps fi : N -! Q[c;* * c-1] V which become an isomorphism when c is inverted. It is often helpful to examine* * any algebraic construction in this case before passing to the general case H = F, a* *nd we shall adopt this policy for the more intricate bits of algebra. Before we study the category A and its derived category, it will be useful to* * import information into A from categories we understand. For comparison with the model* * DQ of F-contractible spectra, we let Q*-mod denote the category of graded rational ve* *ctor spaces and define the functor e : Q*-mod -! A by i j e(V ) = tF* V -1! tF* V : Objects of A isomorphic to ones in the image of e will be called torsion free o* *bjects. For comparison with the model of F-spectra we define the functor f : torsOfF-mod-! A by f(N) = (N -! 0): Objects of A isomorphic to ones in the image of f will be called torsion object* *s. It is easy to see that both e and f are right adjoints. Lemma 5.3.3. For any object M -! tF* U of A, any graded vector space V and any F-finite torsion OF-module N we have natural isomorphisms (i) Hom A((M -! tF* U); e(V )) = HomQ(U; V ): 5.3. THE STANDARD MODEL CATEGORY. 61 (ii) Hom A ((M -! tF* U); f(N)) = Hom OF(M; N): __|_| Corollary 5.3.4.(i) The functors e and f are full and faithful embeddings. (ii) For any graded vector space W and any F-finite torsion OF-module N Hom A(f(N); e(W )) = 0: __|_| We can also import injective objects, and they will prove to be sufficient fo* *r homological algebra. Lemma 5.3.5.(i) The object e(V ) = (tF*V -1! tF*V ) of A is injective for any* * graded vector space V . (ii) The object f(I) = (I -! 0) of A is injective if I is an F-finite torsion i* *njective. Proof: This follows from 5.3.3 together with the obvious fact that the functors* * (M -! tF* V ) 7-! M and (M -! tF* V ) 7-! V are exact. __|_| This is enough for us to prove that A is a split 1-dimensional abelian catego* *ry. Theorem 5.3.6. The category A is a split one dimensional graded abelian categ* *ory in the sense of Section 4.2. Proof: We must verify that A is abelian, that every object admits an injective * *resolution of length 1, and that every object X splits as a sum X+ X- of pure pieces so t* *hat Hom(Xffi; Yffl) = 0 and Ext(Xffi; Yffl) = 0 if ffi 6= ffl. First we make an observation about F-finite torsion OF-modules. Lemma 5.3.7.If 0 -! M0 -! M -! M00-! 0 is an exact sequence of OF-modules then M is F-finite and torsion if and only if M0and M00are both F-finite and to* *rsion. __|_| Now, considering the map in the square displayed in Definition 5.3.1 above, w* *e obtain two short exact sequences: 0 -! K -! M -! M0 - ! 0 # fi # j # 0 -! tF* U -! tF* V -! tF* V 0- ! 0 and 0 -! M0 -! N -! C -! 0 j # fl # oe # 0 -! tF* V 0-! tF* W -! tF* X -! 0 By hypothesis, the central verticals have kernel and cokernel F-finite and to* *rsion. It follows from the first diagram that ker() and cok(j) are F-finite and torsion, * *and from the second that ker(j) and cok(oe) are F-finite and torsion. Hence j : M0 -! tF* ** V 0is an object of A. Now we may use the Snake Lemma and closure of F-finite torsion * *spectra 62 5. ASSEMBLY AND THE STANDARD MODEL. under extensions to deduce that cok() and ker(oe) are also F-finite and torsion* *. This completes the proof that A is abelian. To see that fi : M -! tF* V has a resolution of length 1, we recall that 5.3.* *5 allows us to import injectives from Q*-mod and torsOfF-mod. Now we proceed to constru* *ct an injective resolution of length 1. First note that ker(fi) is the torsion submod* *ule T M of M, and that there is a resolution 0 -! T M -! I0 -! J0 -! 0 of T M by objects I0 a* *nd J0 which are injective as F-finite torsion OF-modules. Furthermore if Q is the * *image of then J00= (tF* V )=Q is divisible and an F-finite torsion module, and hence inj* *ective in the same sense. In the usual way we form the diagram, 0 0 0 # # # 0 -! T M -! M -! Q -! 0 # # # 0 -! I0 -! I0 (tF* V ) -! tF* V -! 0 # # # 0 -! J0 -! J0 J00 -! J00 -! 0 # # # 0 0 0 and hence the diagram 0 -! M -! I0 (tF* V ) -! J0 J00 -! 0 # # # 0 -! tF* V -! tF* V -! 0 -! 