EQUIVARIANT AND NONEQUIVARIANT MODULE SPECTRA

J. P. MAY

Abstract. Let G be a compact Lie group, let R_{G} be a commutative algebra over the sphere G-spectrum S_{G}, and let R be its underlying nonequivariant algebra over the sphere spectrum S. When R_{G} is split as an algebra, as holds for example for R_{G} = MU_{G}, we show how to “extend scalars” to construct a split R_{G}-module M_{G} = R_{G} ∧_{R} M from an R-module M. We also show how to compute the coefficients M_{∗}^{G} in terms of the coefficients R_{∗}^{G}, R_{∗}, and M_{∗}. This allows the wholesale construction of highly structured equivariant module spectra from highly structured nonequivariant module spectra. In particular, it applies to construct MU_{G}-modules from MU-modules and therefore gives conceptual constructions of equivariant Brown-Peterson and Morava K-theory spectra.

1. Introduction

We enrich the theory of highly structured modules over highly structured ring spectra that was developed in [4] by showing how to construct highly structured equivariant modules from highly structured nonequivariant modules. Throughout, we let G be a compact Lie group. As pointed out in its introduction, although [4] is written nonequivariantly, its theory applies verbatim equivariantly; an equivariant exposition will be given in [10]. The equivariant ring spectra we are interested in are the algebras over the equivariant sphere spectrum S_{G}. These are essentially equivalent to the earlier E_{∞} ring G-spectra of [9, VII§2]. The prime examples are S_{G} itself and the spectrum MU_{G} of stabilized equivariant complex cobordism; MU_{G} is proven to be a commutative S_{G}-algebra in [8]. In [5], Elmendorf and I showed how to construct examples from nonequivariant S-algebras. For finite groups G, a great many other examples are known.

In [4], we gave new constructions of the Brown-Peterson spectra BP, the connective and periodic Morava K-theory spectra k(n) and K(n), and all of the other spectra that are usually constructed from MU by means of the Baas-Sullivan theory of manifolds with singularities by killing some of the generators of MU_{∗} and inverting others. The new constructions give all of these spectra module structures over the commutative S-algebra MU. We also analyzed their multiplicative structure as “MU-ring spectra”, which are the analogs in the derived category of MU-modules of classical ring spectra in the stable homotopy category.

In [8], a localization and completion theorem is proven that relates M_{∗}^{G} to M_{∗}(BG) and M_{G}^{∗} to M^{∗}(BG) for any split MU_{G}-module M_{G} with underlying nonequivariant MU-module M. To apply this theorem to a given MU-module M, such as M = BP, M = k(n), or M = K(n), we must be able to construct a split MU_{G}-module M_{G} with underlying MU-module M. We shall accomplish this by means of a kind of “extension of scalars” functor that transforms MU-modules into split MU_{G}-modules. This construction commutes with smash products and so carries MU-ring spectra to MU_{G}-ring G-spectra.

We let ℳ_{R} denote the category of modules over an S-algebra R and let Gℳ_{R_{G}} denote the category of modules over an S_{G}-algebra R_{G}. We let 𝒟_{R} and G𝒟_{R_{G}} denote the respective derived categories; these are obtained from the respective homotopy categories by adjoining inverses to the weak equivalences or, equivalently, by passing to approximations by weakly equivalent cell modules. As usual, we write E_{∗}^{G} for the homotopy groups of the G-fixed point spectrum of a G-spectrum E_{G}.

Theorem 1.1. There is a monoidal functor MU_{G} ∧_{MU} (?) : ℳ_{MU} ⟶ Gℳ_{MU_{G}}. If M is a cell MU-module, then M_{G} ≡ MU_{G} ∧_{MU} M is split as an MU_{G}-module with underlying nonequivariant MU-module M. The functor MU_{G} ∧_{MU} (?) induces a derived monoidal functor 𝒟_{MU} ⟶ G𝒟_{MU_{G}}. Therefore, if M is an MU-ring spectrum, then MU_{G} ∧_{MU} M is an MU_{G}-ring G-spectrum. Moreover, there is a strongly convergent natural spectral sequence

Tor_{p,q}^{MU_{∗}}(MU_{∗}^{G}, M_{∗}) ⟹ M_{p+q}^{G}

of differential MU_{∗}-modules. In particular, if G is Abelian, then

M_{∗}^{G} ≅ MU_{∗}^{G} ⊗_{MU_{∗}} M_{∗}.

The term “split as a module” will be given a precise meaning below. The last statement holds since MU_{∗}^{G} is a free MU_{∗}-module when G is Abelian [2]. This is true in a few other cases, but little is known in general about the algebraic structure of MU_{∗}^{G}. A special case of the theorem solves a problem raised by Carlsson [1]:

“Define and compute equivariant Morava K-theory spectra.”

There is nothing in the literature on this subject. Theorem 1.1 allows the definitions

k(n)_{G} = MU_{G} ∧_{MU} k(n) and K(n)_{G} = MU_{G} ∧_{MU} K(n),

and it computes the coefficients when G is Abelian. Similarly, we can define equivariant Brown-Peterson spectra by

BP_{G} = MU_{G} ∧_{MU} BP,

and so on for all of the other standard MU-modules in current use. We point out an anomaly in one of the few familiar cases: the equivariant form k_{G} = MU_{G} ∧_{MU} k of the MU-module k that represents connective K-theory cannot represent the usual “connective equivariant K-theory” since, as observed by Greenlees, the latter theory does not take values in modules over the RO(G)-graded ring MU_{∗}^{G}. We do not know whether or not the equivariant form MU_{G} ∧_{MU} K of periodic K-theory that we construct represents equivariant K-theory, but we conjecture that it does.

