COMPLEXES OF INJECTIVE kG-MODULES

DAVE BENSON AND HENNING KRAUSE

Abstract. Let G be a finite group and k be a field of characteristic p. We investigate the homotopy category K(Inj kG) of the category C(Inj kG) of complexes of injective (= projective) kG-modules. If G is a p-group, this category is equivalent to the derived category D_{dg}(C^{∗}(BG; k)) of the cochains on the classifying space; if G is not a p-group it has better properties than this derived category. The ordinary tensor product in K(Inj kG) with diagonal G-action corresponds to the E_{∞} tensor product on D_{dg}(C^{∗}(BG; k)).

We show that K(Inj kG) can be regarded as a slight enlargement of the stable module category StMod kG. It has better formal properties inasmuch as the ordinary cohomology ring H^{∗}(G, k) is better behaved than the Tate cohomology ring Ĥ^{∗}(G, k).

It is also better than the derived category D(Mod kG), because the compact objects in K(Inj kG) form a copy of the bounded derived category D^{b}(mod kG), whereas the compact objects in D(Mod kG) consist of just the perfect complexes.

Finally, we develop the theory of support varieties and homotopy colimits in K(Inj kG).

1. Introduction

Let k be a field and G a finite group. The purpose of this paper is to develop the properties of K(Inj kG), the homotopy category of complexes of injective kGmodules.

For any ring Λ, we write C(Inj Λ) for the category whose objects are the chain complexes of injective Λ-modules and whose arrows are the degree zero morphisms of chain complexes. We write K(Inj Λ) for the category with the same objects, but where the maps are the homotopy classes of degree zero maps of chain complexes. We write K_{ac}(Inj Λ) for the full subcategory whose objects are the acyclic chain complexes of injective Λ-modules.

We investigate a recollement relating K(Inj kG) to the stable module category StMod kG and the derived category D(Mod kG):

StMod kG ≃ K_{ac}(Inj kG) \begin{matrix}Hom_{k}(tk,−) \\ ⟵ \\ ⟶ \\ ⟵ \\ −⊗_{k}tk \end{matrix} K(Inj kG) \begin{matrix}Hom_{k}(pk,−) \\ ⟵ \\ ⟶ \\ ⟵ \\ −⊗_{k}pk \end{matrix} D(Mod kG).

The compact objects in these categories are

stmod kG ⟵ D^{b}(mod kG) ⟵ D^{b}(proj kG).

The research of the first author was undertaken during visits to Paderborn in Germany, supported by a Forschungspreis from the Humboldtstiftung.

This means that K(Inj kG) can be regarded as the appropriate “big” category for D^{b}(mod kG), whereas D(Mod kG) has too few compact objects for this purpose. In this sense, K(Inj kG) is a nicer category to work in than D(Mod kG).

From the point of view of algebraic topology, what K(Inj kG) does for us is provide an algebraic replacement for the derived category of the differential graded algebra of singular cochains on the classifying space, D_{dg}(C^{∗}(BG; k)). Namely, if G is a p-group then there is an equivalence of categories

K(Inj kG) ≃ D_{dg}(C^{∗}(BG; k)).

We prove that the tensor product over k of complexes in K(Inj kG) corresponds under this equivalence to the left derived tensor product over C^{∗}(BG; k) coming from the fact that the latter is E_{∞}, or “commutative up to all higher homotopies” (see Theorem 7.9 and the remarks after Theorem 4.1).

If G is not a p-group, then there is more than one simple kG-module, and the only one C^{∗}(BG; k) “sees” is the trivial kG-module. In this sense, K(Inj kG) is nicer to work in than D_{dg}(C^{∗}(BG; k)), even though it is not necessarily equivalent to it. Writing ik for an injective resolution of the trivial module, what we obtain in general is an equivalence between D_{dg}(C^{∗}(BG; k)) and the localizing subcategory of K(Inj kG) generated by ik.

In the work of Dwyer, Greenlees and Iyengar [14], a close relationship was established between D(Mod kG) and D_{dg}(C^{∗}(BG; k)). For a general finite group, the relationship between K(Inj kG) and D_{dg}(C^{∗}(BG; k)) is much closer, and provides some sort of context for understanding what is going on in [14]. Traces of arguments from that paper can be seen from time to time in this paper.

We develop the theory of support varieties for objects in K(Inj kG), extending the theory developed by Benson, Carlson and Rickard [4]. The extra information not included in StMod kG is reflected in the fact that the maximal ideal 𝔪 of positive degree elements in H^{∗}(G, k) becomes relevant in the variety theory. Thus K(Inj kG) can be regarded as a slight enlargement of StMod(kG) in which one more prime ideal 𝔪 of the cohomology ring is reflected. We also construct objects with injective cohomology, extending the work of [6]; the theory in K(Inj kG) is easier than in StMod kG because one does not have to compare ordinary and Tate cohomology.

Homotopy colimits in K(Inj kG) are harder to deal with than in StMod kG or than in D(Mod kG), so we conclude with a section describing how the theory works in this case. The main theorem here is that localizing subcategories of K(Inj kG) are closed under filtered colimits in C(Inj kG), in spite of the fact that the compact objects in K(Inj kG) do not lift to compact objects in C(Inj kG).

We would like to thank John Greenlees, Srikanth Iyengar and Jeremy Rickard for stimulating discussions related to this work. We would also like to thank the referee for some constructive comments.

2. K(Inj kG) is compactly generated

Let Λ be a Noetherian ring. We consider the category Mod Λ of Λ-modules and denote by mod Λ the full subcategory which is formed by all finitely generated modules. The injective Λ-modules form a subcategory Inj Λ of Mod Λ which is closed under taking arbitrary coproducts. This implies that K(Inj Λ) is a triangulated category which admits arbitrary coproducts.

We need to recall some definitions. Let T be a triangulated category with arbitrary coproducts. An object X of T is called compact if the functor Hom_{T}(X, −) into the category of abelian groups preserves all coproducts. We denote by T^{c} the full subcategory which is formed by all compact objects of T and observe that T^{c} is a thick subcategory. The triangulated category T is compactly generated if the isomorphism classes of objects of T^{c} form a set and if T coincides with its smallest triangulated subcategory containing T^{c} and closed under all coproducts.

Well known examples of compactly generated triangulated categories include the stable module category StMod Λ provided that Λ is self-injective, and the derived category D(Mod Λ) for any ring Λ. Note that the inclusion functors stmod Λ → StMod Λ and proj Λ → Mod Λ induce equivalences

stmod Λ ⟶^{≃} (StMod Λ)^{c} and D^{b}(proj Λ) ⟶^{≃} D(Mod Λ)^{c}.

Proposition 2.1. The triangulated category K(Inj Λ) is compactly generated. Let K^{c}(Inj Λ) denote the full subcategory which is formed by all compact objects. Then the canonical functor K(Inj Λ) → D(Mod Λ) induces an equivalence

K^{c}(Inj Λ) ⟶^{≃} D^{b}(mod Λ).

Proof. See Proposition 2.3 in [18]. □

Remark 2.2. The canonical functor Q: K(Inj Λ) → D(Mod Λ) has a right adjoint sending a complex X to its semi-injective resolution iX (the definition can be found just before Corollary 6.2). This right adjoint induces an equivalence D^{b}(mod Λ) ⟶^{≃} K^{c}(Inj Λ) which is a quasi-inverse for the equivalence K^{c}(Inj Λ) → D^{b}(mod Λ) induced by Q. For details of this construction we refer to Section 6.

3. K(Inj kG) is a derived category

Given two chain complexes X and Y in Mod Λ, we define the chain complex Hom_{Λ}(X, Y). The nth component is

∏_{p∈ℤ}Hom_{Λ}(X_{p}, Y_{n+p})

and the differential is defined so that

(d(f))(x) = d(f(x)) − (−1)^{|f|}f(d(x)).

Note that

H_{n} Hom_{Λ}(X, Y) ≅ Hom_{K(Mod Λ)}(X, Y [n]).

Composition of maps gives

End_{Λ}(X) = Hom_{Λ}(X, X)

the structure of a differential graded algebra, over which Hom_{Λ}(X, Y) is a differential graded module. Given a DG algebra Γ, we denote by D_{dg}(Γ) the derived category of DG Γ-modules.

Proposition 3.1. Let C be an object of K^{c}(Inj Λ) ≃ D^{b}(mod Λ) and let Γ = End_{Λ}(C). Denote by C the smallest full triangulated subcategory of K(Inj Λ) closed under all coproducts and containing C. Then the functor

Hom_{Λ}(C, −): K(Inj Λ) ⟶ D_{dg}(Γ)

induces an equivalence C ⟶^{≃} D_{dg}(Γ) of triangulated categories.

Proof. We begin by defining Hom_{Λ}(C, −) as a functor from C(Inj Λ) to the category of differential graded Γ-modules. This functor is exact, and the composite to D_{dg}(Γ) takes homotopic maps to the same place. So we obtain a well defined exact functor from K(Inj Λ) to D_{dg}(Γ) (cf. Keller [17, §4.3, bottom of p. 77]). To see that it preserves coproducts, fix a family of objects X_{i} in K(Inj Λ). Then we have for every n ∈ ℤ

H_{n} ∐_{i} Hom_{Λ}(C, X_{i}) ≅ ∐_{i} H_{n} Hom_{Λ}(C, X_{i}) ≅ ∐_{i} Hom_{K(Inj Λ)}(C, X_{i}[n])

≅ Hom_{K(Inj Λ)}(C, ∐_{i} X_{i}[n]) ≅ H_{n} Hom_{Λ}(C, ∐_{i} X_{i}[n])

since C is compact in K(Inj Λ). Thus the canonical map

∐_{i} Hom_{Λ}(C, X_{i}) ⟶ Hom_{Λ}(C, ∐_{i} X_{i})

is an isomorphism. Furthermore, the functor induces bijections

Hom_{K(Inj Λ)}(C, C[n]) ≅ H_{n} Hom_{Λ}(C, C) ≅ Hom_{D_{dg}(Γ)}(Γ, Γ[n]).

Thus the class D of objects in K(Inj Λ) such that the induced map

Hom_{K(Inj Λ)}(X, Y) ⟶ Hom_{D_{dg}(Γ)}(Hom_{Λ}(C, X), Hom_{Λ}(C, Y))

is bijective for all X, Y in D contains C. The functor Hom_{Λ}(C, −) is, up to isomorphism, surjective on objects since the image contains the free Γ-module Γ which generates D_{dg}(Γ). □

Remark 3.2. (1) The functor Hom_{Λ}(C, −) admits left and right adjoints. This is a consequence of Brown representability (see Neeman [22]) because the functor preserves (co)products. Thus Hom_{Λ}(C, −) induces a recollement of the form

Ker Hom_{Λ}(C, −) \begin{matrix}⟵ \\ ⟶ \\ ⟵ \end{matrix} K(Inj Λ) \begin{matrix}⟵ \\ ⟶ \\ ⟵ \end{matrix} D_{dg}(Γ).

Here, Ker Hom_{Λ}(C, −) denotes the full subcategory of K(Inj Λ) formed by all objects X with Hom_{Λ}(C, X) = 0. The functors between Ker Hom_{Λ}(C, −) and K(Inj Λ) are the inclusion together with its left and right adjoints.

(2) If the object C generates K^{c}(Inj Λ), that is, there is no proper thick subcategory containing C, then C = K(Inj Λ) and the functor Hom_{Λ}(C, −) is an equivalence.

In the case where G is a finite p-group, one choice for the compact generator C of Proposition 3.1 is ik, an injective resolution of k. For a more general finite group, we may take the sum of the injective resolutions of the simple modules. We write E_{G} for the differential graded algebra End_{kG}(ik) whether or not G is a p-group.

4. The Rothenberg–Steenrod construction

We now relate the category K(Inj kG) to the classifying space BG. For general background references on classifying spaces of groups, see for example [2, 10]. The basic link between K(Inj kG) and the derived category of C^{∗}(BG; k) is achieved through the Rothenberg–Steenrod construction [26], which we now make precise. For any path-connected space X, this construction gives a quasi-isomorphism of differential graded algebras from the derived endomorphisms of k over the chains on the loop space and the cochains on X:

ℝEnd_{C_{∗}(ΩX;k)}(k) ≃ C^{∗}(X; k).

In the case where X is the classifying space BG, ΩX is equivalent to G, and C_{∗}(ΩX; k) is equivalent as a differential graded algebra to the group algebra kG in degree zero. So in this case the left hand side is just E_{G} = End_{kG}(ik), and so we obtain

D_{dg}(E_{G}) ≃ D_{dg}(C^{∗}(BG; k)).

The purpose of this section is to investigate this equivalence algebraically.

We begin by remarking that End_{kG}(pk) and End_{kG}(ik) are quasi-isomorphic differential graded algebras. To see this, choose a quasi-isomorphism pk → ik. Then we have quasi-isomorphisms

End_{kG}(pk) → Hom_{kG}(pk, ik) ← End_{kG}(ik).

The middle object is not a differential graded algebra, but the pullback of this pair of maps

\xymatrix{X \ar[r] \ar[d] & End_{kG}(ik) \ar[d] \\ End_{kG}(pk) \ar[r] & Hom_{kG}(pk, ik)}

is a differential graded algebra

X = End_{kG}(pk) ×_{Hom_{kG}(pk,ik)} End_{kG}(ik)

that comes with quasi-isomorphisms

End_{kG}(pk) ⟵^{≃} X ⟶^{≃} End_{kG}(ik).

Thus we obtain equivalences of derived categories

D_{dg}(End_{kG}(pk)) ≃ D_{dg}(X) ≃ D_{dg}(End_{kG}(ik)).

Similarly, iff p′k is another projective resolution then there is a comparison map pk → p′k, and hence there are homomorphisms

End_{kG}(pk) → Hom_{kG}(pk, p′k) ← End_{kG}(p′k).

The pullback of this pair of maps is a differential graded algebra

Y = End_{kG}(pk) ×_{Hom_{kG}(pk,p′k)} End_{kG}(p′k)

that comes with quasi-isomorphisms

End_{kG}(pk) ⟵^{≃} Y ⟶^{≃} End_{kG}(p′k).

Thus we obtain equivalences of derived categories

D_{dg}(End_{kG}(pk)) ≃ D_{dg}(Y) ≃ D_{dg}(End_{kG}(p′k)).

It follows that D_{dg}(End_{kG}(pk)) is, up to natural equivalence, independent of choice of projective resolution, and is also equivalent to D_{dg}(End_{kG}(ik)).

The augmentation map ε: pk → k gives a quasi-isomorphism of complexes

End_{kG}(pk) ≃ Hom_{kG}(pk, k).

Suppose that pk is a resolution supporting a strictly coassociative and counital diagonal Δ: pk → pk ⊗_{k} pk, meaning that the following diagrams commute:

\xymatrix{pk \ar[r]^{Δ} \ar[d]^{Δ} & pk ⊗_{k} pk \ar[d]^{1⊗Δ} &  & pk \ar[dl]_{≅} \ar[d]^{Δ} \ar[dr]^{≅} \\ pk ⊗_{k} pk \ar[r]^{Δ⊗1} & pk ⊗_{k} pk ⊗_{k} pk & pk ⊗_{k} k & pk ⊗_{k} pk \ar[l]_{ε⊗1} \ar[r]^{1⊗ε} & k ⊗_{k} pk}

This happens, for example, when pk is the bar resolution, and when pk is equal to the singular cochains on EG. Then there is a multiplication on Hom_{kG}(pk, k) given as follows. If α, β: pk → k then α.β is given by the composite

pk ⟶^{Δ} pk ⊗_{k} pk ⟶^{α⊗β} k ⊗_{k} k ⟶^{≅} k.

The fact that Δ is coassociative and counital implies that this multiplication is associative and unital.

We claim that there is a quasi-isomorphism of differential graded algebras

Hom_{kG}(pk, k) → End_{kG}(pk)

given by sending α: pk → k to the map α̃: pk → pk given by the composite pk ⟶^{Δ} pk ⊗_{k} pk ⟶^{1⊗α} pk ⊗_{k} k ⟶^{≅} pk. Since Δ is counital, we have ε ∘ α̃ = α, so that α ↦ α̃ is a quasi-isomorphism. The commutative diagram

[figure: pk; α̃∘β̃; pk; ε; pk; pk; β̃; Δ; pk; α; α̃; k; pk ⊗_{k} pk; 1⊗β; α⊗β; pk ⊗_{k} k; α⊗1; ≅; k ⊗_{k} k; ≅]

shows that the map α ↦ α̃ preserves multiplication.

Using this, we see that we have quasi-isomorphisms of differential graded algebras

End_{kG}(pk) ≃ Hom_{kG}(pk, k) ≃ Hom_{kG}(C_{∗}(EG; k), k))
≅ Hom_{k}(C_{∗}(BG; k), k) ≅ C^{∗}(BG; k).

Now suppose that H is a subgroup of G. Then EG can be used as a model for EH. In particular, C_{∗}(EG; k) is another model of pk in K(Inj kH) with a strictly coassociative and counital diagonal map. Restricting resolutions for G to the subgroup H gives us resolutions for H, so we have a restriction map of differential graded algebras res_{G,H}: E_{G} → E_{H}. We also have a restriction map res_{G,H}: C^{∗}(BG; k) → C^{∗}(BH; k). Naturality of the Rothenberg–Steenrod construction gives us the following theorem.

Theorem 4.1. There are equivalences of categories

D_{dg}(E_{G}) = D_{dg}(End_{kG}(ik)) ≃ D_{dg}(End_{kG}(pk)) ≃ D_{dg}(C^{∗}(BG; k)).

The equivalence D_{dg}(E_{G}) ≃ D_{dg}(C^{∗}(BG; k)) is natural, in the sense that if H is a subgroup of G then the square

\xymatrix{D_{dg}(E_{H}) \ar[r]^{≃} \ar[d]^{_{res{∗ \atop G,H}}} & D_{dg}(C^{∗}(BH; k)) \ar[d]^{_{res{∗ \atop G,H}}} \\ D_{dg}(E_{G}) \ar[r]^{≃} & D_{dg}(C^{∗}(BG; k))}

commutes up to natural isomorphism. □

Next, we discuss the tensor product − ⊗_{C^{∗}(BG;k)} − on D_{dg}(C^{∗}(BG; k)). It is convenient at this stage to be able to pass back and forth between differential graded algebras and S-algebras (S here is the sphere spectrum). The point of this formalism is to have a category of spectra with a smash product that is commutative and associative up to coherent natural isomorphism, and not just up to all higher homotopies. In the 1990s, several sets of authors produced such categories.

We will work with the formalism of S-algebras introduced by Elmendorf, Kříž, Mandell and May [15].

We make use of the paper of Shipley [27] to translate between the language of Salgebras and the language of differential graded algebras. Shipley shows that if R is a discrete commutative ring, with associated Eilenberg–Mac Lane spectrum HR, then the model categories of differential graded R-algebras and S-algebras over HR are Quillen equivalent. In particular, their homotopy categories are equivalent as triangulated categories. It would be possible to work directly in the category of E_{∞} differential graded algebras, but we would need to be working over an E_{∞} operad such as the surjection operad of McClure and Smith [21] and then transfer to an E_{∞} operad satisfying the Hopkins lemma of [15]. Alternatively, one could work directly with the formalism of Hovey, Shipley and Smith [16] and use the algebraic analogue of symmetric spectra. Further comments on the relationships between E_{∞} algebras and singular cochains on spaces can be found in Mandell [19].

In any case, the upshot of the discussion is that if X and Y are objects in D_{dg}(C^{∗}(BG; k)) then so is the left derived tensor product X ⊗_{C^{∗}(BG;k)} Y. This tensor product is symmetric monoidal; so there are coherent natural isomorphisms

X ⊗_{C^{∗}(BG;k)} Y ≅ Y ⊗_{C^{∗}(BG;k)} X
(X ⊗_{C^{∗}(BG;k)} Y) ⊗_{C^{∗}(BG;k)} Z ≅ X ⊗_{C^{∗}(BG;k)} (Y ⊗_{C^{∗}(BG;k)} Z).

In the case where G is a p-group, we can compare D_{dg}(Γ) with D_{dg}(C^{∗}(BG; k)) as in the following theorem. If G is not a p-group, then D_{dg}(C^{∗}(BG; k)) is not equivalent to the whole of K(Inj kG), but just the part generated by ik. This is because there is more than one simple kG-module, and C^{∗}(BG; k) only “sees” what is generated by the trivial module; in particular, non-principal blocks of kG are invisible to C^{∗}(BG; k).

Theorem 4.2. Let G be a finite group. Then we have functors

K(Inj kG) ⟶^{Hom_{kG}(ik,−)} D_{dg}(E_{G}) ≃ D_{dg}(C^{∗}(BG; k))

whose composite we denote by Φ. If G is a finite p-group, this gives an equivalence of categories

Φ: K(Inj kG) ⟶^{≃} D_{dg}(C^{∗}(BG; k)).

Proof. This is proved by combining Proposition 3.1 and Theorem 4.1. As remarked above, in the case of a p-group, we can take ik as a generator for K^{c}(Inj kG), so that the differential graded algebra Γ of Proposition 3.1 is equal to E_{G}. □

Remark 4.3. An explicit right adjoint Ψ: D_{dg}(C^{∗}(BG; k)) → K(Inj kG) to Φ is described just before Lemma 7.5; see also Remark 3.2. The functor Ψ satisfies Φ ∘ Ψ ≅ Id_{D_{dg}(C^{∗}(BG;k))}.

5. K(Inj kG) is a tensor category

If G_{1} and G_{2} are groups then there is a natural isomorphism of group algebras k(G_{1} × G_{2}) ≅ kG_{1} ⊗_{k} kG_{2}. Taking the tensor product of complexes gives an external tensor product

C(Mod kG_{1}) × C(Mod kG_{2}) → C(Mod k(G_{1} × G_{2}))

and hence also

K(Mod kG_{1}) × K(Mod kG_{2}) → K(Mod k(G_{1} × G_{2})).

If G = G_{1} = G_{2}, then restricting the external tensor product via the diagonal embedding of G in G × G defines an internal tensor product

C(Mod kG) × C(Mod kG) → C(Mod kG)

which induces

K(Mod kG) × K(Mod kG) → K(Mod kG).

Similar arguments show that Hom_{k}(−, −) induces internal products on the categories C(Mod kG) and K(Mod kG). Note that we have a natural isomorphism

(5.1)  Hom_{K(Mod kG)}(X ⊗_{k} Y, Z) ≅ Hom_{K(Mod kG)}(X, Hom_{k}(Y, Z))

for all X, Y, Z in K(Mod kG).

The subcategories K(Inj kG) and K_{ac}(Inj kG) inherit tensor products from the category K(Mod kG) because they are tensor ideals. This follows from the next lemma.

Lemma 5.2. Let X, Y be complexes of kG-modules.

(i) If X is a complex of injective kG-modules, then X ⊗_{k} Y and Hom_{k}(X, Y) are complexes of injective kG-modules.

(ii) If X is an acyclic complex, then X ⊗_{k} Y and Hom_{k}(X, Y) are acyclic complexes.

Proof. The first assertion is clear since M ⊗_{k} N and Hom_{k}(M, N) are injective for any pair of kG-modules M, N provided that one of them is injective. The second assertion follows from the fact that the tensor product and Hom are computed over k. □

Proposition 5.3. The unit for the tensor product on K(Inj kG) is the injective resolution ik of the trivial representation k.

Proof. For any object X in K(Inj kG), the map of complexes k → ik induces the following chain of isomorphisms:

Hom_{K(Mod kG)}(ik ⊗_{k} X, −) ≅ Hom_{K(Mod kG)}(ik, Hom_{k}(X, −))
≅ Hom_{K(Mod kG)}(k, Hom_{k}(X, −))
≅ Hom_{K(Mod kG)}(k ⊗_{k} X, −).

Here we use (5.1) and that k → ik induces an isomorphism

Hom_{K(Mod kG)}(ik, Y) ≅ Hom_{K(Mod kG)}(k, Y)

for all Y in K(Inj kG) by [18, Lemma 2.1]. Thus the map of complexes k ⊗_{k} X → ik ⊗_{k} X is an isomorphism in K(Inj kG). □

Definition 5.4. If X is an object in K(Inj kG), we define

H^{∗}(G, X) = Hom_{K(Inj kG)}^{∗}(ik, X)

where the nth component is Hom_{K(Inj kG)}(ik, X[n]). This is a graded module for the cohomology ring H^{∗}(G, k) = Hom_{K(Inj kG)}^{∗}(ik, ik).

6. A recollement for K(Inj kG)

Let Λ be a Noetherian ring. We have seen that K(Inj Λ) is compactly generated and this fact has some interesting consequences. For instance, any exact functor K(Inj Λ) → T into a triangulated category T admits a right adjoint if it preserves coproducts and a left adjoint if it preserves products. We apply this consequence of Brown representability (see Neeman [22]) to the canonical functor

Q: K(Inj Λ) ⟶^{inc} K(Mod Λ) ⟶^{can} D(Mod Λ)

and obtain the following result [18, Corollary 4.3].

Proposition 6.1. The pair of canonical functors

K_{ac}(Inj Λ) ⟶^{I} K(Inj Λ) ⟶^{Q} D(Mod Λ)

induces a recollement

K_{ac}(Inj Λ) \begin{matrix}I_{ρ} \\ ⟵ \\ ⟶ \\ ⟵ \\ I_{λ} \end{matrix} K(Inj Λ) \begin{matrix}Q_{ρ} \\ ⟵ \\ ⟶ \\ ⟵ \\ Q_{λ} \end{matrix} D(Mod Λ).

More precisely, the functors I and Q admit left adjoints I_{λ} and Q_{λ} as well as right adjoints I_{ρ} and Q_{ρ} such that the following adjunction morphisms

I_{λ} ∘ I ⟶^{≃} Id_{K_{ac}(Inj Λ)} ⟶^{≃} I_{ρ} ∘ I and Q ∘ Q_{ρ} ⟶^{≃} Id_{D(Mod Λ)} ⟶^{≃} Q ∘ Q_{λ}

are isomorphisms.

Recall from Avramov, Foxby and Halperin [1] (see also Spaltenstein [28]) that for any differential graded algebra Γ, a DG Γ-module X is said to be semiprojective if Hom_{Γ}(X, −) carries surjective quasi-isomorphisms to surjective quasiisomorphisms. Similarly, X is semi-injective if Hom_{Γ}(−, X) carries injective quasiisomorphisms to surjective quasi-isomorphisms. A semi-projective resolution of a DG Γ-module X is a quasi-isomorphism pX → X with pX semi-projective, and a semi-injective resolution of X is a quasi-isomorphism X → iX with iX semiinjective. If Γ is a ring, these definitions are applied by regarding Γ as a DG algebra concentrated in degree zero, so that a DG module is just a complex of Γ-modules.

Note that the recollement provides two embeddings of D(Mod Λ) into K(Inj Λ). The more familiar one is the fully faithful functor Q_{ρ}: D(Mod Λ) → K(Inj Λ) which sends a complex X of Λ-modules to a semi-injective resolution iX. The less familiar embedding is the fully faithful functor Q_{λ}: D(Mod Λ) → K(Inj Λ) which identifies D(Mod Λ) with the localizing subcategory of K(Inj Λ) generated by iΛ. If Λ is self-injective, then Q_{λ} sends a complex X of Λ-modules to a semi-projective resolution pX.

We summarize this discussion as follows.

Corollary 6.2. Let Λ be a Noetherian ring, and let X be a complex of injective Λ-modules. Then the following are equivalent.

(i) X is semi-injective.

(ii) X ≅ Q_{ρ}Y for some Y in D(Mod Λ).

(iii) I_{ρ}X ≅ 0. If Λ is selfinjective (so that projective and injective Λ-modules coincide), then the following are equivalent.

(i) X is semi-projective.

(ii) X ≅ Q_{λ}Y for some Y in D(Mod Λ).

(iii) I_{λ}X ≅ 0.

In the case where Λ = kG, we have StMod kG ≃ K_{ac}(Inj kG), and the adjoints in the recollement take the form

(6.3) StMod kG ≃ K_{ac}(Inj kG) \begin{matrix}Hom_{k}(tk,−) \\ ⟵ \\ ⟶ \\ ⟵ \\ −⊗_{k}tk \end{matrix} K(Inj kG) \begin{matrix}Hom_{k}(pk,−) \\ ⟵ \\ ⟶ \\ ⟵ \\ −⊗_{k}pk \end{matrix} D(Mod kG).

Here, we write ik for a semi-injective resolution, pk for a semi-projective resolution, and tk for a Tate resolution of the trivial kG-module k. Note that these resolutions fit into an exact triangle

pk ⟶ ik ⟶ tk ⟶ pk[1]

in K(Inj kG). This triangle induces for each object X of K(Inj kG) the following exact triangles:

X ⊗_{k} pk ⟶ X ⊗_{k} ik ⟶ X ⊗_{k} tk ⟶ X ⊗_{k} pk[1],
Hom_{k}(tk, X) ⟶ Hom_{k}(ik, X) ⟶ Hom_{k}(pk, X) ⟶ Hom_{k}(tk[−1], X).

The first two maps in each triangle are the obvious adjunction morphisms which are induced by the recollement. This becomes clear once we observe that the canonical map k → ik induces isomorphisms

X = X ⊗_{k} k ⟶^{≃} X ⊗_{k} ik and Hom_{k}(ik, X) ⟶^{≃} Hom_{k}(k, X) = X;

see Proposition 5.3.

Thus K(Inj kG) is a sort of intermediary between StMod kG and D(Mod kG), and in some ways is better behaved than either of them. The problem with StMod kG is that the graded endomorphisms of the trivial module form a usually non-Noetherian ring (the Tate cohomology ring). The problem with D(Mod kG), on the other hand, is that k is usually not a compact object.

The compact objects in the three categories in the recollement give the perhaps more familiar sequence of categories and functors

stmod kG ⟵ D^{b}(mod kG) ⟵ D^{b}(proj kG).

Note that only the left adjoints in the recollement preserve compact objects.

7. The dictionary between K(Inj kG) and D_{dg}(C^{∗}(BG; k))

Let G be a finite group. Then by Theorem 4.1 we have functors

(7.1)  K(Inj kG) ⟶^{Hom_{kG}(ik,−)} D_{dg}(E_{G}) ≃ D_{dg}(C^{∗}(BG; k)).

(where E_{G} = End_{kG}(ik)), which in the case of a p-group give an equivalence of triangulated categories

Φ: K(Inj kG) → D_{dg}(C^{∗}(BG; k)).

In this section, we investigate the functor Φ further, and we develop a dictionary for translating between K(Inj kG) and D_{dg}(C^{∗}(BG; k)).

First we deal with external tensor products. Now if R_{1} and R_{2} are commutative S-algebras over k, then R_{1} ⊗_{k} R_{2} is also a commutative S-algebra over k by VII.1.6 of [15]. If X and Y are spaces then C^{∗}(X; k) ⊗_{k} C^{∗}(Y; k) is equivalent to C^{∗}(X × Y; k) as S-algebras over k. If δ: X → X × X is the diagonal map, then the composite

C^{∗}(X; k) ⊗_{k} C^{∗}(X; k) ≃ C^{∗}(X × X; k) ⟶^{δ^{∗}} C^{∗}(X; k)

is the multiplication map, and is a map of commutative S-algebras over k.

In particular, if X = BG_{1} and Y = BG_{2} then X × Y = B(G_{1} × G_{2}), and we get the equivalence of C^{∗}(BG_{1}; k) ⊗_{k} C^{∗}(BG_{2}; k) with C^{∗}(B(G_{1} × G_{2}); k). This means that if X and Y are modules over C^{∗}(BG_{1}; k) and C^{∗}(BG_{2}; k) respectively, we have an external tensor product X ⊗_{k} Y as a module over C^{∗}(B(G_{1} × G_{2}); k).

If Δ: G → G × G is the diagonal map, then the composite

C^{∗}(BG; k) ⊗_{k} C^{∗}(BG; k) ≃ C^{∗}(BG × BG; k)

= C^{∗}(B(G × G); k) ⟶^{BΔ^{∗}} C^{∗}(BG; k)

is the multiplication map on C^{∗}(BG; k).

Theorem 7.2. The functor Φ takes the external tensor product over k discussed in Section 5 to the external tensor product described above.

Proof. If ik_{G_{1}} and ik_{G_{2}} are injective resolutions of k for G_{1} and G_{2}, then the external tensor product ik_{G_{1}} ⊗_{k} ik_{G_{2}} is an injective resolution of k for G_{1} × G_{2}.

We have a commutative diagram

\xymatrix{K(Inj kG_{1}) × K(Inj kG_{2}) \ar[r]^{Hom_{kG_{1}}(ik_{G_{1}},−)×Hom_{kG_{2}}(ik_{G_{2}},−)} \ar[d]^{⊗_{k}} & D_{dg}(E_{G_{1}}) × D_{dg}(E_{G_{2}}) \ar[d]^{⊗_{k}} \\ K(Inj k(G_{1} × G_{2})) \ar[r]^{Hom_{k(G_{1}×G_{2})}(ik_{G_{1}}⊗_{k}ik_{G_{2}},−)} & D_{dg}(E_{G_{1}} ⊗_{k} E_{G_{2}})}

We combine this with the commutative diagram

\xymatrix{D_{dg}(E_{G_{1}}) × D_{dg}(E_{G_{2}}) \ar[r]^{≃} \ar[d]^{⊗_{k}} & D_{dg}(C^{∗}(BG_{1}, k) × D_{dg}(C^{∗}(BG_{2}; k)) \ar[d]^{⊗_{k}} \\ D_{dg}(E_{G_{1}} ⊗_{k} E_{G_{2}}) \ar[r]^{≃} & D_{dg}(C^{∗}(BG_{1}; k) ⊗_{k} C^{∗}(BG_{2}; k))}

and the equivalence

D_{dg}(C^{∗}(BG_{1}; k) ⊗_{k} C^{∗}(BG_{2}; k)) ≃ D_{dg}(C^{∗}(B(G_{1} × G_{2}); k))

to prove the theorem. □

Next we deal with subgroups.

Lemma 7.3. If H is a subgroup of G then the following diagram commutes up to natural isomorphism:

\xymatrix{K(Inj kH) \ar[r]^{Hom_{kH}(ik,−)} \ar[d]_{_{ind_{H,G}}} & D_{dg}(E_{H}) \ar[d]^{_{res{∗ \atop G,H}}} \\ K(Inj kG) \ar[r]^{Hom_{kG}(ik,−)} & D_{dg}(E_{G})}

Proof. This follows from the Frobenius reciprocity (or Eckmann–Shapiro) isomorphism

Hom_{kG}(ik, ind_{H,G}(X)) ≅ Hom_{kH}(ik, X).  □

Theorem 7.4. The functor Φ takes induction from kH-modules to kG-modules to restriction from C^{∗}(BH; k)-modules to C^{∗}(BG; k)-modules.

Proof. By Theorem 4.1 and Lemma 7.3, the following diagram commutes up to natural isomorphisms:

\xymatrix{K(Inj kH) \ar[r]^{Hom_{kH}(ik,−)} \ar[d]_{_{ind_{H,G}}} & D_{dg}(E_{H}) \ar[d]^{_{res{∗ \atop G,H}}} \ar[r]^{≃} & D_{dg}(C^{∗}(BH; k)) \ar[d]^{_{res{∗ \atop G,H}}} \\ K(Inj kG) \ar[r]^{Hom_{kG}(ik,−)} & D_{dg}(E_{G}) \ar[r]^{≃} & D_{dg}(C^{∗}(BG; k))}

□

The corresponding statement for restriction from K(Inj kG) to K(Inj kH) requires more preparation. We begin by defining a functor

− ⊗^{𝕃}_{E_{G}} ik: D_{dg}(E_{G}) → K(Inj kG)

as the left adjoint of Hom_{kG}(ik, −). The existence of such a left adjoint follows from Brown’s representability theorem (see Neeman [22]) since Hom_{kG}(ik, −) preserves products. Alternatively, we construct this functor explicitly by tensoring over E_{G} a semi-projective resolution (for the definition, see Section 6) of the given differential

graded E_{G}-module with ik. It is clear from the construction that −⊗^{𝕃}_{E_{G}} ik identifies E_{G} with ik.

Lemma 7.5. Let X be an object in D_{dg}(E_{G}). Then the natural map

X → Hom_{kG}(ik, X ⊗^{𝕃}_{E_{G}} ik)

is an isomorphism in D_{dg}(E_{G}).

Proof. This is obviously true for X = E_{G}. The functor on the right preserves triangles and direct sums in the variable X because ik is compact. So the assertion is true for any object in the localizing subcategory generated by E_{G}, which is all of D_{dg}(E_{G}). □

Remark 7.6. The functor − ⊗^{𝕃}_{E_{G}} ik identifies D_{dg}(E_{G}) with the localizing subcategory Loc(ik) of K(Inj kG) generated by ik. In particular, for each object Y in K(Inj kG), the natural map

η_{Y} : Hom_{kG}(ik, Y) ⊗^{𝕃}_{E_{G}} ik → Y

is the best left approximation of Y by objects in Loc(ik). More precisely, the object

Hom_{kG}(ik, Y) ⊗^{𝕃}_{E_{G}} ik belongs to Loc(ik) and the induced map Hom_{K(Inj kG)}(X, η_{Y}) is bijective for all X in Loc(ik).

Lemma 7.7. Suppose we have given a diagram of functors

\xymatrix{S \ar[d]^{F} \ar[r]^{H} & T \ar[d]^{G} \\ S′ \ar[r]^{H′} & T′}

which is commutative up to isomorphism such that F, G, H′ admit right adjoints F_{ρ}, G_{ρ}, H′_{ρ}, and H admits a left adjoint H_{λ}. Suppose in addition that Id_{T} ≅ H∘H_{λ} and H′_{ρ} ∘ H′ ≅ Id_{S′}. Then the following diagram of functors commutes up to isomorphism.

\xymatrix{S′ \ar[d]^{F_{ρ}} \ar[r]^{H′} & T′ \ar[d]^{G_{ρ}} \\ S \ar[r]^{H} & T}

Proof. We have

G ≅ G ∘ H ∘ H_{λ} ≅ H′ ∘ F ∘ H_{λ}.

Taking right adjoints, we obtain

G_{ρ} ≅ H ∘ F_{ρ} ∘ H′_{ρ}

and this implies

G_{ρ} ∘ H′ ≅ H ∘ F_{ρ} ∘ H′_{ρ} ∘ H′ ≅ H ∘ F_{ρ}.  □

Theorem 7.8. Let G be a finite p-group and let H be a subgroup of G. Then the functor Φ takes restriction from kG-modules to kH-modules to coinduction from C^{∗}(BG; k)-modules to C^{∗}(BH; k)-modules.

Proof. We claim that the following diagram commutes.

\xymatrix{K(Inj kG) \ar[r]^{Hom_{kG}(ik,−)} \ar[d]_{_{res_{G,H}}} & D_{dg}(E_{G}) \ar[d]^{ℝHom_{E_{G}}(E_{H},−)} \ar[r]^{≃} & D_{dg}(C^{∗}(BG; k)) \ar[d]^{ℝHom_{C^{∗}(BG;k)}(C^{∗}(BH;k),−)} \\ K(Inj kH) \ar[r]^{Hom_{kH}(ik,−)} & D_{dg}(E_{H}) \ar[r]^{≃} & D_{dg}(C^{∗}(BH; k))}

For the right hand square this is clear. For the left hand square, this follows from Lemma 7.3, 7.5 and 7.7. The assumption on G to be a p-group is needed for Hom_{kG}(ik, −) to be an equivalence. □

Theorem 7.9. Let G be a finite p-group. Then the functor Φ takes the internal tensor product with diagonal G-action to the E_{∞} tensor product discussed at the end of Section 3.

Proof. The internal tensor product in K(Inj kG) is given by external tensor product to K(Inj k(G × G)) followed by restriction to the diagonal copy of G. Using Theorems 7.2 and 7.8, we see that

Hom_{kG}(ik, (X ⊗_{k} Y)↓_{G}^{G×G}) ≅ ℝHom_{E_{G×G}}(E_{G}, Hom_{k(G×G)}(ik, X ⊗_{k} Y))

≅ ℝHom_{E_{G}⊗_{k}E_{G}}(E_{G}, Hom_{kG}(ik, X) ⊗_{k} Hom_{kG}(ik, Y))

Applying the equivalence with D_{dg}(C^{∗}(BG; k)) to the latter, we obtain

ℝHom_{C^{∗}(BG;k)⊗_{k}C^{∗}(BG;k)}(C^{∗}(BG; k), Φ(X) ⊗_{k} Φ(Y)).

which is isomorphic to

Φ(X) ⊗_{C^{∗}(BG;k)} Φ(Y)

with the E_{∞} tensor product. □

Theorems 7.4 and 7.8 above can be thought of as saying that the roles of restriction and (co)induction are reversed by the equivalence. So it makes sense that the roles of the trivial representation and the regular representation should also be reversed.

It is easy to see that ik in K(Inj kG) corresponds to the regular representation of C^{∗}(BG; k), and that the regular representation kG corresponds to the trivial representation k of C^{∗}(BG; k).

We summarize all this information in the following table.

K(Inj kG)  D_{dg}(C^{∗}(BG; k))
external − ⊗_{k} −  external − ⊗_{k} −
internal − ⊗_{k} −  − ⊗_{C^{∗}(BG;k)} −
diagonal G-action  E_{∞} tensor product
Induction  Restriction
−_{H}↑^{G}  via C^{∗}(BG; k) → C^{∗}(BH; k)
Restriction  Coinduction
−_{G}↓_{H}  Hom_{C^{∗}(BG;k)}(C^{∗}(BH; k), −)
ik  C^{∗}(BG; k)
kG  k

8. K(Inj Λ) is a derived invariant

The classical Morita theory for derived categories (Rickard [23, 24]) can be extended to complexes of injective modules as follows.

Proposition 8.1. Let Λ and Γ be Noetherian algebras over a commutative ring k. