Rational Equivariant S-Modules

Abstract

We take the work of Splitting Rational G-Spectra and apply it to the category of equivariant S-modules.

Contents

1 The Basics of Equivariant S-modules. 1

2 A Rational Sphere Spectrum 3

3 A (non-equivariant) rational sphere ring spectrum. 4

4 The Equivariant Case 9

5 Another Rational Ring Spectrum 12

6 Comparisons and Splitting 13

Index 15

References 16

1 The Basics of Equivariant S-modules.

Following the notation of [MM02] we denote the category of G-equivariant S-modules by GM, where G is a compact Lie group, when we wish to draw attention to the universe we are using we shall add a superscript.

There are two model structures on this category: the cellular model structure (classical, see [EKMM97]) and the generalised cellular. We use the later as it relates well to the model category of G-equivariant orthogonal spectra GI𝒮 . This uses the generalised q-cofibrations, restricted q-fibrations and π_{∗}-isomorphisms, see [MM02, IV, Section 2], this is a cofibrantly generated, G-topological, monoidal model category satisfying the monoid axiom.

In particular we have the following strong monoidal Quillen equivalences

ℕ : GI𝒮 \overrightarrow{⟵}GM : ℕ^{#} 1 : GM\overrightarrow{⟵}GM : 1

left adjoints are on top and from left to right we are using the following model structures from [MM02]: positive stable, generalised cellular, cellular, generalised cellular.

We remind the reader that the unit of the smash product S is not cofibrant in any of the above four model structures. We now have the usual collection of results on pushouts, weak equivalences, colimits of sequences of h-cofibrations etc see [MM02, IV 2.8, I.4.11 and III 3.5]. We write out some standard results below.

(i). Consider the generalised cellular model structure on GM; a generating cofibration has the form E → CE = E ∧ I and a generating trivial cofibration has the form E ∧ I → E ∧ I ∧ I_{+} for E a generalised sphere S-module. All of these maps are h-cofibrations, since S^{0} → I and S^{0} → I_{+} are h-cofibrations. Now recall that h-cofibrations are preserved by forming relative cell complexes and retracts.

(ii). A cofibration of commutative ring spectra or a cofibration of R-modules for a ring spectrum R is an h-cofibration (using the generalised cellular model structure on GM). This follows since ℂ and R∧− preserve h-cofibrations and define the generating cofibrations for these model categories. Now recall that h-cofibrations are preserved by forming relative cell complexes and retracts.

(iii). In the generalised cellular model structure on GM every object is fibrant. We must check that following diagram has a lift.

\xymatrix{CE \ar[r]^{f} \ar[d]_{i_{0}} & X \ar[d] \\ CE ∧ I_{+} \ar[r] & ∗}

This is trivial to show since we have a retraction map r: CE ∧ I_{+} → CE such that r ∘ i_{0} = 1 and we take the lift to be f ∘ r

(iv). For any trivial cofibration X → Y and any spectrum Z, the map X∧Z → Y ∧Z is an h-cofibration and a π_{∗}-isomorphism. This follows by exactly the same proof as in [MMSS01, 12.5].

(v). If a spectrum X is generalised cofibrant, then the functor X ∧ − preserves weak equivalences.

Lemma 1.1 If a spectrum X is generalised cofibrant, then the functor X ∧ − preserves weak equivalences.

Proof So the spectrum X is a retract of a generalised cell complex A. That is: we have the following diagram X → A → X with composite map the identity. We show that we only need show the lemma for generalised cell complexes. If the lemma holds for A then it holds for X by the following: take f: Y → Z a weak equivalence then we have the diagram below

\xymatrix{X ∧ Y \ar[d] \ar[r] & A ∧ Y \ar[r] \ar[d]^{∼} & X ∧ Y \ar[d] \\ X ∧ Z \ar[r] & A ∧ Z \ar[r] & X ∧ Z}

hence X∧Y → X∧Z is a retract of a weak equivalence, therefore a weak equivalence.

We can reduce further by following the arguments in [MM02, III 7.3]. All that is needed from here to prove the lemma is to show that if X is a spectrum with trivial homotopy groups, then X ∧ E has trivial homotopy groups for E a generalised sphere S-module. Fix n and a G-indexing space V, let E = S ∧ 𝕃Σ_{V}^{∞}(G/H_{+} ∧ S^{n}) a generalised sphere S-module. Since E is strongly dualisable ([LMSM86, Chapter III] or [May96, Section 7, Chapter XVI]) we see (by the conditions on X)

π_{n}^{K}(X ∧ E) = [S^{n}, X ∧ E]^{K} ≅ [S^{n} ∧ F (E, S), X]^{K} = 0.

2 A Rational Sphere Spectrum

Since [S, S]^{G} ≅ A(G) we can choose (for each integer n) a map n: ĉS → ĉS representing n (ĉ is the cofibrant replacement in the category of G-equivairant S-modules). Following the method in our earlier paper, we obtain maps

ĉS → ĉ(ĉS × ĉS) → ĉS ∨ ĉS → ĉS

which allow us to add maps, in turn we then construct S_{ℚ} homotopy pushout:

⋁_{j} ĉS → ⋁_{i} ĉS → S_{ℚ}.