The theorem is a special case of one that applies to all S_{G}-algebras that are “split as algebras”, in a sense that we shall make precise below.

Theorem 1.2. Let R_{G} be a commutative S_{G}-algebra and assume that R_{G} is split as an algebra with underlying nonequivariant S-algebra R. Then there is a monoidal functor R_{G} ∧_{R} (?) : ℳ_{R} ⟶ Gℳ_{R_{G}}. If M is a cell R-module, then M_{G} ≡ R_{G} ∧_{R} M is split as an R_{G}-module with underlying nonequivariant R-module M. The functor R_{G} ∧_{R} (?) induces a derived monoidal functor 𝒟_{R} ⟶ G𝒟_{R_{G}}. Therefore, if M is an R-ring spectrum, then R_{G} ∧_{R} M is an R_{G}-ring G-spectrum. Moreover, there is a strongly convergent natural spectral sequence

Tor_{p,q}^{R_{∗}}(R_{∗}^{G}, M_{∗}) ⟹ M_{p+q}^{G}

of differential R_{∗}-modules. In particular, if either R_{∗}^{G} or M_{∗} is a flat R_{∗}-module, then

M_{∗}^{G} ≅ R_{∗}^{G} ⊗_{R_{∗}} M_{∗}.

To justify the application to MU_{G}, we need to know that it is split as an algebra. This is a special case of a general criterion for R_{G} to be split as an algebra. The notion of a global ℐ_{∗}-FSP was defined in [8], and it was shown there that the sphere and cobordism functors provide examples.

Theorem 1.3. If T is a global ℐ_{∗}-FSP, then its associated commutative S_{G}-algebra is split as an algebra for every compact Lie group G.

Although the basic idea and construction predate the writing of [5], this paper is best understood as a sequel to that one, and we shall freely use its notations and results. The reader is referred to [9, 6, 10] for the relevant background on equivariant stable homotopy theory.

2. Change of universe and operadic smash products

The functor R_{G} ∧_{R} M that we shall construct depends on an extension of the operadic smash product

∧_{ℒ} : G𝒮 U[𝕃] × G𝒮 U[𝕃] ⟶ G𝒮 U[𝕃]

of [4, Ch I] that incorporates the change of universe functors

I_{U′}^{U} : G𝒮 U′[𝕃′] ⟶ G𝒮 U[𝕃]

of [5, 2.1]. Recall from [5, 1.1] that the functors I_{U′}^{U} are monoidal equivalences of categories.

Definition 2.1. Let U, U′, and U′′ be G-universes. For an 𝕃′-spectrum M and an 𝕃′′-spectrum N, define an 𝕃-spectrum M ∧_{ℒ} N by

M ∧_{ℒ} N = I_{U′}^{U}M ∧_{ℒ} I_{U′′}^{U}N.

Obviously, the formal properties of this product can be deduced from those of the functors I_{U′}^{U} together with those of the operadic smash product for the fixed universe U. In particular, since the functor I_{U′}^{U} takes S_{U′}-modules to S_{U}-modules and the smash product over S_{U} is the restriction to S_{U}-modules of the smash product over ℒ, we have the following observation. Here S_{U} denotes the sphere G-spectrum indexed on U.

Lemma 2.2. The functor ∧_{ℒ} : G𝒮 U′[𝕃′] × G𝒮 U′′[𝕃′′] ⟶ G𝒮 U[𝕃] restricts to a functor

∧_{S_{U}} : Gℳ_{S_{U′}} × Gℳ_{S_{U′′}} ⟶ Gℳ_{S_{U}} .

There is an alternative description of this product that makes its structure more apparent. It depends on the following generalization of [4, I.5.4], which in fact is implied by that result; compare [5, 2.2].

Lemma 2.3. Assume given universes U, U′, U′′, U′_{r} for 1 ≤ r ≤ i, and U′′_{s} for 1 ≤ s ≤ j, where i ≥ 1 and j ≥ 1. Then the following diagram is a split coequalizer of spaces and therefore a coequalizer of G-spaces; the maps γ are given by sums and compositions of linear isometries.

[figure: ℐ (U′ ⊕ U′′, U) × ℐ (U′, U′) × ℐ (U′′, U′′) × ℐ (⊕_{r=1}^{i}U′_{r}, U′) × ℐ (⊕_{s=1}^{j}U′′_{s}, U′′); γ×id; id ×γ; ℐ (U′ ⊕ U′′, U) × ℐ (⊕_{r=1}^{i}U′_{r}, U′) × ℐ (⊕_{s=1}^{j}U′′_{s}, U′′); γ; ℐ ((⊕_{r=1}^{i}U′_{r}) ⊕ (⊕_{s=1}^{j}U′′_{s}), U).]

Lemma 2.4. There is a natural isomorphism

M ∧_{ℒ} N ⟶ ℐ (U′ ⊕ U′′, U) ⋉_{ℐ (U′,U′)×ℐ (U′′,U′′)} M ∧ N,

where ∧ on the right is the external smash product G𝒮 U′×G𝒮 U′′ ⟶ G𝒮 (U′⊕U′′).