0 This is the required resolution; injectivity of the last term was 5.3.5 (i), an* *d injectivity of the middle term follows since it is an extension of two things known to be inje* *ctive by 5.3.5. Finally we just need to observe that any object X = (M -! tF* V ) splits as a* * sum X+ X- where X+ is the even graded part and X- is the odd graded part. It is th* *en clear that Hom(Xffi; Yffl) = 0 if ffi 6= ffl, and the fact that Ext(Xffi; Yffl) = 0 i* *f ffi 6= ffl follows since the resolution of Yfflconstructed above is entirely in parity ffl. __|_| Now that we know the derived category DA exists by 4.1.3 we may observe that* * the functors e and f induce functors on derived categories. Corollary 5.3.8. The functors e : Q*-mod -! A and f : torsOfF-mod-! A induce maps of derived categories. Proof: The functors e and f induce functors on categories of differential grade* *d objects, and, by the universal property of the category of fractions, it is enough to sh* *ow that the functors e and f take homology isomorphisms to homology isomorphisms. Since e a* *nd f are obviously exact functors, this is clear. __|_| 5.4. THE ALGEBRAICIZATION OF RATIONAL T-SPECTRA. 63 5.4.The algebraicization of rational T-spectra. We have constructed a split one dimensional abelian category A , and it has a* * derived category DA by 4.1.3; we shall show in this section that A is the linearization* * of the category of T-spectra, and hence that DA is an algebraic model of the category of T-spe* *ctra: this is the main theorem of Part I. Theorem 5.4.1. (Algebraicization of rational T-spectra.) There are equivalenc* *es of tri- angulated categories T-Spec ' DA : Proof: Once again we aim to apply Theorem 4.2.6. It remains to show that T-Spec* * has linearization A, and that it arises as the homotopy category of a model categor* *y. It follows from the construction of Lewis-May that T-Spec is the homotopy cat* *egory of a model category. It remains to describe a functor T-Spec -! A, and show it is * *essentially surjective and a linearization. For this we define ssA*(X) := ssT*(X ^ DEF+ -! X ^ "EF ^ DEF+) Lemma 5.4.2.The above definition gives a functor ssA*: T-Spec -! A: Proof: The definition is clearly natural, so the only point is to verify that s* *sA*(X) is an object of A. First we note that ssT*(X ^ "EF ^ DEF+) ~=tF* ssT*(TX); and we hav* *e the natural inclusion ss*(T X) ~=ssT*(X ^E"F) -! ssT*(X ^E"F ^DEF+) ~=tF*ssT*(TX):* * This shows that the second term has the correct form. Secondly we must show that the* * kernel K and cokernel C of ssT*(X ^ DEF+) -! ssT*(X ^ "EF ^ DEF+) are F-finite and tor* *sion. However the fibre of X ^ DEF+ -! X ^ "EF ^ DEF+ is X ^ EF+ ^ DEF+ ' X ^ EF+, whose homotopy is F-finite and torsion. Since there is an extension 0 -! -1C -! ssT*(X ^ EF+) -! K -! 0 the result follows. __|_| For special types of T-spectra we can go further, and it is useful to relate * *ssA*to the functors we already understand; this makes it rather easy to calculate ssA*(X) * *when X is either an F-spectrum or F-contractible. Lemma 5.4.3.On the subcategories of F-contractible spectra and F-spectra the * *functor ssA*factorizes through ssT*in the sense that the following two diagrams commute: ssT* ssT* f T-Spec=T -! DQ T-Spec=H - ! D(torsOF) # # e and # # f ssA* ssA* T-Spec -! DA T-Spec - ! DA : 64 5. ASSEMBLY AND THE STANDARD MODEL. Proof: For the first diagram, note that, if Y is an F-contractible spectrum, th* *en Y ^EF+ ' *, and hence the map Y ^ DEF+ -! Y ^ "EF ^ DEF+ is an equivalence. We have seen that ssT*(Y ^ "EF ^ DEF+) = tF* ssT*(Y ), so ssA*(Y ) = e(ssT*(Y )) as required. For the second diagram, we use the facts that if X is an F-spectrum then X ^ * *"EF ' * and X ^ DEF+ ' X, so ssA*(X) = (ssT*(X ^ DEF+) -! 0) = f(ssT*(X)). __|_| Next we need to observe that the Whitehead Theorem holds; this lemma was a cr* *itical point in the recognition of the standard model. Lemma 5.4.4. (Karlsruhe Lemma). If f : X -! Y is a map of T-spectra for which ssA*(f) is an isomorphism, then f is an equivalence. Proof: Using exactness of ssA*it is enough to consider the cofibre Z of f, for * *which ssA*(Z) = 0, and deduce Z ' *. The hypothesis states that (a) ssT*(Z ^ DEF+) = 0 and (b) ssT*(Z ^ "EF ^ DEF+* *) = 0. Now we have remarked that quite generally ssT*(X ^ "EF ^ DEF+) = e(ss*(TX)), so* * we conclude from (b) that TZ ' *. Thus Z is an F-spectrum, and Z ^ DEF+ ' Z, and by 3.3.2 it follows that Z ' *. __|_| Next we observe that enough injectives are realizable. We have seen in the p* *roof of Theorem 5.3.6 that there are enough injectives amongst those of form f(I) = (I * *-! 