AN ALGEBRAIC MODEL FOR RATIONAL S^{1}-EQUIVARIANT STABLE HOMOTOPY THEORY

BROOKE SHIPLEY

Abstract. Greenlees defined an abelian category A whose derived category is equivalent to the rational S^{1}-equivariant stable homotopy category whose objects represent rational S^{1}- equivariant cohomology theories. We show that in fact the model category of differential graded objects in A models the whole rational S^{1}-equivariant stable homotopy theory. That is, we show that there is a Quillen equivalence between dgA and the model category of rational S^{1}-equivariant spectra, before the quasi-isomorphisms 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.

1. Introduction

In [5], Greenlees defined an abelian category A and showed that its derived category is equivalent 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 A and the model category on differential graded objects in A, dgA, are recalled in Section 2.

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 A has injective dimension one, calculations of these higher order structures are quite practical, as 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 geometric 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, D(A), is equivalent to the homotopy category of rational 𝕋-equivariant spectra, Ho(𝕋-spectra), rely heavily on the fact that the circle is rank one. The arguments used here though can be modified to 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].

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, [22, 6.1.3] or [23, 1.3]. The following statement is proved as Corollary 3.9 below. Definition 3.5 recalls the notion of a Quillen equivalence.

Date: September 25, 2000; 2000 AMS Math. Subj. Class.: 55P62, 55P91, 55P42, 55N91, 18E30.

Research partially supported by an NSF grant.

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 E(B_{t}) with one object for each closed subgroup in 𝕋.

𝕋-spectra ≃_{Q} Mod-E(B_{t})

Similarly, the model category of differential graded objects in A is Quillen equivalent to (right) modules over a rational differential graded ring E(B_{a}) with one object for each closed subgroup in 𝕋; see Corollary 3.7 below. Since quasi-isomorphisms of differential graded rings (with many objects) induce Quillen equivalences of the associated categories of modules by [21, 4.3], see also [22, A.1.2] or [23, A.1], Theorem 1.1 follows given a zig-zag of quasi-isomorphisms between E(B_{a}) and E(B_{t}). This is carried out in Sections 4, 5 and 6.

Theorem 1.3. The model category of differential graded objects in A is Quillen equivalent to modules over the differential graded ring E(B_{a}). Also, there is a zig-zag of quasi-isomorphisms between E(B_{a}) and E(B_{t}) which induces a zig-zag of Quillen equivalences between the associated model categories. Hence we have a chain of Quillen equivalences

𝕋-spectra ≃_{Q} Mod-E(B_{t}) ≃_{Q} Mod-E(B_{a}) ≃_{Q} dgA.

In more detail, in Section 4 we explicitly describe E(B_{a}) and a quasi-isomorphic sub-ring E_{a}. In Section 5, we modify E(B_{t}) to define a more amenable ring E_{t}. Then in Section 6 we construct a quasi-isomorphism φ: E_{a} ⟶ E_{t}. This construction involves Massey products for differential graded rings with many objects which are defined in Definition 4.8.

In particular, we use the fact that the higher order products for E_{a} and E_{t} must agree because there are Quillen equivalences

Mod-E_{a} ≃_{Q} Mod-E(B_{a}) ≃_{Q} dgA

and

Mod-E_{t} ≃_{Q} Mod-E(B_{t}) ≃_{Q} 𝕋-spectra

and Ho(𝕋-spectra) and D(A) are equivalent as triangulated categories by [5, 5.6.1]. This invariance of higher order products 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: T ⟶ T′ is an exact equivalence of triangulated categories then the Toda brackets for T and T′ agree.

Acknowledgments: I would like to thank Bill Dwyer for a discussion which sparked my interest in this project. I would also like to thank John Greenlees and Jeff Smith for many conversations throughout this project. Finally, I would like to 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 Ch_{ℚ}-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 F be the set of finite subgroups of 𝕋. Let O_{F} be the ring of operations, Π_{H∈F}ℚ[c_{H}] with c_{H} in degree −2. Let e_{H} be the idempotent which is projection onto the factor corresponding to H. Let c be the total Chern class with e_{H}c = c_{H}. Let E be the multiplicative set of all Euler classes, {c^{v} | v: F ⟶ ℤ_{≥0} with finite support}, where e_{H}c^{v} = c_{H}^{v(H)}. Let t_{∗}^{F} = E^{−1}O_{F}. As a vector space (t_{∗}^{F})_{2n} is Π_{H∈F}ℚ for n ≤ 0 and is ⨁_{H∈F} ℚ for n > 0.

We now define the abelian category A. The category of differential graded objects in A is then the standard algebraic model of rational 𝕋-equivariant spectra. Definition 2.2. [5, 5.4.1] The objects of A are maps N ⟶^{β} t_{∗}^{F} ⊗ V of O_{F}-modules with V a graded vector space such that E^{−1}β is an isomorphism. N is referred to as the nub and V is the vertex. Morphisms are commutative squares

M ⟶^{θ} N
β↓  ↓^{β′}
t_{∗}^{F} ⊗ U ⟶_{1⊗ϕ} t_{∗}^{F} ⊗ V.

The condition that E^{−1}β is an isomorphism is equivalent to requiring that the kernel K and cokernel C of β are F-finite torsion modules [5, 4.5.1, 4.5.2, 4.6.6]. That is, they are torsion modules with respect to the total Chern class and they decompose as direct sums, K = ⨁_{H} e_{H}K and C = ⨁_{H} e_{H}C.

Greenlees, in [5, 5.6.1], shows that the derived category of differential graded objects in A is equivalent to the homotopy category of rational 𝕋-equivariant spectra. Here we recall the model category structure on dgA. The associated homotopy category is equivalent to D(A). Proposition 2.3. [5, Appendix B] The category dgA of differential graded objects in A with cofibrations the monomorphisms, weak equivalences the quasi-isomorphisms, and fibrations determined by the right lifting property is a model category.

The model category dgA is also tensored, cotensored and enriched over rational chain complexes in a way that is compatible with the model category structure. We call such a model category a Ch_{ℚ}-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 Ch_{ℚ} 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 Ch_{ℚ}-model category is a complete and cocomplete model category C which is tensored, cotensored and enriched (denoted Hom_{C}) over the category Ch_{ℚ} and satisfies the following compatibility axiom (CM): (CM) For every cofibration A ⟶^{i} B and every fibration X ⟶^{p} Y in C the induced map

Hom_{C}(i, p): Hom_{C}(B, X) ⟶ Hom_{C}(A, X) ×_{Hom_{C}(A,Y)} Hom_{C}(B, Y)

is a fibration in Ch_{ℚ}. If in addition one of the maps i or p is a weak equivalence, then Hom_{C}(i, p) is a trivial fibration. We use the notation K ⊗ X and X^{K} to denote the tensors and cotensors for X in C and K a chain complex.

For example, Ch_{ℚ} is itself a Ch_{ℚ}-model category [9, 4.2.13] as is the projective model category defined in [22, A.1.2] 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 dgA is a Ch_{ℚ}-model category.

Proof. For K ∈ Ch_{ℚ} and A ∈ dgA define K ⊗ A in dgA by

K ⊗ A = ⨁_{n} Σ^{n}(K_{n} ⊗_{ℚ} A)

with differential d(k_{n} ⊗a) = dk_{n} ⊗a+(−1)^{n}k_{n} ⊗da. Let ℚ[n] denote the rational chain complex with ℚ in degree n. Then for A, B in dgA define Hom_{C}(A, B) in Ch_{ℚ} by Hom_{C}(A, B)_{n} = A(ℚ[n] ⊗ A, B) where A(−, −) is the rational vector space of maps of underlying graded objects in A (ignoring the differential). The differential for Hom_{C}(A, B) is given by df_{n} = d_{B}f_{n} + (−1)^{n}f_{n}d_{A}. The cotensor is defined as the right adjoint of the tensor. So we have the following isomorphisms

Ch_{ℚ}(K, Hom_{C}(A, B)) ≅ dgA(K ⊗ A, B) ≅ dgA(A, B^{K}).