Proof. Expanding definitions, we see that M ∧_{ℒ} N is

ℐ (U ⊕ U, U) ⋉_{ℐ (U,U)×ℐ (U,U)} [(ℐ (U′, U) ⋉_{ℐ (U′,U′)} M) ∧ (ℐ (U′′, U) ⋉_{ℐ (U′′,U′′)} N)].

Formal properties of the twisted half-smash product allow us to rewrite this as

[ℐ (U ⊕ U, U) ×_{ℐ (U,U)×ℐ (U,U)} ℐ (U′, U) × ℐ (U′′, U)] ⋉_{ℐ (U′,U′)×ℐ (U′′,U′′)} M ∧ N.

The previous lemma gives a homeomorphism

ℐ (U ⊕ U, U) ×_{ℐ (U,U)×ℐ (U,U)} ℐ (U′, U) × ℐ (U′′, U) ⟶ ℐ (U′ ⊕ U′′, U)

of G-spaces over ℐ (U′ ⊕ U′′, U), and the conclusion follows.

Similarly, as in the proof of [4, I.5.5 and I.5.6], Lemma 2.3 implies the following associativity property of our generalized operadic smash products and therefore, upon restriction, of our generalized smash products over sphere G-spectra.

Lemma 2.5. Let M ∈ G𝒮 U′[𝕃′], P ∈ G𝒮 U′′[𝕃′′], and N ∈ G𝒮 U′′′[𝕃′′′]. Then both (M ∧_{ℒ′} P) ∧_{ℒ} N and M ∧_{ℒ} (P ∧_{ℒ′′′} N) are canonically isomorphic to

ℐ (U′ ⊕ U′′ ⊕ U′′′, U) ⋉_{ℐ (U′,U′)×ℐ (U′′,U′′)×ℐ (U′′′,U′′′)} M ∧ P ∧ N,

which in turn is canonically isomorphic to

I_{U′}^{U}M ∧_{ℒ} I_{U′′}^{U}P ∧_{ℒ} I_{U′′′}^{U}N.

Using change of universe explicitly or, via the previous lemmas, implicitly, we can define modules indexed on one universe over algebras indexed on another.

Definition 2.6. Let R ∈ Gℳ_{S_{U′′}} be an S_{U′′}-algebra and let M ∈ Gℳ_{S_{U′}}. Say that M is a right R-module if it is a right I_{U′′}^{U′}R-module, and similarly for left modules.

It is quite clear how one must define smash products over R in this context.

Definition 2.7. Let R ∈ Gℳ_{S_{U′′}} be an S_{U′′}-algebra, let M ∈ Gℳ_{S_{U′}} be a right R-module and let N ∈ Gℳ_{S_{U′′′}} be a left R-module. Define

M ∧_{R} N = I_{U′}^{U}M ∧_{I_{U′′}^{U}R} I_{U′′′}^{U}N.

Here we have used that I_{U′′}^{U} ≅ I_{U′}^{U}I_{U′′}^{U′} and that I_{U′}^{U}M is therefore an I_{U′′}^{U}R-module, and similarly for N. Expanding definitions and using the associativity isomorphism of Lemma 2.5, we obtain the following more explicit description.

Lemma 2.8. M ∧_{R} N is the coequalizer displayed in the diagram

ℐ (U′ ⊕ U′′ ⊕ U′′′, U) ⋉_{ℐ (U′,U′)×ℐ (U′′,U′′)×ℐ (U′′′,U′′′)} M ∧ R ∧ N

ℐ (U′ ⊕ U′′′, U) ⋉_{ℐ (U′,U′)×ℐ (U′′′,U′′′)} M ∧ N

M ∧_{R} N,

where the parallel arrows are induced by the actions of R on M and on N.

Evidently these smash products inherit good formal properties from those of the smash products of R-modules studied in [4]. Similarly, their homotopical properties can be deduced from the homotopical properties of the smash product of R-modules and the homotopical properties of the I_{U′}^{U}, which were studied in [5].

3. S_{G}-algebras and their underlying S-algebras

We are concerned with genuine G-spectra and their comparison with naive Gspectra. Recall that these are indexed respectively on a complete G-universe U and its G-fixed point universe U^{G}. We write S_{G} for the sphere G-spectrum indexed on U and S for the sphere spectrum indexed on U^{G}. We regard nonequivariant spectra such as S as G-spectra with trivial G-action. We have the forgetful change of universe functor i^{∗} : G𝒮 U ⟶ G𝒮 U^{G} obtained by forgetting those indexing spaces of U that are not contained in U^{G}. The underlying nonequivariant spectrum E of a Gspectrum E_{G} is defined to be i^{∗}E_{G}, with its action by G ignored. Said another way, let U^{#} denote U with its action by G ignored and let E_{G}^{#} denote the nonequivariant spectrum indexed on U^{#} that is obtained from E_{G} by forgetting the action of G. Then E = i^{∗}E_{G}^{#}.

