AN ALGEBRAIC MODEL FOR RATIONAL S1-EQUIVARIANT STABLE HOMOTOPY THEORY
BROOKE SHIPLEY
Abstract. Greenlees defined an abelian category whose derived category is equivalent to the rational S1-equivariant stable homotopy categoryA whose objects represent rational S1- equivariant cohomology theories. We show that in fact the model category of differential graded objects in models the whole rational S1-equivariant stable homotopy theory. That is, we show that thereA is a Quillen equivalence between dg and the model category of ra- tional S1-equivariant spectra, before the quasi-isomorphismsA or stable equivalences have been inverted. This implies that all of the higher order structures such as mapping spaces, function spectra and homotopy (co)limits are reflected in the algebraic model. The construction of this equivalence involves calculations with Massey products. In an appendix we show that Toda brackets, and hence also Massey products, are determined by the derived category.
1. Introduction In [5], Greenlees defined an abelian category and showed that its derived category is equiv- A alent to the rational T-equivariant stable homotopy category where T is the circle group. We strengthen this result by showing that this derived equivalence can be lifted to an equivalence on the underlying model categories, before the quasi-isomorphisms or stable equivalences have been inverted.
Theorem 1.1. The model category of rational T-equivariant spectra is Quillen equivalent to the model category of differential graded objects in . A The definition of and the model category on differential graded objects in , dg , are recalled in Section 2. A A A As mentioned in [5, 16.1], one of the applications of this stronger equivalence is that the algebraic models for the smash product and function spectra given in [5, Part IV] are natural. This also shows that the higher order structures such as mapping spaces and homotopy (co)limits are captured by the algebraic model [2, 3]. Since has injective dimension one, calculations of these higher order structures are quite practical, asA demonstrated here and in [5]. [5, Part III] discusses many motivations for studying rational T-equivariant spectra, including its connection to algebraic K-theory, equivariant topological K-theory and Tate cohomology. This algebraic model has also been used to define a model for rational T-equivariant elliptic cohomology [6]. Another reason for studying rational T-equivariant spectra is that the circle is the simplest infinite compact Lie group. The arguments used in [5, 5.6.1] to show that the derived category, ( ), is equivalent to the homotopy category of rational T-equivariant spectra, D A Ho(T-spectra), rely heavily on the fact that the circle is rank one. The general approach here though, using Quillen model categories and Morita equivalences, does apply to compact Lie groups of higher rank. Algebraic models of rational T r-equivariant spectra, for tori of any rank, are considered in [7].
Date: May 3, 2001; 2000 AMS Math. Subj. Class.: 55P62, 55P91, 55P42, 55N91, 18E30. Research partially supported by an NSF grant. 1 2 BROOKE SHIPLEY
The main new ingredient here is a Morita equivalence for stable model categories considered in [22], see Section 3. In general, this Morita equivalence models a stable model category with a set of small generators by modules over a ring spectrum (with many objects). For rational stable model categories, this can be simplified further as (unbounded) modules over a rational differential graded ring with many objects; see Definition 3.1 or [23, 1.3]. The following statement is proved as Corollary 3.9 below. Definition 3.5 recalls the notion of a Quillen equivalence.
Theorem 1.2. The model category of rational T-equivariant spectra is Quillen equivalent to the model category of (right) modules over a rational differential graded ring ( t) with one object E B foreachclosedsubgroupinT.
T-spectra Q od- ( t) ' M E B This gives an algebraic model for rational T-equivariant spectra, but ( t) is large and difficult to make explicit. Hence this algebraic model is not very practical.E Instead,B it is useful as a stepping stone to create a Quillen equivalence between T-spectra and differential graded objects in . As another stepping stone we apply this same process to dg . The model category of differentialA graded objects in is Quillen equivalent to (right) modulesA over a rational differential A graded ring ( a) with one object for each closed subgroup in T; see Corollary 3.7 below. We E B then show that these two differential graded rings (with many objects) ( t)and ( a) are quasi-isomorphic. Since quasi-isomorphisms of differential graded ringsE (withB manyE objects)B induce Quillen equivalences of the associated categories of modules by [21, 4.3], see also [22, A.1.1] or [23, A.1], Theorem 1.1 follows given a zig-zag of quasi-isomorphisms between ( a) E B and ( t). This is carried out in Sections 4, 5 and 6. E B Theorem 1.3. The model category of differential graded objects in is Quillen equivalent to A modules over the differential graded ring ( a). Also, there is a zig-zag of quasi-isomorphisms E B between ( a) and ( t) which induces a zig-zag of Quillen equivalences between the associated model categories.E B HenceE B we have a chain of Quillen equivalences
T-spectra Q od- ( t) Q od- ( a) Q dg . ' M E B ' M E B ' A In more detail, in Section 4 we explicitly describe ( a) and a quasi-isomorphic sub-ring a. E B E In Section 5, we modify ( t) to define a more amenable ring t. Then in Section 6 we construct E B E η ϕ an intermediary differential graded ring and two quasi-isomorphisms a t. This construction involves Massey products forS differential graded rings with manyE ←−S objects−→E which are defined in Definition 4.8. In particular, the proof of Theorem 1.3 uses the fact that the higher order products for a E and t agree. These products agree because there are Quillen equivalences E od- a Q od- ( a) Q dg M E ' M E B ' A and
od- t Q od- ( t) Q T-spectra M E ' M E B ' and Ho(T-spectra) and ( ) are equivalent as triangulated categories by [5, 5.6.1]. This in- variance of higher orderD productsA does not seem to appear in the literature, so we include it in an appendix on Toda brackets. A related statement, that quasi-isomorphisms preserve Massey products, appears in [18, 1.5]. Specifically, we prove the following as Theorem A.3.
Theorem 1.4. If ϕ: 0 is an exact equivalence of triangulated categories then the Toda T−→T brackets for and 0 agree. T T AN ALGEBRAIC MODEL FOR RATIONAL S1-EQUIVARIANT STABLE HOMOTOPY THEORY 3
Acknowledgments: I thank Bill Dwyer for a discussion which sparked my interest in this project. I also thank John Greenlees and Jeff Smith for many conversations throughout this project. Finally, I thank Mike Mandell and Peter May for sharing their equivariant expertise and Mark Mahowald for sharing his understanding of Toda brackets.
2. Algebraic Model
In this section we recall Greenlees’ algebraic model for rational T-equivariant spectra; for more detail see [5, Chapters 4, 5; Appendix E.1]. We then show that this model is a ChQ-model category; see Definition 2.4. Finally we recall certain objects called the algebraic basic cells. First we recall some preliminary definitions. Definition 2.1. [5, 4.5, 4.6, 5.2] Let be the set of finite subgroups of T. Let be the ring F OF of operations, ΠH Q[cH ]withcH in degree 2. Let eH be the idempotent which is projection ∈F − onto the factor corresponding to H. Let c be the total Chern class with eH c = cH . Let v E be the multiplicative set of all Euler classes, c v : Z 0 with finite support ,where ≥ v v(H) 1 { | F−→ } eH c = cH . Let tF = − . As a vector space (tF )2n is ΠH Q for n 0 and is H Q for n>0. ∗ E OF ∗ ∈F ≤ ∈F L We now define the abelian category . The category of differential graded objects in is A A then the standard algebraic model of rational T-equivariant spectra. β Definition 2.2. [5, 5.4.1] The objects of are maps N tF V of -modules with V a 1 A −→ ∗ ⊗ OF graded vector space such that − β is an isomorphism. N is referred to as the nub and V is the vertex. Morphisms are commutativeE squares
M θ N −−−−→
β β0 tF U tF V. ∗ y⊗ −−−−1 φ→ ∗ y⊗ ⊗ 1 The condition that − β is an isomorphism is equivalent to requiring that the kernel K and cokernel C of β are E-finite torsion modules [5, 4.5.1, 4.5.2, 4.6.6]. That is, they are torsion F modules with respect to the total Chern class and they decompose as direct sums, K = H eH K and C = H eH C. Greenlees, in [5, 5.6.1], shows that the derived category of differential graded objectsL in is L A equivalent to the homotopy category of rational T-equivariant spectra. Here we recall the model category structure on dg . The associated homotopy category is equivalent to ( ). Proposition 2.3. [5, AppendixA B] The category dg of differential graded objectsD A in with cofibrations the monomorphisms, weak equivalences theA quasi-isomorphisms, and fibrationsA de- termined by the right lifting property is a model category. The model category dg is also tensored, cotensored and enriched over rational chain com- plexes in a way that is compatibleA with the model category structure. We call such a model category a ChQ-model category. This is the analogue of a simplicial model category [20, II.2] with simplicial sets replaced by rational chain complexes. See [9, Ch. 4] for a more general definition. Here ChQ is the category of unbounded rational chain complexes with the projective model category structure [9, 2.3.3] and the standard closed symmetric monoidal structure [9, 4.2.13]. For definitions of tensor, cotensor and enriched see [13, 1.2, 3.7].