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_{ℚ} and g: A ⟶ B in dgA, define f □ g: K ⊗ B ⨿_{K⊗A} L ⊗ A ⟶ L ⊗ B. Then dgA is a Ch_{ℚ}-model category; if f, g are cofibrations, then f □ g is a cofibration which is trivial if f or g is. Since Ch_{ℚ} 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 D^{n} be the acyclic rational chain complex with ℚ in degrees n and n − 1. Then the generating cofibrations of Ch_{ℚ} are the maps i_{n}: ℚ[n − 1] ⟶ D^{n} for n an integer and the generating trivial cofibrations are the maps j_{n}: 0 ⟶ D^{n} for n an integer [9, 2.3.3]. The source of the map i_{n} ⊗ g is the cofiber of g and the target is D^{n} ⊗ B. So for g a monomorphism, i_{n} □ g is a monomorphism because it is g on one summand and id_{B} on the other. Since D^{n} ⊗ B is always acyclic and the cofiber of a quasi-isomorphism is acyclic, if g is a trivial cofibration, then i_{n} □g is a quasi-isomorphism. Also, j_{n} □g: D^{n} ⊗A ⟶ D^{n} ⊗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 dgA. The natural building blocks for 𝕋-equivariant spectra are the G-cells 𝕋_{+} ∧_{H} S^{n}. Rationally, these can be simplified by using idempotents in the Burnside ring for a finite group, A(H) = [S^{0}, S^{0}]^{H} ≅ Π_{K≤H}ℚ. Following Greenlees, we suppress the rationalization in our notation for spectra. Let e_{K} 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 σ_{𝕋}^{0} = S^{0} and σ_{H}^{0} = 𝕋_{+} ∧_{H} e_{H}S^{0} for H a finite subgroup of 𝕋.

Via the equivalences between D(A) and rational 𝕋-equivariant spectra, these geometric basic cells correspond to the following algebraic basic cells. Definition 2.7. [5, 5.8.1] For H a finite subgroup, let ℚ(H) denote the ℚ[c_{H}]-module ℚ regarded as an O_{F}-module. Then the algebraic basic cells are the objects L_{H} = (Σℚ(H) ⟶ 0) and L_{𝕋} = (O_{F} ⟶ t_{∗}^{F}) in A.

We need fibrant replacements of these algebraic basic cells which are inclusions into their respective injective hulls. The injective dimension of A is one, so the injective hulls are simple to construct. First we define the basic injective objects. Definition 2.8. [5, 2.4.3, 3.4.1, 5.2.1, 5.5.1] Let 𝕀(H) = Σ^{−2}ℚ[c_{H}, c_{H}^{−1}]/ℚ[c_{H}]. Let 𝕀 = ⨁_{H} 𝕀_{H} ≅ Σ^{−2}t_{∗}^{F}/O_{F}. The underlying vector spaces are 𝕀(H)_{2n} = ℚ for n ≥ 0 and 𝕀_{2n} = ⨁_{H} ℚ for n ≥ 0. Then f(𝕀) = (𝕀 ⟶ 0), f(𝕀(H)) = (𝕀(H) ⟶ 0), and e(V) = (t_{∗}^{F} ⊗ V ⟶ t_{∗}^{F} ⊗ V) for any graded vector space V are injective objects in A.

The construction of fibrant approximations in [5, Appendix B] produces injective hulls for L_{H} and L_{𝕋} which we denote by I_{H} and I_{𝕋} respectively. Let P (I) = I ⋉ Σ^{−1}I be the acyclic path object in dgA with underlying graded object I ⊕ Σ^{−1}I. For a finite subgroup H of 𝕋, let Σ𝕀(H) ⋉ Σ^{2}𝕀(H) be the pullback in the following square.

Σ𝕀(H) ⋉ Σ^{2}𝕀(H) ⟶ P (Σ^{3}𝕀(H))
↓  ↓
Σ𝕀(H)  ⟶ Σ^{3}𝕀(H)

The bottom map is zero in degree one and an isomorphism in all higher degrees. Let I_{H} denote the object (Σ𝕀(H) ⋉ Σ^{2}𝕀(H) ⟶ 0) in dgA. The inclusion L_{H} ⟶ I_{H} is a quasi-isomorphism and I_{H} is fibrant since it is built from standard injectives.

For 𝕋 itself, let t_{∗}^{F} ⋉ Σ𝕀 be the pullback in the following square

t_{∗}^{F} ⋉ Σ𝕀 ⟶ P (Σ^{2}𝕀)
↓  ↓
t_{∗}^{F} ⟶ Σ^{2}𝕀

The bottom map is zero in degrees n ≤ 0 and an isomorphism in all positive degrees. Let I_{𝕋} denote the object (t_{∗}^{F} ⋉ Σ𝕀 ⟶ t_{∗}^{F}) in dgA. The inclusion L_{𝕋} ⟶ I_{𝕋} is a quasi-isomorphism and I_{𝕋} is fibrant since it is built from standard injectives.

Proposition 2.9. I_{H} and I_{𝕋} are fibrant replacements of L_{H} and L_{𝕋} 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 E-modules for some ring E (with many objects). See Definition 3.1 below. The ring E in question may be taken to be the endomorphism ring of the projective generators; this is naturally 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 G is an Ab-category with objects G. This is also a category enriched over abelian groups, [13, 1.2] or an Ab-module, [9, 4.1.6]. A ring is then an Ab-category with one object. Similarly, R, a rational differential graded ring with objects G, is a category enriched over Ch_{ℚ}, or a Ch_{ℚ}-category or Ch_{ℚ}-module. A (right) R-module is a Ch_{ℚ}-functor M: R^{op} ⟶ Ch_{ℚ}.

Given R, a rational differential graded ring with objects G, and objects G, G′, G′′ in G there is a morphism chain complex Hom_{R}(G, G′) ∈ Ch_{ℚ} and coherently associative and unital composition maps

Hom_{R}(G′, G′′) ⊗ Hom_{R}(G, G′) ⟶ Hom_{R}(G, G′′).

An R-module M is then determined by chain complexes M(G) for each object G ∈ G and coherently associative and unital action maps

M(G′) ⊗ Hom_{R}(G, G′) ⟶ M(G).

For each object G in G there is a free R-module, F_{G}^{R}, defined by F_{G}^{R}(G′) = R(G′, G). The projective model structure on the category of R-modules, Mod-R, is established in [22, A.1.2] or [23, A.1] using [21, 2.3]. The weak equivalences are the quasi-isomorphisms and the fibrations are the epimorphisms. With the standard action of Ch_{ℚ}, Mod-R is a Ch_{ℚ}-model category.

Definition 3.2. Let G be a set of objects in a Ch_{ℚ}-model category C. We denote by E(G) the full Ch_{ℚ}-subcategory of C with objects G, i.e., E(G)(G, G′) = Hom_{C}(G, G′) is a differential graded ring with many objects, one for each element of G. We let

Hom_{C}(G, −) : C ⟶ Mod-E(G)

denote the functor given by Hom_{C}(G, Y)(G) = Hom_{C}(G, Y).

Note that, if G = {G} has a single element, then E(G) is determined by the single differential graded ring, End(G) = Hom_{C}(G, G).

For our enriched variant of Gabriel’s theorem it is crucial that a Ch_{ℚ}-model category is a stable model category, that is, a pointed model category where the suspension functor is a selfequivalence 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 C is a Ch_{ℚ}-model category then C is a stable model category. If X is a cofibrant object in C and Y is a fibrant object in C, then there is a natural isomorphism of graded abelian groups H_{∗} Hom_{C}(X, Y) ≅ [X, Y]_{∗}^{Ho(C)}.

Proof. First, C is pointed since Ch_{ℚ} is pointed [9, 4.2.19]. For a cofibrant object X in C, a cylinder object [20, I.1] is given by (D^{1} ⊕ ℚ[0]) ⊗ X. It is a cylinder object because X ⨿ X ≅ (ℚ[0] ⊕ ℚ[0]) ⊗ X ⟶^{i} (D^{1} ⊕ ℚ[0]) ⊗ X is a cofibration, by an adjoint form of the compatibility axiom (CM), see [20, II.2, SM7(b)], and (D^{1} ⊕ ℚ[0]) ⊗ X ⟶ ℚ[0] ⊗ X ≅ X is a quasiisomorphism. Hence the cofiber of i, ℚ[1] ⊗ X, represents ΣX. But ℚ[1] is invertible with inverse ℚ[−1]. Since the action of Ch_{ℚ} on C is associative up to coherent isomorphism, this shows that Σ is a self-equivalence on Ho(C). Thus C is a stable model category.

Since a cylinder object in Ch_{ℚ} is also given by tensoring with D^{1} ⊕ℚ[0], we have the following natural isomorphisms for a cofibrant object X in C and a fibrant object Y in C.

