LOCALIZATION AND DUALITY IN TOPOLOGY AND MODULAR REPRESENTATION THEORY

DAVID J. BENSON AND J. P. C. GREENLEES

Abstract. We develop a duality theory for localizations in the context of ring spectra in algebraic topology. We apply this to prove a theorem in the modular representation theory of finite groups.

Let G be a finite group and k be an algebraically closed field of characteristic p. If 𝔭 is a homogeneous nonmaximal prime ideal in H^{∗}(G, k), then there is an idempotent module κ_{𝔭} which picks out the layer of the stable module category corresponding to 𝔭, and which was used by Benson, Carlson and Rickard [9] in their development of varieties for infinitely generated kGmodules. Our main theorem states that the Tate cohomology Ĥ^{∗}(G, κ_{𝔭}) is a shift of the injective hull of H^{∗}(G, k)/𝔭 as a graded H^{∗}(G, k)-module. Since κ_{𝔭} can be constructed using a version of the stable Koszul complex, this can be viewed as a statement of localized Gorenstein duality in modular representation theory. Various consequences of this theorem are given, including the statement that the stable endomorphism ring of the module κ_{𝔭} is the 𝔭-completion of cohomology H^{∗}(G, k)𝔭^{∧}, and the statement that κ_{𝔭} is a pure injective kG-module.

In the course of proving the theorem, we further develop the framework introduced by Dwyer, Greenlees and Iyengar [18] for translating between the unbounded derived categories D(kG) and D(C^{∗}(BG; k)). We also construct a functor Ψ: D(kG) → StMod(kG) to the full stable module category, which extends the usual functor D^{b}(kG) → stmod(kG) and which preserves Tate cohomology. The main theorem is formulated and proved in D(C^{∗}(BG; k)), and then translated to D(kG) and finally to StMod(kG).

The main theorem in D(C^{∗}(BG; k)) can be viewed as stating that a version of Gorenstein duality holds after localizing at a prime ideal in H^{∗}(BG; k). This version of the theorem holds more generally for a compact Lie group satisfying a mild orientation condition. This duality lies behind the local cohomology spectral sequence of Greenlees and Lyubeznik for localizations of H^{∗}(BG; k).

In a companion paper [6], a more recent and shorter proof of the main theorem is given. The more recent proof seems less natural, and does not say anything about localization of the Gorenstein condition for compact Lie groups.

The first author was partly supported by a grant from the NSF. During the beginning of the preparation of this paper, he was a guest of the Mathematical Sciences Research Institute (MSRI) in Berkeley, California, and is grateful for their hospitality. Both authors are grateful to MSRI for providing the opportunity for them to meet.

Contents.

1. Introduction.

2. Statement of results in modular representation theory.

3. Derived categories of rings and 𝕊-algebras.

4. Derived and stable categories of kG-modules.

5. Some objects in D(kG).

6. Endomorphism 𝕊-algebras.

7. Cochains on BG.

8. The stable Koszul complex.

9. Local duality.

10. Duality for localizations of S.

11. Completion, and local duality for C^{∗}(BG; k)_{𝔭}.

12. The main theorems.

1. Introduction

The purpose of this paper is to develop machinery for translating problems from modular representation theory into algebraic topology. As an application, we prove a conjecture of the first author in the modular representation theory of finite groups. The statement of the theorem is given in §2. The proof involves a marriage of ideas from algebraic topology, commutative algebra, and group representation theory. Roughly speaking, we develop a version of Grothendieck’s theory of local duality (see §3.5 of Bruns and Herzog [16]) in the context of 𝕊algebras, a notion from algebraic topology (Elmendorf, Kříž, Mandell and May [19]). This is done in the framework developed by Dwyer, Greenlees and Iyengar [18] for unifying various dualities in algebra and topology, such as Poincaré duality for manifolds, Gorenstein duality in commutative algebra, and Benson– Carlson duality for cohomology rings of finite groups. It is the last of these which gives the application to modular representation theory.

The proof given in this paper was the first proof we found for the conjecture in modular representation theory. An outline for this proof was developed when the authors met at MSRI in the Spring of 2003. In November 2005, while this paper was in the publication process, the first author found a much shorter proof without going through algebraic topology. This proof appears in a companion paper to this one [6]. We should mention, though, that not only was the shorter proof inspired by this one, but also it seems in many ways less natural. Furthermore, the shorter proof does not generalize to compact Lie groups (see Section 11).

We feel that the general framework is at least as important as the specific theorem proved in this paper. The background required for understanding this framework may seem daunting. For this reason, we have included more background material than would normally be included in a research paper.

The framework consists of the following diagram of categories and functors.

\xymatrix{Mod(kG) \ar[r] \ar[dr] & D(kG) \ar[r]_{Hom_{kG}(k,−)} \ar[d]_{Ψ} & D(End_{kG}(P (k))) \ar[l]_{−⊗_{R}k} \ar[r]^{≃} & D(C^{∗}(BG; k)) \ar[l] \\  & StMod(kG)} (1.1)

From the point of view of the modular representation theorist, the motivation for the framework may be explained as follows. Usually, the objects of study in modular representation theory are the module category Mod(kG), the stable module category StMod(kG), the derived category of kG-modules, D(kG), and their “small” versions mod(kG), stmod(kG) and D^{b}(kG). The idea is that if P (k) is a projective resolution of k as a kG-module, then R = End_{kG}(P (k)) is a differential graded algebra (DGA) which is equivalent to the cochains on the classifying space C^{∗}(BG; k), and whose cohomology is H^{∗}(G, k). Equivalent DGAs have equivalent derived categories, and in fact R will denote a suitable form of C^{∗}(BG; k). The derived category of R is closely related to that of kG. Although R is not commutative as a differential graded algebra, it is commutative “up to all higher homotopies.” In the language of homotopy theory, it is E_{∞}. This is almost as good as commutativity, and allows many of the constructions of commutative algebra to go through with suitable modification. The details of the modification are what involves us in the theory of algebras over the sphere spectrum 𝕊. Just as a ℤ-algebra is the same as a ring in conventional algebra, an 𝕊-algebra is the same as a ring spectrum in the sense of [19] or [30].

DGAs give examples of 𝕊-algebras (Shipley [38]). But when the term “commutative” is used for an 𝕊-algebra, it corresponds to E_{∞} commutativity for the DGA. There are two main uses made of this kind of commutativity in our work. The first is that if M and N are two modules over an 𝕊-algebra R, then M ⊗_{R} N is again an R-module with all the expected properties. This will be essential for understanding stable Koszul complexes. The second is the ability to localize with respect to a multiplicative subset of the homotopy. We shall use this to localize with respect to a prime ideal in H^{∗}(G, k).

It should be stressed at this point that it is not good enough to deal with spectra in the old-fashioned sense (see for example Adams [1] or May [32]). This is because the tensor product is only commutative and associative up to homotopy, so that we can only define rings and modules up to homotopy; the category of modules up to homotopy is not triangulated, and we cannot form tensor products over a ring up to homotopy. What is necessary is to work in a category of spectra with a smash product which is strictly commutative and associative. The details of the construction of such a category of spectra have been worked out in the 1990s by various groups of people such as Elmendorf, Kříž, Mandell and May [19], and Hovey, Shipley and Smith [30]. Each version has its advantages and disadvantages. For definiteness we work with the version of [19], because the homotopy theory rings and modules, of tensor products, localization, and so on, has been worked out in detail for this version. Furthermore, it turns out that rings in this category are the same as E_{∞}-ring spectra in the classical sense.

The reader who does not have a strong background in algebraic topology can think of DGAs whenever we mention 𝕊-algebras, and think of homology whenever we mention homotopy. However, constructions based on commutativity will then have to be taken on trust, because there is no easy way to perform the required constructions entirely within the world of DGAs. Algebraists may want to look at the article [25], which gives a description of spectra and 𝕊-algebras intended for commutative algebraists.

Dwyer, Greenlees and Iyengar [18] have developed a definition of what it means for an 𝕊-algebra to be Gorenstein. They show that if R is a commutative 𝕊-algebra satisfying certain mild extra hypotheses, one may construct a stable Koszul complex for the computation of local cohomology. For a commutative Gorenstein 𝕊-algebra this gives rise to a spectral sequence whose E_{2} page is the local cohomology of π_{∗}R and converging to the local cohomology of R. The 𝕊-algebra R = End_{kG}(P (k)) is an example of a commutative Gorenstein 𝕊algebra satisfying these extra hypotheses, and the resulting spectral sequence is Benson–Carlson duality [7] in the form developed by Greenlees [24].

Greenlees and Lyubeznik [26] have shown how to apply Grothendieck’s dual localization to obtain a version of of this spectral sequence for the localization of R at a prime ideal in π_{∗}R. We shall need to perform this dual localization at the level of 𝕊-algebras. The construction of Greenlees and Lyubeznik in our context is merely an algebraic shadow of our construction (see Corollary 12.3).

An essential ingredient in our setup is the following. A finite group (or more generally a compact Lie group) G may be embedded in some special unitary group SU(n). The cochains S = C^{∗}(BSU(n); k) form an example of a regular commutative 𝕊-algebra whose homotopy is a polynomial ring in the Chern classes c_{2}, …, c_{n}. This allows us to give a very explicit description of Grothendieck’s local duality for a localization of R = C^{∗}(BG; k) over the corresponding localization of S (Theorem 11.5).

If 𝔭 is a homogeneous nonmaximal prime ideal in π_{∗}R = H^{∗}(BG; k), let k(𝔭) be the field of fractions of the quotient H^{∗}(BG; k)/𝔭. What we would like to do is build an 𝕊-algebra with homotopy k(𝔭) and a map from the localization R_{𝔭} to this, and then claim that this map is Gorenstein in the sense of [18]. Unfortunately, it is not always possible even to produce a R_{𝔭}-module with this homotopy, let alone an algebra and a homomorphism. This makes the formulation of Gorenstein duality for R_{𝔭} tricky, so our main theorem is not formulated in terms of the definition of Gorenstein in [18], but rather as the equivalence

Σ^{d−s}Cell_{K_{𝔭}}(R_{𝔭}) ≃ I_{𝔭}  (1.2)

of Theorem 12.1, which is the duality statement that would have followed from the Gorenstein property. Here, d is the dimension of the prime 𝔭, s is the dimension of the compact Lie group G, K_{𝔭} is the Koszul complex, which stands in place for the residue field that does not exist, and I_{𝔭} is built as a representing object for Matlis duality on homotopy. There is a mild orientability condition, which is satisfied if G is finite or if G is connected. The equivalence (1.2) is the statement which is fed through the diagram (1.1) of categories and functors to give the results in modular representation theory described in the next section.

We deduce the equivalence (1.2) from the following statement of local duality over the corresponding localization S_{𝔮} of S = C^{∗}(BSU(n); k):

Hom_{R_{𝔭}}(Cell_{K_{𝔭}}(R_{𝔭}), I_{𝔭}) ≃ Σ^{−(n^{2}−d−1)}Hom_{S_{𝔮}}(R_{𝔭}, Ŝ_{𝔮}),

see Theorem 11.5. This should be compared with the statement of Grothendieck’s local duality given for example in Theorem 3.5.8 of Bruns and Herzog [16].

2. Statement of results in modular representation theory

We begin by describing the setup. Let G be a finite group and let k be a field of characteristic p. We write Mod(kG) for the category of kG-modules and homomorphisms, and mod(kG) for the full subcategory of finitely generated kG-modules.

We say that a map M → N of kG-modules is projective if it factors through a projective kG-module. The projective maps from M to N form a linear subspace PHom_{kG}(M, N) of Hom_{kG}(M, N), and the quotient is written

\underline{Hom}_{kG}(M, N) = Hom_{kG}(M, N)/PHom_{kG}(M, N).

The stable module category StMod(kG) has the same objects as Mod(kG), but the arrows from M to N are given by \underline{Hom}_{kG}(M, N). We write stmod(kG) for the full subcategory of finitely generated kG-modules. The fact that projective and injective kG-modules coincide (see for example Faith and Walker [21]) can be used to show that StMod(kG) and stmod(kG) are triangulated categories (see for example Theorem I.2.6 of Happel [29] or Section 5 of Carlson [17]).

We write ΩM for the kernel of a surjective map from a projective module onto M, and Ω^{−1}M for the cokernel of an injective map from M into an injective kGmodule. These operations are not well defined on Mod(kG), but they are well defined and functorial inverse equivalences on StMod(kG), and Ω^{−1} gives the translation for the triangulated structure on StMod(kG).