Definition 2.4. A ChQ-model category is a complete and cocomplete model category which is tensored and enriched (denoted Hom ) over the category Ch , has cotensors withC finite C Q 4 BROOKE SHIPLEY dimensional complexes in ChQ and satisfies the following compatibility axiom (CM): p (CM) For every cofibration A i B and every fibration X Y in the induced map −→ −→ C Hom (i, p): Hom (B,X) Hom (A, X) Hom (A,Y ) Hom (B,Y ) C C −−−−→ C × C C is a fibration in ChQ. If in addition one of the maps i or p is a weak equivalence, then Hom (i, p) is a trivial fibration. We use the notation K X and XK to denote the tensors and cotensorsC for X in and K a chain complex. ⊗ C
For example, ChQ is itself a ChQ-model category [9, 4.2.13] as is the projective model category defined in [22, A.1.1] or [23, A.1] of unbounded differential graded modules over any rational differential graded ring (with many objects); see Definition 3.1. Proposition 2.5. The model category on dg is a Ch -model category. A Q Proof. For K Ch and A dg define K A in dg by ∈ Q ∈ A ⊗ A n K A = Σ (Kn Q A) ⊗ n ⊗ M n with differential d(kn a)=dkn a+( 1) kn da. Let Q[n] denote the rational chain complex ⊗ ⊗ − ⊗ with Q in degree n.ThenforA, B in dg define Hom (A, B)inCh by Hom (A, B)n = A C Q C (Q[n] A, B)where ( , ) is the rational vector space of maps of underlying graded objects A ⊗ A − − in (ignoring the differential). The differential for Hom (A, B)isgivenbydf n = dB fn + An+1 C K ( 1) fndA.ForK a finite dimensional complex, define A as HomCh (K, [0]) A. This Q Q cotensor− is the right adjoint of the tensor K . So we have the following isomorphisms⊗ (where defined) ⊗− Ch (K, Hom (A, B)) = dg (K A, B) = dg (A, BK ). Q C ∼ A ⊗ ∼ A As in [20, II.2 SM7(b)] or [9, 4.2.1, 4.2.2], the compatibility axiom in Definition 2.4 has an equivalent adjoint form involving the tensor. For f : K L in Ch and g : A B in dg , −→ Q −→ A define f g : K B K A L A L B. Then Axiom (CM) from Definition 2.4 is equivalent ⊗ q ⊗ ⊗ −→ ⊗ to requiring that if f and g are cofibrations, then f g is a cofibration which is trivial if f or g is. Since ChQ is a cofibrantly generated model category we only need to check this property when f is one of the generating cofibrations or generating trivial cofibrations [21, 3.5]. Let Dn be the acyclic rational chain complex with Q in degrees n and n 1. Then the n − generating cofibrations of ChQ are the inclusions in : Q[n 1] D for n an integer and the n − −→ generating trivial cofibrations are the maps jn :0 D for n an integer [9, 2.3.3]. The source −→ n of the map in g is the cofiber of g and the target is D B.Soforg a monomorphism, in g ⊗ ⊗ n is a monomorphism because it is g on one summand and idB on the other. Since D B is always acyclic and the cofiber of a quasi-isomorphism is acyclic, if g is a trivial cofibration,⊗ then n n in g is a quasi-isomorphism. Also, jn g : D A D B is a quasi-isomorphism between acyclic complexes for any monomorphism g. ⊗ −→ ⊗
For the Morita equivalence discussed in the next section we need to define the algebraic cells which can be used to build any object in dg . The natural building blocks for T-equivariant n A spectra are the G-cells T+ H S . Rationally, these can be simplified by using idempotents in the ∧ 0 0 H Burnside ring for a finite group, A(H)=[S ,S ] = ΠK H Q. Following Greenlees, we suppress ∼ ≤ the rationalization in our notation for spectra. Let eK be the idempotent which is projection onto the Kth factor. Definition 2.6. [5, 2.1.2] The geometric basic cells are the rational T spectra σ0 = S0 and 0 0 T σH = T+ H eH S for H a finite subgroup of T. ∧ AN ALGEBRAIC MODEL FOR RATIONAL S1-EQUIVARIANT STABLE HOMOTOPY THEORY 5
Via the equivalences between ( ) and rational T-equivariant spectra, these geometric basic cells correspond to the followingD algebraicA basic cells. Definition 2.7. [5, 5.8.1] For H a finite subgroup, let Q(H)denotetheQ[cH ]-module Q regarded as an -module. Then the algebraic basic cells are the objects LH =(ΣQ(H) 0) OF −→ and LT =( tF )in . OF −→ ∗ A We need fibrant replacements of these algebraic basic cells which are inclusions into their respective injective hulls. The injective dimension of is one, so the injective hulls are simple to construct. First we define the basic injective objects.A 2 1 Definition 2.8. [5, 2.4.3, 3.4.1, 5.2.1, 5.5.1] Let I(H)=Σ− Q[cH ,cH− ]/Q[cH ]. Let I = 2 H IH = Σ− tF / . The underlying vector spaces are I(H)2n = Q for n 0andI2n = H Q ∼ ∗ OF ≥ for n 0. Then (I 0), (I(H) 0), and (tF V tF V ) for any graded vector space LV are≥ injective objects−→ in . −→ ∗ ⊗ −→ ∗ ⊗ L A The construction of fibrant approximations in [5, Appendix B] produces injective hulls for 1 LH and LT which we denote by IH and IT respectively. Let P (I)=I n Σ− I be the acyclic 1 path object in dg with underlying graded object I Σ− I. For a finite subgroup H of T,let 2 A ⊕ ΣI(H) n Σ I(H) be the pullback in the following square. ΣI(H) n Σ2I(H) P (Σ3I(H)) −−−−→
3 ΣI(H) Σ I(H) y −−−−→ y The bottom map is zero in degree one and an isomorphism in all higher degrees. Let IH denote 2 the object (ΣI(H) n Σ I(H) 0) in dg . The inclusion LH IH is a quasi-isomorphism −→ A −→ and IH is fibrant since it is built from standard injectives. For T itself, let tF n ΣI be the pullback in the following square ∗ 2 tF n ΣI P (Σ I) ∗ −−−−→
2 tF ΣI y∗ −−−−→ y The bottom map is zero in degrees n 0 and an isomorphism in all positive degrees. Let I ≤ T denote the object (tF n ΣI tF )indg . The inclusion LT IT is a quasi-isomorphism and ∗ −→ ∗ A −→ IT is fibrant since it is built from standard injectives.
Proposition 2.9. IH and IT are fibrant replacements of LH and LT respectively. 3. The Morita equivalence Gabriel [4] proved a Morita theorem which shows that any cocomplete abelian category with a set of finitely generated projective generators is a category of -modules for some ring (with many objects). See Definition 3.1 below. The ring in questionE may be taken to beE the endomorphism ring of the projective generators; this is naturallyE a ring with one object for each generator. An enriched variant of this Morita equivalence was considered in [22]. Here we consider another enriched variant which shows that a model category with a set of small generators which is compatibly enriched over chain complexes is Quillen equivalent to a category of differential graded modules over a differential graded ring (with many objects). Definition 3.1. A ring with objects is an Ab-category with object set . This is also a category enriched over abelian groups,G [13, 1.2] or an Ab-module, [9, 4.1.6].G A ring is then an Ab-category with one object. Similarly, ,arational differential graded ring with objects , R G 6 BROOKE SHIPLEY is a category enriched over ChQ,oraChQ-category or ChQ-module. A (right) -module is a Ch -functor M : op Ch . R Q R −→ Q Given , a rational differential graded ring with objects , and objects G, G0,G00 in there is a R G G morphism chain complex Hom (G, G0) ChQ and coherently associative and unital composition maps R ∈
Hom (G0,G00) Hom (G, G0) Hom (G, G00). R ⊗ R −→ R An -module M is then determined by chain complexes M(G) for each object G and coherentlyR associative and unital action maps ∈G
M(G0) Hom (G, G0) M(G). ⊗ R −→ For each object G in there is a free -module, FGR, defined by FGR(G0)= (G0,G). The projective model structureG on the categoryR of -modules, od- , is establishedR in [22, A.1.1] or [23, A.1] using [21, 2.3]. The weak equivalencesR are the quasi-isomorphismsM R and the fibrations are the epimorphisms. With the standard action of Ch , od- is a Ch -model category. Q M R Q Definition 3.2. Let be a set of objects in a Ch -model category .Wedenoteby ( ) G Q C E G the full ChQ-subcategory of with objects , i.e., ( )(G, G0)=Hom (G, G0) is a differential graded ring with many objects,C one for eachG elementE ofG .Welet C G Hom ( , ): od- ( ) C G − C−−−−→M E G denote the functor given by Hom ( ,Y)(G)=Hom (G, Y ). C G C Note that, if = G has a single element, then ( ) is determined by the single differential graded ring, End(G G)=Hom{ } (G, G). E G C For our enriched variant of Gabriel’s theorem it is crucial that a ChQ-model category is a stable model category, that is, a pointed model category where the suspension functor is a self- equivalence on the homotopy category [22]. The homotopy category of a stable model category is triangulated [9, 7.1.1, 7.1.6], [25]. The suspension and loop functors are inverse equivalences which induce the shift functor. The cofiber and fiber sequences agree up to a sign and induce the triangles.