Since this category is G-topological and S^{0} → I is a cofibration of based G-spaces S_{ℚ} will be cofibrant (and as with all objects of this category, fibrant).

Definition 2.1 Apply [MM02, IV 6.3] to the object S_{ℚ} to obtain the model category of rational G-equivariant S-modules GM_{ℚ} := L_{S_{ℚ}}GM.

Lemma 2.2 For X an S-module, π_{∗}^{H}(S_{ℚ} ∧ X) ≅ π_{∗}^{H}(X) ⊗ ℚ. The map π_{∗}^{H}(X) → π_{∗}^{H}(S_{ℚ} ∧ X) acts as x → x ⊗ 1.

Remark 2.3 In the positive model structure on GI𝒮 we have a corresponding result. We build 𝕊_{ℚ}^{+} from cofibrant-fibrant replacement of the sphere spectrum, this object is weakly equivalent to the usual rational sphere spectrum (𝕊_{ℚ}) and so in the stable structure localising with respect to either of these gives the same model structure. We also know the following: a positive cofibrant spectrum is certainly cofibrant, thus smashing with a positive cofibrant spectrum preserves weak equivalences. It is useful to know that X ∧ ĉ𝕊 → X ∧ 𝕊 is a weak equivalence for any X, so consider:

X ∧ ĉ𝕊 ⟶ X ∧ 𝕊
↑  ↑
cX ∧ ĉ𝕊 ⟶ cX ∧ 𝕊

where c denotes cofibrant replacement in the stable structure and ĉ denotes cofibrant replacement in the positive model structure on GI𝒮 . The vertical arrows are weak equivalences, as is the lower arrow, hence this diagram consists of weak equivalences (this is a step in the proof of the above lemma).

Lemma 2.4 The two rational sphere spectra ℕ𝕊_{ℚ}^{+} and S_{ℚ} are weakly equivalent.

Proof We use f̂ and ĉ for fibrant and cofibrant replacements in either GI𝒮 _{+} or GM. Since ℕ is a monoidal left adjoint, we know that the map ℕĉ𝕊 → ℕ𝕊 ≅ S is a weak equivalence and of course the map ℕĉ𝕊 → ℕf̂ĉ𝕊 is a trivial cofibration. We can then (via lifting properties) construct a weak equivalence ℕf̂ĉ𝕊 → S, we use this to create a weak equivalence ℕf̂ĉ𝕊 → ĉS. Thus we have a homotopy commuting square

\xymatrix{⋁_{j} ℕfc𝕊 \ar[d] \ar[r] & ⋁_{i} ℕfc𝕊 \ar[d] \\ ⋁_{j} ĉS \ar[r] & ⋁_{i} ĉS}

and the vertical maps are weak equivalences, hence we have our result.

Proposition 2.5 We have a strong monoidal Quillen equivalence of model categories

ℕ : GI𝒮 _{ℚ}\overrightarrow{⟵}L_{ℕ𝕊_{ℚ}^{+}}GM ≡ L_{S_{ℚ}}GM : ℕ^{#}

Proof The first part is standard, see Theorem 4.1, the second is automatic since an ℕ𝕊_{ℚ}^{+}-equivalence is an S_{ℚ}-equivalence - they are both the same as rational homotopy equivalences. Note that GI𝒮 _{ℚ} is the localisation of the positive model structure with respect to the cofibrant object 𝕊_{ℚ}^{+}.

3 A (non-equivariant) rational sphere ring spectrum.

We have H ℚ a ring object in the category of symmetric spectra, see [HSS00, 1.2.5] and [Sch04, 4.14] (we are using positive model structure on this category).

Lemma 3.1 The symmetric spectrum H ℚ is a level-wise Kan complex and an Ωspectrum, hence H ℚ is fibrant in the category of symmetric spectra.

Proof The underlying symmetric sequence of H ℚ is (ℚ̃S^{0}, ℚ̃S^{1}, ℚ̃S^{2}, …) and the structure maps are

((x_{1}, …, x_{p}), (q, y_{1}, …, y_{q})) ↦ (q, x_{1}, …, x_{p}, y_{1}, …, y_{q})

where (x_{1}, …, x_{p}) ∈ S^{p} and (q, y_{1}, …, y_{q}) ∈ ℚ̃S^{q}. From this the first statement is clear, each level ℚ̃S^{q} is has the structure of a simplicial abelian group and hence is a Kan complex. Note that the structure maps give isomorphisms Σℚ̃S^{q} → ℚ̃S^{q+1} for all q.

For each non zero rational number a we have an obvious inclusion S^{q} ↪ ℚ̃S^{q} sending a k-simplex x to (a, x). This gives an isomorphism π_{q}ℚ̃S^{q} ≅ ℚ, and it is clear that these maps are ‘preserved’ by suspension so that Σ: π_{q}ℚ̃S^{q} → π_{q+1}ℚ̃S^{q+1} is an isomorphism.