The homomorphisms in a triangulated category form a ℤ-graded object. In the case of StMod(kG), this is the Tate cohomology (see for example Section 6 of Carlson [17])

Ex̂t_{kG}^{n}(M, N) = \underline{Hom}_{kG}(M, Ω^{−n}N) ≅ \underline{Hom}_{kG}(Ω^{n}M, N)

which can be computed as follows. A Tate resolution P̂(M) is defined to be a doubly infinite exact sequence of projective kG-modules

⋯ → P̂_{2} → P̂_{1} → P̂_{0} → P̂_{−1} → P̂_{−2} → ⋯

with the property that the image Z_{−1} of P̂_{0} → P̂_{−1} is isomorphic to M. The cohomology of the complex Hom_{kG}(P̂(M), N) is equal to Ex̂t_{kG}^{∗}(M, N). We write Ĥ^{∗}(G, M) for Ex̂t_{kG}^{∗}(k, M). Note that Ex̂t_{kG}^{∗}(M, N) can also be computed using a Tate resolution of N, but there is a shift in degree; namely, it is the cohomology of the complex Hom_{kG}(M, ΣP̂(N)). To summarize,

Ex̂t_{kG}^{j}(M, N) ≅ H^{j} Hom_{kG}(P̂(M), N) ≅ H^{j} Hom_{kG}(M, ΣP̂(N)).  (2.1)

Since the nonnegative part of a Tate resolution is a projective resolution, it follows that there is a natural map Ext_{kG}^{∗}(M, N) → Ex̂t_{kG}^{∗}(M, N) which is an isomorphism in positive degrees, an epimorphism in degree zero, and zero in negative degrees.

Let Tate(kG) be the category of Tate resolutions, whose objects are the exact chain complexes of projective kG-modules and whose arrows are homotopy classes of maps. This is a triangulated category, in which the distinguished triangles are isomorphic to those coming from short exact sequences of complexes.

Lemma 2.2. Two maps of Tate resolutions f, g: P → Q are homotopic, i.e., equal in Tate(kG), if and only if f_{−1} − g_{−1} : Z_{−1}(P) → Z_{−1}(Q) factors through a projective module.

Proof. This is a diagram chase, using the fact that the modules P_{i} and Q_{i} are both projective and injective. □

Applying the lemma to the identity map and the zero map of a Tate resolution, it follows that an object in Tate(kG) is isomorphic to the zero object if and only if it is a Tate resolution of a projective module; in other words, the kernel of each of the maps is a projective module.

Theorem 2.3. We have an equivalence of triangulated categories

\xymatrix{StMod(kG) \ar[r]_{−⊗_{k}P̂(k)} & Tate(kG) \ar[l]_{Z_{−1}}}

given as follows. If P is an object in Tate(kG) then Z_{−1}(P) consists of the cycles of degree minus one. If M is an object in StMod(kG), then the functor − ⊗_{k} P̂(k) sends it to the Tate resolution M ⊗_{k} P̂(k) of M.

Proof. Lemma 2.2 shows that the functor Z_{−1} is well defined and reflects isomorphisms. The existence of Tate resolutions shows that Z_{−1} is essentially surjective. A map of kG-modules lifts to a map of Tate resolutions, so Z_{−1} is full. It follows that Z_{−1} is an equivalence of categories. Since Z_{−1}(M ⊗_{k} P̂(k)) ≅ M, it follows that − ⊗_{k} P̂(k) is an inverse equivalence. □

We write H^{∗}(G, k) for the cohomology ring of G with coefficients in k. For the algebraist this is Ext_{kG}^{∗}(k, k), and for the topologist it is the cohomology H^{∗}(BG; k) of the classifying space BG.

The cohomology ring H^{∗}(G, k) is a finitely generated graded commutative ring (Evens [20], Venkov [39]). Its maximal ideal spectrum V_{G} was studied in great detail by Quillen [33, 34]. We recall from Benson, Carlson and Rickard [9] that the thick subcategories of stmod(kG) which are tensor-ideal (in the sense of being closed under tensor product with an arbitrary finitely generated module; this condition is automatic if G is a p-group) are classified by collections 𝒱 of closed homogeneous irreducible subvarieties of V_{G} which satisfy V ∈ 𝒱, W ⊆ V ⇒ W ∈ 𝒱. Corresponding to such a thick subcategory, there is a triangle of Rickard idempotent modules

E_{𝒱} → k → F_{𝒱}

where E_{𝒱} is a filtered colimit of modules M with V_{G}(M) ∈ 𝒱, and

\underline{Hom}_{kG}(M, F_{𝒱}) = 0

for all such modules M.

This means that stmod(kG) is stratified by layers corresponding to the homogeneous nonmaximal prime ideals in H^{∗}(G, k). If 𝔭 is such a prime ideal, then there is an idempotent module κ_{𝔭} picking out the corresponding stratum. This module is written κ_{V} in [8], where V is the closed homogeneous irreducible subvariety V_{G} corresponding to 𝔭. It is usually an infinitely generated module, and it is constructed as a tensor product of two Rickard idempotent modules [36], one to include only the subvarieties in V, and the other to exclude the proper subvarieties of V. In more detail, if ζ ∈ H^{∗}(G, k) then F_{ζ} is the homotopy colimit in StMod(kG) of the sequence

k ⟶^{ζ} Ω^{−n}k ⟶^{ζ} Ω^{−2n}k ⟶^{ζ} ⋯

and we complete k → F_{ζ} to a triangle

E_{ζ} → k → F_{ζ}

in StMod(kG). This is the Rickard triangle for the thick subcategory of modules satisfying V_{G}(M) ⊆ V_{G}⟨ζ⟩, the hypersurface corresponding to ζ.

If ζ_{1}, …, ζ_{s} ∈ H^{∗}(G, k) are homogeneous elements forming a sequence of parameters for H^{∗}(G, k)_{𝔭} then

E_{V} = E_{ζ_{1}} ⊗_{k} ⋯ ⊗_{k} E_{ζ_{s}}

is the Rickard idempotent corresponding to the set of subvarieties of V. We write

E(𝔭) → k → F (𝔭)

for the Rickard triangle corresponding to the thick subcategory of stmod(kG) consisting of modules M such that V_{G}(M) ⊉ V. Then

κ_{𝔭} = E_{V} ⊗_{k} F (𝔭) = E_{ζ_{1}} ⊗_{k} ⋯ ⊗_{k} E_{ζ_{s}} ⊗_{k} F (𝔭).

This was not quite the original definition of κ_{𝔭}, but it is equivalent.

Part of the point of the paper [8] was to construct a theory of varieties for infinitely generated modules. The definition given there is that 𝒱_{G}(M) is the set of closed homogeneous irreducible subvarieties of V_{G} such that if κ_{𝔭} is the corresponding kappa module then M ⊗_{k} κ_{𝔭} is not projective.

One way to formulate our main theorem is the following statement, which is proved in Section 12.

Theorem 2.4. The Tate cohomology Ĥ^{∗}(G, κ_{𝔭}) of the kappa module is isomorphic to the injective hull I_{𝔭} of H^{∗}(G, k)/𝔭 in the category of graded H^{∗}(G, k)- modules, shifted by the Krull dimension d of H^{∗}(G, k)/𝔭:

Ĥ^{∗}(G, κ_{𝔭}) ≅ I_{𝔭}[d].  (2.5)

There is a canonical construction described in Benson and Krause [13] for producing a kG-module T (I) whose Tate cohomology Ĥ^{∗}(G, T (I)) is isomorphic to a given injective H^{∗}(G, k)-module I. It is constructed as a representing object for the functor taking a kG-module M to Hom_{H^{∗}(G,k)}(Ĥ^{∗}(G, M), I), and it is uniquely determined up to isomorphism in StMod(kG) by this property. In Benson [5], it was conjectured that

T (I_{𝔭}) ≅ Ω^{−d}κ_{𝔭}

in StMod(kG). It was also shown there that this isomorphism follows from the isomorphism (2.5), and that the consequences of this are given by the following theorem, which can be deduced from Theorem 2.4 using [5].

Theorem 2.6. Let G be a finite group and k a field of characteristic p. Suppose that 𝔭 is a homogeneous nonmaximal prime ideal of dimension d in H^{∗}(G, k). Then the following hold.

(i) The Tate cohomology of κ_{𝔭}, Ĥ^{∗}(G, κ_{𝔭}) is isomorphic in the category of graded H^{∗}(G, k)-modules to the injective module I_{𝔭}[d].

(ii) The modules T (I_{𝔭}) and Ω^{−d}κ_{𝔭} are stably isomorphic.

(iii) The module κ_{𝔭} is a pure injective kG-module. In other words, the inclusion into the double dual κ_{𝔭} ↪ κ_{𝔭}^{∗∗} is a split monomorphism.

(iv) There are no phantom maps into κ_{𝔭} (for the definition of phantom maps and relationship with pure injectivity see [10, 11, 23]).

(v) The ring Ex̂t_{kG}^{∗}(κ_{𝔭}, κ_{𝔭}) is isomorphic to the completion H^{∗}(G, k)𝔭^{∧} of the localized cohomology Ĥ^{∗}(G, k)_{𝔭} with respect to the maximal ideal 𝔭_{𝔭}.

(vi) The formula T (I_{𝔭}) ⊗_{k} T (I_{𝔭}) ≅ Ω^{−d}T (I_{𝔭}) holds in the stable module category. □

3. Derived categories of rings and 𝕊-algebras

As mentioned in the introduction, we shall be working in the language of spectra in the sense of Elmendorf, Kříž, Mandell and May [19]. This is a rigidification of the notion of spectrum (May [32]), in which the smash product is strictly commutative and associative, not just up to homotopy. We could equally well work in another good category of spectra, such as that of Hovey, Shipley and Smith [30]. With such a smash product we may define ring spectra, and modules over them. Commutative ring spectra turn out to be the same as E_{∞} ring spectra in the classical sense of [32]. In particular the sphere spectrum 𝕊 is the initial example of a ring spectrum, all spectra are 𝕊-modules and ring spectra are 𝕊-algebras.

If R is an 𝕊-algebra and M is an R-module then π_{∗}M is a π_{∗}R-module. A map M → N of R-modules is said to be a quasiisomorphism or a weak equivalence if the induced map π_{∗}M → π_{∗}N is an isomorphism. The derived category D(R) is obtained from the category of R-modules and homomorphisms by adjoining formal inverses for the quasiisomorphisms. If M is an R-module, we write Σ^{i}M (i ∈ ℤ) for the ith suspension of M, so that π_{n+i}Σ^{i}M ≅ π_{n}M. The translation functor in D(R) is Σ, so that the triangles take the form

X → Y → Z → ΣX.

We write R^{op} for the opposite 𝕊-algebra of R, so that D(R^{op}) is the derived category of right R-modules.

If R is a ring in the classical sense then we also write R for the corresponding Eilenberg–Mac Lane spectrum, which is an 𝕊-algebra. But we need to be careful about terminology for R-modules and R-algebras. Regarding R as an 𝕊-algebra, an R-module is essentially the same thing as a chain complex of R-modules in the classical sense. So D(R) is the usual unbounded derived category in this case. Similarly, an R-algebra is essentially the same as a DGA over R in the classical sense, so for example a ℤ-algebra is just a DGA. This is made more precise in Shipley [38], where it is shown that there is a functor from associative DGAs to 𝕊-algebras compatible with tensor products, with various other good properties. In particular the derived category of modules over a DGA is equivalent to the derived category of modules over the corresponding 𝕊-algebra.

When it comes to commutative objects, the situation is more complicated. If R is commutative as a ring, then it is also commutative as an 𝕊-algebra. But as mentioned in the introduction, a commutative 𝕊-algebra is E_{∞} as a ring spectrum in the sense of May [32], which means that it is commutative “up to all higher homotopies.” So the comparison does not behave well for commutativity, because a commutative R-algebra is not necessarily commutative as a DGA. For example, the conventional cochain complex of a space X, when viewed as a DGA, is not commutative, but there is a commutative 𝕊-algebra which plays the corresponding role. We write C^{∗}(X; k) for the function spectrum of maps from X into the commutative 𝕊-algebra associated to k; this is a commutative 𝕊-algebra with

π_{i}C^{∗}(X; k) ≅ H^{−i}(X; k),

so that it is a suitable strictly commutative substitute for the usual cochain complex.

The ith homotopy of an R-module or an R-algebra is the ith homology of the corresponding differential graded object. This means that our differential always decreases degree. So for example for the DGA described in the introduction R = End_{kG}(P (k)), π_{∗}R is the group cohomology, which is concentrated in negative degrees under the usual convention π_{−j}R ≅ H^{j}(G, k).

If R is a commutative 𝕊-algebra and M and N are left R-modules, then so is

the left derived tensor product in D(R), sometimes written M ⊗^{𝕃}_{R} N, but which for notational brevity we write as M ⊗_{R} N; there can be no confusion, as the ordinary tensor product does not make sense on D(R). This gives the category of left R-modules a symmetric monoidal structure, and is one of the main uses made of commutativity in this paper. In particular, the construction of the stable Koszul complex (see Theorem 8.6) and the associated local cohomology spectral sequence depend on the left derived tensor product, viewed in this way.

Given two objects M and N in D(R), we write Hom_{R}(M, N) for the spectrum of homomorphisms from M to N. We also write End_{R}(M) for the 𝕊-algebra Hom_{R}(M, M). In the case where R is a DGA, Hom_{R}(M, N) can be thought of as the chain complex of homomorphisms from M to a semiinjective resolution of N, or from a semiprojective resolution of M to N, and is sometimes written ℝHom_{R}(M, N) in this context, because it is the total right derived Hom functor. In particular, if R is a ring concentrated in degree zero, then

π_{i}Hom_{R}(M, N) = Ext_{R}^{−i}(M, N).