Proposition 3.3. If is a ChQ-model category then is a stable model category. If X is a cofibrant object in andC Y is a fibrant object in , thenC there is a natural isomorphism of graded C Ho( ) C abelian groups H Hom (X, Y ) = [X, Y ] C . ∗ C ∼ ∗
Proof. First, is pointed since ChQ is pointed [9, 4.2.19]. For a cofibrant object X in ,a C 1 C cylinder object [20, I.1] is given by (D Q[0]) X. It is a cylinder object because X X = i ⊕ ⊗ q ∼ (Q[0] Q[0]) X (D1 Q[0]) X is a cofibration, by an adjoint form of the compatibility ⊕ ⊗ −→ ⊕ ⊗ 1 axiom (CM), see [20, II.2, SM7(b)], and (D Q[0]) X Q[0] X = X is a quasi- ⊕ ⊗ −→ ⊗ ∼ isomorphism. Hence the cofiber of i, Q[1] X,representsΣX.ButQ[1] is invertible with ⊗ inverse Q[ 1]. Since the action of ChQ on is associative up to coherent isomorphism, this shows that− Σ is a self-equivalence on Ho( ). ThusC is a stable model category. C C 1 Since a cylinder object in ChQ is also given by tensoring with D Q[0], we have the following natural isomorphisms for a cofibrant object X in and a fibrant object⊕ Y in . C C Ho(Ch ) Ho( ) n Ho( ) Hn Hom (X, Y ) = [Q[n], Hom (X, Y )] Q = [Q[n] X, Y ] C = [Σ X, Y ] C C ∼ C ∼ ⊗ ∼
We also need the following definitions from triangulated categories, see [11, 1.1]. AN ALGEBRAIC MODEL FOR RATIONAL S1-EQUIVARIANT STABLE HOMOTOPY THEORY 7
Definition 3.4. An object X in a stable model category is small if in Ho( ) the natural C C map i[X, Yi] [X, iYi] is an isomorphism for any set of objects Yi . A subcategory of a triangulated category−→ isqlocalizing if it is closed under cofiber sequences,{ retracts} and coproducts. A setL of objects in is a set of (weak) generators if the only localizing subcategory of Ho( ) which contains Gis Ho(C ). C G C The following enriched variant of Gabriel’s theorem is stated rationally, but holds integrally as well with ChQ replaced by ChZ. A similar statement, on the derived category level, can be found in [12]. Here though we consider the underlying model categories, before inverting the quasi-isomorphisms, to construct a Quillen equivalence. Definition 3.5. A pair of adjoint functors between model categories is a Quillen adjoint pair if the right adjoint preserves fibrations and trivial fibrations. A Quillen adjoint pair induces adjoint total derived functors between the homotopy categories [20, I.4]. A Quillen adjoint pair is a Quillen equivalence if the total derived functors are adjoint equivalences of the homotopy categories. This is equivalent to the usual definition by [9, 1.3.13].
Theorem 3.6. Let be a ChQ-model category and a set of small, cofibrant and fibrant gen- erators. The functorC G Hom ( , ): od- ( ) C G − C−−−−→M E G is the right adjoint of a Quillen equivalence. The left adjoint is denoted ( ) . −⊗E G G
Proof. The functor Hom ( , ) preserves fibrations and trivial fibrations by the compatibility axiom (CM) of DefinitionC G 2.4,− since all objects of are cofibrant. So together with its left adjoint it forms a Quillen pair. Denote the associatedG total derived functors on the homotopy L categories by RHom ( , )and ( ) . Since Ho( ) is a triangulated category both total C derived functors preserveG − shifts and−⊗ triangles;E G G they areC exact functors of triangulated categories by [20, I.4 ]. Since L is a left adjoint it commutes with colimits. The right adjoint also −⊗ ( ) G commutes with coproductsE G since each object of is small. This follows as in [22, 3.10.3(ii)]. We consider the full subcategories of those MG Ho( od- ( )) and X Ho( ) respectively for which the unit of the adjunction ∈ M E G ∈ C L η : M RHom ( ,M ( ) ) −−−−→ C G ⊗E G G or the counit of the adjunction L ν :RHom ( ,X) ( ) X X C G ⊗E G −−−−→ are isomorphisms. Since both derived functors are exact and preserve coproducts, these are localizing subcategories. For every G the ( )-module Hom ( ,G) is isomorphic to the ( ) ∈G( ) E G C G free module FGE G by inspection and FGE G ( ) is isomorphic to G since they represent the same functor on .Sothemapη is an isomorphism⊗E G G for every free module, and the map ν is C ( ) an isomorphism for every object of . Since the free modules FGE G generate the homotopy category of ( )-modules and the objectsG of generate , the localizing subcategories are the whole categories.E G So the derived functors are inverseG equivalencesC of the homotopy categories.
Since dg is a ChQ-model category by Proposition 2.5, we can apply Theorem 3.6. Trian- gulated equivalencesA preserve generators so we identify generators of dg via the triangulated A equivalence πA :Ho(T-spectra) ( ) from the homotopy category of T-spectra to the ho- motopy category of dg , or the−→ derived D A category of , by [5, 5.6.1]. Note that composition of A A πA with homology gives the functor πA which appears in [5, 5.6.2]. The geometric basic cells 0 ∗ t0 = σH H for H any closed subgroup of T form a set of small generators for T-spectra. B { } ≤T 8 BROOKE SHIPLEY
This set is labeled t0 because we will soon consider its image t in another Quillen equivalent B H B 0 category. They detect stable equivalences because π (X)=[ K H σ ,X]T for H finite and ≤ πT(X)=[σ0,X]T by [5, 2.1.5]. It follows by [22, 2.2.1]∗ or part∨ of [11, 2.3.2]∗ that is a set of T t0 ∗ ∗ 0 B small generators. Hence, the set of images πA(σH )=LH H form a set of small generators { } ≤T for dg . Let a = IH H be the set of their fibrant replacements. Then a is a set of A B { } ≤T B small, cofibrant and fibrant generators. Let ( a) be the endomorphism Ch -category as in E B Q Definition 3.2. In Section 4 we explicitly describe ( a). E B Corollary 3.7. The model category dg is Quillen equivalent to the model category of differ- A ential graded modules over ( a). E B To prepare for constructing a Quillen equivalence between dg and the model category of A rational T-equivariant spectra, T-spectra, we show that there is an equivalence between T-spectra and modules over a rational differential graded ring with many objects. A Morita equivalence similar to Theorem 3.6 in [22] considers SpΣ-model categories, or spectral model categories,model categories compatibly enriched over symmetric spectra. [22, 3.10.3] shows that any spectral model category with a set of small generators is equivalent to modules over a ring spectrum with many objects. In this paper we are interested in particular in rational stable model categories, Ho( ) stable model categories where [X, Y ] C is a rational vector space for any objects X and Y in . In this case the Gabriel equivalence produces a rational ring spectrum which can be replaced byC a differential graded ring with many objects. Theorem 3.8. [23, 1.3] Let be a spectral model category. If is rational and has a set of small generators, then is QuillenC equivalent (via a chain of severalC Quillen equivalences) to the model category of modulesC over a rational differential graded ring with many objects.