The G-fixed point spectrum of E_{G} is obtained by taking the spacewise fixed points of i^{∗}E_{G}. We say that E_{G} is split if there is a map E ⟶ (E_{G})^{G} of spectra indexed on U^{G} whose composite with the inclusion of (E_{G})^{G} in E is an equivalence. As observed in [7, 0.4], E_{G} is split if and only if there is a map of G-spectra i_{∗}E ⟶ E_{G} that is a nonequivariant equivalence, where i_{∗} : G𝒮 U^{G} ⟶ G𝒮 U is the left adjoint of i^{∗}. In either form, the notion of a split G-spectrum is essentially a homotopical one. More precisely, it is a derived category notion: its purpose is to allow the comparison of equivariant and nonequivariant homology and cohomology theories, which are defined on derived categories. Thus we could have used weak equivalences in the definitions just given, and we shall use the term equivalence to mean weak equivalence of underlying spectra or G-spectra in what follows; we will add the adjective “weak” in cases where we would not expect to have a homotopy equivalence in general.

We must modify these definitions in the context of highly structured ring and module spectra. This point was left implicit in both [6] and [8], where reference was made to “the underlying S-algebra R of an S_{G}-algebra R_{G}”: in fact there is no obvious way to give R = i^{∗}R_{G}^{#} a structure of S-algebra. The point becomes clear when one thinks back to the underlying E_{∞} ring structures. We are given G-maps

ℐ (U^{j}, U) ⋉ R_{G}^{j} ⟶ R_{G}, (3.1)

and there is no obvious way to obtain induced nonequivariant maps

ℐ ((U^{G})^{j}, U^{G}) ⋉ R^{j} ⟶ R. (3.2)

We therefore think of i^{∗}R_{G}^{#} as only prescribing the appropriate weak homotopy type of the underlying S-algebra of an S_{G}-algebra R_{G}.

Definition 3.3. An underlying S-algebra of an S_{G}-algebra R_{G} is an S-algebra whose underlying spectrum is weakly equivalent to i^{∗}R_{G}^{#}.

There are several natural ways to construct such an S-algebra. First, forgetting the G-actions and regarding the maps (3.1) as maps of nonequivariant spectra, we obtain a nonequivariant S_{U}^{#}-algebra R_{G}^{#} indexed on U^{#} from our equivariant S_{U}- algebra R_{G}. We may choose a nonequivariant linear isomorphism f : U^{G} ⟶ U^{#}. By conjugation of linear isometries by f, we obtain an isomorphism between the nonequivariant linear isometries operads of U^{G} and of U^{#}, and we see immediately that f^{∗}R_{G}^{#} is an S_{U}^{G}-algebra. By Lemma 3.4 below, it is weakly equivalent to i^{∗}R_{G}^{#}.

Second, instead of making an arbitrary choice of an isomorphism f, we can follow the philosophy of [4] and consider the twisted function spectrum F [ℐ (U^{G}, U^{#}), R_{G}^{#}); Lemma 3.4 shows that F [ℐ (U^{G}, U^{#}), R_{G}^{#}) is also weakly equivalent to i^{∗}R_{G}^{#}. As in [4, I.7.5], one can construct a weakly equivalent operadic modification of the spectrum F [ℐ (U^{G}, U^{#}), R_{G}^{#}) that is an S-algebra. However, there is a simpler way to arrive at the cited operadic modification: it turns out to be given by a functor that is right adjoint to the functor I_{U^{G}}^{U^{#}} : 𝒮 U^{G}[𝕃] ⟶ 𝒮 U^{#}[𝕃], and [5, 2.3] shows that the right adjoint of I_{U^{G}}^{U^{#}} is I_{U^{#}}^{U^{G}}. By [5, 1.1], I_{U^{#}}^{U^{G}}R_{G}^{#} is an S-algebra. By Lemma 3.4 and the specialization to G = e of Lemma 3.5 below, I_{U^{#}}^{U^{G}}R_{G}^{#} is weakly equivalent to i^{∗}R_{G}^{#}.

Lemma 3.4. For nonequivariant spectra F ∈ 𝒮 U^{#}, there are natural weak equivalences between i^{∗}F and f^{∗}F and between i^{∗}F and F [ℐ (U^{G}, U^{#}), F).

Proof. Choose a path h : I ⟶ ℐ (U^{G}, U^{#}) connecting i to f. For spectra E ∈ 𝒮 U^{G}, there result natural maps i_{∗}E ⟶ h ⋉ E ⟵ f_{∗}E in 𝒮 U^{#}, and these are homotopy equivalences if E is tame, for example if E has the homotopy type of a CW spectrum [4, I.2.5]. Similarly, the natural map i_{∗}E ⟶ ℐ (U^{G}, U^{#}) ⋉ E is a homotopy equivalence when E is tame. Conjugation from left to right adjoints gives the conclusions since simple diagram chases show that the conjugate natural maps induce isomorphisms of homotopy groups.

Lemma 3.5. Let f : U ⟶ U′ be an isomorphism of G-universes. Then there are natural isomorphisms

f_{∗}E ≅ I_{U}^{U′}E and f^{∗}E′ ≅ I_{U′}^{U}E′

for E ∈ G𝒮 U[𝕃] and E′ ∈ G𝒮 U′[𝕃′].

Proof. Regard f as a G-map from a point into ℐ (U, U′). Then the following composite is a homeomorphism of G-spaces over ℐ (U, U′):

[figure: {∗} × ℐ (U, U); f×id; ℐ (U, U′) × ℐ (U, U); ∘; ℐ (U, U′).]

By [9, VI.3.1(iii)], there results a natural isomorphism

f_{∗}(ℐ (U, U) ⋉ E) ≅ ℐ (U, U′) ⋉ E.