Thus the 𝕊-algebra R described in the introduction, which we have been writing as End_{kG}(P (k)), we shall from now on write as End_{kG}(k). This may seem confusing at first, but it is not hard to get used to it.

4. Derived and stable categories of kG-modules

If G is a finite group and k is a field, it is well known that the stable category stmod(kG) of finitely generated kG-modules is a quotient of the derived category D^{b}(kG) of bounded complexes of finitely generated kG-modules by the perfect complexes. A perfect complex is one which is quasiisomorphic to a bounded complex of finitely generated projective modules. We write ψ: D^{b}(kG) → stmod(kG) for the quotient functor. Roughly speaking, the functor ψ is applied by resolving beyond the homology of the complex and then dimension shifting the kernel back to degree zero. See Rickard [35] for details. We shall extend this to a functor Ψ from the full derived category D(kG) to the stable category StMod(kG) of all kG-modules. The construction we give is a simplification due to Rickard of our original construction, and is closely related to constructions described in Krause [31].

The reason why we want to use the full derived category and not the bounded derived category is that the objects we wish to discuss, namely the analogs in D(kG) of the modules T (I) and κ_{𝔭} in StMod(kG), are constructed using Brown representability. For this purpose, it is essential to have infinite direct sums available, and so the bounded version does not work.

We begin with a discussion of semiinjective resolutions of complexes of modules over a ring R, following Avramov, Foxby and Halperin [2]. The discussion works more generally for DG modules over a DG algebra, but we do not need this extra generality in this section.

Let R be a ring. A complex I of R-modules is said to be homotopically injective if Hom_{R}(−, I) preserves quasiisomorphisms. A complex I is said to be semiinjective if it transforms injective quasiisomorphisms into surjective quasiisomorphisms. It is proved in [2] that I is semiinjective if and only if it is homotopically injective and the modules in I are injective R-modules.

A semiinjective resolution of a complex C is a quasiisomorphism C → I with I semiinjective. A semiinjective resolution is strict if C → I is an injective map of complexes. The existence of strict semiinjective resolutions is proved in [2].

A map of complexes extends to a map of semiinjective resolutions, unique up to homotopy. A quasiisomorphism between semiinjective complexes is a homotopy equivalence. If follows that the homotopy category Ho Inj (R) of semiinjective complexes is equivalent to the derived category D(R).

We define a functor

Φ: Ho Inj (kG) → Tate(kG)

as follows. Given a semiinjective complex I, tensor with a Tate resolution P̂(k) of k, and define

Φ(I) = I ⊗_{k} P̂(k).

Composing, we obtain a functor from D(kG) to StMod(kG) as follows.

\xymatrix{D(kG) \ar[r]^{≃} & Ho Inj (kG) \ar[l] \ar[r]^{Φ} & Tate(kG) \ar[r]^{≃}_{Z_{−1}} & StMod(kG) \ar[l]_{−⊗_{k}P̂(k)}}

Here, the last equivalence is the one described in Theorem 2.3. We write

Ψ: D(kG) → StMod(kG)

for the composite.

Lemma 4.1. Let C be a finite length exact complex of kG-modules. Then C ⊗_{k} P̂(k) is a Tate resolution of a projective module, and hence isomorphic to the zero object of Tate(kG).

Proof. Since P̂(k) is exact, so is C ⊗_{k} P̂(k), so it is a Tate resolution of the module M = Z_{−1}(C ⊗_{k} P̂(k)). So

E_{0}^{∗∗} = Hom_{kG}(C ⊗_{k} P̂(k), M)

is a double complex with the property that cohomology of its total complex is Ex̂t_{kG}^{∗}(M, M). The columns C ⊗_{k} P̂_{i} of the double complex C ⊗_{k} P̂(k) are split exact, so in the spectral sequence of the double complex E_{0}^{∗∗}, the E_{1} page is zero. It follows that Ex̂t_{kG}^{∗}(M, M) = 0. In particular, the identity element of Ex̂t_{kG}^{0}(M, M) is equal to zero, so the identity endomorphism of M factors through a projective module. So M is a direct summand of a projective module, and hence is projective. □

Lemma 4.2. Let C be an exact complex of kG-modules satisfying C_{i} = 0 for i ≫ 0. Then C ⊗_{k} P̂(k) is a Tate resolution of a projective module, and hence isomorphic to the zero object of Tate(kG).

Proof. Let C^{(n)} denote the truncation with cokernel,

⋯ → 0 → ⋯ → C_{n+1} ⟶^{δ_{n+1}} C_{n} → Coker δ_{n+1} → 0.

Then C^{(n)} is an exact complex of finite length, so we can apply Lemma 4.1 to C^{(n)} ⊗_{k} P̂(k). Now C ⊗_{k} P̂(k) is the union as n → −∞ of the subcomplexes C^{(n)} ⊗_{k} P̂(k), each of which is exact, and so C ⊗_{k} P̂(k) is exact. Moreover, Z_{−1}(C ⊗_{k} P̂(k)) is a filtered colimit of projective modules, and hence flat. But flat kG-modules are projective by a theorem of Bass [3]. □

Lemma 4.3. Let f: C → D be a quasiisomorphism of complexes of kG-modules which are bounded above. Then the induced map

C ⊗_{k} P̂(k) → D ⊗_{k} P̂(k)

is an isomorphism in Tate(kG).

Proof. Let E be the mapping cone of f. Then the triangle of chain complexes

C ⟶^{f} D → E → ΣC

gives a triangle

C ⊗_{k} P̂(k) → D ⊗_{k} P̂(k) → E ⊗_{k} P̂(k) → Σ(C ⊗_{k} P̂(k)).

Since f: C → D is a quasiisomorphism of bounded above complexes, E is exact and bounded above. So by Lemma 4.2, E ⊗_{k} P̂(k) is isomorphic to the zero object in Tate(kG). It follows that C ⊗_{k} P̂(k) → D ⊗_{k} P̂(k) is an isomorphism in Tate(kG). □

Theorem 4.4. The following diagram of categories and functors commutes up to natural isomorphism.

\xymatrix{Mod(kG) \ar[r]^{i} \ar[dr] & D(kG) \ar[d]^{Ψ} \\  & StMod(kG)}

Here, the diagonal arrow is the obvious functor from Mod(kG) to StMod(kG). The functor i sends a module M to the complex consisting of M in degree zero and 0 in nonzero degrees.

Proof. Let M is a kG-module, regarded as a chain complex concentrated in degree zero. Let I be an injective resolution of M, so that the map of complexes M → I is a quasiisomorphism. Then M and I are bounded above, so by Lemma 4.3, the induced map M ⊗_{k} P̂(k) → I ⊗_{k} P̂(k) is an isomorphism in Tate(kG). Applying Z_{0}, Theorem 2.3 shows that

M = Z_{−1}(M ⊗_{k} P̂(k)) → Z_{−1}(I ⊗_{k} P̂(k)) = Ψ(M)

is an isomorphism in StMod(kG). □

Remarks. The functor Ψ preserves triangles, but does not preserve products or coproducts. There can be no such functor preserving coproducts. If there were, then the direct sum of the truncations

P (k)^{(n)} : 0 → P_{n} → ⋯ → P_{1} → P_{0} → 0

of a projective resolution P (k) of the trivial kG-module k would be taken to zero, because projective modules are zero in the stable category. Then the homotopy colimit triangle

⨁_{n=0}^{∞} P (k)^{(n)} → ⨁_{n=0}^{∞} P (k)^{(n)} → P (k)

would show that P (k) is taken to zero. But P (k) is equivalent to k, which is supposed to go to k, not to zero.

Tate cohomology. We define ordinary and Tate cohomology for the derived category as follows. If X is an object in D(kG), we define

H^{j}(G, X) = π_{−j} Hom_{kG}(k, X),
Ĥ^{j}(G, X) = lim_{i^{⟶}} π_{−j} Hom_{kG}(Ω^{−i}k, Σ^{i}X).

The maps in this directed system come from the natural maps

Ω^{−i−1}k → ΣΩ^{−i}k

as in the following diagram

[figure: ⋯; ≃; 0; 0; Ω^{−i−1}k; 0; ⋯; ⋯; 0; Ω^{−i}k; P̂_{−i}; 0; ⋯; ⋯; 0; Ω^{−i}k; 0; 0; ⋯]

If Y is another object in D(kG), we define

Ext_{kG}^{j}(Y, X) = H^{j}(G, Hom_{k}(Y, X)) ≅ π_{−j} Hom_{kG}(Y, X)

and

Ex̂t_{kG}^{j}(Y, X) = Ĥ^{j}(G, Hom_{k}(Y, X)).

If Y is a finitely generated kG-module M, we can rewrite this as

Ex̂t_{kG}^{j}(M, X) ≅ lim_{i^{⟶}} π_{−j} Hom_{kG}(Ω^{−i}k ⊗_{k} M, Σ^{i}X).

Theorem 4.5. If X is in D(kG) then Ĥ^{j}(G, X) ≅ Ĥ^{j}(G, Ψ(X)). Furthermore, for any finitely generated kG-module M we have

Ex̂t_{kG}^{j}(M, X) ≅ Ex̂t_{kG}^{j}(M, Ψ(X)).

Proof. We shall prove the second statement, as the first is just the special case M = k.

Write P̂(k) as a filtered colimit as i → ∞ and j → −∞ of complexes U_{i,j}, where U_{i,j} is the truncated complex

⋯ → 0 → P̂_{i−1} → P̂_{i−2} → … → P̂_{j} → Ω^{j}k → 0 → ⋯

if i > j, zero if i < j, and Σ^{i−1}Ω^{i}k if i = j. Tensor product commutes with filtered colimit, so letting I be a semiinjective resolution of X, we have

I ⊗_{k} P̂(k) ≅ lim_{i,j^{⟶}} I ⊗_{k} U_{i,j}.

Now the map of complexes Σ^{i−1}Ω^{i}k → U_{i,j} given by the following diagram

\xymatrix{⋯ \ar[r] & 0 \ar[r] \ar[d] & Ω^{i}k \ar[r] \ar[d] & 0 \ar[r] \ar[d] & ⋯ \ar[r] & 0 \ar[r] \ar[d] & 0 \ar[r] \ar[d] & ⋯ \\ ⋯ \ar[r] & 0 \ar[r] & P̂_{i−1} \ar[r] & P̂_{i−2} \ar[r] & ⋯ \ar[r] & Ω^{j}k \ar[r] & 0 \ar[r] & ⋯}

is a quasiisomorphism as long as i ≥ j. Dualizing and shifting, the map

Σ^{−1}U_{i,j}^{∗} → Σ^{−i}Ω^{−i}k

is also a quasiisomorphism. Since I is semiinjective, it is homotopically injective. So using equation (2.1) for Tate cohomology, it follows that

Ex̂t_{kG}^{m}(M, Ψ(X)) ≅ π_{−m} Hom_{kG}(M, Σ(I ⊗ P̂(k)))

≅ π_{−m} Hom_{kG}(M, lim_{i,j^{⟶}} I ⊗_{k} ΣU_{i,j})

≅ lim_{i,j^{⟶}} π_{−m} Hom_{kG}(M, I ⊗_{k} ΣU_{i,j})

(since M is finitely generated)

≅ lim_{i,j^{⟶}} π_{−m} Hom_{kG}(M ⊗_{k} Σ^{−1}U_{i,j}^{∗}, I)

≅ lim_{i^{⟶}} π_{−m} Hom_{kG}(M ⊗_{k} Σ^{−i}Ω^{−i}k, I)

≅ lim_{i^{⟶}} π_{−m} Hom_{kG}(Ω^{−i}M, Σ^{i}I) = Ex̂t_{kG}^{m}(M, X).  □

From this theorem, we can understand what objects are sent to zero by Ψ. We begin with a lemma.

Lemma 4.6. If N is a kG-module, then the following are equivalent. (i) N is zero in StMod(kG).

(ii) Ex̂t_{kG}^{∗}(M, N) = 0 for all kG-modules M.

(iii) Ex̂t_{kG}^{∗}(S, N) = 0 for all simple kG-modules S.

(iv) Ext_{kG}^{1}(S, N) = 0 for all simple kG-modules S.

Proof. The implications (i) ⇒ (ii) ⇒ (iii) ⇒ (iv) are clear. If (iv) holds, then the beginning of a minimal injective resolution of N

0 → I_{0} → I_{1} → …

has Hom_{kG}(S, I_{1}) = 0 for all simple modules S, and so I_{1} = 0. It follows that N is injective, hence projective, and zero in StMod(kG). □

Theorem 4.7. The objects in D(kG) that are sent to zero under the functor Ψ are precisely the complexes X satisfying Ex̂t_{kG}^{∗}(S, X) = 0 for all simple kGmodules S.