Before rationalization, the category of T-equivariant spectra is a cellular, simplicial, left proper, stable model category [16, III.4.2]. In particular, a G-topological model category [16, III.1.14] is a topological model category which in turn is a simplicial model category via the sin- gular and realization functors. Also, a compactly generated model category is a stricter notion than a cofibrantly generated model category. One can check that T-equivariant spectra is in fact cellular, see [10, A.1] or [8, 14.1.1]. Since localizations preserve these properties by [8, 4.1.1], the model category of rational T-equivariant spectra is also a cellular, simplicial, left proper, stable model category. Then Hovey in [10, 8.11,9.1] shows that T-spectra is Quillen equivalent to a spectral model category, 0. Since T-spectra is also a simplicial, cofibrantly generated, proper, stable model category, [22,C 3.9.2] also shows it is Quillen equivalent to a spectral model category. Since rational T-equivariant spectra has a set of small generators, t0, the geometric basic cells, B 0 is also rational and has a set of small generators because these properties are determined on C the homotopy category level. Thus, by Theorem 3.8, 0 is Quillen equivalent to od- for some C M R rational differential graded ring with many objects, . Let t be a set of cofibrant and fibrant R B replacements of the images of the basic cells in od- .Then t is a set of small generators in od- . Since od- is a Ch -model category,M TheoremR 3.6B shows that od- is Quillen M R M R Q M R equivalent to od- ( t)where ( t) is the endomorphism ring on the generators t. M E B E B B Corollary 3.9. T-spectra is Quillen equivalent to od- ( t) which is the endomorphism ring M E B on a set of small generators t in an intermediate model category od- . B M R T-spectra Q od- Q od- ( t) ' M R' M E B This does give an algebraic model for T-spectra, but ( t) is large and not explicit. This algebraic model is nonetheless useful because we can constructE B a zig-zag of quasi-isomorphisms between ( t)and ( a) which induce Quillen equivalences on the respective module categories. E B E B AN ALGEBRAIC MODEL FOR RATIONAL S1-EQUIVARIANT STABLE HOMOTOPY THEORY 9
4. The endomorphism ring for dg A In this section we first explicitly calculate ( a), the endomorphism ring associated to dg by E B A Corollary 3.7. Then in Definition 4.6 we define a sub-ring a which is quasi-isomorphic to ( a). E E B In Section 6, we then use this simpler ring a to construct a zig-zag of Quillen equivalences E between dg and T-spectra. For this construction, we need to understand the Massey products A in H a, so we end this section with a definition of Massey products for differential graded rings ∗E with many objects and a calculation of the products and Massey products in H a. Since a ∗E E and ( a) are quasi-isomorphic this calculation also applies to H ( a) by [18, 1.5]. E B ∗E B First we calculate the underlying rational chain complexes of ( a). E B Proposition 4.1. For any distinct finite subgroups H, K of T the rational chain complexes of homomorphisms are as follows: n 1. Homdg (IH ,IH )=( n 0 D ) Q[0] Q[1], A ≤ ⊕ ⊕ 2. Homdg (IH ,IK )=0, A L 2n 1 3. Homdg (IH ,IT)=( n 0 D − ) Q[0], A ≤ 2n+1 ⊕ m 4. Homdg (IT,IH )=( n 0 D ) ( m 0 D ) Q[1] and A L ≥ ⊕ ≤ ⊕ 2m 5. Homdg (I ,I )=( n 0 Q [2n +1]) ( m 0(D Q )) Q[0] where Q is the A T T L F L⊕ ⊗ F ⊕ F rational vector space with≥ basis , the set of finite≤ subgroups of T. L F L Proof. In the first three cases, since the vertex of the source is trivial the maps are determined by considering graded -module maps of the nubs, denoted by ( , ). As an underlying OF 2 OF∗ − − -module, the nub of IH is ΣI(H) Σ I(H). As a graded vector space (I(H), I(H)) = OF ⊕ OF∗ n 0 Q[2n]. This determines the underlying graded vector space of Homdg (IH ,IH ). Let ≤ A 2 m2n n 0 be a basis for (ΣI(H), ΣI(H)), m2n+1 n 0 be a basis for (ΣI(H), Σ I(H)), L{ } ≤ OF∗2 2 { } ≤ OF∗ 2 l2n n 0 be a basis for (Σ I(H), Σ I(H)) and l2n 1 n 0 be a basis for (Σ I(H), ΣI(H)). When{ } ≤ necessary, theseOF∗ bases and those below are{ decorated− } ≤ with an HO.F∗ The differential is determined by dm2n = m2n 1, dm2n+1 =0,dl2n = m2n 1 and dl2n 1 = l2n 2 m2n 2. Since − − − − − − (I(H), I(K)) = 0, it follows that Homdg (IH ,IK )=0. OF∗ A As an underlying -module, the nub of IT is tF ΣI. As a graded vector space (I(H), I)= OF ∗ ⊕ OF∗ n 0 Q[2n]and (I(H),tF ) = 0; this determines the underlying graded vector space of ≤ OF∗ ∗ Homdg (IH ,IT). Let f2n n 0 be a basis for (ΣI(H), ΣI)and f2n 1 n 0 be a basis for L A2 { } ≤ OF∗ { − } ≤ (Σ I(H), ΣI). The differential is determined by df 2n 1 = f2n 2 and df 2n =0. OF∗For the fourth case the vertex of the target is trivial,− so− again− the maps are determined by the nubs. Also, note that (t , (H)) = [2n]. Let g2n+1 n be a basis for F I n Z Q Z OF∗ ∗ 2 ∈ { } ∈ (tF , ΣI(H)), g2n n Z be a basis for (tF , Σ I(H)), h2n n 0 be a basis for (ΣI(H), ΣI(H)) OF∗ ∗ { } ∈ OF∗ 2 ∗ L { } ≤ OF∗ and h2n+1 n 0 be a basis for (ΣI(H), Σ I(H)). The differential is determined by dg2n+1 = { } ≤ OF∗ g2k, dg2n =0,dh2n = h2n 1 + g2n 1 and dh2n+1 = g2n. The last case is the only− one where− we must consider− the vertex. This restricts the maps from tF Q to itself to maps of the form 1 φ. Hence (tF ,tF ) contributes only Q[0] to ∗ ⊗ ⊗ OF∗ ∗ ∗ Homdg (I ,I ). Also, note that (t , )= [2n]and ( , )= [2n]. Let T T F I n Z Q I I n 0 Q H A OF∗ ∗ ∈H F OF∗ ≤ F i2n+1 n Z,H be a basis for (tF , ΣI(H)), j2n n 0,H be a basis for (ΣI(H), ΣI(H)) { } ∈ ∈F OF∗ ∗ L{ } ≤ ∈F H OLF∗H H and idT the identity map. The differential is determined by di2n+1 =0,dj 2n = i2n 1 and − d idT =0. As we note below in Proposition 4.4, for the rest of this paper we only need to understand the product structure on the ( 1)-connected cover of ( a). This greatly simplifies the structure, see Proposition 4.5. So only− the non-negative dimensionalE B products in the following proposition are needed and these products are displayed in Definition 4.6. 10 BROOKE SHIPLEY
Proposition 4.2. The following is a list of all of the non-trivial products, or compositions, of the bases chosen for ( a) in Proposition 4.1. E B m2ng2k+1 = g2n+2k+1 m2n+1g2k+1 = g2n+2k+2 l2ng2k = g2n+2k l2n 1g2k = g2n+2k 1 − − f2n 1g2k = i2n+2k 1 f2ng2k+1 = i2n+2k+1 l2nh2k = g2n+2k l2n 1h2k = g2n+2k 1 − − − − l2nh2k+1 = h2n+2k+1 l2n 1h2k+1 = h2n+2k h2nf2k = m2n+2k h2nf2k 1 = l2n+2k 1 − − − h2n+1f2k = m2n+2k+1 h2n+1f2k 1 = l2n+2k f2nh2k = j2n+2k f2n 1h2k+1 = j2n+2k − − h2nj2k = h2n+2k h2n+1j2k = h2n+2k+1 h2ni2k+1 = g2n+2k+1 h2n+1i2k+1 = g2n+2k+2 f2nm2k = f2n+2k f2n 1m2k+1 = f2n+2k f2n 1l2k = f2n+2k 1 f2nl2k 1 = f2n+2k 1 − − − − − j2nf2k = f2n+2k j2nf2k 1 = f2n+2k 1 m2nm2k = m2n+2k m2n+1m2k = m2n+2k+1 − − m2nl2k 1 = l2n+2k 1 m2n+1l2k 1 = l2n+2k l2nm2k+1 = m2n+2k+1 l2n 1m2k+1 = m2n+2k − − − H H H H−H H l2nl2k = l2n+2k l2n 1l2k = l2n+2k 1 j2nj2k = j2n+2k j2ni2k+1 = i2n+2k+1 − − Also, x idT = x where x = i2n+1,j2n, idT or fn and idT y = y where y = i2n+1,j2n, idT,gn or hn.