Passing to coequalizers, we obtain

f_{∗}E ≅ f_{∗}(ℐ (U, U) ⋉_{ℐ (U,U)} E) ≅ ℐ (U, U′) ⋉_{ℐ (U,U)} E ≡ I_{U}^{U′}E.

Since f^{∗} = f_{∗}^{−1}, the second isomorphism follows from the first.

We can now define the terms “split as an algebra” and “split as a module” that appear in Theorem 1.2.

Definition 3.6. A commutative S_{G}-algebra R_{G} is split as an algebra if there is a commutative S-algebra R and a map η : I_{U^{G}}^{U}R ⟶ R_{G} of S_{G}-algebras such that η is a (nonequivariant) equivalence of spectra and the natural map α : i_{∗}R ⟶ I_{U^{G}}^{U}R is an (equivariant) equivalence of G-spectra.

Since the composite η ∘ α : i_{∗}R ⟶ R_{G} is a nonequivariant equivalence and the natural map R ⟶ i^{∗}i_{∗}R is a weak equivalence (provided that R is tame, [9, II.1.8] and [4, I.2.5]), R is weakly equivalent to i^{∗}R_{G}^{#}. Of course, R_{G} is split as a G-spectrum with splitting map η∘α. By abuse, we refer to R as “the” rather than “an” underlying nonequivariant S-algebra of R_{G}.

Definition 3.7. Let R_{G} be a commutative S_{G}-algebra that is split as an algebra with underlying S-algebra R and let M_{G} be an R_{G}-module. Regard M_{G} as an I_{U^{G}}^{U}R-module by pullback along η. Then M_{G} is split as an R_{G}-module if there is an R-module M and a map χ : I_{U^{G}}^{U}M ⟶ M_{G} of I_{U^{G}}^{U}R-modules such that χ is a (nonequivariant) equivalence of spectra and the natural map α : i_{∗}M ⟶ I_{U^{G}}^{U}M is an (equivariant) equivalence of G-spectra.

Again, M is weakly equivalent to i^{∗}M_{G}^{#} and M_{G} is split as a G-spectrum with splitting map χ∘α : i^{∗}M ⟶ M_{G}. By abuse, we call M the underlying nonequivariant R-module of M_{G}.

The ambiguity that we allow in the notion of an underlying object is quite useful: it allows us to arrange the condition on α in the definitions if we have succeeded in arranging the condition on η. The proof of this depends on the closed model category structures on all categories in sight that is given in [4, VII§4].

Lemma 3.8. Let R_{G} be a commutative S_{G}-algebra and M_{G} be an R_{G}-module.

(i) Suppose given a commutative S-algebra R′ and a map η′ : I_{U^{G}}^{U}R′ ⟶ R_{G} of S_{G}- algebras such that η′ is a (nonequivariant) equivalence of spectra. Let γ : R ⟶ R′ be a weak equivalence of S-algebras, where R is a q-cofibrant commutative S-algebra, and define η = η′ ∘ I_{U^{G}}^{U}γ : I_{U^{G}}^{U}R ⟶ R_{G}. Then R_{G} is split as an algebra with underlying nonequivariant S-algebra R and splitting map η. (ii) Suppose given an R′-module M′ and a map χ′ : I_{U^{G}}^{U}M′ ⟶ M_{G} of I_{U^{G}}^{U}R′-modules such that χ′ is a (nonequivariant) equivalence of spectra. Regard M′ as an R-module by pullback along γ, let ϑ : M ⟶ M′ be a weak equivalence of R-modules, where M is a q-cofibrant R-module, and define χ = χ′ ∘ I_{U^{G}}^{U}ϑ : I_{U^{G}}^{U}M ⟶ M_{G}. Then M_{G} is split as an R_{G}-module with underlying nonequivariant R-module M and splitting map χ.

Proof. It is immediate from [5, 1.4] that α : i_{∗}R ⟶ I_{U^{G}}^{U}R and α : i_{∗}M ⟶ I_{U^{G}}^{U}M are equivalences of G-spectra. Thus we need only observe that, ignoring the G-action, the maps I_{U^{G}}^{U^{#}}γ and I_{U^{G}}^{U^{#}}ϑ are weak equivalences since it is immediate from the case G = e of Lemma 3.5 that the functor I_{U^{G}}^{U^{#}} preserves weak equivalences.

Proof of Theorem 1.2. As observed in [4, VII.1.3], the splitting map η : I_{U^{G}}^{U}R ⟶ R_{G} of our given split commutative S_{G}-algebra R_{G} is the unit of a structure of I_{U^{G}}^{U}Ralgebra on R_{G}. The composite

[figure: R_{G} ∧_{S_{G}} I_{U^{G}}^{U}R; id ∧η; R_{G} ∧_{S_{G}} R_{G}; ϕ; R_{G}]

gives R_{G} a structure of right R-module in the sense prescribed in Definition 2.6, and R_{G} is an (R_{G}, R)-bimodule with left action of R_{G} induced by the product ϕ of R_{G}. Therefore, for an R-module M, we can take U = U′ and U^{G} = U′′ = U′′′ in Definition 2.7 and define

M_{G} = R_{G} ∧_{R} M.

Clearly M_{G} is an R_{G}-module with action induced by the left action of R_{G} on itself. The functor I_{U^{G}}^{U} : ℳ_{R} ⟶ Gℳ_{I_{U^{G}}^{U}R} is monoidal by [5, 1.3]. The functor

R_{G} ∧_{I_{U^{G}}^{U}R} (?) : Gℳ_{I_{U^{G}}^{U}R} ⟶ Gℳ_{R_{G}}

is monoidal since R_{G} ≅ R_{G} ∧_{R_{G}} R_{G} and

(R_{G} ∧_{R_{G}} R_{G}) ∧_{I_{U^{G}}^{U}R} (M ∧_{I_{U^{G}}^{U}R} N) ≅ (R_{G} ∧_{I_{U^{G}}^{U}R} M) ∧_{R_{G}} (R_{G} ∧_{I_{U^{G}}^{U}R} N)

by a comparison of coequalizer diagrams. Therefore the functor R_{G}∧_{R}(?) is monoidal.

As observed in Lemma 3.8, we can assume that our given underlying nonequivariant S-algebra R is q-cofibrant as an S-algebra. Let M be a cell R-module. Then M is q-cofibrant and, by [5, 1.4], α : i_{∗}M ⟶ I_{U^{G}}^{U}M is an equivalence. To prove that M_{G} is split as an R_{G}-module, define

χ = η ∧ id : I_{U^{G}}^{U}M ≅ I_{U^{G}}^{U}R ∧_{I_{U^{G}}^{U}R} I_{U^{G}}^{U}M ⟶ R_{G} ∧_{I_{U^{G}}^{U}R} I_{U^{G}}^{U}M = M_{G}.

Clearly χ is a map of I_{U^{G}}^{U}R-modules, and we must prove that it is an equivalence of spectra. Recall from [4, III§1] that we have a free functor F_{R} from spectra to R-modules given by

F_{R}X = R ∧_{S} (S ∧_{ℒ} 𝕃X) ≅ R ∧_{ℒ} 𝕃X;

here ℒ and 𝕃 refer to the universe U^{G}, but we have a similar free functor F_{R_{G}} from G-spectra to R_{G}-modules based on use of the linear isometries operad for U, and similarly for I_{U^{G}}^{U}R. If we forget about G-actions and compare definitions, we find by use of an isomorphism f : U^{G} ⟶ U^{#} that, nonequivariantly,

I_{U^{G}}^{U^{#}}F_{R}X ≅ F_{I_{U^{G}}^{U^{#}}R}X.

Recalling the definition of cell R-modules from [4, III.2.1], we see that cell R-modules are built up via pushouts and sequential colimits from the free R-modules generated by sphere spectra and their cones. The functor I_{U^{G}}^{U} is a left adjoint, whether we interpret it equivariantly or nonequivariantly. We conclude that this functor carries cell R-modules to I_{U^{G}}^{U}R-modules that, with G-actions ignored, are nonequivariant cell I_{U^{G}}^{U^{#}}R-modules. Now [4, III.3.8] gives that χ is an equivalence of spectra since η is an equivalence of spectra.

The passage to derived categories is immediate, modulo one slight subtlety: our functor on modules was constructed as the composite of two functors, but, as we saw in [5, §3], it does not follow that the induced functor on derived categories factors as a composite. The solution is simple: we ignore the intermediate category G𝒟_{I_{U^{G}}^{U}R}, as it is of no particular interest to us. Since cell approximation of R-modules commutes up to equivalence with smash products, passage to derived categories preserves smash products.

Finally, we must construct the spectral sequence claimed in Theorem 1.2. By definition, R_{∗}^{G} = π_{∗}((R_{G})^{G}), where (R_{G})^{G} = (i^{∗}R_{G})^{G}. Similarly,

M_{∗}^{G} = π_{∗}((R_{G} ∧_{R} M)^{G}), where (R_{G} ∧_{R} M)^{G} = (i^{∗}(R_{G} ∧_{R} M))^{G}.

Assuming as we may that all given algebras and modules are suitably cofibrant, we obtain the same groups if we replace the functor i^{∗} by the more structured change of universe functor I_{U}^{U^{G}}. In view of the following lemma, the desired spectral sequence is a special case of the spectral sequence

Tor_{∗,∗}^{R_{∗}}(M_{∗}, N_{∗}) ⟹ π_{∗}(M ∧_{R} N)

constructed in [4, IV.4.1] for an S-algebra R and R-modules M and N.

Lemma 3.9. Let R_{G} be split as an algebra with underlying S-algebra R. For Rmodules M, there is a natural isomorphism of R-modules

(I_{U}^{U^{G}}R_{G})^{G} ∧_{R} M ≅ (I_{U}^{U^{G}}(R_{G} ∧_{R} M))^{G}.

Proof. Passage to G-fixed points commutes with smash products in the category of naive G-spectra. The same is true in the category Gℳ_{S} of equivariant S-modules and therefore in the the category Gℳ_{R} for any G-trivial S-algebra R. In our situation, G acts trivially on both R and M, and it follows that

(I_{U}^{U^{G}}R_{G})^{G} ∧_{R} M ≅ ((I_{U}^{U^{G}}R_{G}) ∧_{R} M)^{G}. (3.10)

Since R ≅ I_{U}^{U^{G}}I_{U^{G}}^{U}R, M ≅ I_{U}^{U^{G}}I_{U^{G}}^{U}M, and the functor I_{U}^{U^{G}} is monoidal, we have

(I_{U}^{U^{G}}R_{G}) ∧_{R} M ≅ I_{U}^{U^{G}}(R_{G} ∧_{I_{U^{G}}^{U}R} I_{U^{G}}^{U}M) ≡ I_{U}^{U^{G}}(R_{G} ∧_{R} M). (3.11)

We obtain the desired isomorphism by composing (3.10) with the isomorphism obtained from (3.11) by passage to G-fixed point spectra.

4. Global I_{∗}-functors and split S_{G}-algebras

We must prove Theorem 1.3. The notion of a global ℐ_{∗}-FSP, or 𝒢 ℐ_{∗}-FSP, was defined in [8, §§5,6]. We shall only sketch the definition here, referring the reader to [8] for more details. In fact, we only need a tiny fraction of the structure that is present on S_{G}-algebras that arise from 𝒢 ℐ_{∗}-FSP’s. In what follows, we could work with either real or complex inner product spaces, and of course the complex case is the one relevant to complex cobordism; see [8, 6.5].

Let 𝒢 ℐ_{∗} be the category of pairs (G, V) consisting of a compact Lie group G and a finite dimensional G-inner product space V; the morphisms (α, f) : (G, V) ⟶ (G′, V′) consist of a homomorphism α : G ⟶ G′ of Lie groups and an α-equivariant linear isomorphism f : V ⟶ V′. Let 𝒢 𝒯 be the category of pairs (G, X), where G is a compact Lie group and X is a based G-space; the morphisms (α, f) : (G, X) ⟶ (G′, X′) consist of a homomorphism α : G ⟶ G′ and an α-equivariant based map f : X ⟶ X′. Let S^{•} : 𝒢 ℐ_{∗} ⟶ 𝒢 𝒯 be the functor that sends a pair (G, V) to the based G-space S^{V}.

A 𝒢 ℐ_{∗}-FSP T is a continuous functor T : 𝒢 ℐ_{∗} ⟶ 𝒢 𝒯 over the category 𝒢 of compact Lie groups together with continuous natural transformations

η : S^{•} ⟶ T and ω : T ∧ T ⟶ T ∘ ⊕

such that the appropriate unity, associativity, and commutativity diagrams commute. Since T is a functor over 𝒢, we may write T (G, V) = (G, T V), and we require that

T (α, id) = (α, id) : (G, T V′) ⟶ (G′, T V′) (4.1)

for a homomorphism α : G ⟶ G′ and a G′-inner product space V′ regarded by pullback along α as a G-inner product space. Henceforward, we abbreviate notation by writing T (G, V) = T V on objects. For each object (G, V), we are given a G-map

η : S^{V} ⟶ T V.

For each pair of objects (G, V) and (G′, V′), we are given a G × G′-map

ω : T V ∧ T V′ ⟶ T (V ⊕ V′);

by pullback along the diagonal, we regard ω as a G-map when G = G′.

We insert some observations that show the power of condition (4.1) and are at the heart of our work. Let e denote the trivial group and let ι : e ⟶ G and ε : G ⟶ e be the unique homomorphisms.

Lemma 4.2. If V has trivial G-action, then T V also has trivial G-action. For a general G-inner product space V, if V^{#} denotes V regarded as an e-space, then T V^{#} is the space T V with its action by G ignored.

Proof. For the first statement, the functor T carries the morphism (ε, id) : (G, V) ⟶ (e, V) of 𝒢 ℐ_{∗} to the morphism (ε, id) of 𝒢 𝒯 , so that the identity map on T V must be ε-equivariant. For the second statement, the functor T carries the morphism (ι, id) : (e, V) ⟶ (G, V) to the identity map on the space T V.

For a compact Lie group G and a G-universe U, we obtain a G-prespectrum T_{(G,U)} indexed on U with V th G-space T V. The structural maps are given by the composites

[figure: T V ∧ S^{W −V}; id ∧η; T V ∧ T (W − V); ω; T W]

for V ⊂ W. Write R_{(G,U)} for the G-spectrum LT_{(G,U)}, where L is the spectrification functor of [9, I.2.2].

Now suppose given G-universes U and U′. Then there is a canonical map of Gspectra indexed on U′

ζ : ℐ (U, U′) ⋉ R_{(G,U)} ⟶ R_{(G,U′)}. (4.3)

Indeed, if f : U ⟶ U′ is a linear isometry and V is an indexing space in U, then the maps T f : T V ⟶ T f(V) specify a map of prespectra T_{(G,U)} ⟶ f^{∗}T_{(G,U′)} indexed on U. By adjunction, T f gives a map of prespectra ζ(f) : f_{∗}T_{(G,U)} ⟶ T_{(G,U′)} indexed on U′; see [9, p.58]. We record the following observation for later reference.

Lemma 4.4. If f : U ⟶ U′ is an isomorphism, then ζ(f) : f_{∗}T_{(G,U)} ⟶ T_{(G,U′)} is an isomorphism; if f is a G-map, then ζ(f) is a G-map.

Intuitively, the twisted half-smash product ℐ (U, U′)⋉R_{(G,U)} is obtained by gluing together the spectrifications f_{∗}R_{(G,U)} of the f_{∗}T_{(G,U)}, and the maps ζ(f) glue together to give the G-map ζ. This sort of argument first appeared in [11, IV.1.6, IV.2.2], before the twisted half-smash product was invented, and it was formalized in current terminology in [9, VI.2.17].