Proof. This follows from Theorem 4.5, together with the equivalence of (i) and (iii) in Lemma 4.6. □

Theorem 4.8. If there exists an element ζ ∈ H^{n}(G, k) with n > 0, such that multiplication by ζ induces an equivalence X → Σ^{n}X, then the natural map

H^{∗}(G, X) → Ĥ^{∗}(G, X)

is an isomorphism. Furthermore, for any finitely generated kG-module M, the map

Ext_{kG}^{∗}(M, X) → Ex̂t_{kG}^{∗}(M, X)

is an isomorphism.

Proof. We shall prove the second statement, as the first is just the special case M = k. Choose a map ζ̂: k → Ω^{−n}k representing ζ. We have a commutative diagram

\xymatrix{Hom_{kG}(Ω^{−n}k ⊗_{k} M, X) \ar[r]^{ζ̂^{∗}} \ar[d]^{ζ}_{≅} & Hom_{kG}(k ⊗_{k} M, X) \ar[dl] \ar[d]^{ζ}_{≅} \\ Hom_{kG}(Ω^{−n}k ⊗_{k} M, Σ^{n}X) \ar[r]_{ζ̂^{∗}} & Hom_{kG}(k ⊗_{k} M, Σ^{n}X).}

The diagonal map, which is the map in the directed system defining Tate cohomology, has a left inverse and a right inverse, so it is an isomorphism. Applying this with powers of ζ shows that the map from ordinary to Tate cohomology is an isomorphism. □

5. Some objects in D(kG)

Rickard [36] showed how to use techniques of Bousfield localization to produce idempotent modules in StMod(kG) whose importance has become apparent in [4, 8, 9]. We would like to construct corresponding objects in D(kG), but there is a problem. Writing D^{b}(kG) for the bounded derived category of finitely generated kG-modules, it is unfortunately not true that for X in D^{b}(kG), morphisms from X commute with coproducts, or with homotopy colimits. This obstructs us from using Rickard’s construction to produce idempotent objects in D(kG) analogous to the modules κ_{𝔭} using thick subcategories of D^{b}(kG). Instead, we produce these objects by using localization with respect to the prime ideal 𝔭 in H^{∗}(G, k) followed by a stable Koszul construction.

If 𝔭 is a nonmaximal homogeneous prime ideal in H^{∗}(G, k), we can form the category of fractions of D(kG) by inverting the maps ζ: X → Σ^{n}X induced by elements ζ ∈ H^{n}(G, k) not in 𝔭 (for all n > 0) on objects X in D(kG). We write D(kG)_{𝔭} for the category of fractions, and j_{𝔭}: D(kG) → D(kG)_{𝔭} for the corresponding functor. Similarly, we can form the category of fractions StMod(kG)_{𝔭} of StMod(kG) by inverting the maps ζ: M → Ω^{−n}M in StMod(kG) induced by elements ζ ∈ H^{n}(G, k) not in 𝔭.

The image of a projective kG-module P under the composite

Mod(kG) → D(kG) ⟶^{j_{𝔭}} D(kG)_{𝔭}

is zero because every element of positive degree in H^{∗}(G, k) acts as zero on P. So we obtain a well defined functor

StMod(kG) → D(kG)_{𝔭}.

This functor takes ζ: M → Ω^{−n}M to ζ: M → Σ^{n}M, which has an inverse, and hence defines a functor i_{𝔭}: StMod(kG)_{𝔭} → D(kG)_{𝔭}. It follows from Theorems 4.5 and 4.8 that we obtain an equivalence of categories

\xymatrix{D(kG)_{𝔭} \ar[r]_{Ψ_{𝔭}} & StMod(kG)_{𝔭}. \ar[l]_{i_{𝔭}}}

Since i_{𝔭} preserves coproducts, so does Ψ_{𝔭}, in spite of the fact that Ψ does not.

If Y is an object in D(kG)_{𝔭}, consider the functor from D(kG) to vector spaces which sends an object X to the k-vector space of morphisms in D(kG)_{𝔭} from X to Y. This functor takes coproducts to products, and triangles to long exact sequences, and so by the Brown representability theorem there is an object r_{𝔭}(Y) representing this functor. By the Yoneda lemma, r_{𝔭} can be made into a functor from D(kG)_{𝔭} to D(kG) which is right adjoint to the functor j_{𝔭}:

\xymatrix{D(kG) \ar[r]_{j_{𝔭}} & D(kG)_{𝔭}. \ar[l]_{r_{𝔭}}}

The composite j_{𝔭}r_{𝔭}: D(kG)_{𝔭} → D(kG)_{𝔭} is equivalent to the identity. So the functor Ψ preserves coproducts and homotopy colimits on the image of r_{𝔭}.

It follows from Theorems 4.5 and 4.8 that if M is any finitely generated kGmodule then

Ex̂t_{kG}^{∗}(M, Ψ_{𝔭}r_{𝔭}(X)) ≅ Ext_{kG}^{∗}(M, r_{𝔭}(X)) ≅ Ext_{kG}^{∗}(M, X)_{𝔭}.

Lemma 5.1. The module r_{𝔭}(k) is stably isomorphic to the Rickard idempotent module F (𝔭) corresponding to the collection of subvarieties W satisfying W ⊉ V, where V ⊆ V_{G}(k) is the variety determined by 𝔭.

Proof. If M is any finitely generated kG-module then we have a commutative square

\xymatrix{Ex̂t_{kG}^{∗}(M, k) \ar[r] \ar[d] & Ex̂t_{kG}^{∗}(M, F (𝔭)) \ar[d] \\ Ex̂t_{kG}^{∗}(M, r_{𝔭}(k)) \ar[r] & Ex̂t_{kG}^{∗}(M, r_{𝔭}(k) ⊗_{k} F (𝔭))}

in which the three terms other than the upper left hand one are isomorphic to Ex̂t_{kG}^{∗}(M, k)_{𝔭}, and the right hand vertical arrow and the lower horizontal arrow are isomorphisms. It follows using Lemma 4.6 that the maps r_{𝔭}(k) → r_{𝔭}(k) ⊗_{k} F (𝔭) ← F (𝔭) are stable isomorphisms. □

Now if ζ ∈ H^{n}(G, k) and X is an object in D(kG), then we can form a sequence of modules and maps

X ⟶^{ζ} Σ^{n}X ⟶^{ζ} Σ^{2n}X → ⋯

We write X[ζ^{−1}] for the homotopy colimit of this sequence. There is a natural map X → X[ζ^{−1}]. Complete this map to a triangle in D(kG) and write K_{∞}(X; ζ) for the third vertex of the triangle:

K_{∞}(X; ζ) → X → X[ζ^{−1}].

Repeating this construction for a sequence ζ_{1}, …, ζ_{s}, we obtain an object

K_{∞}(X; ζ_{1}, …, ζ_{s}) = K_{∞}(… K_{∞}(X; ζ_{1}); …; ζ_{s})

which can be viewed as the total complex of a stable Koszul complex

0 → X → ⨁_{i} X[ζ_{i}^{−1}] → ⨁_{i<j} X[(ζ_{i}ζ_{j})^{−1}] → ⋯ → X[(ζ_{1} … ζ_{s})^{−1}] → 0.

This is the representation theoretic analogue of the construction in commutative algebra described for example in Section 3.5 of Bruns and Herzog [16].

Let ζ_{1}, …, ζ_{s} be elements of H^{∗}(G, k) of degrees n_{1}, …, n_{s}, forming a homogeneous sequence of parameters for H^{∗}(G, k)_{𝔭}. Then we define

K_{𝔭} = K_{∞}(r_{𝔭}(k); ζ_{1}, …, ζ_{s})

as an object of D(kG).

Lemma 5.2. We have Ψ(K_{𝔭}) ≅ κ_{𝔭}.

Proof. Since Ψ preserves homotopy colimits on the image of r_{𝔭}, it follows from Lemma 5.1 that for ζ ∈ H^{∗}(G, k), the image under Ψ of the triangle

K_{∞}(r_{𝔭}(k); ζ) → r_{𝔭}(k) → r_{𝔭}(k)[ζ^{−1}]

is the triangle

E_{ζ} ⊗_{k} F (𝔭) → F (𝔭) → F_{ζ} ⊗_{k} F (𝔭).

Repeating this argument for ζ_{1}, …, ζ_{s}, it follows that the image under Ψ of K_{𝔭} = K_{∞}(r_{𝔭}(k); ζ_{1}, …, ζ_{s}) is κ_{𝔭} = E_{ζ_{1}} ⊗_{k} ⋯ ⊗_{k} E_{ζ_{s}} ⊗_{k} F (𝔭). □

Next, we construct a functor T from injective modules over H^{∗}(G, k) to D(kG). This is analogous to the functor T from injective modules over Ĥ^{∗}(G, k) to StMod(kG) constructed in [13], but easier. Given an injective H^{∗}(G, k)- module I, consider the functor from D(kG) to vector spaces which takes a complex X to Hom_{H^{∗}(G,k)}(H^{∗}(G, X), I). This functor takes triangles to long exact sequences, and direct sums to direct products. So by Brown representability, there is an object T (I), unique up to canonical isomorphism in D(kG), such that

Hom_{H^{∗}(G,k)}(H^{∗}(G, X), I) ≅ π_{0}Hom_{kG}(X, T (I)).

The assignment extends by Yoneda’s lemma to a functor

T : Inj H^{∗}(G, k) → D(kG).

The analog of lemma 3.2 of [13] is as follows.

Lemma 5.3. There is an isomorphism of H^{∗}(G, k)-modules

Hom_{H^{∗}(G,k)}^{∗}(H^{∗}(G, X), I) ≅ π_{∗}Hom_{kG}(X, T (I)).

Proof. For i ∈ ℤ, we have

Hom_{H^{∗}(G,k)}^{i}(H^{∗}(G, X), I) ≅ Hom_{H^{∗}(G,k)}(H^{∗−i}(G, X), I)
≅ Hom_{H^{∗}(G,k)}(H^{∗}(G, Σ^{−i}X), I)

≅ π_{0}Hom_{kG}(Σ^{−i}X, T (I)) ≅ π_{−i}Hom_{kG}(X, T (I)).

This isomorphism is functorial in X, so it gives rise to an isomorphism of Hom_{kG}(X, X)-modules. The action of H^{∗}(G, k) on both sides factors through the natural map H^{∗}(G, k) → Hom_{kG}(X, X). □

Lemma 5.4. There is an isomorphism of H^{∗}(G, k)-modules H^{∗}(G, T (I)) ≅ I.

Proof. This follows from Lemma 5.3 by putting X = k, and the observation that π_{∗}Hom_{kG}(k, −) is the same as H^{∗}(G, −). □

Theorem 5.5. Let 𝔭 be a nonmaximal homogeneous prime ideal in H^{∗}(G, k), and let I_{𝔭} be the injective hull of H^{∗}(G, k)/𝔭 as an H^{∗}(G, k)-module. Then

Ψ(T (I_{𝔭})) ≅ T (I_{𝔭}).

Proof. Since 𝔭 is nonmaximal, we can choose an element ζ ∈ H^{n}(G, k), n > 0, such that ζ ∉ 𝔭. Then multiplication by ζ is an isomorphism on I_{𝔭}, and so ζ induces an equivalence T (I_{𝔭}) → Σ^{n}T (I_{𝔭}). So if M is a finitely generated kG-module, we have

Ex̂t_{kG}^{j}(M, Ψ(T (I_{𝔭}))) ≅ Ex̂t_{kG}^{j}(M, T (I_{𝔭}))  by Theorem 4.5
≅ Ext_{kG}^{j}(M, T (I_{𝔭}))  by Theorem 4.8
≅ Hom_{H^{∗}(G,k)}^{j}(H^{∗}(G, M), I_{𝔭})  by Lemma 5.3
≅ Hom_{H^{∗}(G,k)}^{j}(Ĥ^{∗}(G, M), I_{𝔭})  by Prop. 2.9 of [13].

The case j = 0 of this isomorphism is the defining property of T (I_{𝔭}). □

6. Endomorphism 𝕊-algebras

Let R be an 𝕊-algebra, and let k be an object of D(R). A map U → V in D(R) is a k-equivalence if the induced map [Σ^{i}k, U]_{R} → [Σ^{i}k, V]_{R} is an isomorphism for all i ∈ ℤ. An object U is k-trivial if 0 → U is a k-equivalence; in other words if [Σ^{i}k, U]_{R} = 0 for all i ∈ ℤ. An R-module M is said to be k-torsion if any k-equivalence U → V induces an isomorphism [M, U]_{R} → [M, V]_{R}.

The localizing subcategory generated by k is defined to be the smallest subcategory of D(R) satisfying the following conditions.

(i) It is closed under shifts and isomorphism,

(ii) it contains k (and hence all Σ^{i}k),

(iii) it is closed under coproducts,

(iv) it is closed under retracts (i.e., taking direct summands), and

(v) if two vertices of a triangle are in then so is the third.

If instead of (iii), we only demand that it is closed under finite coproducts, then this is the definition of the thick subcategory generated by k.