H_{n} Hom_{C}(X, Y) ≅ [ℚ[n], Hom_{C}(X, Y)]^{Ho(Ch_{ℚ})} ≅ [ℚ[n] ⊗ X, Y]^{Ho(C)} ≅ [Σ^{n}X, Y]^{Ho(C)}

We also need the following definitions from triangulated categories, see [11, 1.1].

Definition 3.4. An object X in a stable model category C is small if in Ho(C) the natural map ⨁_{i}[X, Y_{i}] ⟶ [X, ⨿_{i}Y_{i}] is an isomorphism for any set of objects {Y_{i}}. A subcategory of a triangulated category is localizing if it is closed under cofiber sequences, retracts and coproducts. A set of objects G in C is a set of (weak) generators if the only localizing subcategory of Ho(C) which contains G is Ho(C).

The following enriched variant of Gabriel’s theorem is stated rationally, but holds integrally as well with Ch_{ℚ} replaced by Ch_{ℤ}. 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 C be a Ch_{ℚ}-model category and G a set of small, cofibrant and fibrant generators. The functor

Hom_{C}(G, −) : C ⟶ Mod-E(G)

is the right adjoint of a Quillen equivalence. The left adjoint is denoted − ⊗_{E(G)} G.

Proof. The functor Hom_{C}(G, −) preserves fibrations and trivial fibrations by the compatibility axiom (CM) of Definition 2.4, since all objects of G are cofibrant. So together with its left adjoint it forms a Quillen pair. Denote the associated total derived functors on the homotopy categories by RHom_{C}(G, −) and − ⊗_{E(G)}^{L} G. Since Ho(C) is a triangulated category both total derived functors preserve shifts and triangles; they are exact functors of triangulated categories by [20, I.4 ]. Since − ⊗_{E(G)}^{L} G is a left adjoint it commutes with colimits. The right adjoint also commutes with coproducts since each object of G is small. This follows as in [22, 3.10.3(ii)].

We consider the full subcategories of those M ∈ Ho(Mod-E(G)) and X ∈ Ho(C) respectively for which the unit of the adjunction

η : M ⟶ RHom_{C}(G, M ⊗_{E(G)}^{L} G)

or the counit of the adjunction

ν : RHom_{C}(G, X) ⊗_{E(G)}^{L} X ⟶ X

are isomorphisms. Since both derived functors are exact and preserve coproducts, these are localizing subcategories. For every G ∈ G the E(G)-module Hom_{C}(G, G) is isomorphic to the free module F_{G}^{E(G)} by inspection and F_{G}^{E(G)} ⊗_{E(G)} G is isomorphic to G since they represent the same functor on C. So the map η is an isomorphism for every free module, and the map ν is an isomorphism for every object of G. Since the free modules F_{G}^{E(G)} generate the homotopy category of E(G)-modules and the objects of G generate C, the localizing subcategories are the whole categories. So the derived functors are inverse equivalences of the homotopy categories.

Since dgA is a Ch_{ℚ}-model category by Proposition 2.5, we can apply Theorem 3.6. Triangulated equivalences preserve generators so we identify generators of dgA via the triangulated equivalence π_{∗}^{A}: Ho(𝕋-spectra) ⟶ D(A) from the homotopy category of 𝕋-spectra to the homotopy category of dgA, or the derived category of A, by [5, 5.6.1]. The geometric basic cells B′_{t} = {σ_{H}^{0}}_{H≤𝕋} for H any closed subgroup of 𝕋 form a set of small generators for 𝕋-spectra. They detect stable equivalences because π_{∗}^{H}(X) = [∨_{K≤H}σ_{K}^{0}, X]_{∗}^{𝕋} for H finite and π_{∗}^{𝕋}(X) = [σ_{𝕋}^{0}, X]_{∗}^{𝕋} by [5, 2.1.5]. It follows by [22, 2.2.1] or part of [11, 2.3.2] that B′_{t} is a set of small generators. Hence, the set of images {π_{∗}^{A}(σ_{H}^{0}) = L_{H}}_{H≤𝕋} form a set of small generators for dgA. Let B_{a} = {I_{H}}_{H≤𝕋} be the set of their fibrant replacements. Then B_{a} is a set of small, cofibrant and fibrant generators. Let E(B_{a}) be the endomorphism Ch_{ℚ}-category as in Definition 3.2. In Section 4 we explicitly describe E(B_{a}).

Corollary 3.7. The model category dgA is Quillen equivalent to the model category of differential graded modules over E(B_{a}).

To prepare for constructing a Quillen equivalence between dgA and the model category of 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 Sp^{Σ}-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, stable model categories where [X, Y]^{Ho(C)} is a rational vector space for any objects X and Y in C. In this case the Gabriel equivalence produces a rational ring spectrum which can be replaced by a differential graded ring with many objects. Theorem 3.8. [22, 6.1.3] Let C be a spectral model category. If C is rational and has a set of small generators, then C is Quillen equivalent (via a chain of several Quillen equivalences) to the category of modules 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 singular 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, 7.11,8.1] shows that 𝕋-spectra is Quillen equivalent to a spectral model category, C′. Since rational 𝕋-equivariant spectra has a set of small generators, B′_{t}, the geometric basic cells, C′ is also rational and has a set of small generators because these properties are determined on the homotopy category level. Thus, by Theorem 3.8, C′ is Quillen equivalent to Mod-R for some rational differential graded ring with many objects, R. Let B_{t} be a set of cofibrant and fibrant replacements of the images of the basic cells in Mod-R. Then B_{t} is a set of small generators in Mod-R. Since Mod-R is a Ch_{ℚ}-model category, Theorem 3.6 shows that Mod-R is Quillen equivalent to Mod-E(B_{t}) where E(B_{t}) is the endomorphism ring on the generators B_{t}.

Corollary 3.9. 𝕋-spectra is Quillen equivalent to Mod-E(B_{t}) which is the endomorphism ring on a set of small generators B_{t} in an intermediate model category Mod-R.

𝕋-spectra ≃_{Q} Mod-R ≃_{Q} Mod-E(B_{t})

This does give an algebraic model for 𝕋-spectra, but E(B_{t}) is large and not explicit. This algebraic model is nonetheless useful because we can construct a zig-zag of quasi-isomorphisms between E(B_{t}) and E(B_{a}) which induce Quillen equivalences on the respective module categories.

4. The endomorphism ring for dgA

In this section we first explicitly calculate E(B_{a}), the endomorphism ring associated to dgA by Corollary 3.7. Then in Definition 4.6 we define a sub-ring E_{a} which is quasi-isomorphic to E(B_{a}). In Section 6, we then use this simpler ring E_{a} to construct a zig-zag of Quillen equivalences between dgA and 𝕋-spectra. For this construction, we need to understand the Massey products in H_{∗}E_{a}. So we end this section with a definition of Massey products for differential graded rings with many objects and a calculation of the products and Massey products in H_{∗}E_{a}. Since E_{a} and E(B_{a}) are quasi-isomorphic this calculation also applies to H_{∗}E(B_{a}) by [18, 1.5].

First we calculate the underlying rational chain complexes of E(B_{a}).

Proposition 4.1. For any distinct finite subgroups H, K of 𝕋 the rational chain complexes of homomorphisms are as follows:

1. Hom_{dgA}(I_{H}, I_{H}) = (⨁_{n≤0} D^{n}) ⊕ ℚ[0] ⊕ ℚ[1],

2. Hom_{dgA}(I_{H}, I_{K}) = 0,

3. Hom_{dgA}(I_{H}, I_{𝕋}) = (⨁_{n≤0} D^{2n−1}) ⊕ ℚ[0],

4. Hom_{dgA}(I_{𝕋}, I_{H}) = (⨁_{n≥0} D^{2n+1}) ⊕ (⨁_{m≤0} D^{m}) ⊕ ℚ[1] and

5. Hom_{dgA}(I_{𝕋}, I_{𝕋}) = (⨁_{n≥0} ℚF[2n + 1]) ⊕ (⨁_{m≤0}(D^{2m} ⊗ ℚF)) ⊕ ℚ[0] where ℚF is the rational vector space with basis F, the set of finite subgroups of 𝕋.