Fixing G and U, a precisely similar argument, formalized in [9, VI.5.5, VII.2.4, and VII.2.6], shows that the maps

ξ_{j}(f) : T V_{1} ∧ ⋯ T V_{j}⟶^{ω}T (V_{1} ⊕ ⋯ ⊕ V_{j})⟶^{T f}T f(V_{1} ⊕ ⋯ ⊕ V_{j})

for linear isometries f : U^{j} ⟶ U give rise to maps

ξ_{j} : ℒ (j) ⋉ (R_{(G,U)})^{j} ⟶ R_{(G,U)}

that give R_{(G,U)} a structure of ℒ G-spectrum. When the universe U is complete, so that its linear isometries operad ℒ is an E_{∞} operad of G-spaces, this means that R_{(G,U)} is an E_{∞} ring G-spectrum. The map ξ_{1} gives R_{(G,U)} an action of ℒ (1) = ℐ (U, U), and functoriality implies that ξ_{2} factors through the coequalizer that defines the operadic smash product, giving

ξ : R_{(G,U)} ∧_{ℒ} R_{(G,U)} = ℒ (2) ⋉_{ℒ (1)^{2}} (R_{(G,U)})^{2} ⟶ R_{(G,U)}. (4.5)

As explained in [4, II.3.3 and II§4], the maps ξ_{j} for j ≥ 3 can be reconstructed from the maps for j = 1 and j = 2.

Lemma 4.6. The map ζ of (4.3) factors through the coequalizer to give a map

ζ : I_{U}^{U′}R_{(G,U)} = ℐ (U, U′) ⋉_{I(U,U)} R_{(G,U)} ⟶ R_{(G,U′)},

and ζ is a map of ℒ′ G-spectra, where ℒ′ is the linear isometries operad of U′.

Proof. The factorization is clear from functoriality. To check that ζ is a map of ℒ′ G-spectra, we must show that the following diagram commutes:

[figure: ℒ′(2) ⋉_{ℒ′(1)^{2}} (I_{U}^{U′}R_{(G,U)})^{2}; id ⋉ζ^{2}; ξ; ℒ′(2) ⋉_{ℒ′(1)^{2}} (R_{(G,U′)})^{2}; ξ; I_{U}^{U′}R_{(G,U)}; ζ; R_{(G,U′)}.]

Using Lemma 2.3 to identify the upper left corner of the diagram and chasing through the definitions, we see that both composites coincide with the following one:

[figure: ℐ (U ⊕ U, U′) ⋉_{ℐ (U,U)^{2}} (R_{(G,U)})^{2}; id ⋉ω; ℐ (U ⊕ U, U′) ⋉_{ℐ (U⊕U,U⊕U)} R_{(G,U⊕U)}; ζ; R_{(G,U)};]

here ω is induced by passage to spectra from the evident map of prespectra.

Returning to our fixed G and a complete G-universe U, we consider R_{(e,U^{G})} and R_{(G,U)}. We deduce from Lemma 4.2 that

R_{(e,U^{G})} = R_{(G,U^{G})} and R_{(G,U)}^{#} = R_{(e,U^{#})}. (4.7)

That is, R_{(G,U^{G})} is R_{(e,U^{G})} regarded as a G-trivial G-spectrum indexed on the Gtrivial universe U^{G}, and R_{(G,U)} regarded as a nonequivariant spectrum indexed on U^{#} is R_{(e,U^{#})}

The first of these equalities allows us to specialize the map ζ to obtain a map of E_{∞} ring G-spectra

ζ : I_{U^{G}}^{U}R_{(e,U^{G})} = ℐ (U^{G}, U) ⋉_{I(U^{G},U^{G})} R_{(G,U^{G})} ⟶ R_{(G,U)}. (4.8)

The second of these equalities allows us to identify the target of the underlying map ζ^{#} of nonequivariant spectra with R_{(e,U^{#})}.

Lemma 4.9. The map ζ^{#} is an isomorphism of spectra.

Proof. Choose an isomorphism f : U^{G} ⟶ U^{#}. It is immediate that the composite

[figure: f_{∗}R_{(e,U^{G})}; ≅; I_{U^{G}}^{U^{#}}R_{(e,U^{G})}; ζ^{#}; R_{(e,U^{#})}]

coincides with the isomorphism ζ(f) of Lemma 4.4; here the unlabelled isomorphism is given by the case G = e of Lemma 3.5.

To pass to S_{G}-algebras, we let R be the S-algebra S ∧_{ℒ} R_{(e,U^{G})} and R_{G} be the S_{G}-algebra S_{G} ∧_{ℒ} R_{(G,U)} (where ℒ refers respectively to U^{G} and to U). By [5, 2.4], we have an isomorphism of S_{G}-algebras

I_{U^{G}}^{U}R ≅ S_{G} ∧_{ℒ} I_{U^{G}}^{U}R_{(e,U^{G})},

and this allows us to define an isomorphism of S_{G}-algebras

η = id ∧ζ : I_{U^{G}}^{U}R ⟶ R_{G}. (4.10)

At this level of generality, we cannot expect to prove that α : i_{∗}R ⟶ I_{U^{G}}^{U}R is an equivalence of G-spectra, although it seems plausible that this holds in the examples of interest. However, we can appeal to q-cofibrant approximation, as in Lemma 3.8, to complete the proof of Theorem 1.3, thereby losing that η is an isomorphism in order to make sure that α is an equivalence.

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Department of Mathematics, The University of Chicago, Chicago, IL 60637, USA E-mail address: may@math.uchicago.edu