We write Cell(D(R), k) for the localizing subcategory generated by k, and we say that an R-module M is k-cellular or built from k if M is in Cell(D(R), k). It is clear that k-cellular implies k-torsion.

Since a homotopy colimit can be constructed as the total object of a simplicial object formed from the colimit system (Bousfield and Kan [15]), it has a countable filtration with subquotients which are coproducts of suspensions of the terms in the colimit system. It follows that if a diagram consists of k-cellular objects, then its homotopy colimit is k-cellular.

We say that a map N → M is a k-cellular approximation to M if N is kcellular and the map is a k-equivalence. Using the model structure on categories of modules over R one may show that k-cellular approximations always exist (Farjoun [22]). They are essentially unique and we write Cell_{k}(M) → M for the k-cellular approximation. Applying this to k-torsion modules, we see that k-torsion implies k-cellular, so that these two notions are equivalent.

An R-module M is said to be finitely built from k if it is in the thick subcategory generated by k. We say that k is small if Hom_{R}(k, −) commutes with arbitrary coproducts. It is easy to see that this holds if k is finitely built from R. The converse is also true, but we shall not use this fact. We say that k is proxy-small if there exists another R-module K, called a Koszul complex, with the properties that K is finitely built from R, K is finitely built from k, and k is built from K. The point of this definition is the following proposition.

Proposition 6.1. Suppose that k is a proxy-small R-module, with Koszul complex K, and set E = End_{R}(k). Then the adjunction

\xymatrix{D(R) \ar[r]_{Hom_{R}(k,−)} & D(E^{op}) \ar[l]_{−⊗_{E}k}}

has the following properties.

(i) The functor −⊗_{E} k preserves triangles, direct sums and homotopy colimits. The functor Hom_{R}(k, −) preserves triangles; if k is small then it also preserves direct sums and homotopy colimits.

(ii) For every object X in D(E^{op}), X ⊗_{E} k is k-cellular.

(iii) These functors pass down to an equivalence of categories

\xymatrix{Cell(D(R), k) \ar[r]_{Hom_{R}(k,−)} & Cell(D(E^{op}), J). \ar[l]_{−⊗_{E}k}}

where J = Hom_{R}(k, K).

Note that if K = k then the right hand side is the whole of D(E^{op}).

(iv) The counit of the adjunction

Hom_{R}(k, M) ⊗_{E} k → M

is a k-cellular approximation of M.

Proof. This follows from Lemma 4.4 and Theorem 4.10 of [18]. □

A homomorphism of 𝕊-algebras R → k is said to be regular if k is finitely built from R (i.e., small) as an R-module, and coregular if R is finitely built from k. For example, if R and k are rings concentrated in degree zero, then regularity is equivalent to saying that k has a finite resolution by finitely generated projective R-modules (R → k has type FP). If R is local and k is the residue field, Serre’s theorem shows this agrees with normal usage.

A homomorphism of 𝕊-algebras R → k is said to be proxy-regular if k is proxy-small as an R-module. For example, if R is a commutative Noetherian local ring with residue field k then letting K be the usual Koszul complex shows that R → k is proxy-regular. In this case, the total right derived functor ℝΓ_{𝔪} of torsion with respect to the maximal ideal 𝔪 gives the k-cellular approximation of M so that ℝΓ_{𝔪}M = Cell_{k}(M). By part (iii) of Proposition 6.1, whenever R → k is proxy-regular the k-cellular approximation may be constructed explicitly

Cell_{k}M = Hom_{R}(k, M) ⊗_{E} k.  (6.2)

Definition 6.3. The double centralizer of R is defined to be the ring R̂ = End_{E}(k) (see §4.16 of [18]). Left multiplication induces a homomorphism R → R̂, and the pair (R, k) is said to be dc-complete if this homomorphism is an equivalence.

7. Cochains on BG

This section summarizes and expands on some facts about the 𝕊-algebra C^{∗}(BG; k) proved in [18], where G is a compact Lie group.

If X is a pointed connected space and k is a commutative ring of coefficients then the Rothenberg–Steenrod construction gives an equivalence

End_{C_{∗}(ΩX;k)}(k) ≃ C^{∗}(X; k).  (7.1)

If X is simply connected, or if k is a field of characteristic p and π_{1}(X) is a finite p-group, then the Eilenberg–Moore construction gives an equivalence

End_{C^{∗}(X;k)}(k) ≃ C_{∗}(ΩX; k).  (7.2)

So in this case, both C^{∗}(X; k) and C_{∗}(ΩX; k) are dc-complete (Definition 6.3). See §4.22 and §5.7 of [18] for further details of the equivalences (7.1) and (7.2).

The following theorem describes the situation for the Bousfield–Kan 𝔽_{p}-com-pletion BGp^{∧} of the classifying space BG of a compact Lie group G.

Theorem 7.3. Let G be a compact Lie group and let k be a field of characteristic

p. Set R = C^{∗}(BG; k) ≃ C^{∗}(BGp^{∧}; k), E_{R} = End_{R}(k) and I_{R} = C_{∗}(BG; k).

Then the following hold.

(a) BG is 𝔽_{p}-good.

(b) π_{1}(BGp^{∧}) is a finite p-group (in fact it is π_{0}(G)/O^{p}π_{0}(G)).

(c) E_{R} ≃ C_{∗}(Ω(BG_{p}^{∧}); k).

(d) π_{0}(E_{R}) is a finite p-group.

(e) End_{E_{R}}(k) ≃ R, and both (R, k) and (E_{R}, k) are dc-complete (Definition 6.3).

(f) (Uniqueness of Matlis lift; cf. §6 of [18]) There is a unique right E_{R}-module structure on k extending the left k-module structure.

(g) Giving k the right E_{R}-module structure of (f), we have I_{R} ≃ k ⊗_{E_{R}} k.

(h) π_{∗}(I_{R}) ≅ I_{π_{∗}R}(k), the injective hull of the field k as a module for π_{∗}R = H^{∗}(BG; k).

(i) If π_{0}(G) is a finite p-group then E_{R} ≃ C_{∗}(Gp^{∧}; k).

Suppose that the conjugation action of G on Lie(G) preserves orientation,^{1} or that k has characteristic two, and let s be the dimension of G as a manifold. Then the following additional properties hold.

1This condition is satisfied if G is finite or connected.

(j) Both R and E_{R} are Gorenstein with shift s, in the sense of §8 of [18].

(k) Cell_{k}(R) ≃ Σ^{s}I_{R}.

(l) End_{R}(I_{R}) ≃ End_{R}(Cell_{k}(R)) ≃ R.

(m) Hom_{R}(Cell_{k}(R), I_{R}) ≃ Σ^{−s}R.

Proof. Part (a) follows from Proposition VII.5.1 of Bousfield and Kan [15]. Part (b) follows from Proposition VII.4.3 of [15]. Part (c) follows from (b) and (7.2). Part (d) follows from (b) and (c). Part (e) follows from (c) and (7.1). For part (f), it follows from (d) that the only homomorphism π_{0}(E_{R}) → k^{×} is the trivial one. So there is a unique π_{0}(E_{R})-module structure on k. Uniqueness of right E_{R}-module structure on k now follows from Proposition 3.9 of [18]. Part (g) follows from §5.5 of [18]. Part (h) follows from the duality between H_{∗}(BG; k) and H^{∗}(BG; k). For part (i), we apply the nilpotent fibration lemma II.4.8 of Bousfield–Kan [15] to the fibration EG → BG to deduce that Ω(BGp^{∧}) ≃ Gp^{∧}, and then use (c).

Part (j) is proved in §10.3 of [18], but without computing the shift. But in particular, this holds with G = SU(n) and s = n^{2} − 1. Choose an embedding of G as a closed subgroup of SU(n) for some value of n. Since Lie(SU(n)) ≅ Lie(G) ⊕ T (SU(n)/G) as representations of G, and SU(n) is connected, the conjugation action of G on Lie(G) preserves orientation if and only if the manifold SU(n)/G is orientable. So by Proposition 8.12 of [18], under this condition C^{∗}(M; k) is Gorenstein with shift equal to minus its dimension, namely −(n^{2} − s − 1). The same holds even if M is not orientable, provided that k has characteristic two. Applying Proposition 8.10 of [18] to the homomorphism C^{∗}(BG; k) → C^{∗}(BSU(n); k), and noting that the proof of that proposition allows us to compute the shift, we see that C^{∗}(BG; k) is Gorenstein with shift equal to (n^{2} − 1) − (n^{2} − s − 1) = s. Proposition 8.13 of [18] now shows that E_{R} is Gorenstein with the same shift.

Part (k) follows from parts (g) and (j), using Remark 8.2 of [18]. The first equivalence of part (l) follows from (k). To prove the second, by (j) we have Hom_{R}(k, R) ≃ k. So by Proposition 6.1 (iv), Cell_{k}(R) ≃ k ⊗_{E_{R}} k. So we have

End_{R}(Cell_{k}(R)) ≃ Hom_{R}(k ⊗_{E_{R}} k, R)
≃ Hom_{E_{R}}(k, Hom_{R}(k, R)) ≃ End_{E_{R}}(k) = R̂.

By (e), R̂ = R.

Part (m) follows from (k) and (l). □

Remarks. If G is a finite group then C_{∗}(G; k) ≅ kG. So if G is a finite p-group then parts (e) and (i) of Theorem 7.3 show that kG is dc-complete.

On the other hand, if G is an arbitrary finite group then by (7.1),

End_{kG}(k) ≃ C^{∗}(BG; k) ≃ C^{∗}(BGp^{∧}; k).

So for finite groups which are not p-groups, part (c) of Theorem 7.3 shows that kG is not dc-complete, as

C_{∗}(Ω(BGp^{∧}); k) ≄ kG.

For an arbitrary finite group G, it is shown in §5.9 of [18] that we can form a Koszul complex K for kG by choosing an embedding G → SU(n) and taking K = C_{∗}(SU(n); k). Since R = C^{∗}(BG; k) is commutative, we have R ≃ R^{op}. It follows from Proposition 6.1 that there is an equivalence of categories

\xymatrix{Cell(D(kG), k) \ar[r]_{Hom_{kG}(k,−)} & Cell(D(R), J), \ar[l]_{−⊗_{R}k}}  (7.4)

where J = Hom_{kG}(k, K).

Next, for an arbitrary compact Lie group G and field k, we set R = C^{∗}(BG; k) as above, and we discuss the analog in D(R) of the modules T (I_{𝔭}). If 𝔭 is a homogeneous prime ideal in π_{∗}R ≅ H^{∗}(BG; k), and I_{𝔭} is the injective hull of the π_{∗}R-module π_{∗}R/𝔭, then we write I_{𝔭} for the representing object in StMod(kG) for the functor Hom_{π_{∗}R}(π_{∗}(−), I_{𝔭}), whose existence is guaranteed by the Brown representability theorem. So we have functorial isomorphisms for M in D(R)

Hom_{π_{∗}R}(π_{∗}M, I_{𝔭}) ≅ π_{∗}Hom_{R}(M, I_{𝔭}).  (7.5)

Lemma 7.6. (i) We have π_{∗}I_{𝔭} ≅ I_{𝔭}.

(ii) If X is any R-module satisfying π_{∗}X ≅ I_{𝔭} then X ≃ I_{𝔭}.

(iii) Let 𝔪 be the maximal ideal of positive degree elements in H^{∗}(BG; k). Then I_{𝔪} ≃ I_{R} = C_{∗}(BG; k).

Proof. (i) This follows by putting M = R in (7.5).

(ii) Let ϕ: X → I_{𝔭} be a map corresponding to an isomorphism between π_{∗}X and I_{𝔭} under (7.5). Then ϕ_{∗}: π_{∗}X → I_{𝔭} is this isomorphism, and so ϕ is an equivalence.

(iii) This follows from (ii), together with part (g) of Theorem 7.3. □

Theorem 7.7. The objects T (I_{𝔭}) in D(kG) and I_{𝔭} in D(R) correspond under the equivalence (7.4).

Proof. It follows from Lemma 5.4 that

π_{∗}Hom_{kG}(k, T (I_{𝔭})) ≅ H^{∗}(G, T (I_{𝔭})) ≅ I_{𝔭}.

Now apply Lemma 7.6. □

8. The stable Koszul complex

In this section, we describe the stable Koszul complex, which under suitable circumstances gives another model for cellular approximation. The arguments in this section are mostly taken from §9 of Dwyer, Greenlees and Iyengar [18].

Let R be a commutative 𝕊-algebra (recall that commutative in this context really means E_{∞}; for the purpose of this section, the point of this hypothesis is that we can tensor together R-modules and have the result be an R-module).

Given an element x ∈ π_{−i}R, we can look at the map given by right multiplication by x,

R ≃ R ∧ 𝕊 ⟶^{1∧x} R ∧ Σ^{i}R ⟶^{mult} Σ^{i}R,

which we write as R ⟶^{x} Σ^{i}R. Similarly, if M is an R-module we define M ⟶^{x} Σ^{i}M

to be the map R ⊗_{R} M ⟶^{x⊗1} Σ^{i}R ⊗_{R} M.