Proof. In the first three cases, since the vertex of the source is trivial the maps are determined by considering graded O_{F}-module maps of the nubs, denoted by O_{F∗}(−, −). As an underlying O_{F}-module, the nub of I_{H} is Σ𝕀(H) ⊕ Σ^{2}𝕀(H). As a graded vector space O_{F∗}(𝕀(H), 𝕀(H)) = ⨁_{n≤0} ℚ[2n]. This determines the underlying graded vector space of Hom_{dgA}(I_{H}, I_{H}). Let {m_{2n}}_{n≤0} be a basis for O_{F∗}(Σ𝕀(H), Σ𝕀(H)), {m_{2n+1}}_{n≤0} be a basis for O_{F∗}(Σ𝕀(H), Σ^{2}𝕀(H)), {l_{2n}}_{n≤0} be a basis for O_{F∗}(Σ^{2}𝕀(H), Σ^{2}𝕀(H)), and {l_{2n−1}}_{n≤0} be a basis for O_{F∗}(Σ^{2}𝕀(H), Σ𝕀(H)). When necessary, these bases and those below are decorated with an H. The differential is determined by dm_{2n} = m_{2n−1}, dm_{2n+1} = 0, dl_{2n} = m_{2n−1}, and dl_{2n−1} = l_{2n−2} − m_{2n−2}. Since O_{F∗}(𝕀(H), 𝕀(K)) = 0, it follows that Hom_{dgA}(I_{H}, I_{K}) = 0.

As an underlying O_{F}-module, the nub of I_{𝕋} is t_{∗}^{F}⊕Σ𝕀. As a graded vector space O_{F∗}(𝕀(H), 𝕀) = ⨁_{n≤0} ℚ[2n] and O_{F∗}(𝕀(H), t_{∗}^{F}) = 0, this determines the underlying graded vector space of Hom_{dgA}(I_{H}, I_{𝕋}). Let {f_{2n}}_{n≤0} be a basis for O_{F∗}(Σ𝕀(H), Σ𝕀) and {f_{2n−1}}_{n≤0} be a basis for O_{F∗}(Σ^{2}𝕀(H), Σ𝕀). The differential is determined by df_{2n−1} = −f_{2n−2} and df_{2n} = 0.

For the fourth case the vertex of the target is trivial, so again the maps are determined by the nubs. Also, note that O_{F∗}(t_{∗}^{F}, 𝕀(H)) = ⨁_{n∈ℤ} ℚ[2n]. Let {g_{2n+1}}_{n∈ℤ} be a basis for O_{F∗}(t_{∗}^{F}, Σ𝕀(H)), {g_{2n}}_{n∈ℤ} be a basis for O_{F∗}(t_{∗}^{F}, Σ^{2}𝕀(H)), {h_{2n}}_{n≤0} be a basis for O_{F∗}(Σ𝕀(H), Σ𝕀(H)) and {h_{2n+1}}_{n≤0} be a basis for O_{F∗}(Σ𝕀(H), Σ^{2}𝕀(H)). The differential is determined by dg_{2n+1} = g_{2k}, dg_{2n} = 0, dh_{2n} = h_{2n−1} + g_{2n−1}, and dh_{2n+1} = −g_{2n}.

The last case is the only one where we must consider the vertex. This restricts the maps from t_{∗}^{F} ⊗ ℚ to itself to maps of the form 1 ⊗ ϕ. Hence O_{F∗}(t_{∗}^{F}, t_{∗}^{F}) contributes only ℚ[0] to Hom_{dgA}(I_{𝕋}, I_{𝕋}). Also, note that O_{F∗}(t_{∗}^{F}, 𝕀) = ⨁_{n∈ℤ} ℚF[2n] and O_{F∗}(𝕀, 𝕀) = ⨁_{n≤0} ℚF[2n]. Let {i_{2n+1}^{H}}_{n∈ℤ,H∈F} be a basis for O_{F∗}(t_{∗}^{F}, Σ𝕀(H)), {j_{2n}^{H}}_{n≤0,H∈F} be a basis for O_{F∗}(Σ𝕀(H), Σ𝕀(H)), and id_{𝕋} the identity map. The differential is determined by di_{2n+1}^{H} = 0, dj_{2n}^{H} = i_{2n−1}^{H}, and d id_{𝕋} = 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 E(B_{a}). This greatly simplifies the structure, see Proposition 4.5. We include the following proposition for completeness.

Proposition 4.2. The following is a list of all of the non-trivial products, or compositions, of the bases chosen for E(B_{a}) in Proposition 4.1.

\begin{matrix}m_{2n}g_{2k+1} = g_{2n+2k+1} & m_{2n+1}g_{2k+1} = g_{2n+2k+2} & l_{2n}g_{2k} = g_{2n+2k} & l_{2n−1}g_{2k} = g_{2n+2k−1} \\ f_{2n−1}g_{2k} = i_{2n+2k−1} & f_{2n}g_{2k+1} = i_{2n+2k+1} & l_{2n}h_{2k} = g_{2n+2k} & l_{2n−1}h_{2k} = g_{2n+2k−1} \\ l_{2n}h_{2k+1} = h_{2n+2k+1} & l_{2n−1}h_{2k+1} = h_{2n+2k} & h_{2n}f_{2k} = m_{2n+2k} & h_{2n}f_{2k−1} = l_{2n+2k−1} \\ h_{2n+1}f_{2k} = m_{2n+2k+1} & h_{2n+1}f_{2k−1} = l_{2n+2k} & f_{2n}h_{2k} = j_{2n+2k} & f_{2n−1}h_{2k+1} = j_{2n+2k} \\ h_{2n}j_{2k} = h_{2n+2k} & h_{2n+1}j_{2k} = h_{2n+2k+1} & h_{2n}i_{2k+1} = g_{2n+2k+1} & h_{2n+1}i_{2k+1} = g_{2n+2k+2} \\ f_{2n}m_{2k} = f_{2n+2k} & f_{2n−1}m_{2k+1} = f_{2n+2k} & f_{2n−1}l_{2k} = f_{2n+2k−1} & f_{2n}l_{2k−1} = f_{2n+2k−1} \\ j_{2n}f_{2k} = f_{2n+2k} & j_{2n}f_{2k−1} = f_{2n+2k−1} & m_{2n}m_{2k} = m_{2n+2k} & m_{2n+1}m_{2k} = m_{2n+2k+1} \\ m_{2n}l_{2k−1} = l_{2n+2k−1} & m_{2n+1}l_{2k−1} = l_{2n+2k} & l_{2n}m_{2k+1} = m_{2n+2k+1} & l_{2n−1}m_{2k+1} = m_{2n+2k} \\ l_{2n}l_{2k} = l_{2n+2k} & l_{2n−1}l_{2k} = l_{2n+2k−1} & j_{2n}^{H}j_{2k}^{H} = j_{2n+2k}^{H} & j_{2n}^{H}i_{2k+1}^{H} = i_{2n+2k+1}^{H} \end{matrix}

Also, x id_{𝕋} = x where x = i_{2n+1}, j_{2n}, id_{𝕋} or f_{n} and id_{𝕋} y = y where y = i_{2n+1}, j_{2n}, id_{𝕋}, g_{n} or h_{n}.

Next we consider simpler sub-rings of E(B_{a}).

Definition 4.3. The (−1)-connected cover R⟨−1⟩ of a differential graded ring R is defined by R⟨−1⟩_{n} = 0 for n < 0, R⟨−1⟩_{n} = R_{n} for n > 0 and R⟨−1⟩_{0} = Z_{0}R where Z_{0}R is the kernel of the differential d: R_{0} ⟶ R_{−1}. Since the product of two cycles is a cycle, the product on R induces a product on R⟨−1⟩. Moreover the inclusion R⟨−1⟩ ⟶ R is a map of differential graded rings which induces an isomorphism in homology in non-negative degrees. This definition generalizes to a differential graded rings with many objects, R, by applying this to each morphism complex R(o, o′).

Since the homology of E(B_{a}) is concentrated in non-negative degrees it is quasi-isomorphic to its (−1)-connected cover. So by [21, 4.3] the module categories are Quillen equivalent.

Proposition 4.4. The map E(B_{a})⟨−1⟩ ⟶ E(B_{a}) is a quasi-isomorphism of differential graded rings with objects B_{a}. Thus, there is a Quillen equivalence between the respective categories of modules.

Mod-E(B_{a})⟨−1⟩ ≃_{Q} Mod-E(B_{a})

Next we explicitly describe E(B_{a})⟨−1⟩.