Next we consider simpler sub-rings of ( a). E B Definition 4.3. The ( 1)-connected cover R 0 of a differential graded ring R is defined by − h i R 0 n =0forn<0, R 0 n = Rn for n>0andR 0 0 = Z0R where Z0R is the kernel of the h i h i h i differential d: R0 R 1. Since the product of two cycles is a cycle, the product on R induces a product on R 0−→. Moreover− the inclusion R 0 R is a map of differential graded rings which induces anh isomorphismi in homology in non-negativeh i−→ degrees. This definition generalizes to a differential graded ring with many objects, , by applying this to each morphism complex R (o, o0). R Since the homology of ( a) is concentrated in non-negative degrees it is quasi-isomorphic to its ( 1)-connected cover.E ByB [21, 4.3], the module categories are Quillen equivalent. − Proposition 4.4. The map ( a) 0 ( a) is a quasi-isomorphism of differential graded E B h i−→EB rings with objects a. Thus, there is a Quillen equivalence between the respective categories of modules. B
od- ( a) 0 Q od- ( a) M E B h i' M E B Next we explicitly describe ( a) 0 . E B h i Proposition 4.5. The rational chain complexes underlying the differential graded ring ( a) 0 are given by: E B h i 1. ( a) 0 (IH ,IH )=Q[0] Q[1] with generators idH = m0 + l0 and m1, E B h i ⊕ 2. ( a) 0 (IH ,IK )=0, E B h i 3. ( a) 0 (IH ,IT)=Q[0] with generator f0, E B h i 2n+1 4. ( a) 0 (IT,IH )=( n 0 D ) Q[1] with generators gn for degrees n 2, g1 + h1 E B h i ≥ ⊕ 1 ≥ generating the cycles in degree 1 and h1 and g0 generating D and L H 5. ( a) 0 (IT,IT)=( n 0 Q [2n +1]) Q[0] with generators i2n+1 and idT with n 0. E B h i ≥ F ⊕ ≥ The products are determined by the products in ( a). L E B We make one more modification to the ring ( a) 0 by considering the sub-ring a. E B h i E Definition 4.6. Define the differential graded ring a with objects a as the sub-ring of ( a) 0 1 E B E B h i produced by deleting the D generated by h1 and g0. Denote the element g1 + h1 in ( a) 0 by E B h i g1 in a. To ease notation in Section 6, the element g1 will be renamed g1 to match the family E of elements g2n+1. Then the rational chain complexes underlying a are given by: H H E H e 1. a(IH ,IH )=Q[0] Q[1] with generators idH =em0 + l0 and m1 , E ⊕ 2. a(IH ,IK )=0, E AN ALGEBRAIC MODEL FOR RATIONAL S1-EQUIVARIANT STABLE HOMOTOPY THEORY 11
H 3. a(IH ,IT)=Q[0] with generator f0 , E 2n+1 H H 4. a(IT,IH )=( n 1 D ) Q[1] with generators g1 , gn for n 2, and E ≥ ⊕ H ≥ 5. a(IT,IT)=( n 0 Q [2n +1]) Q[0] with generators i2n+1 and idT with n 0. E L ≥ F ⊕ ≥ The non-trivial products of these chosen bases, except thee obvious products with id and id L H T are: H H H 1. g1 f0 = m1 , H H H 2. f0 g2n+1 = i2n+1 for n 0and H H H H ≥ 3. ge1 f0 g2n+1 = g2n+2 for n 0. ≥ H H When n = 0, the second line should be read as f0g1 = i1 and the third line should be read as H H H H H H g1 f0 eg1 = g2 . The products in the third line can also be rewritten in terms of m1 or i2n+1 using the first two lines. e e e Another description of a is that it is the differential graded ring generated by the elements H H H E H H H H H f0 , g1 and g2n+1 for n>0 with relations generated by f0 g1 f0 =0,g2n+1f0 =0forn>0 and xH yK = 0 whenever H = K. These relations are equivalent to requiring (I ,I ) = [0], a H T ∼ Q (I ,I ) [0] [1] and6 (I ,I )=0. E a He H ∼= Q Q a H K e E Since quasi-isomorphisms⊕ induceE Quillen equivalences by [21, 4.3], we have the following statement.
Proposition 4.7. The inclusion of differential graded rings a ( a) 0 which sends g1 E −→ E B h i to g1 + h1 is a quasi-isomorphism. Hence the composite a ( a) 0 ( a) is also E −→ E B h i−→EB a quasi-isomorphism. Thus, there is a Quillen equivalence between the respective categoriese of modules.
od- a Q od- ( a) M E ' M E B We now consider higher order products. Here the higher order products we consider are Massey products. We generalize the definition of Massey products for a differential graded algebra due to Massey [17] to a differential graded ring with many objects; see also [19, 8.3.2]. Since there is no real difference between the homological and cohomological versions we still refer to these as Massey products instead of the ‘Yessam’ products of Stasheff [24]. Notice that with this definition the indeterminacy may be smaller for a differential graded ring with many objects than for the associated differential graded matrix ring on only one object. To deal with signs here we define (a)=( 1)deg a+1a instead of using the usual bar notation. − Definition 4.8. Let be a differential graded ring with objects = Gi i I . Suppose that R G { } ∈ γ1,...,γn are classes in H with γi Hpi (Gi,Gi 1). A defining system, associated to ∗R ∈ R − γ1,...,γn ,isasetofelementsai,j for 1 i j n with (i, j) =(1,n) such that h i ≤ ≤ ≤ 6 1. ai,j (Gj ,Gi 1)pi+ +pj +j i, ∈R − ··· − 2. ai,i is a cycle representative of γi and j 1 deg a+1 3. d(ai,j )=Σ− (ai,r)ar+1,j with (a)=( 1) a. r=i − To each defining system associate the cycle n 1 Σr=1− (a1,r)ar+1,n (Gn,G0)p1+...pn+n 2 ∈R − The n-fold Massey product, γ1,...,γn , is the set of all homology classes of cycles associated to all possible defining systems.h i Since Massey products are preserved by quasi-isomorphisms by [18, 1.5], we can use the simpler ring a to calculate these products for ( a). E E B Proposition 4.9. The homology of a (and hence also of ( a) and ( a) 0 ) is given by E E B E B h i 12 BROOKE SHIPLEY
H 1. H a(IH ,IH )=Q[0] Q[1] with generators [idH ] and [m1 ], ∗E ⊕ 2. H a(IH ,IK )=0, ∗E H 3. H a(IH ,IT)=Q[0] with generator [f0 ], ∗E H 4. H a(IT,IH )=Q[1] with generator [g1 ],and ∗E H 5. H a(IT,IT)=( n 0 Q [2n +1]) Q[0] with generators [i2n+1] and [idT]. ∗E ≥ F ⊕ The non-trivial products, except the obviouse ones with [id ] and [id ], and the non-trivial Massey L H T products are given by H H H 1. [g1 ][f0 ]=[m1 ], H H H 2. [f0 ][g1 ]=[i1 ],and 3. [f H ], [mH ],...,[mH ], [gH ] = [ iH ] . he 0 1 1 1 i { − 2n+1 } e n H H e Note that [g|1 ]and[{z f0 ]} generate all of the elements in homology via products and Massey products. e Proof. The products follow from the fact that g1f0 = m1 and f0g1 = i1. Next consider the H H H H H triple product [f0 ], [m1 ], [g1 ] . The cycle representatives here are unique. Since f0 m1 =0, h H i H H H H a(IH ,I )2 =0andg is the unique element such that dg = g = m g , this determines E T 3 e 3 2e 1 1 the only defining system ase the following matrix. H f0 0 e H H m1 g3 H g1 The associated cycle is f H gH +0gH= iH ,so [f H ], [mH ], [gH ] = [ iH ] . − 0 3 1 − 3 h 0 1 1 i { − 3 } The higher order products follow inductively. At eache stage because a(IH ,IH )n and a(IH ,I )n E E T are trivial for n>1and a(IT,IH )neis generated by one element,e there are no choices and hence E H H H no indeterminacy. Set a1,1 = f0 , ai,i = m1 for 2 i n +1,and an+2,n+2 = g1 . Since H H H H ≤ ≤ f m =0andm m = 0, the only choice for ai,j is 0 for i 5. The endomorphism ring for T-spectra In this section we consider the endomorphism ring ( t) associated to T-spectra in Corol- E B lary 3.9. As with ( a), we modify ( t) to define a quasi-isomorphic ring t. One property E B E B E of t is that t(H, T) is concentrated in degree zero. This forces certain relations to hold and E E thus makes it easier to construct a zig-zag of quasi-isomorphisms between a and t.Another property of is that in degree zero ( ) H . This gives a place to startE the constructionE t t 0 ∼= 0 t of the maps.E This is carried out in SectionE 6. E We first modify ( t) by taking its ( 1)-connected cover ( t) 0 as in Definition 4.3. We E B − E B h i next modify ( t) 0 so that degree zero agrees with its homology. This uses a Postnikov approximationE (DefinitionB h i 5.3); its general properties are stated in Propositions 5.4 and 5.5 and are applied to this specific case in Proposition 5.6. Finally, we define t as a sub-ring of E the resulting differential graded ring with many objects. This ring t then has the properties mentioned above. E We first