0) for an F-finite torsion injective I and e(V ) = (1 : tF* V -! tF* V ) for a gra* *ded vector space V . Objects of the first type occur as f(ssT*(X)) = ssA*(X) for F-spectra* * X by 3.4.2. Those of the second type occur as e(ssT*(Y )) = ssA*(Y ) for F-contractible spe* *ctra Y . Once we have established an Adams short exact sequence, essentially surjectivity wil* *l follow by 4.2.2. Theorem 5.4.5. For any rational T-spectra, there is a natural short exact seq* *uence 0 -! ExtA(ssA*(X); ssA*(Y )) -! [X; Y ]T*-! Hom A(ssA*(X); ssA*(Y )) -! 0; and it splits unnaturally. Proof: We are constructing an Adams spectral sequence, so we use the standard o* *perating procedure, and resolve all spectra by injectives for which the theorem holds. Firstly, we observe that ssA*gives a natural transformation : [X; Y ]T*-! Hom A(ssA*(X); ssA*(Y )): Next, we note that is an isomorphism if Y is an injective F-spectrum or an F-c* *ontractible spectrum. Lemma 5.4.6. If Y ' Y ^ "EF is F-contractible then is an isomorphism. 5.4. THE ALGEBRAICIZATION OF RATIONAL T-SPECTRA. 65 Proof: First note that, by 5.4.3, ssA*(Y ) = e(ssT*(Y )) = e(ss*(TY )). Now * *use the commu- tative diagram [X; Y ^ "EF]T*-! HomA (ssA*(X); e(ss*(TY ))) ~=# #~= ~= T T [TX; TY ]* -! HomQ(ss*( X); ss*( Y )): The right hand vertical is an isomorphism by obstruction theory, and the lef* *t hand vertical is an isomorphism by 5.3.3; the bottom isomorphism is the well known algebra* *icization of non-equivariant rational spectra. __|_| Lemma 5.4.7.If Y ' Y ^ EF+ is an injective F-spectrum then is an isomorph* *ism. Proof: This is a little harder since there are non-zero maps from torsion fr* *ee objects to torsion objects. Note first that if Y is an F-spectrum then ssA*(Y ) = f(ssT* **(Y )) by 5.4.3. We now deal separately with various types of spectrum X. If X is an F-spectrum, ssA*(X) = f(ssT*(X)) by 5.4.3 and the lemma follows* * from 3.1.1, using commutativity of the diagram [X; Y ^ EF+]T*-! Hom A(f(ssT*(X)); f(ssT*(Y ))) & #~= ~= Hom OF (ssT*(X); ssT*(Y )): The right hand vertical is an isomorphism by 5.3.3. If X is F-contractible ssA*(X) = e(ssT*(X)) by 5.4.3 and the lemma follows* * from 5.2.4 by commutativity of the diagram [X; Y ^ EF+]T*-! Hom A(e(ssT*(X)); f(ssT*(Y ))) & #~= ~= Hom OF(tF* ssT*(X); ssT*(Y )): The right hand vertical is an isomorphism by 5.3.3. The general case follows by applying the 5-lemma to the diagram, [X ^ EF+; Y ]T* -! [X ^ "EF; Y ]T* -! [X; Y ]T* ~=# ~=# # Hom A(ssA*(X ^ EF+); ssA*(Y-))!Hom A(ssA*(X ^ "EF); ssA*(Y-))!HomA(ssA*(X); ss* *A*(Y )) -! [X ^ EF+; Y ]T* -! [X ^ -1"EF; Y ]T* ~=# ~=# -! Hom A(ssA*(X ^ EF+); ssA*(Y-))!HomA(ssA*(X ^ -1"EF); ssA*(Y )): Since X ^ EF+ -! X -! X ^ "EF is a cofibre sequence the first row is exact, * *and there is a long exact sequence in ssA*. Since ssA*(Y ) is injective, the second ro* *w is also exact. __|_| 66 5. ASSEMBLY AND THE STANDARD MODEL. Next we observe that any T-spectrum Y admits a convergent Adams resolution fo* *r ssA* by imported injectives. More precisely, we choose a standard resolution 0 -! ssA*(Y ) -! I -! J -! 0 of ssA*(Y ) as constructed in 5.3.6 above. Since both I and J are sums of injec* *tives of the form e(V ) or f(I), there are spectra I(Y ) and J(Y ) realizing them. From the * *injective case proved above for I(Y ) and J(Y ), we may lift the maps in the resolution to for* *m a sequence Y -! I(Y ) -! J(Y ) which gives the resolution once ssA*is applied. In fact it is a cofibre sequenc* *e. Indeed, since the composite is null from the injective case, the map Y -! I(Y ) factors thro* *ugh the fibre F of I(Y ) -! J(Y ), and the map Y -! F is an isomorphism of ssA*; theref* *ore Y is equivalent to the fibre by the Whitehead Theorem 5.4.4. The theorem follows by applying [X; .]T*to Y -! I(Y ) -! J(Y ). __|_| It is worth recording here the compatibility of the models with the inclusion* *s of categories of spectra. Corollary 5.4.8. The functors e and f induce full and faithful embeddings e : DQ - ! DA andf : D(torsOfF)-! DA of derived categories. Furthermore, these are compatible with the geometric inc* *lusions in the sense that the diagram f f DQ -e! DA - D(torsOF) '# '# '# T-Spec=T -! T-Spec - T-Spec=F commutes. Proof: The fact that e and f induce full and faithful embeddings of derived cat* *egories follows from the Adams spectral sequences together with 5.3.3 and the fact that* * they preserve injectives. The compatibility with geometry follows from 5.4.3 and the universal properti* *es of cat- egories of fractions. __|_| 5.5. Notation. In this section we provide notation to cover our needs, and make a convenient* * fibrant approximation explicit: this will be useful in consideration of Mackey functor * *valued ho- motopy groups in Section 13.1. On the one hand we want to refer to an object of A by a single letter like L.