Proposition 4.5. The rational chain complexes underlying the differential graded ring E(B_{a})⟨−1⟩ are given by:

1. E(B_{a})⟨−1⟩(I_{H}, I_{H}) = ℚ[0] ⊕ ℚ[1] with generators id_{H} = m_{0} + l_{0} and m_{1},

2. E(B_{a})⟨−1⟩(I_{H}, I_{K}) = 0,

3. E(B_{a})⟨−1⟩(I_{H}, I_{𝕋}) = ℚ[0] with generator f_{0},

4. E(B_{a})⟨−1⟩(I_{𝕋}, I_{H}) = (⨁_{n≥0} D^{2n+1}) ⊕ ℚ[1] with generators g_{n} for degrees n ≥ 2, g_{1} + h_{1} generating the cycles in degree 1 and h_{1} and g_{0} generating D^{1} and

5. E(B_{a})⟨−1⟩(I_{𝕋}, I_{𝕋}) = (⨁_{n≥0} ℚF[2n + 1]) ⊕ ℚ[0] with generators i_{2n+1}^{H} and id_{𝕋}. The products are determined by the products in E(B_{a}).

We make one more modification to the ring E(B_{a})⟨−1⟩ by considering the sub-ring E_{a}.

Definition 4.6. Define the differential graded ring E_{a} with objects B_{a} as the sub-ring of E(B_{a})⟨−1⟩ produced by deleting the D^{1} generated by h_{1} and g_{0}. Also denote the element g_{1} + h_{1} by x_{1}. Then the rational chain complexes underlying E_{a} are given by:

1. E_{a}(I_{H}, I_{H}) = ℚ[0] ⊕ ℚ[1] with generators id_{H} = m_{0} + l_{0} and m_{1},

2. E_{a}(I_{H}, I_{K}) = 0,

3. E_{a}(I_{H}, I_{𝕋}) = ℚ[0] with generator f_{0},

4. E_{a}(I_{𝕋}, I_{H}) = (⨁_{n≥1} D^{2n+1}) ⊕ ℚ[1] with generators g_{n} for degrees n ≥ 2 and x_{1}, and

5. E_{a}(I_{𝕋}, I_{𝕋}) = (⨁_{n≥0} ℚF[2n + 1]) ⊕ ℚ[0] with generators i_{2n+1} and id_{𝕋}. The non-trivial products of these chosen bases, except the obvious products with id_{H} and id_{𝕋} are:

1. x_{1}f_{0} = m_{1}.

2. f_{0}x_{1} = i_{1}

3. m_{1}x_{1} = g_{2},

4. m_{1}g_{2n+1} = g_{2n+2} for n ≥ 1,

5. f_{0}g_{2n+1} = i_{2n+1} for n ≥ 1 and

6. x_{1}i_{2n+1} = g_{2n+2} for n ≥ 1.

Then, since quasi-isomorphisms induce Quillen equivalences by [21, 4.3], we have the following statement.

Proposition 4.7. The inclusion of differential graded rings E_{a} ⟶ E(B_{a})⟨−1⟩ which sends x_{1} to g_{1} + h_{1} is a quasi-isomorphism. Hence the composite E_{a} ⟶ E(B_{a})⟨−1⟩ ⟶ E(B_{a}) is also a quasi-isomorphism. Thus, there is a Quillen equivalence between the respective categories of modules.

Mod-E_{a} ≃_{Q} Mod-E(B_{a})

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.

Definition 4.8. Let R be a differential graded ring with objects G = {G_{i}}_{i∈I}. Suppose that γ_{1}, …, γ_{n} are classes in H_{∗}R with γ_{i} ∈ H_{p_{i}}R(G_{i}, G_{i−1}). A defining system, associated to ⟨γ_{1}, …, γ_{n}⟩, is a set of elements a_{i,j} for 1 ≤ i ≤ j ≤ n with (i, j) ≠ (1, n) such that

1. a_{i,j} ∈ R(G_{j}, G_{i−1})_{p_{i}+···+p_{j}+j−i},

2. a_{i,i} is a cycle representative of γ_{i} and

3. d(a_{i,j}) = Σ_{r=i}^{j−1}ϵ(a_{i,r})a_{r+1,j} with ϵ(a) = (−1)^{deg a+1}a. To each defining system associate the cycle

Σ_{r=1}^{n−1}ϵ(a_{1,r})a_{r+1,n} ∈ R(G_{n}, G_{0})_{p_{1}+...p_{n}+n−2}

The n-fold Massey product, ⟨γ_{1}, …, γ_{n}⟩, is the set of all homology classes of cycles associated to all possible defining systems.

Since Massey products are preserved by quasi-isomorphisms by [18, 1.5], we can use the simpler ring E_{a} to calculate these products for E(B_{a}).

Proposition 4.9. The homology of E_{a} (and hence also of E(B_{a}) and E(B_{a})⟨−1⟩) is given by 1. H_{∗}E_{a}(I_{H}, I_{H}) = ℚ[0] ⊕ ℚ[1] with generators 1_{H} and η^{H},

2. H_{∗}E_{a}(I_{H}, I_{K}) = 0,

3. H_{∗}E_{a}(I_{H}, I_{𝕋}) = ℚ[0] with generator ρ^{H},

4. H_{∗}E_{a}(I_{𝕋}, I_{H}) = ℚ[1] with generator t^{H}, and

5. H_{∗}E_{a}(I_{𝕋}, I_{𝕋}) = (⨁_{n≥0} ℚF[2n + 1]) ⊕ ℚ[0] with generators y_{2n+1}^{H} and 1_{𝕋}. The non-trivial products, except the obvious ones with 1_{H} and 1_{𝕋}, and the non-trivial Massey products are given by

1. t^{H}ρ^{H} = η^{H},

2. ρ^{H}t^{H} = y_{1}^{H}, and

3. ⟨ρ^{H}, \underbrace{η^{H}, …, η^{H}}_{n}, t^{H}⟩ = \{−y_{2n+1}^{H}\}.

Note that t^{H} and ρ^{H} generate all of the elements in homology via products and Massey products.

Proof. The products follow from the fact that x_{1}f_{0} = m_{1} and f_{0}x_{1} = i_{1}. Next consider the triple product ⟨ρ^{H}, η^{H}, t^{H}⟩. The only cycle representatives are ρ^{H} = [f_{0}^{H}], η^{H} = [m_{1}^{H}] and t^{H} = [x_{1}^{H}]. Since f_{0}^{H}m_{1}^{H} = 0, E_{a}(I_{H}, I_{𝕋})_{2} = 0 and g_{3}^{H} is the unique element such that dg_{3}^{H} = g_{2}^{H} = m_{1}^{H}x_{1}^{H}, this determines the only defining system as the following matrix.

( \begin{matrix}f_{0}^{H} & 0 \\ & m_{1}^{H} & g_{3}^{H} \\ & & x_{1}^{H} \end{matrix} )

The associated cycle is −f_{0}^{H}g_{3}^{H} + 0x_{1}^{H} = −i_{3}^{H}, so ⟨ρ^{H}, η^{H}, t^{H}⟩ = {[−i_{3}^{H}]} = {−y_{3}^{H}}.

The higher order products follow inductively. At each stage because E_{a}(I_{H}, I_{H})_{n} and E_{a}(I_{H}, I_{T})_{n} are trivial for n > 1 and E_{a}(I_{T} , I_{H})_{n} is generated by one element, there are no choices and hence no indeterminacy. Set a_{1,1} = f_{0}^{H}, a_{i,i} = m_{1}^{H} for 2 ≤ i ≤ n + 1, and a_{n+2,n+2} = x_{1}^{H}. Since f_{0}^{H}m_{1}^{H} = 0 and m_{1}^{H}m_{1}^{H} = 0, the only choice for a_{i,j} is 0 for i < j and j ≠ n + 2. Hence the only non-zero entries in the defining system are a_{i,i} and a_{i,n+2} for i = 1, …, n + 2. Then the only choice for a_{i,n+2} is g_{2n−2i+5} for i > 1. Since −a_{1,1}a_{2,n+2} = −f_{0}^{H}g_{2n+1}^{H} = −i_{2n+1}^{H} and [i_{2n+1}^{H}] = y_{2n+1}^{H}, the conclusion follows.

5. The endomorphism ring for 𝕋-spectra

In this section we consider the endomorphism ring E(B_{t}) associated to 𝕋-spectra in Corollary 3.9. As with E(B_{a}), we modify E(B_{t}) to define a quasi-isomorphic ring E_{t}. In Section 6 we then construct a quasi-isomorphism from E_{a} to E_{t}. We first modify E(B_{t}) by taking its (−1)- connected cover E(B_{t})⟨−1⟩ as in Definition 4.3. We next modify E(B_{t})⟨−1⟩ so that degree zero agrees with the the zeroth homology. This uses the zeroth Postnikov approximation (Definition 5.3) and its general properties stated in Propositions 5.4 and 5.5 which are applied to this specific case in Proposition 5.6. Finally, we define E_{t} as a sub-ring of the resulting differential graded ring with many objects.