We now prove that H ℚ is an Ω-spectrum. We have the commutative diagram

\xymatrix{π_{k}ℚ̃S^{q} \ar[d]^{Σ} \ar[r]^{f^{∗}} & π_{k}Ωℚ̃S^{q+1} \ar[d]_{≅}^{ϕ} \\ π_{k+1}Σℚ̃S^{q} \ar[r]^{g^{∗}}_{≅} & π_{k+1}ℚ̃S^{q+1}}

Where f: ℚ̃S^{q} → Ωℚ̃S^{q+1} is the adjoint of the structure map g: Σℚ̃S^{q} → ℚ̃S^{q+1}. This diagram commutes since we have the following identifications: let ε: ΣΩ → 1 be the counit, then ϕ = εΣ and g = εΣf. As mentioned above suspension is an isomorphism as is g, thus the maps f (for varying q) are weak equivalences of simplicial sets, so we have an Ω spectrum.

We would like to transport this ring object to the category of G-equivariant Smodules, we use the commutative diagram of [SS03, 7.1]. The first step is to take a cofibrant replacement of this symmetric spectrum in the category of commutative ring spectra, c H ℚ. Next we apply geometric realisation and the functor Λ to obatin a ring spectrum in GM which we denote by Λc H ℚ.

It is very important that we are using the cofibrant replacement c H ℚ (which is also fibrant) as this gives control over the homotopical information and allows us to calculate the homotopy groups of Λc H ℚ. We can apply [Sch01, 4.3] or the opening paragraph of Section 4 to see that c H ℚ is S-cofibrant. Hence we can use [Sch01, 3.1] to see that c H ℚ → ΦΛc H ℚ is a positive stable equivalence between positive fibrant symmetric spectra (recall that H ℚ is fibrant as a symmetric spectrum) and hence a π_{∗} isomorphism (Lemma 3.6). Thus since Φ preserves homotopy groups we know the homotopy groups of Λc H ℚ. In turn c H ℚ and H ℚ are positive stably equivalent and so π^{∗}-isomorphic (see Lemmas 3.5 and 3.6). We have proven the following.

Lemma 3.2 The map c H ℚ → H ℚ is a π^{∗}-isomorphism, Λc H ℚ has its homotopy groups concentrated in degree zero, where the value is ℚ.

Now we use [Sch01, 5.1 (iv)] to obtain a Quillen equivalence between c H ℚ-modules and Λc H ℚ-modules. An application of [MMSS01, 12.1(vi)] (Quillen invariance of modules) shows that we have a Quillen equivalence between c H ℚ-modules and H ℚmodules. Thus the results of [Shi04b] tell us that we have the ‘right’ category.

Proposition 3.3 There is a zig-zag of weak monoidal Quillen equivalences between the model categories of Λc H ℚ-modules and rational differential graded algebras.

Remark 3.4 Note that we have a cofibration of commutative symmetric ring spectra S → c H ℚ (since S is the initial object of this category) and hence a cofibration of S-algebras S → Λc H ℚ, and equivariant S-algebras S → I_{U^{G}}^{U}ε^{∗}Λc H ℚ. All of these maps are also h-cofibrations.

Lemma 3.5 In a simplicial model category a cofibration of fibrant objects is trivial if and only if it is a strong deformation retract.

Proof This is [Qui67, II p. 2.5], we give the ‘only if’ proof, since this is needed below. Take the map i: A → B as above, then we can construct r and h by taking the lift in each of the following diagrams respectively.

\xymatrix{A \ar[r]^{1_{A}} \ar[d]_{i} & A \ar[d] & A \ar[r]^{si} \ar[d]_{i} & B^{I} \ar[d]^{(j_{0},j_{1})} \\ B \ar[r] & ∗ & B \ar[r]_{(1_{B},ir)} & B × B}

Thus we have a strong deformation retract r, that is, we have r: B → A and h: B ⊗ Δ[1] → B such that ri = 1_{A}, h_{0} = 1_{B}, h_{1} = ir and h(i ⊗ 1) = iσ (where σ: A ⊗ Δ[1] → A is the contraction).

Lemma 3.6 A stable equivalence between positive stable fibrant symmetric spectra is a positive level equivalence and hence a π_{∗}-isomorphism.

Proof We follow the similar proof [HSS00, 4.2.6]. Factor such a map f: X → Y as f = pi, with p a positive stable fibration and i a positive trivial cofibration. Since i is a positive trivial cofibration between positive fibrant objects it is a strong deformation retract, see [Qui67, II p. 2.5]. Since this is a simplicial model category with the action given levelwise this must be a level equivalence.

We know that p is a stable equivalence, (two-out-of-three-property) and we show that the positive stable trivial fibrations are positive level trivial fibrations c.f. [HSS00, 3.4.5]. Factor p as qj with q ∈ F I^{+}-inj and j ∈ F I^{+}-cof, so q is a positive level trivial fibration and j is a positive stable cofibration. Since j has the LLP with respect to g it follows ([HSS00, 3.2.4]) that g is a retract of q, and hence also a positive level trivial fibration.

Clearly a positive level equivalence will give a isomorphism of homotopy groups, since these are defined as a colimit.

Definition 3.7 For a symmetric spectrum X we define X_{ℚ} to be the cofibre of the following map, which is defined in the same way as in the construction of 𝕊_{ℚ}^{+}.