* * We shall write fiL F L = NL -! t* VL : 5.5. NOTATION. 67 We call NL the nub of L and VL the vertex of L. The map fiL is called the basin* *g map. By definition KL = ker(rL) and CL = cok(rL) are F-finite torsion modules, and CL is also divisible and hence injective. We also use the notation kL : KL -! NL * *and cL : tF* VL -! CL for the inclusion and projection. Since all short exact sequ* *ences 0 -! -1CL -! T -! KL -! 0 therefore split, it is reasonable to define TL := -1CL KL; and call it the torsion part of L. With this notation, corresponding to the cof* *ibre sequence qX X -! X ^ "EF -! X ^ EF+ we have a cofibre sequence in the derived category qL L -! e(VL) -! f(TL): To see this, note that there is a map L -! e(VL) which is an isomorphism on ver* *tices, since e is right adjoint to the passage-to-vertex functor. It is clear that the kerne* *l and cokernel are f(KL) and f(CL) respectively, and hence, since CL is injective, the cofibre* * is f(TL). Explicitly 0 1 q0L B tF* VL -! TL C (qL : e(VL) -! f(TL))= B@ # # CA tF* VL -! 0 If TL is injective then q0Lis simply a homomorphism, but in general it is an ob* *ject of a derived category of OF-modules that we have not formally introduced. We shal* *l be completely explicit about q0Lin 5.5.3 below. Now, given a T-spectrum X, we obtain the object LX = ssA*(X), but we would li* *ke abbreviations for its terms, so we take NX = NssA*(X)= ssT*(X ^ DEF+) and VX = VssA*(X)= ss*(TX): Similarly, rX : NX -! tF* VX is the map ssT*(X ^ DEF+ -! X ^ DEF+ ^ "EF). Final* *ly we let TX = TssA*(X); this has the geometric significance one expects. Lemma 5.5.1. TX = ssT*(X ^ EF+): Proof: Since EF+ ^ DEF+ ' EF+ we have a cofibre sequence X ^ EF+ -! X ^ DEF+ -! X ^ DEF+ ^ "EF: __|_| 68 5. ASSEMBLY AND THE STANDARD MODEL. Example 5.5.2.Some examples of particular interest arise from the basic cells* *. Let us write LH := ssA*(oe0H), and similarly for the other associated invariants. (i) For the group T itself we have i j LT = OF -! tF* NT = OF; VT = Q andTT = I (ii) If H is a finite subgroup, and we let Q(H) denote the Q[cH-]module Q, rega* *rded as a OF-module in the natural way, then LH = (Q(H) -! 0) NH = Q(H); VH = 0 andTH = Q(H): __|_| In Section 13.1 we use this to calculate the homotopy Mackey functor of an ar* *bitrary object of A. It is also worth being completely explicit about some of the above constructi* *ons. We must begin by talking about injective representations of objects. We use the convent* *ion that if M is an arbitrary object then ^Mdenotes an injective object with a homology iso* *morphism M -! ^M. Of course there is usually no canonical choice of M^. Thus, if T is an* *y F-finite torsion OF-module, with injective resolution 0 -! T -! I -! J -! 0 then we may * *take T^= fibre(I -! J). Of course e(V ) is already injective, and [f(T )may be taken* * to be f(T^) by 5.3.5. Thus for an arbitrary object L = (NL -! tF* VL) with zero diffe* *rential we may take ^Lto be the fibre of qL : e(VL) -! f(T^L). In what follows we shall* * use the notation A n B to denote a dg module which is additively a direct sum of A and * *B, and which has B as a submodule and A as a quotient. However the sum is twisted by g* *iving the differential a component A -! B. There is thus a short exact sequence 0 -! B -! A n B -! A -! 0 of dg modules. Lemma 5.5.3. For an object L of A (with zero differential) an injective equiv* *alent to L in DA is given by 0 1 tF* VL n -1CL ^KL L^= B@ # # CA tF* VL 0 where the differential in (tF* VL) n (-1CL ^KL) is -q0L: tF* VL -! CL K^Lgiven explicitly by q0L= {q0C; q0K} where q0C= cL and q0K2 Ext(tF* VL; KL) lifts the * *class of the extension [NL] = (0 -! KL -! NL -! im(rL) -! 0) : Furthermore there is a fibre sequence ^L-! e(VL) -qL!f(TL) 5.5. NOTATION. 69 where 0 1 q0L ^ B tF* VL -! CL KL C (qL : e(VL) -! f(TL))= B@ # # CA tF* VL -! 0 Proof: It is enough to construct a homology isomorphism L -! ^Lsince the displa* *yed ^L is visibly injective, and is the fibre of qL. To simplify the picture in the re* *ader's mind we now suppose L is even. We shall construct a diagram NL iL-!