recall the homology of ( t) from [5, 2.1.6]. Of course, it is isomorphic as a ring E B to the homology of ( a) stated in Proposition 4.9 because the triangulated equivalence of E B Ho(T-spectra) and ( ) from [5, 5.6.1] can be chosen in such a way that the geometric basic D A cells in t are taken to the algebraic basic cells in a. Because of this isomorphism we use the B B AN ALGEBRAIC MODEL FOR RATIONAL S1-EQUIVARIANT STABLE HOMOTOPY THEORY 13 notation x to denote the element in H ( t) corresponding to [x]inH a. We also delete most ∗E B ∗E of the superscripts H.Welabeltheelementsin t by their corresponding subgroup in T. B Propositionb 5.1. The homology of ( t) is given by E B 1. H ( t)(H, H)=Q[0] Q[1] with generators idH and m1, ∗E B ⊕ 2. H ( t)(H, K)=0, ∗E B 3. H ( t)(H, T)=Q[0] with generator f0, d c ∗E B 4. H ( t)(T,H)=Q[1] with generator g1 and ∗E B b [H 5. H ( t)(T, T)=( n 0 Q [2n +1]) Q[0] with generators i2n+1 and idT. ∗E B ≥ F ⊕b The non-trivial products, except the obvious ones with id and id are given by L H T c 1. gH f H = mH , 1 0 1 d c 2. f H gH = iH ,and c0 c1 d1 The next step is to consider the ( 1)-connected cover of ( t). Since the homology of ( t) c c c − E B E B is concentrated in non-negative degrees, ( t) 0 ( t) is a quasi-isomorphism. E B h i−→E B Proposition 5.2. The map ( t) 0 ( t) is a quasi-isomorphism of differential graded E B h i−→EB rings with objects t. Thus, there is a Quillen equivalence between the respective categories of modules. B od- ( t) 0 Q od- ( t) M E B h i' M E B The purpose of the next step is to construct a ring quasi-isomorphic to ( t) 0 such that H ( ). This involves a Postnikov approximationR which is obtained byE killingB h i homology 0 ∼= 0 groupsR aboveR degree zero. The next definition gives a functorial definition of this approximation which uses the small object argument; see [3, 7.12]. Consider the functor Ev(H,K) from rational differential graded rings with objects t to rational B chain complexes which is evaluation at (H, K). Denote the left adjoint of Ev(H,K) by T(H,K). T(H,H)M is the tensor algebra on M at (H, H), Q[0] at (K, K)forH = K and trivial everywhere else. 6 Definition 5.3. Given a rational differential graded ring with objects t a Postnikov approx- i R B imation of is given by the factorization P0 0 produced by the small object argu- R R −→ n+1R−→ ment for the set of maps I = T(H,K)(Q[n] D ) for n 1 and all pairs (H, K) t t. { −→ } ≥ ∈B ×B Since the map i: P0 is constructed by pushouts of the maps in I and P0 0has the right lifting propertyR−→ withR respect to the maps in I, these maps have the followingR−→ properties. i Proposition 5.4. The map P0 is an isomorphism in degrees less than or equal to one R −→ R and hence induces an isomorphism in homology in non-positive degrees. The map P0 0 induces an isomorphism in homology in positive degrees. R−→ i By applying the small object argument one more time we factor the map P0 .Con- ¯i p¯ R −→ R struct the factorization P0 by using the small object argument with respect to R −→n R −→ R the set J = T(H,K)(0 D ) for n 2 and all pairs (H, K) t t. Again by construction, since the maps{ in J are−→ injective} quasi-isomorphisms≥ and isomorphisms∈B ×B in non-positive degrees i and the map P0 has the above listed properties, ¯i andp ¯ have the following properties. R −→ R ¯ Proposition 5.5. The map i is a quasi-isomorphism which is an isomorphism in p¯ R −→ R non-positive degrees. P0 is an epimorphism and in non-positive degrees is a quasi- isomorphism and an isomorphism.R −→ R 14 BROOKE SHIPLEY Proof. Sincep ¯ has the lifting property with respect to J,Ev(H,K) p¯ is an epimorphism in degrees i greater than one. Since P0 is an isomorphism in degrees less than two and i =¯p¯i,it follows thatp ¯ is also an epimorphismR −→ R in degrees less than two. We now apply these general constructions to ( t) 0 . For ease of notation, let 0 stand E B h i Et for ( t) 0 . Since 0 is ( 1)-connected, it is trivial in negative degrees and its homology is E B h i Et − concentrated in non-negative degrees. Hence, P0 t0 is trivial in negative degrees and its homology ¯i Ep¯ is concentrated in degree zero. Let 0 P0 0 be the factorization constructed above Et −→ Et0 −→ Et for = 0. Consider H0 0 = H0 ( t) as a differential graded ring with objects t which is R Et Et ∼ E B B concentrated in degree zero. From Proposition 5.1, H0 ( t)isthefreeQ-algebra with objects E B t generated by a one dimensional vector space of morphisms from H to T for each H . B ∈F Since H0 ( t) = H0 0 = H0P0 0, one can construct a quasi-isomorphism of differential graded E B ∼ Et ∼ Et rings q : H0 t0 P0 t0. Define t00 as the differential graded ring with objects t which is the pullback ofpE ¯ and−→q. E E B q0 00 Et −−−−→ Et0 p¯ H0 t0 P0 t0 yE −−−−q → yE Proposition 5.6. The map q0 : 00 is a quasi-isomorphism. Moreover, ( 00)0 = H0 00 = Et −→ Et0 Et ∼ Et ∼ H0 0 and ( 00)n =0for n<0. Et Et Proof. 00(H, K) is the pullback of differential graded modules of the maps Ev(H,K) p¯ and Et Ev(H,K) q. Each map Ev(H,K) p¯ is an epimorphism by Proposition 5.5 and each map Ev(H,K) q is a quasi-isomorphism. The model category of differential graded modules is right proper since every object is fibrant. So t00(H, K) t0(H, K) is a quasi-isomorphism because it is the pullback of a weak equivalenceE across−→ a fibration.E Sincep ¯ is an isomorphism in non-positive degrees by Proposition 5.5, ( 00)n is isomorphic to (H0 0)n for n 0. Note that since q0 is a Et Et ≤ quasi-isomorphism this also implies that d1 :( 00)1 ( 00)0 is trivial. Et −→ Et We make one more modification to 00. Define t as the sub-differential graded ring with Et E objects t which differs from t00 only by setting t(H, T)= t00(H, T)0 concentrated in degree B E E E zero for each H and t(H, K)=0forH, K distinct finite subgroups. This defines a ∈F E differential graded ring since t00(H, T)0 is a retract of t00(H, T) because d1 is trivial. It follows E E that the map t 00 is a quasi-isomorphism and t has the properties listed in Proposition 5.6 E −→ E t E for 00. Combining this with Propositions 5.2 and 5.5 gives the following. Et Proposition 5.7. The composite map t 00 is a quasi-isomorphism of differential E −→ E t −→ Et0 graded rings with objects t. Combining this with the quasi-isomorphisms ( t) 0 = 0 B E B h i ∼ Et −→ Et0 and ( t) 0 ( t) gives a zig-zag of quasi-isomorphisms between t and ( t). Thus, there is aE zig-zagB h i−→E of QuillenB equivalences between the respective categories ofE modules.E B od- t Q od- ( t) M E ' M E B Moreover, ( ) H ( ), ( ) =0for n<0, (H, K)=0and (H, ) is concentrated in t 0 ∼= 0 t t n t t T degree zero.E E E E E 6. The quasi-isomorphism between a and t E E In this section we construct a differential graded ring with objects a and a zig-zag of η S B quasi-isomorphisms ϕ: a t. Note that there is an obvious bijection between the E ←−S−→E AN ALGEBRAIC MODEL FOR RATIONAL S1-EQUIVARIANT STABLE HOMOTOPY THEORY 15 object sets a and t which is the first condition required for such a quasi-isomorphism of rings with many objects.B B In this section we label the elements of both these sets by the corresponding H H subgroup of T. Recall that a is generated as an algebra by the elements f0 and g2n+1 with the E relations forced by a(H, K) being trivial, a(H, T) being trivial above degree zero and a(H, H) E E E being trivial above degree one. Since t(H, H) does not necessarily share this property, we must E use the auxiliary ring which is basically constructed from a by removing the relations forced S E by a(H, H) being trivial above degree one. E Proposition 6.1. There is a differential graded ring with objects a and quasi-isomorphisms η ϕ S B id a t of differential graded rings which induce the obvious bijections a a t. E ←−S−→E B ←−B−→ B Combining Propositions 4.7, 5.7 and 6.1 and Corollaries 3.7 and 3.9 gives our main result. Corollary 6.2. There is a zig-zag of Quillen equivalences between rational T-equivariant spectra and differential graded objects in . A T-spectra Q dg ' A We first define the intermediary ring and the map η : a. As mentioned above, S S−→E H H (H, H) is not required to be trivial above degree one; so we remove the relation g2k+1f0 =0. S H Then we must add generators si,j to ensure that these products are still trivial in homology. Definition 6.3. Define the differential graded ring with objects a as the ring generated by elements S B H H 1. f0 (H, T)0 with d(f0 )=0, H ∈S H H H H H 2. g2k+1 (T,H)2k+1 for k 0withd(g1 )=0andd(g2k+1)=g1 f0 g2k 1 for k>0and H ∈S ≥ H − H H 3. s2m+1,2n (H, H)2n for m 1andn m +1 with d(s2m+1,2m+2)=g2m+1f0 and H ∈S H H H≥ ≥ d(s2m+1,2n)=s2m+1,2n 2g1 f0 for n>m+1 − H K H H with relations generated by x y = 0 for any elements with H = K, f0 s2m+1,2n =0and H H H 6 f0 g f0 = 0. The relations are equivalent to setting (H, T) = Q[0] and (H, K)=0. 