We first recall the homology of E(B_{t}) from [5, 2.1.6]. Of course, it is isomorphic to the homology of E(B_{a}) stated in Proposition 4.9 because the functor π_{∗}^{A} which induces the equivalence of Ho(𝕋-spectra) and D(A) takes the geometric basic cells in B_{t} to the algebraic basic cells in B_{a}. We label the elements in B_{t} by their corresponding subgroup in 𝕋.

Proposition 5.1. The homology of E(B_{t}) is given by 1. H_{∗}E(B_{t})(H, H) = ℚ[0] ⊕ ℚ[1],

2. H_{∗}E(B_{t})(H, K) = 0,

3. H_{∗}E(B_{t})(H, 𝕋) = ℚ[0],

4. H_{∗}E(B_{t})(𝕋, H) = ℚ[1] and

5. H_{∗}E(B_{t})(𝕋, 𝕋) = (⨁_{n≥0} ℚF[2n + 1]) ⊕ ℚ[0].

The next step is to consider the (−1)-connected cover of E(B_{t}). Since the homology of E(B_{t}) is concentrated in non-negative degrees, E(B_{t})⟨−1⟩ ⟶ E(B_{t}) is a quasi-isomorphism.

Proposition 5.2. The map E(B_{t})⟨−1⟩ ⟶ E(B_{t}) is a quasi-isomorphism of differential graded rings with objects B_{t}. Thus, there is a Quillen equivalence between the respective categories of modules.

Mod-E(B_{t})⟨−1⟩ ≃_{Q} Mod-E(B_{t})

The purpose of the next step is to construct a ring \overline{R} quasi-isomorphic to E(B_{t})⟨−1⟩ such that \overline{R}_{0} ≅ H_{0}(\overline{R}_{0}). This involves the zeroth Postnikov approximation which is constructed using the small object argument; see [3, 7.12].

Consider the functor Ev_{(H,K)} from rational differential graded rings with objects B_{t} to rational 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) for H ≠ K and trivial everywhere else.

Definition 5.3. Given a rational differential graded ring R with objects B_{t} the zeroth Postnikov approximation of R is given by the factorization R ⟶^{i} P_{0}R ⟶ 0 produced by the small object argument for the set of maps I = {T_{(H,K)}(ℚ[n] ⟶ D^{n+1})} for n ≥ 1 and all pairs (H, K) ∈ B_{t} × B_{t}.

Since the map i: R ⟶ P_{0}R is constructed by pushouts of the maps in I and P_{0}R ⟶ 0 has the right lifting property with respect to the maps in I, these maps have the following properties.

Proposition 5.4. The map R ⟶^{i} P_{0}R is an isomorphism in degrees less than or equal to one and hence induces an isomorphism in homology in non-positive degrees. The map P_{0}R ⟶ 0 induces an isomorphism in homology in positive degrees.

By applying the small object argument one more time we factor the map R ⟶^{i} P_{0}R. Construct the factorization R ⟶^{ī} \overline{R} ⟶^{p̄} P_{0}R by using the small object argument with respect to the set J = {T_{(H,K)}(0 ⟶ D^{n})} for n ≥ 2 and all pairs (H, K) ∈ B_{t} ×B_{t}. Again by construction, since the maps in J are injective quasi-isomorphisms and isomorphisms in non-positive degrees and the map R ⟶^{i} P_{0}R has the above listed properties, ī and p̄ have the following properties.

Proposition 5.5. The map R ⟶^{ī} \overline{R} is a quasi-isomorphism which is an isomorphism in non-positive degrees. \overline{R} ⟶^{p̄} P_{0}R is an epimorphism and in non-positive degrees is a quasiisomorphism and an isomorphism.

Proof. Since p̄ has the lifting property with respect to J, Ev_{(H,K)} p̄ is an epimorphism in degrees

greater than one. Since R ⟶^{i} P_{0}R is an isomorphism in degrees less than two and i = p̄ī, it follows that p̄ is also an epimorphism in degrees less than two.

We now apply these general constructions to E(B_{t})⟨−1⟩. For ease of notation, let E′_{t} stand for E(B_{t})⟨−1⟩. Since E′_{t} is (−1)-connected, it is trivial in negative degrees and its homology is concentrated in non-negative degrees. Hence, P_{0}E′_{t} is trivial in negative degrees and its homology

is concentrated in degree zero. Let E′_{t} ⟶^{ī} \overline{E′_{t}} ⟶^{p̄} P_{0}E′_{t} be the factorization constructed above for R = E′_{t}. Consider H_{0}E′_{t} as a differential graded ring with objects B_{t} which is concentrated in degree zero. Since H_{0}E′_{t} ≅ H_{0}E(B_{t}) ≅ H_{0}P_{0}E′_{t}, which is given above in Proposition 5.1, there is a quasi-isomorphism of differential graded rings q: H_{0}E′_{t} ⟶ P_{0}E′_{t}. This relies on the fact that H_{0}E′_{t} happens to be the free ℚ-algebra with objects B_{t} generated by a one dimensional vector space of morphisms from H to 𝕋 for each H ∈ F. Define E′′_{t} as the differential graded ring with objects B_{t} which is the pullback of p̄ and q.

E′′_{t} ⟶^{q′}  \overline{E′_{t}}
↓  ↓p̄
H_{0}E′_{t} ⟶_{q} P_{0}E′_{t}

Proposition 5.6. The map q′: E′′_{t} ⟶ \overline{E′_{t}} is a quasi-isomorphism. Moreover, (E′′_{t})_{0} ≅ H_{0}E′_{t} and (E′′_{t})_{n} = 0 for n < 0.

Proof. E′′_{t}(H, K) is the pullback of differential graded modules of the maps Ev_{(H,K)} p̄ and Ev_{(H,K)} q. Each map Ev_{(H,K)} p̄ 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 E′′_{t}(H, K) ⟶ E′_{t}(H, K) is a quasi-isomorphism because it is the pullback of a weak equivalence across a fibration. Since p̄ is an isomorphism in non-positive degrees by Proposition 5.5, (E′′_{t})_{n} is isomorphic to (H_{0}E′_{t})_{n} for n ≤ 0. Note that since q′ is a quasi-isomorphism this also implies that d_{1} : (E′′_{t})_{1} ⟶ (E′′_{t})_{0} is trivial.

We make one more modification to E′′_{t}. Define E_{t} as the sub-differential graded ring with objects B_{t} which differs from E′′_{t} only by setting E_{t}(H, 𝕋) = E′′_{t}(H, 𝕋)_{0} concentrated in degree zero for each H ∈ F. This defines a differential graded ring since E′′_{t}(H, 𝕋)_{0} is a retract of E′′_{t}(H, 𝕋) because d_{1} is trivial. It follows that the map E_{t} ⟶ E′′_{t} is a quasi-isomorphism and E_{t} has the properties listed in Proposition 5.6 for E′′_{t}. Combining this with Propositions 5.2 and 5.5 gives the following.

Proposition 5.7. The composite map E_{t} ⟶ E′′_{t} ⟶ \overline{E′_{t}} is a quasi-isomorphism of differential graded rings with objects B_{t}. Combining this with the quasi-isomorphisms E(B_{t})⟨−1⟩ ≅ E′_{t} ⟶ \overline{E′_{t}} and E(B_{t})⟨−1⟩ ⟶ E(B_{t}) gives a zig-zag of quasi-isomorphisms between E_{t} and E(B_{t}). Thus, there is a zig-zag of Quillen equivalences between the respective categories of modules.

Mod-E_{t} ≃_{Q} Mod-E(B_{t})

Moreover, (E_{t})_{0} ≅ H_{0}(E_{t}), (E_{t})_{n} = 0 for n < 0, and E_{t}(H, 𝕋) is concentrated in degree zero.

6. The quasi-isomorphism between E_{a} and E_{t}

In this section we construct a quasi-isomorphism φ: E_{a} ⟶ E_{t}. First note that there is an obvious isomorphism between the object sets B_{a} and B_{t}.

Proposition 6.1. There is a quasi-isomorphism φ: E_{a} ⟶ E_{t} of differential graded rings which induces the obvious isomorphism B_{a} ⟶ B_{t}.

In the proof of this proposition we use the Massey product structure on E_{a} from Proposition 4.9 and its relation to Toda brackets. For R a differential graded ring with objects G = {G_{i}}_{i∈I}, a cycle a ∈ R(G_{j}, G_{i})_{p} induces a map of right R-modules, a: Σ^{p}F_{G_{j}}^{R} ⟶ F_{G_{i}}^{R}, by left multiplication (or post-composition) where F_{G_{i}}^{R}(−) = R(−, G_{i}) is the free right R-module represented by G_{i}. The associated homotopy class of maps [a] is determined by the homology class represented by a in the homology of R. This translates between Massey products, which are homology classes in R, and Toda brackets, which are homotopy classes of right R-module maps. Notice, if a ∈ R(G_{j}, G_{i})_{p} and b ∈ R(G_{i}, G_{k})_{q} are cycles then extra suspensions are required to compose these elements; the associated composition is b(Σ^{q}a): Σ^{p+q}G_{j} ⟶ Σ^{q}G_{i} ⟶ G_{k}.

Starting with the Massey products for E_{a} we construct elements of the Toda brackets for Ho(Mod-E_{a}). See Definition A.2 for the definition of higher Toda brackets. Since Toda brackets are invariant under exact equivalences by Theorem A.3, these Toda brackets agree with those for Ho(Mod-E_{t}). This follows because Quillen equivalences induce triangulated equivalences [20, I.4.3], Mod-E_{t} is Quillen equivalent to 𝕋-spectra by Proposition 5.7 and Corollary 3.9, Mod-E_{a} is Quillen equivalent to dgA by Proposition 4.7 and Corollary 3.7 and Ho(𝕋-spectra) is triangulated equivalent to D(A) by [5, 5.6.1]. These Toda brackets in Ho(Mod-E_{t}) then give us the structure we need to construct φ.

Proof. We construct φ by induction on the degree. Because the homology classes ρ^{H} = [f_{0}^{H}] and t^{H} = [x_{1}^{H}] generate the homology H_{∗}E_{a} via products and Massey products and every element in E_{a} is involved in these products and Massey products, we can exploit the fact that Toda brackets in Ho(Mod-E_{t}) and Ho(Mod-E_{a}) agree to construct φ. At each stage certain products must be non-trivial because of the non-trivial Toda brackets. This then provides targets for φ.

Since π_{∗}^{A} induces an isomorphism between H_{∗}E_{t} and H_{∗}E_{a}, we use the same notation for elements in each homology ring. First, the images of id_{H} and id_{𝕋} in E_{a} are forced to be the respective identity elements in E_{t}. We next determine the images of f_{0}^{H} and x_{1}^{H}. Let a_{0}^{H} be the element in E_{t}(H, 𝕋)_{0} = ℚ such that [a_{0}^{H}] = ρ^{H}. Let b_{1}^{H} be a cycle such that [b_{0}^{H}] = t^{H} in H_{1}E_{t}(𝕋, H) = ℚ. Then set φ(f_{0}^{H}) = a_{0}^{H} and φ(x_{1}^{H}) = b_{1}^{H}. Because of the product structure from Definition 4.6 this forces φ(m_{1}^{H}) = b_{1}^{H}a_{0}^{H} and φ(i_{1}^{H}) = a_{0}^{H}b_{1}^{H}. Since [a_{0}^{H}] = ρ^{H}, [b_{1}^{H}] = t^{H}, t^{H}ρ^{H} = η^{H} and ρ^{H}t^{H} = y_{1}^{H}, these images, φ(m_{1}^{H}) and φ(i_{1}^{H}) are non-trivial cycles so φ induces a quasi-isomorphism in degrees less than two. For ease of notation we drop the superscript H for the rest of the proof, 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.

Starting with the defining system for the Massey product ⟨ρ, η, t⟩ we construct an element of the Toda bracket ⟨ρ, η, Σt⟩. By Definition A.2 such a Toda bracket is the set of homotopy classes of compositions Σ^{3}F_{𝕋}^{E_{a}} ⟶^{g} X ⟶^{h} F_{𝕋}^{E_{a}} where the homotopy type of X is a 2-filtered object in {η}; see Definition A.1. The cofiber of m_{1} is such an object since [m_{1}] = η. Explicitly, set X = Σ^{2}F_{H}^{E_{a}} ⋉_{m_{1}} F_{H}^{E_{a}} with underlying graded object Σ^{2}F_{H}^{E_{a}} ⊕ F_{H}^{E_{a}} and with differential given by d(x, y) = (dx, dy − m_{1}x). Then the defining system found in Proposition 4.9 determines the other two maps as well, h(x, y) = f_{0}y and g(z) = (x_{1}z, g_{3}z). Here we are using the sign convention that d_{ΣC} = −d_{C}. One can check that [σ_{X}g] = [x_{1}] = t and [hσ′_{X}] = [f_{0}] = ρ. Due to the sparseness of elements in E_{a}, one can check that g, h are the unique maps satisfying the definition of the Toda bracket. Since Σ^{3}F_{𝕋}^{E_{a}}, X and F_{𝕋}^{E_{a}} are cofibrant and fibrant E_{a}-modules these are also the unique such homotopy classes. Hence the Toda bracket is the one element set {[f_{0}g_{3}] = y_{3}}. The sign difference here corresponds to the different sign conventions used for Massey products and Toda brackets.

Since Toda brackets agree for Ho(Mod-E_{a}) and Ho(Mod-E_{t}), there must be maps Σ^{3}F_{𝕋}^{E_{t}} ⟶^{g′}

Y ⟶^{h′} F_{𝕋}^{E_{t}} such that [h′g′] = y_{3} is in ⟨ρ, η, Σt⟩ in Ho(Mod-E_{t}). Define Y as the cofiber of b_{1}a_{0}, similar to X above. Then since E_{t}(H, 𝕋)_{n} is trivial for n > 0 the map h′ must be h′(x, y) = a_{0}y. If b_{1}a_{0}b_{1} = 0 then one could set g′(z) = (b_{1}z, 0). Then the composite [h′g′] would be trivial. Since this Toda bracket does not contain the trivial map, this means b_{1}a_{0}b_{1} must be non-trivial. Let b_{3} be an element of E_{t}(𝕋, H)_{3} such that db_{3} = b_{1}a_{0}b_{1}; its existence is assured because H_{2}E_{t}(𝕋, H) = 0. Then set g′(z) = (b_{1}z, b_{3}z). The maps h′ and g′ satisfy the definition of the Toda bracket, so we must have [h′g′] = [a_{0}b_{3}] = y_{3}. Since g_{2} = x_{1}f_{0}x_{1}, we must set φ(g_{2}) = b_{1}a_{0}b_{1} and we may set φ(g_{3}) = b_{3} and φ(i_{3}) = a_{0}b_{3}. This φ induces a quasi-isomorphism through degree 3.

As with the 3-fold Massey product, the higher Massey products also agree with the higher Toda brackets. First, define the iterated cofiber X_{n} = Σ^{2n}F_{H}^{E_{a}} ⋉_{m_{1}} Σ^{2n−2}F_{H}^{E_{a}} ⋉_{m_{1}} … ⋉_{m_{1}} F_{H}^{E_{a}} with differential d(x_{1}, x_{2}, …, x_{n+1}) = (dx_{1}, dx_{2} − m_{1}x_{1}, …, dx_{n+1} − m_{1}x_{n}). Note here d^{2} = 0 because m_{1}^{2} = 0. By Lemma A.4, X_{n} is an element of {η, Ση, …, Σ^{n−1}η}. Then the associated

defining system determines the maps Σ^{2n+1}F_{𝕋}^{E_{a}} ⟶^{g} X_{n} ⟶^{h} F_{𝕋}^{E_{a}} as g(z) = (x_{1}z, g_{3}z, …, g_{2n+1}z)

and h(x_{1}, …, x_{n+1}) = f_{0}x_{n+1}. This shows that the composite, [f_{0}g_{2n+1}] = y_{2n+1} is in the Toda bracket ⟨ρ, η, Ση, …, Σ^{n−1}η, Σ^{n}t⟩. One can check that any other defining system would produce a Toda bracket and vice versa, but again the sparseness of elements in E_{a} forces this to be the only element in the Toda bracket as well as the only defining system. So the higher Toda brackets also only differ by a sign from the Massey products.

Assume by induction that φ(g_{2n+1}) = b_{2n+1}, φ(i_{2n+1}) = a_{0}b_{2n+1}, and φ induces a quasiisomorphism through degree 2n + 1. Since (b_{1}a_{0})^{2} = 0 an E_{t}-module Y_{n} ∈ {η, Ση, …, Σ^{n−1}η} exists, and can be built similarly to X_{n}. Since E_{t}(H, 𝕋)_{n} = 0 for n > 0 the map h′: Y_{n} ⟶ F_{𝕋}^{E_{a}} is determined as h′(y_{1}, …, y_{n+1}) = a_{0}y_{n+1}. As with the 3-fold Toda bracket if b_{1}a_{0}b_{2n+1} were trivial then we could set g′(z) = (b_{1}z, b_{3}z, …, b_{2n+1}z, 0). But then the composite h′g′ would be trivial, which is not allowed. Hence b_{1}a_{0}b_{2n+1} is non-trivial and there must exist an element b_{2n+3} ∈ E_{t}(𝕋, H)_{2n+3} with db_{2n+3} = b_{1}a_{0}b_{2n+1} since H_{2n+2}E_{t}(𝕋, H) = 0. So set g′(z) = (b_{1}z, b_{3}z, …, b_{2n+3}z). Then [h′g′] = [a_{0}b_{2n+3}] is in the Toda bracket ⟨ρ, η, Ση, …, Σ^{n−1}η, Σ^{n}t⟩. So [a_{0}b_{2n+3}] = y_{2n+3}. So we must set φ(g_{2n+2}) = b_{1}a_{0}b_{2n+1} and we may set φ(g_{2n+3}) = b_{2n+3} and φ(i_{2n+3}) = a_{0}b_{2n+3}. With these choices, φ induces a quasi-isomorphism through degree 2n + 3. Hence by induction we have constructed the quasi-isomorphism φ: E_{a} ⟶ E_{t}.

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 dgA.

Appendix A. Higher Toda brackets and Massey products

In this section we show that an exact equivalence of triangulated categories preserves higher Toda brackets. We first make slight modifications to the definitions of higher Toda brackets from [1, 2] so that they apply to an arbitrary triangulated category T . We have kept the same order for the maps in the Toda bracket as in [1], but we have reversed the numbering to match the Massey products numbering.

Definition A.1. An n-filtered object X is an object X in T with maps ∗ ≅ F_{0}X ⟶^{i_{0}} F_{1}X ⟶^{i_{1}} ⋯ ⟶ F_{n−1}X ⟶^{i_{n−1}} F_{n}X ≅ X. Let F_{k+1}X/F_{k}X denote the cofiber of i_{k} so that there are triangles, F_{k}X ⟶^{i_{k}} F_{k+1}X ⟶^{π_{k+1}} F_{k+1}X/F_{k}X ⟶^{δ_{k}} ΣF_{k}X. Given a composable sequence of maps,

A_{n−1} ⟶^{f_{n−1}} A_{n−2} ⟶^{f_{n−2}} ⋯ A_{1} ⟶^{f_{1}} A_{0}, an n-filtered object X is an element of {f_{1}, …, f_{n−1}} if and

only if Σ^{k}A_{k} ≅ F_{k+1}X/F_{k}X and via these isomorphisms Σ^{k}f_{k} is isomorphic to F_{k+1}X/F_{k}X ⟶^{δ_{k}}

ΣF_{k}X ⟶^{Σπ_{k}} Σ(F_{k}X/F_{k−1}X). For X ∈ {f_{1}, …, f_{n−1}}, define σ′_{X} as the composite A_{0} ≅ F_{1}X ⟶

X and define σ_{X} as the composite X ≅ F_{n}X ⟶^{π_{n}} F_{n}X/F_{n−1}X ≅ Σ^{n−1}A_{n−1}.

Definition A.2. Define the (n + 1)-fold Toda bracket ⟨f_{1}, f_{2}, …, f_{n}⟩ for a sequence of maps

A_{n} ⟶^{f_{n}} A_{n−1} ⟶^{f_{n−1}} ⋯ ⟶^{f_{2}} A_{1} ⟶^{f_{1}} A_{0}

as the set of all maps θ ∈ T (Σ^{n−2}A_{n}, A_{0}) such that there is some (n − 1)-filtered object X ∈ {f_{2}, …, f_{n−1}} and maps g: Σ^{n−2}A_{n} ⟶ X and h: X ⟶ A_{0} with Σ^{n−2}f_{n} ≅ σ_{X}g and f_{1} ≅ hσ′_{X} and θ ≅ hg.

\xymatrix{ & A_{1} \ar[d]_{σ′_{X}} \ar[dr]^{f_{1}} \\ Σ^{n−2}A_{n} \ar[r]^{g} \ar[dr]_{Σ^{n−2}f_{n}} & X \ar[d]^{σ_{X}} \ar[r]^{h} & A_{0} \\  & Σ^{n−2}A_{n−1}}

With these definitions it is easy to see that Toda brackets are preserved by exact equivalences of triangulated categories.

Theorem A.3. Given an exact equivalence of triangulated categories ψ: T ⟶ T′, then θ ∈ ⟨f_{1}, …, f_{n}⟩ if and only if ψ(θ) ∈ ⟨ψ(f_{1}), …, ψ(f_{n})⟩.

Proof. Assume given θ ∈ ⟨f_{1}, …, f_{n}⟩. Then θ = hg with g: Σ^{n−2}A_{n} ⟶ X and h: X ⟶ A_{0} with X ∈ {f_{2}, …, f_{n−1}}. Since ψ is exact, ψ(X), with filtrations ψ(F_{k}X), is an element of {ψ(f_{2}, …, ψ(f_{n−1})}. Hence ψ(θ) = ψ(h)ψ(g) is an element of ⟨ψ(f_{1}), …, ψ(f_{n})⟩. For the other direction, apply the same arguments to the inverse equivalence ψ^{−1}.

The following manipulations are useful for constructing filtered objects. Let C_{h} denote the cofiber of the map h.

Lemma A.4. Assume given a composable sequence of maps f_{i}: A_{i} ⟶ A_{i−1}.

1. If X ∈ {f_{2}, …, f_{k}} and h: X ⟶ A_{0}, then C_{h} ∈ {hσ′_{X}, f_{2}, …, f_{k}}.

2. If Y ∈ {f_{1}, …, f_{k−1}} and h: Σ^{k−1}A_{k} ⟶ Y, then C_{h} ∈ {f_{1}, …, f_{k−1}, Σ^{−k+1}σ_{Y} h}.

3. If X ∈ {f_{1}, …, f_{k}}, then F_{k}X ∈ {f_{1}, …, f_{k−1}} and Σ^{−1}(X/F_{1}X) ∈ {f_{2}, …, f_{k}}.

Proof. The first two statements follow by setting n = 1 and m = 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 proposition shows that these are implicit in the above definition.

Proposition A.5. There is an object X in {f_{1}, …, f_{n−1}} if and only if 0 ∈ ⟨f_{1}, …, f_{n−1}⟩.

Proof. First assume X ∈ {f_{1}, …, f_{n−1}}. By Lemma A.4, Σ^{−1}(F_{n−1}X/F_{1}X) ∈ {f_{2}, …, f_{n−2}}. Let g be the composite Σ^{n−3}A_{n−1} ≅ Σ^{−2}(F_{n}X/F_{n−1}X) ⟶ Σ^{−1}F_{n−1}X ⟶ Σ^{−1}(F_{n−1}X/F_{1}X). Let h be the map Σ^{−1}(F_{n−1}X/F_{1}X) ⟶ F_{1}X ≅ A_{0}. Since h is the cofiber of the map Σ^{−1}F_{n−1}X ⟶ Σ^{−1}(F_{n−1}X/F_{1}X), the composite hg is trivial. So hg = 0 ∈ ⟨f_{1}, …, f_{n−1}⟩.

For the other direction, assume θ = 0 ∈ ⟨f_{1}, …, f_{n−1}⟩. Then θ = hg with g: Σ^{n−3}A_{n−1} ⟶ Z and h: Z ⟶ A_{0} with Z ∈ {f_{2}, …, f_{n−2}}. Since hg is trivial, taking cofibers gives a map k: Σ^{n−2}A_{n−1} ⟶ C_{h}. By Lemma A.4, C_{h} ∈ {f_{1}, …, f_{n−2}} and C_{k} ∈ {f_{1}, …, f_{n−1}}.

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Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA

   E-mail address: bshipley@math.purdue.edu