(tF* VL) n (-1CL ^KL) # # tF* VL -1! (tF* VL) 0 where iL = {i0L; 0; i000L} with i0L= fiL; the choice of i000Lneeds more careful* * description. Since the diagram commutes for any choice of i000Land the lower arrow is a homology i* *somorphism, it suffices to choose i000Lso that iL is a map of dg OF-modules and a homology * *isomorphism. For this we display the map iL vertically and find a diagram 0 -! KL -! NL -! im(rL) -! 0 hL # iL # jL # 0 -! K^L -! (tF* VL) n (-1CL ^KL) -! (tF* VL) n -1CL -! 0: We use the natural inclusion KL -! K^L for hL, and jL is the composite im(fiL) * *-! tF* VL -! (tF* VL) n -1CL; both of these are obviously dg maps and homology isomorphisms, and the right hand square commutes for any choice of i000L. By th* *e 5 lemma, it is enough to choose a map i000Lso that iL is a chain map and the left hand s* *quare commutes. For this we refer to the diagram KL -! NL - ! im(iL) -! tF* VL & i000L# [gNL]# . hL I(KL) - ! J(KL) q0K Here [gNL]represents the extension class [NL], and extends to q0Kby injectivity* * of J(KL). Now [NL] is the image of the identity map of KL under the boundary map for the * *short exact sequence 0 -! KL -! NL -! im(fiL) -! 0, and hence there is a map i000Lin the diagram to make the square commute. Finally, from the Snake Lemma construct* *ion of the boundary map, it follows that the left hand composite is the standard in* *clusion hL : KL -! ^KLas claimed. Since the square commutes, the map iL = {i0L; 0; i000L} commutes with differe* *ntials. Since i000Lextends hL the left hand square in the map of short exact sequences commut* *es. __|_| 70 5. ASSEMBLY AND THE STANDARD MODEL. CHAPTER 6 The torsion model. We begin with a section explaining the methods we have actually used for identi* *fying the place of various well known spectra in the standard model: this chapter pla* *ces the method in a proper framework. The methods are closely tied to standard methods* * of equivariant topology, as described in the introduction. Whilst their success ma* *y be sufficient justification, it is desirable to explain what can and cannot be seen by these * *methods. To do this we construct the geometrically natural torsion model in Section 6.2, an* *d show algebraically in Section 6.3 that it is equivalent to the standard model. We do* * not know a direct method for showing the derived category of the torsion model category is* * equivalent to the category of rational T-spectra. The method described in Section 6.1 att* *empts to identify the place of a T-spectrum in the torsion model. The algebraic drawbac* *k of the torsion model is that it is of injective dimension 2, so that the natural Adams* * spectral sequence associated to this construction does not collapse until E3. This means* * that not every homomorphism of objects in the torsion model lifts to a geometric map, an* *d hence an object may not be be formal (ie determined by its homology). It is fortunate th* *at for the usefulness of the easy method presented here, that so many spectra of interest * *are formal. 6.1. Practical calculations. Given a T-spectrum, X we may want to identify its place in our classification* *. In principle this means that we must calculate ssA*(X), but it is often more practical to pr* *oceed in several steps. Thus we know that X is the fibre of qX : X ^ "EF -! X ^ EF+; and hence that the algebraic model ssA*(X) is equivalent to the fibre of the al* *gebraic map ^qX: ssA*(X ^ "EF) -! ssA*(X ^ EF+); corresponding to qX in the derived category DA . Furthermore we have seen in 5.* *4.3 that ssA*(X ^ "EF) = e(ss*(TX)) and ssA*(X ^ EF+) = f(ssT*(X ^ EF+)). Summary 6.1.1. For any T-spectrum X the algebraic model ssA*(X) is the fibre * *of the map ^qX: e(ss*(TX)) -! f(ssT*(X ^ EF+)) 71 72 6. THE TORSION MODEL. in the derived category corresponding to qX : X ^ "EF -! X ^ EF+. __|_| To use this method we need to identify TX, calculate ssT*(X ^ EF+) and identi* *fy the map qX : X ^ "EF -! X ^ EF+ between them, in algebraic terms. Using 5.4.8, this* * may be done by identifying ^qX: tF* ss*(TX) -! ssT*(X ^ EF+) as a map in the derived category. When ssT*(X ^ EF+) is injective this simply involves finding a homomo* *rphism of OF-modules. In practice we begin by calculating ssT*(X ^ EF+) using the spectral sequence* * of the skeletal filtration of EF+, and some ingenuity to obtain the OF-module structur* *e. Next, we note that TX is determined by its homotopy groups ss*(TX) = ssT*(X ^* * "EF). Often the most effective way of calculating this is to use the first step and t* *he long exact homotopy sequence of X ^ EF+ - ! X - ! X ^ "EF. However the best method for calculating TX depends on how X is given. If X is the suspension spectrum of a * *based space then of course TX is the suspension spectrum of the fixed point space. In* * the event that X is equipped with Thom isomorphisms X ^ SV (F)' X ^ S2 (for example if X * *is an F-spectrum or the equivariant K-theory spectrum), then the model "EF = S1V (F)s* *hows ss*(TX) = E-1ssT*(X) where E is the multiplicative set of Euler classes, genera* *ted by the Euler classes of V (H) for finite subgroups H. A variation of this is to use t* *he spectral sequence of the filtration [ S0 SV (F) S2V (F) . . . SkV (F)= S1V (F)= "EF: k In calculating ^qXwe assume again that it is routine to calculate integer gra* *ded ssT*. In any case we will have calculated (qX )* : ss*(TX) -! ssT*(X ^ EF+) during the p* *revious steps; since ^qX(1x) = (qX )*(x) this determines ^qXon all elements o x when o * *2 tF*is of negative degree. Many elements x of ssT*(X ^ E) are uniquely divisible by cH* * , which allows one to determine the images of elements like c-1Hx. However this only ac* *counts for all elements in trivial cases, and it even happen (for example with X = S0) tha* *t (qX )* = 0 whilst the map qX (and hence also ^qX) are highly nontrivial. In this case we n* *eed to do more work to identify the images of elements c-kH x. In fact the definition of * *cH allows us to see that this is in effect the a part of ssT*twisted by a representation. Lemma 6.1.2. For any integer k the map ^qX: Q{c-kH} ss*(TX) -! -2kssT*(X ^ EF+) is the map induced in ssT*by qX^1 -kV (H) 1-2k __ X ^ "EF ' X ^ "EF ^ oe-kV (H)-!X ^ E ^ oe ' X ^ E: |_| 6.2. The torsion model The reader will have noticed that all motivation and all investigations proce* *ed by the following process. First we understand the F-contractible part (i.e. the vertex* *), next the F-free part (i.e. the torsion) and finally we identify the comparison map betwe* *en them. Why then have we not used the abelian category Atof objects (tF* V -! T ), and * *formed 6.2. THE TORSION MODEL 73 a derived category from this? The reason is that At has injective dimension 2, * *and this makes the model much less useful. On the one hand, it makes it hard to get a p* *recise hold on morphisms, and on the other, the objects of the abelian model are not a* *dequate to model all T-spectra. The main purpose of this section is to introduce the torsion model category A* *t, and the purpose of the next is to show that DAt is another model for the category of T-* *spectra. This makes precise the assertion that the extra information that must be added * *to an object of At to specify a T-spectrum is an extension of the cokernel of the str* *ucture map by tF* V ; this is exactly what is encoded by the nub in the standard algebraic* * model. Hence in particular it places our intuition about how to specify a T-spectrum o* *n a firm and practical basis. Furthermore, it will transpire in Part II that certain fu* *nctors (the Lewis-May fixed point functor and the quotient functor) are more approachable o* *n the torsion model. Thus the torsion model is also of theoretical value. Assuming the fact that At has injective dimension 2, and that the derived cat* *egory DAt exists, we explain the consequences in more detail, using the natural Adam* *s spectral sequence. Indeed, we have a functor H* : DAt -! At; obtained by taking homology termwise. Since any object of Atmay be viewed as an* * object of DAt with zero differential, this is certainly surjective on objects. However* * At is only a quadratic approximation to DAt, so has two linked drawbacks. Firstly, two non-i* *somorphic objects of DAt may have isomorphic homology (not all objects are `formal' in th* *e sense of Sullivan), and secondly, maps in DAt are calculated by a spectral sequence * *with a non-trivial d2 differential. In fact there is a filtration 0 F2(X; Y ) F1(X; Y ) F0(X; Y ) = [X; Y ]: Here (F1=F2)(X; Y ) = Ext1(H*(X); H*(Y )) and there is an exact sequence 0 -! (F0=F1)(X; Y ) -! Hom(H*(X); H*(Y )) -d2!Ext2(H*(X); H*(Y )) -! F2(X; Y ) * *-! 0: Thus d2 gives the obstruction to realizing an algebraic map geometrically, and,* * even us- ing the splitting by parity there may be extension problems relating (F0=F1)(X;* * Y ) and F2(X; Y ). We now turn to the justification of our assertions about At. Lemma 6.2.1.(i) For any graded vector space V the object (tF* V -! 0) is inje* *ctive in At. (ii) For any F-finite torsion injective I the object (tF* Hom(tF*; I) -! I); whose structure map is evaluation, is injective in At. 74 6. THE TORSION MODEL. Proof: Part (i) is immediate. For Part (ii) a short calculation ver* *ifies that for any F-finite torsion module T Hom((tF* U -! R); (tF* Hom(tF*; T ) -! T )) = Hom(R; T ):* * __|_| It is plain that the lemma provides enough injectives. Finally, w* *e estimate the injective dimension as follows. For an arbitrary object (tF* V -! T ) we have* * the exact sequence 0 -! 0 -! tF* V -! tF* V -! 0 # # # 0 -! T -! T -! 0 -! 0: Since (tF* V -! 0) is injective it suffices to show (0 -! T ) has d* *imension 2. Suppose then that 0 -! T -! I -! J -! 0 is an injective resolution of F-fin* *ite torsion modules; we may then form the exact sequence 0 -! 0 -! tF* Hom(tF*; I)-! tF* Hom(tF*; I)-! 0 # # # 0 -! T -! I -! J - ! 0; and so it suffices to show (tF* Hom(tF*; I) -! J) has injective dim* *ension 1. For this we have the exact sequence 0 -! tF* Hom(tF*; T )-! tF* Hom(tF*; I)-! tF* Hom(tF*; J)-! tF* Ext(tF*;* * T-)! 0 # # # # 0 -! 0 -! J -! J -! 0 * * -! 0: Finally we note that the bounds can be achieved. For example Ext2((tF*-! 0); (0 -! T )) ~=Ext(tF*; T ); which need not be zero. __|_| We have seen that we may form the derived category DAt of such a* * category by in- verting homology isomorphisms. The construction of an Adams spectra* *l sequence is then routine. We need only know that enough injectives are realizable an* *d that H* : [X; I] -! Hom(H*(X); H*(I)) is isomorphic when H*(I) is injective. Now, for * *any object Y , we construct a sequence Y -! I0 -! I1 -! I2 realizing any specified injective resolution 0 -! H*Y -! H*I0 -! H*I1 -! H*I2 -! 0: Applying [X; .] we obtain a spectral sequence of the form described* * above, which is obviously convergent since it is finite. We recall that an object Y of DAt is said to be formal it is dete* *rmined by its homology in the sense that H*(Y ) ~=H*(Y 0) implies Y ' Y 0. Lemma 6.2.2. If H*(Y ) is of injective dimension 1 then Y is form* *al, and hence in par- ticular Y is formal if the torsion part of H*(Y ) is an injective F* *-finite torsion OF-module. 6.3. EQUIVALENCE OF DERIVED CATEGORIES OF STANDARD AND TORSION MODELS.75 Proof: The formality of one dimensional objects is clear, since the Adams spect* *ral sequence collapses to let us realise an isomorphism OE : H*(Y 0) -! H*(Y ) by a map f : * *Y 0-! Y . Since f* = OE is a homology isomorphism by construction, it is a weak equivalen* *ce. It remains to observe that any object (tF* V -! I) with I injective is of in* *jec- tive dimension 1. However there is an obvious embedding in the injective (tF* * *V - ! 0) tF* Hom(tF*; I) -! I), and the cokernel clearly has zero torsion part. __|* *_| Remark 6.2.3. Later it will be useful to note that the torsion model category* * is isomor- phic to the category of objects V -! Hom(tF*; T ) by the usual adjunction. The * *point to be made is that this is also an abelian category, but epimorphisms need not be * *surjective. In fact a map takes the form f U -! V # # Hom(1;g) F Hom(tF*; S) -! Hom(t*; T ); and its kernel and cokernel are ker(f) -! Hom(1; ker(g)) and cok(f) -! Hom(1; c* *ok(g)). Note also that in this view the basic injectives are of form (V -! 0) and (Ho* *m(tF*; T ) -1! Hom(tF*; T ) for an injective T , which makes them appear more natural. __|_| 6.3.Equivalence of derived categories of standard and torsion models. In this section we establish that the derived category of the torsion model D* *At is equiv- alent to the derived category of thestandard model DA , and hence that it too p* *rovides an algebraic model for rational T-spectra. Theorem 6.3.1. The torsion model category is equivalent to the derived catego* *ry DA of A, and hence we have equivalences DAt ' DA ' T-Spec=T of triangulated categories. Proof: We want to define functors comparing dgAt and dgA by passage to fibre or* * cofibre as the case may be. Indeed we may define fib : dgAt -! dgA by passage to fibre, in the sense that fib(tF* V -s! T ) = (F (s) -! tF* V ): However if we use cofibres and define cof(N -r! tF* V ) = (tF* V -! C(r)) 76 6. THE TORSION MODEL. the image is not in dgAt. Instead we consider the slight variation dgHA t, cons* *isting of dg objects tF* V -! M with the property that H*(M) is an F-finite torsion module. * *Thus we have defined a functor cof : dgA - ! dgHA t: Once we