We define K_{m}(x) by completing R ⟶^{x^{m}} Σ^{im}R to a triangle in D(R)

K_{m}(x) → R ⟶^{x^{m}} Σ^{im}R → ΣK_{m}(x),  (8.1)

and set K(x) = K_{1}(x). If we want to emphasize the ring R, we write K_{m}(R; x), etc. The octahedral axiom gives triangles

Σ^{im−1}K(x) → K_{m}(x) → K_{m+1}(x) → Σ^{im}K(x).  (8.2)

Lemma 8.3. The module K_{m}(x) is self-dual, up to a shift:

Hom_{R}(K_{m}(x), R) ≃ Σ^{1−im}K_{m}(x).

Proof. Apply Hom_{R}(−, Σ^{im}R) to the triangle (8.1). □

Next, for an R-module M we define

M[x^{−1}] = \underrightarrow{Hocolim}(M ⟶^{x} Σ^{i}M ⟶^{x} Σ^{2i}M ⟶^{x} ⋯).

The following lemma is clear from the definition.

Lemma 8.4. We have (i) M[x^{−1}] ≃ R[x^{−1}] ⊗_{R} M. (ii) π_{∗}(M[x^{−1}]) ≅ (π_{∗}M)[x^{−1}]. (iii) If π_{∗}(M)[x^{−1}] = 0 then M[x^{−1}] ≃ ∗.

We define K_{∞}(x) by completing the obvious map R → R[x^{−1}] to a triangle

K_{∞}(x) → R → R[x^{−1}] → ΣK_{∞}(x).

Lemma 8.5. (i) K_{∞}(x) = \underrightarrow{Hocolim}(K_{1}(x) → K_{2}(x) → K_{3}(x) → ⋯).

(ii) The map K_{∞}(x) → R induces an equivalence K(x) ⊗_{R} K_{∞}(x) ≃ K(x).

Proof. (i) This follows from the diagram

\xymatrix{K_{1}(x) \ar[r] \ar[d] & R \ar[r]^{x} \ar[d]^{1} & Σ^{i}R \ar[d]^{Σ^{i}x} \\ K_{2}(x) \ar[r] \ar[d] & R \ar[r]^{x^{2}} \ar[d]^{1} & Σ^{2i}R \ar[d]^{Σ^{2i}x} \\ ⋮ & ⋮ & ⋮}

in which the rows are triangles, and exactness of Hocolim.

(ii) This follows from the diagram

\xymatrix{K(x) ⊗_{R} K_{∞}(x) \ar[r] \ar[d] & K_{∞}(x) \ar[r]^{x} \ar[d] & Σ^{i}K_{∞}(x) \ar[d] \\ K(x) \ar[r] \ar[d] & R \ar[r]^{x} \ar[d] & Σ^{i}R \ar[d] \\ ∗ \ar[r] & R[x^{−1}] \ar[r]^{x}_{≃} & Σ^{i}R[x^{−1}]}

in which the rows and the second and third columns are triangles, and hence so is the first column. □

If x_{1}, …, x_{n} is a finite sequence of elements of π_{∗}R, we set

K = K(x_{1}) ⊗_{R} ⋯ ⊗_{R} K(x_{n})
K_{m} = K_{m}(x_{1}) ⊗_{R} ⋯ ⊗_{R} K_{m}(x_{n})
K_{∞} = K_{∞}(x_{1}) ⊗_{R} ⋯ ⊗_{R} K_{∞}(x_{n}).

The R-module K is called the Koszul complex, and K_{∞} is called the stable Koszul complex. These come with natural maps K → R, K_{m} → R and K_{∞} → R.

Theorem 8.6. The map K_{∞} → R is a K-cellular approximation to R; furthermore, for any R-module M, we have

K_{∞} ⊗_{R} M ≃ Cell_{K}(M).

Proof. An inductive argument using the triangles (8.2) shows that K_{m} is finitely built from K, and hence K_{∞} = \underrightarrow{Hocolim} K_{m} is built from K. Now K_{∞} ⊗_{R} M is built from K_{∞}, and hence from K.

By Lemma 8.5 (ii), the map K_{∞} → R induces an equivalence K ⊗_{R} K_{∞} ≃ K, and hence an equivalence

K ⊗_{R} K_{∞} ⊗_{R} M → K ⊗_{R} M,

so K_{∞} ⊗_{R} M → M is a K-equivalence. □

Corollary 8.7. Suppose that M is an R-module such that (π_{∗}M)[x_{i}^{−1}] = 0 for i = 1, …, n. Then M is K-cellular.

Proof. By Lemma 8.4, we have R[x_{i}^{−1}]⊗_{R}M ≃ ∗. So the map K_{∞}(x_{i})⊗_{R}M → M is an equivalence. Inductively, we see that

K_{∞}(x_{1}) ⊗_{R} ⋯ ⊗_{R} K_{∞}(x_{n}) ⊗_{R} M → M

is an equivalence. By Theorem 8.6, the left hand side is Cell_{K}(M). □

Another consequence of Theorem 8.6 is the following base change theorem.

Theorem 8.8. Let S → R be a map of commutative 𝕊-algebras. If y_{1}, …, y_{n} are elements of π_{∗}S with images x_{1}, …, x_{n} in π_{∗}R, and M is an R-module, then as S-modules we have

(i) K(S) ⊗_{S} R ≃ K(R),

(ii) K_{∞}(S) ⊗_{S} R ≃ K_{∞}(R),

(iii) Cell_{K(S)}(M) ≃ Cell_{K(R)}(M).

Proof. Let y be one of the y_{j}, and let i be its degree. Then we have a commutative diagram

\xymatrix{S ⊗_{S} R \ar[r]^{y⊗1} \ar[d]^{≃} & Σ^{i}S ⊗_{S} R \ar[d]^{≃} \\ R \ar[r]^{x} & Σ^{i}R.}

It follows that K(S, y) ⊗_{S} R ≃ K(R, x). Doing this for each y_{j} and tensoring gives K(S) ⊗_{S} R ≃ K(R). Applying the same argument to the construction of K_{∞} gives K_{∞}(S) ⊗_{S} R ≃ K_{∞}(R). Tensoring over R with M gives

K_{∞}(S) ⊗_{S} M ≃ K_{∞}(R) ⊗_{R} M.

Now apply Theorem 8.6 to obtain Cell_{K(S)}(M) ≃ Cell_{K(R)}(M). □

9. Local duality

In this section, we make use of the fact that a compact Lie group G embeds in a special unitary group SU(n) for some n to prove a version of Grothendieck’s Local Duality Theorem [27, 28] for C^{∗}(BG; k). This enables us to put the dual localization process of Greenlees and Lyubeznik [26] in the context of the theory of Dwyer, Greenlees and Iyengar [18]. We shall use this in the case where G is finite.

Let S = C^{∗}(BSU(n); k). We have

π_{∗}(S) = H^{∗}(BSU(n); k) = k[c_{2}, c_{3}, …, c_{n}]

where c_{i} is the Chern class of degree −2i. Set E_{S} = C_{∗}(SU(n); k) (cf. (7.2)) and I_{S} = C_{∗}(BSU(n); k). By Theorem 7.3 (e) we have End_{S}(k) ≃ E_{S}, End_{E_{S}}(k) ≃ S, and by Theorem 7.3 (h) we have π_{∗}(I_{S}) ≅ I_{π_{∗}S}(k), the injective hull of k as a π_{∗}S-module.

Theorem 9.1 (Local duality for R over S). Suppose that G is a compact Lie group of dimension s, embedded as a closed subgroup of SU(n). Set R = C^{∗}(BG; k) and S = C^{∗}(BSU(n); k). Then

(i) H^{∗}(BG; k) is a finitely generated module over k[c_{2}, …, c_{n}].

(ii) I_{R} ≃ Hom_{S}(R, I_{S}).

(iii) Hom_{R}(Cell_{k}(R), I_{R}) ≃ Σ^{−(n^{2}−1)}Hom_{S}(R, S).

Suppose, furthermore, that the conjugation action of G on Lie(G) preserves orientation, or that k has characteristic two. Then

(iv) R ≃ Σ^{−(n^{2}−s−1)}Hom_{S}(R, S).

Proof. (i) This is a theorem of Venkov [39].

(ii) Using (i) and the definition of I_{S}, we have

π_{∗}Hom_{S}(R, I_{S}) ≅ Hom_{π_{∗}S}(π_{∗}R, I_{π_{∗}S}(k)) ≅ I_{π_{∗}R}(k).

Now use Lemma 7.6.

(iii) The Koszul complex K(S) for S formed using the Chern classes c_{2}, …, c_{n} is equivalent to k. Denoting by ϕ: G → SU(n) the inclusion, the Koszul complex K(R) for R formed using the images of the Chern classes ϕ^{∗}(c_{2}), …, ϕ^{∗}(c_{n}) has the property that π_{∗}K(R) has finite dimension as a vector space over k, and so Cell_{K(R)}(−) is the same as Cell_{k}(−). So it follows from Theorem 8.8 (iii) that for an R-module N, Cell_{k}(N) computed with respect to S agrees with Cell_{k}(N) computed with respect to R.

By Theorem 7.3 (m) and Theorem 8.6, we have

S ≃ Σ^{n^{2}−1}Hom_{S}(Cell_{k}(S), I_{S}) ≃ Σ^{n^{2}−1}Hom_{S}(K_{∞}S, I_{S}).

So

Hom_{S}(R, S) ≃ Σ^{n^{2}−1}Hom_{S}(R, Hom_{S}(K_{∞}S, I_{S}))
≃ Σ^{n^{2}−1}Hom_{S}(K_{∞}S ⊗_{S} R, I_{S})
≃ Σ^{n^{2}−1}Hom_{S}(K_{∞}R, I_{S})  by Theorem 8.8 (ii)
≃ Σ^{n^{2}−1}Hom_{S}(Cell_{k}(R), I_{S})  by Theorem 8.6.
≃ Σ^{n^{2}−1}Hom_{R}(Cell_{k}(R), Hom_{S}(R, I_{S})) by previous paragraph
≃ Σ^{n^{2}−1}Hom_{R}(Cell_{k}(R), I_{R})  by (ii).

(iv) We have

R ≃ Σ^{s}Hom_{R}(Cell_{k}(R), I_{R})  by Theorem 7.3 (m)
≃ Σ^{−(n^{2}−s−1)}Hom_{S}(R, S).  by (iii). □

10. Duality for localizations of S

We begin by discussing localization of 𝕊-algebras. We refer the reader to §V.2 and §VIII.4 of Elmendorf, Kříž, Mandell and May [19] for the general theory.

Theorem 10.1. Let R be a commutative 𝕊-algebra, and let X be a set of elements of π_{∗}R. If A is a commutative R-algebra, then A[X^{−1}] is a commutative R-algebra satisfying

π_{∗}(A[X^{−1}]) ≅ π_{∗}(A)[X^{−1}],

and A → A[X^{−1}] is a map of R-algebras.

Proof. This is proposition V.5.4 of [19]. □

We shall use the above theorem in the case where R = A and X is the set of homogeneous elements outside of a given prime ideal 𝔭 of π_{∗}R.

Theorem 10.2. For any R-module M, the Bousfield localization of M at R[X^{−1}] is given by M ≅ R ⊗_{R} M → R[X^{−1}] ⊗_{R} M.

Proof. This is Proposition VIII.4.1 of [19]. □

Because of the above theorem, we write M[X^{−1}] for R[X^{−1}] ⊗_{R} M. If 𝔭 is a prime ideal in π_{∗}R, in the case where X = π_{∗}R \ 𝔭, we write R_{𝔭} and M_{𝔭} for R[X^{−1}] and M[X^{−1}].

Corollary 10.3. If every element of X acts invertibly on π_{∗}M then the natural map M → M[X^{−1}] is an equivalence. In particular, the action of R on M extends in a unique way to an action of R[X^{−1}]. □

Now let S = C^{∗}(BSU(n); k). Let 𝔮 be a homogeneous prime ideal in π_{∗}S of dimension d; in other words, d is the Krull dimension of the quotient ring π_{∗}S/𝔮. We consider the localization S_{𝔮}, and our first task is to define its 𝕊-module ‘residue field’. We let k(𝔮) denote the conventional residue field of π_{∗}(S_{𝔮}) = (π_{∗}S)_{𝔮}, namely the field of fractions of π_{∗}S/𝔮. The 𝕊-algebra S_{𝔮} is very special in that its homotopy ring is regular. We shall see that this allows us not only to construct an S_{𝔮}-module whose homotopy is the residue field, but also to formulate a version of Gorenstein duality for S_{𝔮}. The latter is expressed in parts (i) and (ii) of Theorem 10.6 below.

Lemma 10.4. Up to equivalence, there is a unique S_{𝔮}-module whose homotopy is isomorphic to k(𝔮) as a π_{∗}(S_{𝔮})-module.