2k+1 S ∼ S As mentioned in Definition 4.6, for ease of notation in this section we denote g1 in a(T,H) E by g1. The element g1 in ( a) does not appear anywhere in this section. Define η : a E B H H H H H S−→E as the ring homomorphism determined by η(f0 )=f0 , η(g2k+1)=g2k+1 and η(se2m+1,2n)=0. Proposition 6.4. The map η : a is a quasi-isomorphism of differential graded rings. S−→E Proof. For ease of notation we delete the superscripts H and implicitly apply these steps for each finite subgroup H. Because there is no interaction between different finite subgroups, this should not cause any confusion. Since η takes the relations in to zero in a, it is a ring homomorphism. S E It also commutes with the differential since in a we have that d(g2k+1)=g2k = g1f0g2k 1 and η E η − η g2k+1f0 =0fork>0. The maps (H, K) a(H, K), (H, T) a(H, T)and (T, T) S −→E S −→E S −→ a(T, T) are isomorphisms. E We next show that H (H, H) = Q[0] Q[1], generated by [idH ]and[g1f0]. Below is a table ∗S ∼ ⊕ of generating elements in degree zero through seven in (H, H)and (T,H). S S 0 1 2 3 4 5 6 7 (H, H) idH g1f0 g3f0 s3,4 s3,4g1f0 s3,6 s3,4g3f0,g7f0 S g5f0 s5,6 s5,6g1f0,s3,6g1f0 (T,H) g1 g1f0g1 g3 g1f0g3 g5 g1f0g5,g3f0g3 g7,s3,4g3 S g3f0g1 s3,4g1 g5f0g1,s3,4g1f0g1 s5,6g1,s3,6g1 These elements have been arranged so that the differential is either zero or takes an element to the element in the corresponding spot one degree below. Here, dn : n n 1 is non-zero for S −→ S − 16 BROOKE SHIPLEY n =4, 6on (H, H)andforn =3, 5, 7on (T,H). In general, because of the relations in S S S any element in (H, H) can be written as a sum of elements of the form w or wg2k+1f0 where S w is a word in the elements of the set s2m+1,2n (including the empty word w =idH .) Again because of the relations, dw has only one{ term: } d(ws2m+1,2m+2)=wg2m+1f0 and d(ws2m+1,2n)=ws2m+1,2n 2g1f0 for n>m+1. − Hence if w is not the empty word, then w is not a cycle. The elements wg2k+1f0 are cycles; if k>0 then it is also a boundary (of ws2k+1,2k+2)orifk =0andw is not idH then again it is a boundary since w = w0s2m+1,2n and d(w0s2m+1,2n+2)=w0s2m+1,2ng1f0 = wg1f0. This leaves only idH and g1f0 as generators of homology. Since m1 = g1f0 in a,weseethat η E (H, H) a(H, H) induces an isomorphism in homology. S −→E ni Finally we show that H (T,H) = Q[1] generated by g1 by showing that (T,H) = ( I D ) ∗S ∼ S ∼ ⊕ Q[1] as a rational chain complex. Every element of (T,H) can be written as a sum of elements S L of the form wgi or wgj f0gk where i, j and k are odd numbers and w is a (possibly empty) word in the s2m+1,2n as above. Except for g1 these module generators fall into certain families w,i,j i indexed by a word, w, and two odd integers, i and j i. The families with i = 1 are exceptional,F − − j+1 w,1,j 1 is the set gj+1,g1f0gj 1 which generates a complex isomorphic to D .Ifi>1and F − { − } w +j+1 j i =2l +1,then w,i,j i generates a complex isomorphic to l + 1 copies of D| | . This set− is defined by F − w,i,j i = wsi,kgj k+1 for k even and i +1 k j, F − { − ≤ ≤ wsi,kg1f0gj k 1 for k even and i +1 k j 2, and wgif0gj i . − − ≤ ≤ − − } Since every module generator except g1 appears in one and only one of these families, this gives the splitting as claimed. Since η(g1)=g1, it follows that η : (T,H) a(T,H)isa quasi-isomorphism. S −→ E To finish the proof of Proposition 6.1 we must construct ϕ: t. This construction uses S−→E Toda brackets in Ho( od- t) which we calculate using Massey products. For a differential M E R graded ring with objects = Gi i I , a cycle a (Gj ,Gi)p induces a map of right -modules, p G { } ∈ ∈R R a:ΣF R F R , by left multiplication (or post-composition) where F R ( )= ( ,Gi) Gj −→ Gi Gi − R − is the free right -module represented by Gi. The associated homotopy class of maps [a]is determined by theR homology class represented by a in the homology of . This translates between Massey products, which are homology classes in , and Toda brackets,R which are R homotopy classes of right -module maps. Notice, if a (Gj,Gi)p and b (Gi,Gk)q are cycles then extra suspensionsR are required to compose these∈R elements; the associated∈R composition q p+q q is b(Σ a): Σ Gj Σ Gi Gk. −→ −→ Starting with the Massey products for a we construct elements of the Toda brackets for E Ho( od- a) in Proposition 6.5. See Definition A.2 for the definition of higher Toda brackets. M E We denote the Toda brackets by T , to differentiate them from the Massey products. Since Toda brackets are invariant underh−i exact equivalences by Theorem A.3, these Toda brackets agree with those for Ho( od- t). This follows because Quillen equivalences induce triangu- M E lated equivalences [20, I.4.3], od- t is Quillen equivalent to T-spectra by Proposition 5.7 and M E Corollary 3.9, od- a is Quillen equivalent to dg by Proposition 4.7 and Corollary 3.7 and M E A Ho(T-spectra) is triangulated equivalent to ( ) by [5, 5.6.1]. The Toda brackets in Ho( od- t) then give us the structure we need to constructD A ϕ. M E Proposition 6.5. The following Toda brackets in Ho( od- a) agree up to sign with the Massey M E products in H a from Proposition 4.9. ∗E H H H n 1 H n H H H H [f ], [m ], Σ[m ],...,Σ − [m ], Σ [g ] T = [f g ] = [i ] h 0 1 1 1 1 i { 0 2n+1 } { 2n+1 } AN ALGEBRAIC MODEL FOR RATIONAL S1-EQUIVARIANT STABLE HOMOTOPY THEORY 17 The sign difference here corresponds to the different sign conventions used for Massey products and Toda brackets. Proof. For ease of notation we fix a finite subgroup H and delete the superscripts H.We first consider the three-fold Toda bracket. Starting with the defining system for the Massey product [f0], [m1], [g1] we construct an element of the Toda bracket [f0], [m1], Σ[g1] T .By h i h i 3 a β Definition A.2 such a Toda bracket is the set of homotopy classes of compositions Σ F E T α a −→ X F E where the homotopy type of X is a 2-filtered object in [m1] ; see Definition A.1. T −→ 2 a { a } The cofiber of m1 is such an object. Explicitly, set X =Σ FHE nm1 FHE with underlying graded 2 a a object Σ F E F E and with differential given by d(x, y)=(dx, dy m1x). Then the defining H ⊕ H − system found in Proposition 4.9 determines the other two maps as well, α(x, y)=f0y and β(z)=(g1z,g3z). Here we are using the sign convention that dΣC = dC . One can check that − [σX β]=[g1]and[ασX0 ]=[f0]. Due to the sparseness of elements in a, one can check that α, β E 3 a a are the unique maps satisfying the definition of the Toda bracket. Since Σ F E ,X and F E are T T cofibrant and fibrant a-modules these are also the unique such homotopy classes. Hence the E Toda bracket is the one element set [f0g3]=[i3] . As with the three-fold Massey product,{ the higher} Massey products also agree up to sign with the higher Toda brackets. First, define the iterated cofiber 2n a 2n 2 a a Xn =Σ FHE nm1 Σ − FHE nm1 ...nm1 FHE 2 with differential d(x1,x2,...,xn+1)=(dx1,dx2 m1x1,...,dxn+1 m1xn). Note here d =0 2 − − n 1 because m = 0. By Lemma A.4, Xn is an element of [m1], Σ[m1],...,Σ − [m1] .Then 1 { } 2n+1 a β α a the associated defining system determines the maps