invert homology isomorphisms the inclusion i : dgAt -! dgHA t becomes an equivalence. It is convenient to introduce a similar variation of A* * : we let dgHA be the category of dg objects N -! tF*V so that the homology is an isomor* *phism modulo F-finite torsion modules. Lemma 6.3.2. The derived categories of dgHA t and dgHA exist and the inclusi* *ons dgAt -! dgHA t and dgA -! dgHA induce isomorphisms of derived categories which preserve the triangulation. Proof: In both cases we have the same formal situation. We have a category A fr* *om which we know how to form the derived category: we first form the category dgA of dg * *objects, and then invert the homology isomorphisms. To establish existence we use the cl* *ass I of objects formed from injective objects of A by a finite number of cofiberings. The category A is a subcategory of a larger category ^A, and we then consider* * the category dgHA of dg objects of a ^Asatisfying the condition that their homology lies in * *A. We show that DA is also the derived category of dgHA by establishing that Condition 4.1* *.1 holds with S being the class of homology isomorphisms. No extra work is necessary for* * Part (ii), and Part (iii) follows by the same proof as given in Section 4.1. It therefore * *remains only to show that any object M of dgHA admits a homology isomorphism M -! ^Mwith ^M2 I. This is sufficient to establish the existence of the derived category of dgHA, * *and the fact that the inclusion dgA -! dgHA induces an equivalence of derived categories. The simpler case is when A = A; we consider it first, and explain the variati* *ons necessary to deal with A = At. First, since A is one dimensional, we may take an injectiv* *e resolution 0 -! H*M -! I -! J -! 0 of H*M. Now view H*M as dg object with zero differentia* *l, and construct a homology isomorphism H*M -! M^, where M^ is the fibre of I -! J, and hence an object of I. From this we obtain the map zM -! H*M -! M^where zM denotes the subobject of cycles. We need to observe that this extends over the * *inclusion zM -! M to give a homology isomorphism. Now we have a surjective map [L; ^M] -! Hom(H*(L); H*(M)); and so we may choose a homotopy class M -! ^Mlifting the projection zM -! H*M; * *this is represented by a function : M -! ^M. The difference between and the given * *function on zM is a map into the injective J, and thus extends to M. Since the different* *ial of M^ is zero on J this extension necessarily commutes with differentials. Adding the* * extension to we obtain a function in the same homotopy class which agrees exactly with t* *he given function on zM as required. To deal with A = At we argue slightly less directly. In fact we note that, by* * the exact sequence (0 -! T ) -! (tF*V -! T ) -! (tF*V -! 0) it is enough to deal with obj* *ects (tF* V -! 0) and (0 -! T ). Those of the first type are already injective, so i* *t is enough 6.3. EQUIVALENCE OF DERIVED CATEGORIES OF STANDARD AND TORSION MODELS.77 to deal with those of the second type. By the above argument applied to the cat* *egory of F-finite torsion modules we obtain a homology isomorphism T -! ^Twith ^Tthe fib* *re of I -! J where 0 -! H*T -! I -! J -! 0 is an injective resolution of H*(T ). Thus* * we have a homology isomorphism (0 -! T ) -! (0 -! ^T), and the codomain is an elem* *ent of dgAt itself, and hence certainly admits a homology isomorphism to an object * *of I. This completes the argument. __|_| Notice by the long exact sequence associated to a cofibre or fibre sequence t* *he functors fib and cof preserve homology isomorphisms, and hence induce maps of derived ca* *tegories. We may now prove the theorem by considering the functors fib0: dgHA t-! dgHA andcof0: dgHA - ! dgHA t; which induce functors on derived categories for the same reasons. The standard * *natural transformations cof0O fib0 -! 1 and 1 -! fib0O cof0 induce homology isomorphism* *s. Hence fib0and cof0 induce an equivalence between derived categories. __|_| We shall have use for one more result. Lemma 6.3.3.The functor fib : dgAt -! dgA takes the class I of fibrant objec* *ts to fibrant objects. Proof: It is obvious that fib takes an F-contractible injective to an injective* * since fib(tF* V -! 0) = (tF* V -! tF* V ). On the other hand fib(tF* Hom(tF*; T ) -! T ) is the fibre of a map -1T -! tF* Hom(tF*; T ) # # 0 -! tF* Hom(tF*; T ): Since fib preserves triangles the result follows. __|_| 78 6. THE TORSION MODEL.