Proof. By a theorem of Serre [37], (π_{∗}S)_{𝔮} = π_{∗}(S_{𝔮}) is a regular ring of dimension n − d − 1, so we can choose a regular sequence of homogeneous elements x_{1}, …, x_{n−d−1} ∈ π_{∗}S such that their images in π_{∗}S_{𝔮} form a minimal generating set for 𝔮_{𝔮}. Now we form the Koszul complex K = K(x_{1}) ⊗_{S} ⋯ ⊗_{S} K(x_{n−d−1}) and note that π_{∗}K_{𝔮} ≅ π_{∗}S_{𝔮}/𝔮_{𝔮} = k(𝔮).

Now if L is another S_{𝔮}-module with π_{∗}(L) = k(𝔮) we may construct a map K_{𝔮} → L which is an isomorphism in homotopy and hence a weak equivalence. Indeed, the triangle (8.1) gives us

Hom_{S_{𝔮}}(K(x)_{𝔮}, L) ≃ L ⊕ Σ^{1−|x|}L

for any x in the maximal ideal of S_{𝔮}, so repeating this with x_{1}, x_{2}, …, x_{n−d−1}, we obtain a splitting of Hom_{S_{𝔮}}(K_{𝔮}, L), and we choose a map from the unit copy of L. □

Henceforth we write K_{𝔮} for the unique S_{𝔮}-module with homotopy k(𝔮). We write E_{𝔮} for End_{S_{𝔮}}(K_{𝔮}), and we write I_{𝔮} for the representing object for the functor Hom_{π_{∗}S}(π_{∗}(−), I_{𝔮}) as described at the end of Section 7, where I_{𝔮} is the injective hull of π_{∗}S/𝔮 as a π_{∗}S-module. By Corollary 10.3 we have π_{∗}I_{𝔮} = I_{𝔮}. So by Lemma 7.6, I_{𝔮} is an S_{𝔮}-module in a unique way. By equation (7.5), for any S_{𝔮}-module M we have

Hom_{π_{∗}S_{𝔮}}(π_{∗}M, I_{𝔮}) ≅ π_{∗}Hom_{S_{𝔮}}(M, I_{𝔮}).  (10.5)

Theorem 10.6. Let S, 𝔮, d, S_{𝔮}, K_{𝔮}, E_{𝔮} and I_{𝔮} be as above.

(i) Cell_{K_{𝔮}}(S_{𝔮}) ≃ Σ^{n^{2}−d−1}I_{𝔮},

(ii) Hom_{S_{𝔮}}(K_{𝔮}, S_{𝔮}) ≃ Σ^{n^{2}−d−1}K_{𝔮},

(iii) End_{S_{𝔮}}(I_{𝔮}) ≃ End_{S_{𝔮}}(Cell_{K_{𝔮}}(S_{𝔮})) ≃ Ŝ_{𝔮},

(iv) Hom_{S_{𝔮}}(Cell_{K_{𝔮}}(S_{𝔮}), I_{𝔮}) ≃ Σ^{−(n^{2}−d−1)}Ŝ_{𝔮}.

Proof. (i) Since π_{∗}S_{𝔮} is a regular ring it is Gorenstein and H_{𝔮}^{∗}(S_{𝔮}) is zero except in degree d, where we have H_{𝔮}^{d}(S_{𝔮}) ≅ Σ^{n^{2}−1}I_{𝔮}. It follows that the stable Koszul complex K_{∞}S_{𝔮} has homotopy Σ^{n^{2}−d−1}I_{𝔮}, so that by Theorem 8.6 and Lemma 7.6 (ii) we have

Cell_{K_{𝔮}}(S_{𝔮}) ≃ K_{∞}S_{𝔮} ≃ Σ^{n^{2}−d−1}I_{𝔮}.

(ii) By (i) we have

Hom_{S_{𝔮}}(K_{𝔮}, S_{𝔮}) ≃ Hom_{S_{𝔮}}(K_{𝔮}, Cell_{K_{𝔮}}(S_{𝔮})) ≃ Hom_{S_{𝔮}}(K_{𝔮}, Σ^{n^{2}−d−1}I_{𝔮}).

By the definition of I_{𝔮}, this has homotopy

Σ^{n^{2}−d−1}Hom_{π_{∗}S_{𝔮}}(k(𝔮), I_{𝔮}) ≅ Σ^{n^{2}−d−1}k(𝔮).

Now apply Lemma 10.4.

(iii) The first equivalence follows from (i). For the second equivalence, we have

End_{S_{𝔮}}(Cell_{K_{𝔮}}(S_{𝔮})) ≃ Hom_{S_{𝔮}}(Cell_{K_{𝔮}}(S_{𝔮}), S_{𝔮})
≃ Hom_{S_{𝔮}}(Hom_{S_{𝔮}}(K_{𝔮}, S_{𝔮}) ⊗_{E_{𝔮}} K_{𝔮}, S_{𝔮})

by Proposition 6.1 (iv)

≃ Hom_{E_{𝔮}}(Hom_{S_{𝔮}}(K_{𝔮}, S_{𝔮}), Hom_{S_{𝔮}}(K_{𝔮}, S_{𝔮}))

≃ End_{E_{𝔮}}(K_{𝔮})  by (ii)
= Ŝ_{𝔮}.

(iv) This follows from (i) and (iii). □

11. Completion, and local duality for C^{∗}(BG; k)_{𝔭}

For this section, we continue to let G be a compact Lie group embedded as a closed subgroup in SU(n), and we set R = C^{∗}(BG; k), S = C^{∗}(BSU(n); k). Write ϕ: G → SU(n) for the inclusion. Suppose further that 𝔭 is a prime ideal of π_{∗}R of dimension d, and set 𝔮 = (ϕ^{∗})^{−1}𝔭, a prime ideal in π_{∗}S. Then the field of fractions k(𝔭) of π_{∗}R_{𝔭}/𝔭_{𝔭} is a finite field extension of k(𝔮).

We are faced with the problem that there is no obvious way to produce an R_{𝔭}-module whose homotopy is k(𝔭). Indeed, it follows from §7 of [14] that for a cyclic group of order three and the minimal prime in cohomology, there is no such module. However, as in the proof of Lemma 10.4, we can use a Koszul complex with respect to a filter-regular sequence of parameters in π_{∗}R_{𝔭} to produce an R_{𝔭}-module K_{𝔭} whose homotopy has finite length as a π_{∗}R_{𝔭}-module. This is good enough for our purposes, because then π_{∗}K_{𝔭} also has finite length as a π_{∗}S_{𝔮}-module via ϕ. So if M is any R_{𝔭}-module regarded as an S_{𝔮}-module via ϕ, then using Theorem 8.8 (iii), we have

Cell_{K_{𝔭}}(M) ≃ Cell_{K_{𝔮}}(M).

Furthermore, if there happens to exist an R_{𝔭}-module k_{𝔭} with homotopy k(𝔭) then K_{𝔭} is built from k_{𝔭} and vice-versa, so that Cell_{k_{𝔭}}(M) ≃ Cell_{K_{𝔭}}(M).

We define E_{𝔭} = End_{R_{𝔭}}(K_{𝔭}). As described near the end of Section 7, we define I_{𝔭} to be the representing object for the functor Hom_{π_{∗}R}(π_{∗}(−), I_{𝔭}). Here, I_{𝔭} is the injective hull of π_{∗}R/𝔭 as a π_{∗}R-module. As in the discussion preceding Theorem 10.6, I_{𝔭} is an R_{𝔭}-module in a unique way, and for any R_{𝔭}-module M we have

Hom_{π_{∗}R_{𝔭}}(π_{∗}M, I_{𝔭}) ≅ π_{∗}Hom_{R_{𝔭}}(M, I_{𝔭}).  (11.1)

Lemma 11.2. We have (i) π_{∗}I_{𝔭} ≅ I_{𝔭}. (ii) I_{𝔭} ≅ Hom_{S_{𝔮}}(R_{𝔭}, I_{𝔮}). (iii) For any R_{𝔭}-module M, Hom_{R_{𝔭}}(M, I_{𝔭}) ≃ Hom_{S_{𝔮}}(M, I_{𝔮}). (iv) If Hom_{R_{𝔭}}(M, I_{𝔭}) ≃ ∗ then M ≃ ∗.

(v) I_{𝔭} is K_{𝔭}-cellular. (vi) The natural map K_{𝔭} → Hom_{R_{𝔭}}(Hom_{R_{𝔭}}(K_{𝔭}, I_{𝔭}), I_{𝔭}) is an equivalence.

Proof. (i) This follows from Lemma 7.6 (i).

(ii) From equation (10.5), we have

π_{∗}Hom_{S_{𝔮}}(R_{𝔭}, I_{𝔮}) ≅ Hom_{π_{∗}S_{𝔮}}(π_{∗}R_{𝔭}, I_{𝔮}) ≅ I_{𝔭}.

Now apply Lemma 7.6 (ii).

(iii) We have

Hom_{R_{𝔭}}(M, I_{𝔭}) ≃ Hom_{R_{𝔭}}(M, Hom_{S_{𝔮}}(R_{𝔭}, I_{𝔮}))
≃ Hom_{S_{𝔮}}(R_{𝔭} ⊗_{R_{𝔭}} M, I_{𝔮})
≃ Hom_{S_{𝔮}}(M, I_{𝔮}).

(iv) If Hom_{R_{𝔭}}(M, I_{𝔭}) ≃ ∗ then from the definition of I_{𝔭} we get

Hom_{π_{∗}R_{𝔭}}(π_{∗}M, I_{𝔭}) = 0.

Since π_{∗}R_{𝔭} is a Noetherian local graded commutative ring, it follows from ordinary commutative algebra that π_{∗}M = 0 and so M ≃ ∗.

(v) This follows from Corollary 8.7 and (i). (vi) This map is an isomorphism in homotopy by applying equation (11.1)

twice, and therefore an equivalence. □

Lemma 11.3. We have End_{R_{𝔭}}(Cell_{K_{𝔭}}(R_{𝔭})) ≃ Hom_{R_{𝔭}}(Cell_{K_{𝔭}}(R_{𝔭}), R_{𝔭}) ≃ R̂_{𝔭}, the dc-completion of R_{𝔭}.

Proof. It follows from the fact that K_{𝔭} is finitely built from R_{𝔭} that

Hom_{R_{𝔭}}(Hom_{R_{𝔭}}(K_{𝔭}, R_{𝔭}), R_{𝔭}) ≃ K_{𝔭}  (11.4)

(this also follows from Lemma 8.3). So we have

End_{R_{𝔭}}(Cell_{K_{𝔭}}(R_{𝔭})) ≃ Hom_{R_{𝔭}}(Cell_{K_{𝔭}}(R_{𝔭}), R_{𝔭})
≃ Hom_{R_{𝔭}}(Hom_{R_{𝔭}}(K_{𝔭}, R_{𝔭}) ⊗_{E_{𝔭}} K_{𝔭}, R_{𝔭})

by Proposition 6.1 (iv)

≃ Hom_{E_{𝔭}}(K_{𝔭}, Hom_{R_{𝔭}}(Hom_{R_{𝔭}}(K_{𝔭}, R_{𝔭}), R_{𝔭}))

≃ Hom_{E_{𝔭}}(K_{𝔭}, K_{𝔭})  by (11.4)

= R̂_{𝔭}.  □

Theorem 11.5 (Local duality for R_{𝔭} over S_{𝔮}). We have

Hom_{R_{𝔭}}(Cell_{K_{𝔭}}(R_{𝔭}), I_{𝔭}) ≃ Σ^{−(n^{2}−d−1)}Hom_{S_{𝔮}}(R_{𝔭}, Ŝ_{𝔮}).

Proof. By Theorem 8.8 (iii), for any R_{𝔭}-module M, we have Cell_{K_{𝔭}}(M) ≃ Cell_{K_{𝔮}}(M). The rest of the proof is essentially the same as the rest of the proof of Theorem 9.1 (iii), but using Theorem 10.6 (iv) and Lemma 11.2 (ii) in place of Theorems 7.3 (m) and 9.1 (ii). □

If M is an S_{𝔮}-module, we define its completion at 𝔮 to be

M 𝔮^{∧} = Hom_{S_{𝔮}}(Cell_{K_{𝔮}}(S_{𝔮}), M).

The dc-completion and the 𝔮-completion of S are equivalent, Ŝ_{𝔮} ≃ (S_{𝔮})𝔮^{∧}, by Theorem 10.6 (iii). So M 𝔮^{∧} is a module for Ŝ_{𝔮}.

We say that M is 𝔮-complete if the natural map Cell_{K_{𝔮}}(S_{𝔮}) → S_{𝔮} induces an equivalence M → M 𝔮^{∧}.

Lemma 11.6. (i) We have (M 𝔮^{∧})𝔮^{∧} ≃ M 𝔮^{∧}.

(ii) An S_{𝔮}-module homomorphism ϕ: M → N between 𝔮-complete modules is induced by a uniquely determined Ŝ_{𝔮}-module homomorphism ϕ̂: M → N.