Σ F E Xn F E as β(z)= T −→ −→ T (g1z,g3z,...,g2n+1z)andα(x1,...,xn+1)=f0xn+1. This shows that the composite, [f0g2n+1]= n 1 n [i2n+1] is in the Toda bracket [f0], [m1], Σ[m1],...,Σ − [m1], Σ [g1] T . One can check that any other defining system would produceh a Toda bracket and vice versa,i but again the sparseness of elements in a forces this to be the only element in the Toda bracket as well as the only defining system. E Proof of Proposition 6.1. We are left with constructing ϕ: t. We proceed by induction S−→E on the degree. We exploit the fact that Toda brackets in Ho( od- t)andHo( od- a) agree to construct ϕ. At each stage certain products must be non-trivialM becauseE of theM non-trivialE Toda H brackets. This then provides non-trivial targets for ϕ of the elements g2k+1 in . The products H H S g2k+1f0 are trivial in homology (for k>0) so they must be boundaries. This produces targets for ϕ of the elements sH in . We can then extend ϕ to a ring homomorphism because 2k+1,2l S the images of the relations in between these generators hold in t as well since t(H, T)is concentrated in degree zero. ByS construction ϕ is then a map of differentialE graded objectsE and we show inductively that it is also a quasi-isomorphism. Again, we mostly delete the superscripts H and implicitly apply these steps for each finite subgroup. Since H t is isomorphic to H ( t), we use the notation for elements in H t from Proposition 5.1. To∗E further ease notation,∗E ifBy is the boundary of x, i.e. dx = y,thenwesay∗E that x is a primitive of y. The images of idH and id in are forced to be the respective identity elements in t.We T S E next determine the images of f0 and g1. Let f00 be an element in t(H, T)0 such that [f00 ]=f0 E in H0 t(H, T). Let g10 be a cycle such that [g10 ]=g1 in H1 t(T,H). Then set ϕ(f0)=f00 and E E ϕ(g1)=g10 . Since t(H, T) is concentrated in degree zero by Proposition 5.7, f00 g10 f00 is trivial.b E Thus the relations between f0 and g1 in also hold in t so we may extend ϕ to products of S b E g1 and f0 using the product structure in t. Since [g0 ][f 0 ]=m1 and [f 0 ][g0 ]=i1 are non-trivial E 1 0 0 1 c b 18 BROOKE SHIPLEY in H t by Propostions 5.1 and 5.7, the images ϕ(g1f0)andϕ(f0g1) must be non-trivial cycles. Hence∗Eϕ induces a quasi-isomorphism in degrees zero and one. Before turning to the general induction step, we consider degrees two and three. It may be useful to refer to the tabulation of elements of (H, H)and (T,H) given in the proof S S of Proposition 6.4. There we also showed the other components of are isomorphic to a, S E see Definition 4.6; as vector spaces (H, K)=0, (H, T) is generated by f0 and (T, T)is S S S generated by the elements idT,f0g1,f0g3,.... As mentioned above, Toda brackets agree for Ho( od- a)andHo( od- t) by Theorem A.3. Since [f0], [m1], Σ[g1] T = [i2n+1] by Propo- M E M E h i { } 3 t β0 α0 t sition 6.5 in Ho( od- a), there must be maps Σ F E X0 F E such that [α0β0]=i3 is in M E T −→ −→ T f0, m1, Σg1 T in Ho( od- t). Here X0 must be a 2-filtered object in m1 , so define X0 as the h i M E { } cofiber of g10 f00 , similar to X in the proof of Proposition 6.5 above. Then since t(H, T)n isb trivial E forb n>c 0themapb α0 must be α0(x, y)=f00 y.Ifg10 f00 g10 = 0 then one couldc set β0(z)=(g10 z,0). Then the composite [α0β0] would be trivial. Since this Toda bracket does not contain the trivial map, this means g10 f00 g10 must be non-trivial. Since H2 t(T,H) = 0 it must also be a boundary, E let g30 be one of its primitives in t(T,H)3.Thensetβ0(z)=(g10 z,g30 z). The maps α0 and β0 sat- E isfy the definition of the Toda bracket, so by Proposition 6.5 we must have [α0β0]=[f00 g30 ]=i3. Set ϕ(g3)=g30 . We next inductively determine the images of s3,2n with n 2. Since f 0 g0 f 0 is in the trivialb ≥ 0 1 0 group t(H, T)1, d(g30 f00 )=(dg30 )f00 = g10 f00 g10 f00 = 0 and the element g30 f00 is a cycle. It is also E a boundary since H3 t(H, H) = 0 by Proposition 5.1 and 5.7; let s0 be one of its primitives E 3,4 and set ϕ(s3,4)=s30 ,4. Assume that elements s30 ,2k have been chosen which are primitives of s30 ,2k 2g10 f00 for 2 Assume by induction that ϕ(g2k+1)=g20 k+1 and ϕ(s2k+1,2l)=s20 k+1,2l for k n and l k + 1, and that the extension of ϕ to the subring generated by these elements induces≤ a quasi-≥ isomorphism through degree 2n + 1. Since g0 f 0 g0 f 0 =0,an t-module 1 0 1 0 E n 1 X0 m1, Σm1,...,Σ − m1 n ∈{ } exists, and can be built similarly to Xn in Proposition 6.5. Since t(H, T)n =0forn>0, the t c c c E map α0 : X0 F E is determined as α0(x1,...,xn+1)=f 0 xn+1. As with the three-fold Toda n −→ T 0 bracket if g10 f00 g20 n+1 were trivial then we could set β0(z)=(g10 z,g30 z,...,g20 n+1z,0). But then the composite α0β0 would be trivial, which is not allowed because the n + 2-fold Toda bracket from Proposition 6.5 does not contain the trivial map. Hence g10 f00 g20 n+1 is non-trivial. Since H2n+2 t(T,H) = 0, it must also be a boundary; let g20 n+3 t(T,H)2n+3 be one of its primitives E ∈E and set β0(z)=(g10 z,g30 z,...,g20 n+3z). Then α0 and β0 satisfy the definition of the Toda Bracket so n 1 n [α0β0]=[f 0 g0 ] f0, m1, Σm1,...,Σ − m1, Σ g1 T = i[2n+3 . 0 2n+3 ∈h i { } Set ϕ(g2n+3)=g20 n+3. b c c c b AN ALGEBRAIC MODEL FOR RATIONAL S1-EQUIVARIANT STABLE HOMOTOPY THEORY 19 Arguments similar to those for s3,2k for k 2 also apply to the elements s2n+3,2k for k n+2, ≥ ≥ n>0. The element g0 f 0 is a cycle and a boundary since H2n+3 t(H, H)=0;lets0 2n+3 0 E 2n+3,2n+4 be one of its primitives and set ϕ(s2n+3,2n+4)=s20 n+3,2n+4. Assume that elements s20 n+3,2k have been chosen which are primitives of s20 n+3,2k 2g10 f00 for n +2 Theorem A.3. Given an exact equivalence of triangulated categories ψ : 0,thenθ T−→T ∈ f1,...,fn T if and only if ψ(θ) ψ(f1),...,ψ(fn) T . h i ∈h i 20 BROOKE SHIPLEY n 2 Proof. Assume given θ f1,...,fn T .Thenθ = αβ with β :Σ− An X and α: X A0 ∈h i −→ −→ with X f2,...,fn 1 . Since ψ is exact, ψ(X), with filtrations ψ(FkX), is an element of ∈{ − } ψ(f2,...,ψ(fn 1) . Hence ψ(θ)=ψ(α)ψ(β)isanelementof ψ(f1),...,ψ(fn) T .Forthe { − } h 1 i other direction, apply the same arguments to the inverse equivalence ψ− . The following manipulations are useful for constructing filtered objects. Let Cα denote the cofiber of the map α. Lemma A.4. Assume given a composable sequence of maps fi : Ai Ai 1. −→ − 1. If X f2,...,fk and α: X A0,thenCα ασX0 ,f2,...,fk . ∈{ } −→k 1 ∈{ } k+1 2. If Y f1,...,fk 1 and α:Σ− Ak Y ,thenCα f1,...,fk 1, Σ− σY α . ∈{ − } −→ ∈{1 − } 3. If X f1,...,fk ,thenFkX f1,...,fk 1 and Σ− (X/F1X) f2,...,fk . ∈{ } ∈{ − } ∈{ } Proof. The first two statements follow by setting n =1andm = 1 respectively in [1, Proposition 2.3]. The third statement follows from [1, Proposition 2.2] and its analogue. Often definitions of Toda brackets require various vanishing conditions. The following propo- sition shows that these are implicit in the above definition. Proposition A.5. There is an object X in f1,...,fn 1 if and only if 0 f1,...,fn 1 T . { − } ∈h − i 1 Proof. First assume X f1,...,fn 1 . By Lemma A.4, Σ− (Fn 1X/F1X) f2,...,fn 2 . ∈{n 3 − }2 1 − ∈{1 − } Let β be the composite Σ − An 1 = Σ− (FnX/Fn 1X) Σ− Fn 1X Σ− (Fn 1X/F1X). 1 − ∼ − −→ − −→ − Let α be the map Σ− (Fn 1X/F1X) F1X = A0. Since α is the cofiber of the map 1 1 − −→ ∼ Σ− Fn 1X Σ− (Fn 1X/F1X), the composite αβ is trivial. So αβ =0 f1,...,fn 1 T . − −→ − ∈h n 3 − i For the other direction, assume θ =0 f1,...,fn 1 T .Thenθ = αβ with β :Σ− An 1 ∈h − i − −→ Z and α: Z A0 with Z f2,...,fn 2 . Since αβ is trivial, taking cofibers gives a map n 2 −→ ∈{ − } γ :Σ− An 1 Cα. By Lemma A.4, Cα f1,...,fn 2 and Cγ f1,...,fn 1 . − −→ ∈{ − } ∈{ − } References [1] J.M.Cohen,The decomposition of stable homotopy, Ann. Math. 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With a preface by Luc Illusie, Edited and with a note by Georges Maltsiniotis, xii+253 pp. Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA E-mail address: [email protected]