⋁_{j} ĉf̂X → ⋁_{i} ĉf̂X

Lemma 3.8 For any symmetric spectrum X we have a pair of natural isomorphisms

π_{∗}(X ∧ 𝕊_{ℚ}) ≅ π_{∗}(X) ⊗ ℚ ≅ π_{∗}(X_{ℚ}).

Proof This follows immediately from [MMSS01, 7.4 (vi)].

Lemma 3.9 In the category of symmetric spectra, the π_{∗}^{ℚ}-isomorphisms are precisely the 𝕊_{ℚ}-equivalences.

Proof The map of homotopy groups π_{∗}(X ∧ 𝕊_{ℚ}) → π_{∗}(X) ⊗ ℚ is induced from the map ℤ → ℚ.

Lemma 3.10 For a symmetric spectrum X, the two forms of rationalisation - X_{ℚ} and X ∧ 𝕊_{ℚ}^{+} are weakly equivalent.

Proof The inclusion map ĉf̂X ∧ 𝕊_{ℚ}^{+} → X_{ℚ} ∧ 𝕊_{ℚ}^{+} is a π_{∗}-isomorphism, and hence a stable equivalence. Since 𝕊_{ℚ}^{+} is cofibrant, ĉf̂X ∧𝕊_{ℚ}^{+} is weakly equivalent to X ∧𝕊_{ℚ}^{+}(by a zigzag of stable equivalences).

Lemma 3.11 We have a zigzag of weak equivalences between the S-modules S_{ℚ} and Λ𝕊_{ℚ}^{+}.

Proof The S-module Λ𝕊_{ℚ}^{+} is the cofibre of the (usual) map

⋁_{j} Λĉf̂𝕊 → ⋁_{i} Λĉf̂𝕊.

We have a stable equivalence ĉf̂𝕊 → ĉ𝕊 between cofibrant objects hence a π_{∗}- isomorphism Λĉf̂𝕊 → Λĉ𝕊. Since Λ is a strong monoidal functor we have a weak equivalence Λĉ𝕊 → Λ𝕊 = S which we lift to a weak equivalence Λĉ𝕊 → ĉS. We then achieve various homotopy commuting diagrams of cofibres with comparison maps that are weak equivalences, so the result follows.

Lemma 3.12 The functor Λ preserves π_{∗}^{ℚ}-isomorphisms between cofibrant symmetric spectra.

Proof Such a map is a 𝕊_{ℚ}^{+}-equivalence, and Λ clearly takes such maps to Λ𝕊_{ℚ}^{+} equivalences. In turn these are exactly the same as S_{ℚ}-equivalences and so π_{∗}^{ℚ}- isomorphisms of S-modules.

Proposition 3.13 The canonical map S → Λc H ℚ is a rational equivalence.

Proof We cannot use the preceding result, since c H ℚ is probably not cofibrant. Begin with 𝕊 → H ℚ which is clearly a π_{∗}^{ℚ}-isomorphism of symmetric spectra, as in the arguments above 𝕊 → c H ℚ is also a π_{∗}^{ℚ}-isomorphism we now want to prove that this is preserved by Λ. By smashing with 𝕊_{ℚ}^{+} we obtain a stable equivalence 𝕊_{ℚ} → c H ℚ ∧ 𝕊_{ℚ}^{+}.

The smash product of two S-cofibrant objects is again S-cofibrant [Shi04a, 2.5]. Thus the object c H ℚ ∧ ĉ𝕊 is ‘good’ in the sense of [Sch01, 3.1], as are wedges of good objects and the cofibre of a map between good objects is also good. So in the commutative diagram below the horizontal maps, which come from the unit of the adjunction, are stable equivalences as is the left vertical by the above.

\xymatrix{𝕊_{ℚ} \ar[r] \ar[d] & ΦΛ(𝕊_{ℚ}^{+}) \ar[d] \\ c H ℚ ∧ 𝕊_{ℚ}^{+} \ar[r] & ΦΛ(c H ℚ ∧ 𝕊_{ℚ}^{+})}

Thus the right hand vertical is a weak equivalence, and since Φ reflects weak equivalences between fibrant objects ([Hov99, 1.3.16]) we have a π_{∗}-isomorphism

Λ𝕊_{ℚ}^{+} → Λ(c H ℚ ∧ 𝕊_{ℚ}^{+})

that is S → Λc H ℚ is a Λ𝕊_{ℚ}^{+}-equivalence and by the above set of lemmas this is an S_{ℚ}-equivalence, which is our desired result.

Proposition 3.14 The rational sphere spectra S_{ℚ} and Λc H ℚ are weakly equivalent.

Proof Let R = Λc H ℚ, we can construct R_{ℚ} from copies of ĉR and we obtain a π_{∗}^{ℚ} isomorphism ĉR → R_{ℚ}, in fact, this is a π_{∗}-isomorphism.

Now use the map ĉS → S → R to create a comparison of cofibres S_{ℚ} → R_{ℚ} (again we only have a comparison up to homotopy, but this is not a problem), which is a π_{∗}^{ℚ}-isomorphism and thus a π_{∗} isomorphism. Combining these we achieve a zigzag of weak equivalences:

S_{ℚ} → R_{ℚ} ← ĉR → R

and R is the only non-cofibrant object above.

Lemma 3.15 A map is an S_{ℚ}-equivalence if and only if it is a ĉR-equivalence. Each of these conditions is exactly equivalent to a map being a π_{∗}^{ℚ}-isomorphism.

Proof Take a map f: X → Y then we have the diagram

\xymatrix{X ∧ S_{ℚ} \ar[r]^{f∧1} \ar[d] & Y ∧ S_{ℚ} \ar[d] \\ X ∧ R_{ℚ} \ar[r]^{f∧1} & Y ∧ R_{ℚ} \\ X ∧ ĉR \ar[r]^{f∧1} \ar[u] & Y ∧ ĉR \ar[u]}

and as usual weak equivalences between cofibrant objects are preserved by smash products.

Lemma 3.16 The homotopy groups of an Λc H ℚ module are rational and all modules are fibrant.

Proof Fibrations of Λc H ℚ modules are defined in terms of their underlying Smodules and all S-modules are fibrant.

Take M, a Λc H ℚ module, and a Λc H ℚ-acyclic S-module A, that is an A such that A∧Λc H ℚ has zero homotopy groups. We use the following standard argument (see [Ada74, 13.1]) any map f: A → M can be factored as follows:

\xymatrix{A \ar[r]^{f} \ar[d]_{1∧η} & M \\ A ∧ Λc H ℚ \ar[r]_{f∧1} & M ∧ Λc H ℚ \ar[u]_{ν}}

(in the above 1∧η includes the unit isomorphism A ≅ A∧S) thus we see that f must be null-homotopic and conclude that [A, M]=0. Similarly we will show that if A is ĉΛc H ℚ-acyclic (where ĉ is cofibrant replacement in the category of S-modules) then [A, M] = 0. Take [f] ∈ [A, M] then [f] is trivial if and only if [f]^{∗}: [M, M] → [A, M] is the trivial map, we draw the following extension of the previous diagram:

\xymatrix{[M, M] \ar[r]^{f^{∗}} \ar[d]_{ν^{∗}} & [A, M] \ar[r]_{≅}^{q^{∗}} & [ĉA, M] \\ [M ∧ Λc H ℚ, M] \ar[r]_{(f∧1)^{∗}} & [A ∧ Λc H ℚ, M] \ar[u]^{(1∧η)^{∗}} \ar[r]_{(q∧1)^{∗}} & [ĉA ∧ Λc H ℚ, M] \ar[d]_{≅} \ar[u]^{(1∧η)^{∗}} \\  &  & [ĉA ∧ ĉΛc H ℚ, M] = 0}

and from this it is clear that [f]^{∗} = 0. Hence all Λc H ℚ modules are Λc H ℚ-local and ĉΛc H ℚ-local.

Since the S_{ℚ}-equivalences are precisely the same ĉΛc H ℚ-equivalences we know that the S_{ℚ}-local objects are precisely the ĉΛc H ℚ-local objects. Finally we know that an S_{ℚ}-local object has rational homotopy groups.

Theorem 3.17 The adjoint pair of functors

F : MS\overrightarrow{⟵}Λc H ℚ –mod : U

pass to a Quillen equivalence

F : MS_{ℚ}\overrightarrow{⟵}Λc H ℚ –mod : U.

Proof As usual (F, U) form an adjoint pair, this is a categorical fact F X = Λc H ℚ∧X and U is the forgetful functor. They also form a strong monoidal Quillen pair (with the usual structure on MS), this is also standard and comes from the construction of a monoidal model structure on Λc H ℚ –mod.

Now we consider the localised case, since the cofibrations are unaffected by localisation, F: MS_{ℚ} ⇆ Λc H ℚ –mod preserves cofibrations. Consider a trivial rational cofibration X → Y, we know that F applied to this gives a cofibration, we must check that it is also a weak equivalence. First we see that X ∧ 𝕊_{ℚ} → Y ∧ 𝕊_{ℚ} is a q-cofibration and a π_{∗}-isomorphism, so in turn X ∧ 𝕊_{ℚ} ∧ Λc H ℚ → Y ∧ 𝕊_{ℚ} ∧ Λc H ℚ is an h-cofibration and a π_{∗}-isomorphism.

This proves that X ∧ Λc H ℚ → Y ∧ Λc H ℚ is a π_{∗}^{ℚ}-isomorphism between Λc H ℚmodules, which we know have rational homotopy groups and thus this map is a π_{∗}-isomorphism.

The right adjoint preserves and detects weak equivalences, since a π_{∗}^{ℚ}-isomorphism between objects with rational homotopy groups is a π_{∗}-isomorphism. So to finish this proof we must show that the map

X → c H ℚ ∧ X → f̂(c H ℚ ∧ X)

is a weak equivalence for all cofibrant S-modules X.

The unit is a π_{∗}^{ℚ}-isomorphism since smashing with a cofibrant object will preserve the π_{∗}^{ℚ}-isomorphism S → Λc H ℚ. Every object in the category of Λc H ℚ-modules is fibrant, and c H ℚ ∧ X → f̂(c H ℚ ∧ X) is certainly a π_{∗}-isomorphism, so we have the above requirement.

Remark 3.18 This model category is proper by [MM02, IV 2.11].

4 The Equivariant Case

Take the object of interest from the previous section, Λc H ℚ, apply inflation (so far we have been working in the fixed point universe U^{G} of a complete G-universe U) to obtain an equivariant spectrum. Finally we change universe to the complete G-universe U, recall that all these functors are strong monoidal. We investigate the properties of this object S^{0}ℚ := I_{U^{G}}^{U}ε^{∗}Λc H ℚ.

This section is similar to the previous, in fact the above proof will suffice for the equivariant case once we know that S → S^{0}ℚ is a rational equivalence of equivariant S-modules. We will delay this result for a while, since it seems appropriate to introduce some results on localising model categories and see how these apply to the standard functors of equivariant spectra (fixed points, inflation and the like). Fix a (closed) subgroup H of G, with inclusion i. Then we consider the adjoint pairs (i^{∗}, F_{H}(G_{+}, −)) and (G_{+} ∧_{H} (−), i^{∗}) and we wish to prove that these pass to Quillen pairs on the rationalised categories.

The fundamental issue is that we have constructions that we can perform in either of H𝒮 or G𝒮 and we want to see how these model categories compare once they are localised. It is hard to use general results, since we are dealing with such a specific case, but we begin with a result that is the most basic to understand.

Theorem 4.1 Take a Quillen adjunction with F strong monoidal

F : 𝒞 \overrightarrow{⟵}𝒟 : G.

Let E be cofibrant in 𝒞, and assume that all model categories mentioned below exist. Then these pass to a monoidal Quillen pair

F : 𝒞_{E}\overrightarrow{⟵}𝒟_{F E} : G.

Furthermore, if (F, G) form a Quillen equivalence, then they pass to a Quillen equivalence of the localised categories.

Proof Since the cofibrations in 𝒞_{E} and 𝒟_{F E} are unchanged F preserves cofibrations. Now take a trivial cofibration in 𝒞 of the form f ∧1: X ∧E → Y ∧E, applying F and using the strong monoidal condition we have a weak equivalence in 𝒟 F f ∧ 1: F X ∧ F E → F Y ∧ F E. Hence F takes E-trivial cofibrations to F E-trivial cofibrations and we have a Quillen pair.

For the second statement we use [Hov99, 1.3.16] and show that F reflects Eequivalences between cofibrant objects and that F Q_{E}GX → X is an E-equivalence for all X fibrant in 𝒟_{F E}. The first follows since F reflects weak equivalences between cofibrant objects, strong monoidality allows us to identify F (f ∧ 1_{E}) and F f ∧ 1_{F E} for a map f in 𝒞. The second condition is equally simple: we know that an Efibrant object is fibrant, and that cofibrant replacement is unaffected by Bousfield localisation hence this map is a weak equivalence.

Corollary 4.2 Thus we see that the equivalence (ℕ, ℕ^{#}) passes to an equivalence of the various localised categories (see Section 6). Furthermore we still have our commutative diagrams relating this comparison to the equivariant structure (taking fixed points etc).

Corollary 4.3 The strong monoidal change of universe functors pass to Quillen equivalences bewteen the localised categories GM_{ℚ}^{V} and GM_{ℚ}^{U} for V ⊆ U.

Note these functors are strong symmetric monoidal and commute with smash products of based G-spaces.

Note that change of model structure functors,

1 : (C, M)\overrightarrow{⟵}(C, M′) : 1

are always strong monoidal functors and hence pass to localised versions. Quillen equivalences of this form are also preserved, since the adjunction acts as the identity on maps, so ϕ(f ∧ 1) = f ∧ 1.

The identity functor is a left adjoint from the positive stable structure to the stable structure on equivariant orthogonal spectra. This is an equivalence and clearly passes to an equivalence of any localised categories.

Lemma 4.4 Given a map α: H → G a homomorphism of groups, the functor α^{∗}: G𝒮 → H𝒮 is strong symmetric monoidal. The first category will be indexed on a universe U, the second on α^{∗}(U). This holds for either of the categories of orthogonal spectra or S-modules.

Proof The proof for the orthogonal spectra case is easy, we know that α^{∗} is a left adjoint, hence it commutes with colimits and coends, it is clearly strong monoidal on the space level and this is all that is needed. We turn to the somewhat more involved case of EKMM S-modules. We begin with an isomorphism of H-spaces, α^{∗}ℐ (U, U′) ≅ α^{∗}ℐ (α^{∗}U, α^{∗}U′) ≅ for any G-vector spaces U and U′. Equally for a map β: A → ℐ (U, U′) we obtain α^{∗}β: α^{∗}A → ℐ (α^{∗}U, α^{∗}U′). We then note that for ℰ ∈ G𝒮 (U′; U) we can define α^{∗}ℰ ∈ G𝒮 (α^{∗}U′; α^{∗}U) in an obvious manner. Recall that α^{∗}(X ∧ E) ≅ α^{∗}X ∧ α^{∗}E for any G-space X and G-spectrum E, thus

α^{∗}(ℰ ∧ E) ≅ α^{∗}ℰ ∧ α^{∗}E

and this is all we need. This also shows that we have a natural isomorphism:

α^{∗}I_{U}^{U′} ≅ I_{α^{∗}U}^{α^{∗}U′}α^{∗}.

The corresponding commutation result also holds for orthogonal spectra, and is easily shown from the definition.

Corollary 4.5 The functor α^{∗} passes to a Quillen functor on the various localised categories and the module categories and the pair of α^{∗} and its right adjoint pass to a Quillen pair as in the theorem.

Remark 4.6 In particular this holds for the forgetful functor i^{∗} and the inflation functor ε^{∗} corresponding to inclusion of a subgroup and the trivial group homomorphisms ε: G → {e}. Thus these functors and their left and right adjoints, (−)/H and (−)^{H} pass to Quillen functors (and Quillen pairs) on the localised categories. Furthermore the pair (G_{+} ∧_{H} (−), α) also pass to a quillen pair between the categories L_{i^{∗}E}G𝒮 and L_{E}H𝒮 (this does not follow from the theorem). As always G_{+} ∧_{H} (−) preserves cofibrations and the natural isomorphism ([MM02, Proposition 2.3])

(G_{+} ∧_{H} (X) ∧ Y ≅ (G_{+} ∧_{H} (X ∧ i^{∗}Y)

when Y = E serves to prove that the left adjoint preserves the trivial cofibrations of the localised category. The case of the pair ((−)/N, ε^{∗}) is less well behaved but this is usual.

Lemma 4.7 Given a change of model structure adjunction, (which is a Quillen pair)

1 : (C, M)\overrightarrow{⟵}(C, M′) : 1

if the weak equivalences agree, then the model categories are Quillen equivalent.

Proof Look at the unit and co-unit conditions, these are weak equivalences since the weak equivalences agree, this is all we need.

Corollary 4.8 The model structure on G𝒮 given by lifting over the right adjoint i^{∗}: G𝒮 → H𝒮 is Quillen equivalent to the model structure given by localisation at E𝔉_{+}.

Proof That is, we are assuming that the subgroups of H form a family. To ensure that the other model structure exists, we note that i^{∗} also has a right adjoint, so preserves colimits, we use fibrant replacement from the localised structure, and the usual path-object construction.

These clearly have the same weak equivalences. The identity map is a right Quillen functor G𝒮_{E𝔉_{+}} → G𝒮_{H} since a fibration in the left-hand structure is also a fibration in H𝒮, since i^{∗} is a right adjoint.

Remark 4.9 The above is just the 𝔉 model structure in the case when the family coincides with the collection of subgroups of a given subgroup. The entirety of the above holds true in the rationalised versions.

Now we can give the main result of this section, which is a triviality.

Theorem 4.10 We have a strong monoidal Quillen equivalence

F : GM_{ℚ}\overrightarrow{⟵}S^{0}ℚ –mod : U.

Proof We prove below that S → S^{0}ℚ is a rational equivalence. With this the proof proceeds in exactly the same way as the non-equivariant case.

Begin with the rational equivalence S → Λc H ℚ now apply the (strong monoidal) inflation functor ε^{∗} to both sides, and consider S → ε^{∗}Λc H ℚ. This is certainly a cofibration in the category of commutative equivariant S-algebras, we would like to know it is a rational equivalence as well. We prove this by using the model category of rational commutative equivariant S-algebras as constructed in the next section.

There is a commutative square of functors, working over the G-trivial universe U^{G} on the left and over U on the right with change of universe notation omitted throughout this proof.

\xymatrix{CRM_{ℚ} \ar[r]^{ε^{∗}} \ar[d]^{𝕌} & CRGM_{ℚ} \ar[l]^{(−)^{G}} \ar[d]^{𝕌} \\ M_{ℚ} \ar[r]^{ε^{∗}} \ar[u]^{ℂ} & GM_{ℚ} \ar[l]^{(−)^{G}} \ar[u]^{ℂ}}

This diagram to proves that the top pair (ε^{∗}, (−)^{G}) is a Quillen pair. Take a (trivial) fibration f in CRGM_{ℚ}, thus 𝕌f is also a (trivial) fibration, as is (𝕌f)^{G} = 𝕌(f^{G}). Thus (f^{G}) is a (trivial) fibration, and we have our Quillen pair.

Now since S → Λc H ℚ is a trivial cofibration in CRM_{ℚ}, S → ε^{∗}Λc H ℚ must be a trivial cofibration in CRGM_{ℚ} and hence a rational equivalence.

5 Another Rational Ring Spectrum

We are working in GMS with the generalised model structure. We localise this with respect to the cofibrant spectrum S_{ℚ}. This is a monoidal model category satisfying the monoid axiom.

Now consider the adjunction (ℂ, U) between GMS and the category of commutative ring spectra. We now want to lift this localised model structure over the right adjoint U. Take the generating cofibrations and trivial cofibrations for GMS_{ℚ}: I and K_{ℚ}, these must satisfy the cofibration hypothesis and every relative K_{ℚ}-cell complex must be an S_{ℚ}-equivalence [MM02, III 8.2,8.3].

The cofibration hypothesis is true, and is verified in [EKMM97, VII Section 4], it is unchanged by localisation. We now turn to proving the second statement, following the proof in [MM02, III 8.3] it suffices to show that that ℂ preserves S_{ℚ}-equivalences between cofibrant objects (since the generating cofibrations are unchanged). Take X → Y was an S_{ℚ}-equivalence between cofibrant S-modules and note that

ℂX ∧ S_{ℚ} ≅ ⋁_{i}(X^{(i)} ∧ S_{ℚ})/Σ_{i}.

Now use an inductive argument:

X^{(i)} ∧ S_{ℚ} → X^{(i−1)} ∧ Y ∧ S_{ℚ}

is a π_{∗} isomorphism, X^{(i−1)} ∧ (−) preserves π_{∗}-isomorphisms.

This gives a model structure for rational S-algebras, hence we can apply fibrant replacement in this and obtain a ring spectrum f_{ℚ}^{CR}S and a rational equivalence that is a cofibration of S-algebras S → f_{ℚ}^{CR}S.

Theorem 5.1 There is a strong monoidal Quillen pair

F : GMS\overrightarrow{⟵}f_{ℚ}^{CR}S –mod : U

such that

F : GMS_{ℚ}\overrightarrow{⟵}f_{ℚ}^{CR}S –mod : U

is a Quillen equivalence.

Proof This is the same as the previous results, all we need to show this theorem is that S → f_{ℚ}^{CR}S is a rational equivalence.

Remark 5.2 We can use the lifting property in the category CRGM_{ℚ} to compare f_{ℚ}^{CR}S and S^{0}ℚ. Both of these objects are cofibrant and fibrant, with trivial cofibrations S → f_{ℚ}^{CR}S and S → S^{0}ℚ. This gives a homotopy equivalence (of ring spectra) between these two ring spectra, we label it f: S^{0}ℚ → f_{ℚ}^{CR}S.

Corollary 5.3 The categories S^{0}ℚ –mod and f_{ℚ}^{CR}S –mod are Quillen equivalent via the restriction of scalers functor f^{∗}.

Proof Since both categories of modules are Quillen equivalent to rational equivariant S-modules, the top map in the commutative diagram below is a Quillen equivalence.

\xymatrix{S^{0}ℚ –mod \ar[rr]^{f_{∗}} \ar[dr]^{𝕌} &  & f_{ℚ}^{CR}S –mod \ar[ll]^{f^{∗}} \ar[dl]^{𝕌} \\  & GM_{ℚ} \ar[ul]^{F} \ar[ur]^{F}}

While it is clear that the right adjoints commute and this is sufficient we note that

f_{∗}(S^{0}ℚ ∧ X) = f_{ℚ}^{CR}S ∧_{S^{0}ℚ} S^{0}ℚ ∧ X ≅ f_{ℚ}^{CR}S ∧ X.

6 Comparisons and Splitting

Now we turn to the splitting theorems from previously, we take the usual assumptions on the existence of an idempotent family F.

Theorem 6.1 We have the following pair of commutative diagrams of monoidal Quillen equivalences.

\xymatrix{GM_{ℚ} \ar[r] \ar[d] & FM_{ℚ} × F̃M_{ℚ} \ar[l] \ar[d] \\ _{GI𝒮 _{ℚ}} \ar[u] \ar[r] & F𝒮 _{ℚ} × F̃𝒮 _{ℚ} \ar[l] \ar[u] \\ S^{0}ℚ –mod \ar[r] \ar[d] & S^{0}ℚ^{F} –mod ×S^{0}ℚF̃ –mod \ar[l] \ar[d] \\ GM_{ℚ} \ar[u] \ar[r] & FM_{ℚ} × F̃M_{ℚ} \ar[l] \ar[u]}

Where S^{0}ℚ^{F} –mod and S^{0}ℚ^{F} –mod are categories of S^{0}ℚ-modules with model structures lifted from FM and F̃M respectively.

Proof The top diagram is simply a previous corollary. The second diagram is straightforward, recall that FM and F̃M are monoidal model categories satisfying the monoid axiom, hence we have categories of modules over S^{0}ℚ in each of these cases. To specify, S^{0}ℚ^{F} –mod is the category of S^{0}ℚ-modules with weak equivalences the underlying EF_{+}-equivalences, fibrations given by underlying fibrations in FM and cofibrations as required.

The usual proof then shows that the free S^{0}ℚ-module functor and its right adjoint are a monoidal Quillen equivalence between FM_{q}q and S^{0}ℚ^{F} –mod. The underlying functor reflects all weak equivalences and the unit map is a weak equivalence, which proves this claim. Obviously the F̃ case is identical.

Remark 6.2 Note that we have a comparison

S^{0}ℚ –mod \overrightarrow{⟵}S^{0}ℚ^{F} –mod

arising from the change of model structure functors 1 : GMS_{ℚ}\overrightarrow{⟵}FMS_{ℚ} : 1.

Remark 6.3 We can construct ℙc H ℚ an orthogonal G-ring spectrum and we obtain the expected comparisons between modules over this spectrum, the Λc H ℚ –mod and c H ℚ –mod. However when we move to the equivariant categories we have to introduce change of universe functors and these do not behave well with respect to the Quillen equivalence of orthogonal spectra and S-modules [MM02, Chapter VI].

Index^{GM, 1 ; GM_{ℚ}, 2 ; 𝕊_{ℚ}^{+}, 3 ; S_{ℚ}, 2 ; S^{0}ℚ, 9 ; X_{ℚ}, 5}

15

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