(iii) We have Hom_{S_{𝔮}}(M, S_{𝔮})𝔮^{∧} ≃ Hom_{S_{𝔮}}(M, Ŝ_{𝔮}).

Proof. (i) We have

Cell_{K_{𝔮}}(S_{𝔮}) ⊗_{S_{𝔮}} Cell_{K_{𝔮}}(S_{𝔮}) ≃ Cell_{K_{𝔮}}(Cell_{K_{𝔮}}(S_{𝔮})) ≃ Cell_{K_{𝔮}}(S_{𝔮}).

Now apply the adjunction between Hom and tensor.

(ii) Define ϕ̂ = Hom_{S_{𝔮}}(Cell_{K_{𝔮}}(S_{𝔮}), ϕ). We get a diagram

\xymatrix{M \ar[r]^{ϕ} \ar[d] & N \ar[d] \\ M 𝔮^{∧} \ar[r]^{ϕ̂} & N 𝔮^{∧}.}

The vertical maps are equivalences since M and N are 𝔮-complete, and ϕ̂ is a homomorphism of modules for (S_{𝔮})𝔮^{∧} ≃ Ŝ_{𝔮}.

(iii) We have

Hom_{S_{𝔮}}(M, S_{𝔮})𝔮^{∧} = Hom_{S_{𝔮}}(Cell_{K_{𝔮}}(S_{𝔮}), Hom_{S_{𝔮}}(M, S_{𝔮}))
≃ Hom_{S_{𝔮}}(M, Hom_{S_{𝔮}}(Cell_{K_{𝔮}}(S_{𝔮}), S_{𝔮}))
≃ Hom_{S_{𝔮}}(M, End_{S_{𝔮}}(Cell_{K_{𝔮}}(S_{𝔮})))

≃ Hom_{S_{𝔮}}(M, Ŝ_{𝔮})  by Theorem 10.6 (iii). □

Similarly, if M is an R_{𝔭}-module, we define its completion at 𝔭 to be

M 𝔭^{∧} = Hom_{R_{𝔭}}(Cell_{K_{𝔭}}(R_{𝔭}), M).  (11.7)

Lemma 11.8. (i) We have (M 𝔭^{∧})𝔭^{∧} ≃ M 𝔭^{∧}.

(ii) An R_{𝔭}-module homomorphism ϕ: M → N between 𝔭-complete modules is induced by a uniquely determined homomorphism ϕ̂: M → N.

(iii) If M is an R_{𝔭}-module, regarded as an S_{𝔮}-module via ϕ, then M 𝔭^{∧} ≃ M 𝔮^{∧}.

Proof. Parts (i) and (ii) are proved in the same way as parts (i) and (ii) of Lemma 11.6. Part (iii) follows from Theorem 8.8 (iii). □

Lemma 11.9. The dc-completion, the completion at 𝔭, and the completion at 𝔮 of R_{𝔭}, and End_{R_{𝔭}}(I_{𝔭}) are all equivalent:

R̂_{𝔭} ≃ (R_{𝔭})𝔭^{∧} ≃ (R_{𝔭})𝔮^{∧} ≃ End_{R_{𝔭}}(I_{𝔭}).

Proof. By (11.7) and Lemma 11.3 we have

(R_{𝔭})𝔭^{∧} = Hom_{R_{𝔭}}(Cell_{K_{𝔭}}(R_{𝔭}), R_{𝔭}) ≃ R̂_{𝔭}.

The second equivalence is a special case of Lemma 11.8 (iii). For the last term, we have

End_{R_{𝔭}}(I_{𝔭}) ≃ Hom_{R_{𝔭}}(Cell_{K_{𝔭}}(I_{𝔭}), I_{𝔭})  by Lemma 11.2 (v)
≃ Hom_{R_{𝔭}}(Hom_{R_{𝔭}}(K_{𝔭}, I_{𝔭}) ⊗_{E_{𝔭}} K_{𝔭}, I_{𝔭}) by Proposition 6.1 (iv)
≃ Hom_{E_{𝔭}}(K_{𝔭}, Hom_{R_{𝔭}}(Hom_{R_{𝔭}}(K_{𝔭}, I_{𝔭}), I_{𝔭}))
≃ Hom_{E_{𝔭}}(K_{𝔭}, K_{𝔭})  by Lemma 11.2 (vi)
= R̂_{𝔭}.  □

Theorem 11.10 (Local duality for R_{𝔭}). There is a natural map of R_{𝔭}-modules

Σ^{d}Hom_{R}(Cell_{k}(R), I_{R})_{𝔭} → Hom_{R_{𝔭}}(Cell_{K_{𝔭}}(R_{𝔭}), I_{𝔭})

which is completion at 𝔭.

Proof. We have

Σ^{d}Hom_{R}(Cell_{k}(R), I_{R})_{𝔭} ≃ Σ^{−(n^{2}−d−1)}Hom_{S}(R, S)_{𝔭}  by Theorem 9.1
≃ Σ^{−(n^{2}−d−1)}Hom_{S_{𝔮}}(R_{𝔭}, S_{𝔮})

completion at 𝔮 ⟶ Σ^{−(n^{2}−d−1)}Hom_{S_{𝔮}}(R_{𝔭}, S_{𝔮})𝔮^{∧}

≃ Σ^{−(n^{2}−d−1)}Hom_{S_{𝔮}}(R_{𝔭}, Ŝ_{𝔮}) by Lemma 11.6 (iii)

≃ Hom_{R_{𝔭}}(Cell_{K_{𝔭}}(R_{𝔭}), I_{𝔭})  by Theorem 11.5.

Finally, by Lemma 11.8 (iii), completion at 𝔮 is equivalent to completion at 𝔭. □

12. The main theorems

The following theorem is a version of Gorenstein duality for a localization of the cochains on a compact Lie group. We shall apply this theorem in the case where G is finite, in which case the orientation condition is trivially satisfied.

Theorem 12.1. Let G be a compact Lie group of dimension s, and let R = C^{∗}(BG; k). Suppose that the conjugation action of G on Lie(G) preserves orientation, or that k has characteristic two. Let 𝔭 be a prime ideal in π_{∗}R = H^{∗}(BG; k) of dimension d. Then we have an equivalence of R_{𝔭}-modules

Σ^{d−s}Cell_{K_{𝔭}}(R_{𝔭}) ≃ I_{𝔭}.

Proof. Under the given hypotheses, by Theorem 7.3 (m) the left hand side in Theorem 11.10 is equivalent to Σ^{d−s}R_{𝔭}. By Lemma 11.9, the 𝔭-completion of this is equivalent to

Σ^{d−s}R̂_{𝔭} ≃ Σ^{d−s}End_{R_{𝔭}}(I_{𝔭}).

So the natural map of Theorem 11.10 gives us an equivalence of R_{𝔭}-modules

Σ^{d−s}Hom_{R_{𝔭}}(I_{𝔭}, I_{𝔭}) ≃ Hom_{R_{𝔭}}(Cell_{K_{𝔭}}(R_{𝔭}), I_{𝔭}).  (12.2)

Both sides are 𝔭-complete, so by Lemma 11.8 (ii), we can view it as an equivalence of R̂_{𝔭}-modules. Choose a map of R_{𝔭}-modules

α: Σ^{d−s}Cell_{K_{𝔭}}(R_{𝔭}) → I_{𝔭}

corresponding to the identity element in π_{d−s} of the left hand side of (12.2), and complete to a triangle

Σ^{d−s}Cell_{K_{𝔭}}(R_{𝔭}) ⟶^{α} I_{𝔭} → X.

Since the left hand side of (12.2) is equivalent to Σ^{d−s}R̂_{𝔭} as an R̂_{𝔭}-module, it follows that composition with α induces a map homotopic to the equivalence (12.2). It follows that

Hom_{R_{𝔭}}(X, I_{𝔭}) ≃ ∗.

and so X ≃ ∗ by Lemma 11.2 (iv). So α is an equivalence. This proves the last statement of the theorem. □

The following corollary can also be obtained by applying the dual localization process from Lemma 7.1 of Greenlees and Lyubeznik [26] to the spectral sequence given in Theorem 1.1 of Benson and Greenlees [12].

Corollary 12.3. Under the conditions of Theorem 12.1, there is a local cohomology spectral sequence

H_{𝔭}^{i,j} H^{∗}(BG; k)_{𝔭} ⇒ (I_{𝔭})_{s−d−i−j}.

where I_{𝔭} is the injective hull of H^{∗}(BG; k)/𝔭 as an H^{∗}(BG; k)-module.

Proof. This can be deduced from Theorem 12.1 by filtering the stable Koszul complex as in the proof of Proposition 9.3 of Dwyer, Greenlees and Iyengar [18]. □

Theorem 12.4. Let G be a finite group and k be a field of characteristic p. Under the equivalence of categories (7.4), Cell_{K_{𝔭}}(R_{𝔭}) in D(R) corresponds to the object K_{𝔭} in D(kG) described in Section 5.

Proof. Let V be the subvariety of V_{G} corresponding to 𝔭, and let F (𝔭) be the Rickard idempotent module introduced in Section 2. Since the map

H^{∗}(G, k) → H^{∗}(G, F (𝔭))

is localization at 𝔭, it follows that the lower horizontal map and the right hand vertical map in the diagram

\xymatrix{R = \ar[d] & Hom_{kG}(k, k) \ar[r] & Hom_{kG}(k, F (𝔭)) \ar[d]^{≃} \\ R_{𝔭} \ar[rr]^{≃} &  & R_{𝔭} ⊗_{R} Hom_{kG}(k, F (𝔭))}

are isomorphisms in homotopy, and therefore equivalences. This shows that the image under (7.4) of k → F (𝔭), namely the upper horizontal map in the above diagram, is R → R_{𝔭}.

If ζ ∈ H^{n}(G, k), for any object X in D(kG) the image of

X ⟶^{ζ} Σ^{n}X ⟶^{ζ} Σ^{2n}X → ⋯

under (7.4) is

Hom_{kG}(k, X) ⟶^{ζ} Σ^{n}Hom_{kG}(k, X) ⟶^{ζ} Σ^{2n}Hom_{kG}(k, X) → ⋯

and so the image of X → X ⊗_{k} F (ζ) is

Hom_{kG}(k, X) → Hom_{kG}(k, X)[ζ^{−1}].

Now let ζ_{1}, …, ζ_{s} be homogeneous elements of H^{∗}(G, k) with

V = V_{G}⟨ζ_{1}⟩ ∩ ⋯ ∩ V_{G}⟨ζ_{s}⟩

so that ζ_{1}, …, ζ_{s} form a homogeneous system of parameters for H^{∗}(G, k)_{𝔭}. For 1 ≤ i ≤ s we define

E_{i} = E_{ζ_{1}} ⊗_{k} ⋯ ⊗_{k} E_{ζ_{i}}

(and E_{0} = k). By induction on i, we can use the triangles

E_{i} → E_{i−1} → E_{i−1} ⊗_{k} F_{ζ_{i}}

to see that the image of E_{i} ⊗_{k} F (𝔭) under (7.4) is the stable Koszul complex

Hom_{kG}(k, E_{i} ⊗_{k} F (𝔭)) ≅ K_{∞}(R_{𝔭}; ζ_{1}, …, ζ_{i}).

The case i = s of this statement shows that the image of K_{𝔭} under (7.4) is

Hom_{kG}(k, E_{s} ⊗_{k} F (𝔭)) ≅ K_{∞}(R_{𝔭}; ζ_{1}, …, ζ_{s}),

which by Theorem 8.6 is equivalent to Cell_{K_{𝔭}}(R_{𝔭}). □

Theorem 12.5. Let G be a finite group and let k be a field of characteristic p. If 𝔭 is a prime ideal in H^{∗}(G, k) of dimension d then in D(kG) we have Σ^{d}K_{𝔭} ≅ T (I_{𝔭}).

Proof. This follows by applying the equivalence of categories (7.4) to the case s = 0 of Theorem 12.1. By Theorem 7.7, T (I_{𝔭}) corresponds to I_{𝔭}, and by Theorem 12.4, Cell_{K_{𝔭}}(R_{𝔭}) corresponds to K_{𝔭}. □

Theorem 12.6. Let G be a finite group and let k be a field of characteristic p. Then there is a stable isomorphism Ω^{−d}κ_{𝔭} ≅ T (I_{𝔭}).

Proof. This follows by applying the functor Ψ to the isomorphism of Theorem 12.5. By Lemma 5.2, we have Ψ(K_{p}) ≅ κ_{𝔭}, while by Theorem 5.5 we have Ψ(T (I_{𝔭})) ≅ T (I_{𝔭}). □

Corollary 12.7. Ĥ^{∗}(G, κ_{𝔭}) ≅ I_{𝔭}[d].

Proof. This follows from Theorem 12.6 using Lemmas 2.1 and 3.3 of [13]. □

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Department of Mathematics, University of Aberdeen, Aberdeen AB24 3EA, UK

Department of Pure Mathematics, University of Sheffield, Hicks Building, Hounsfield Road, Sheffield S3 7RH, UK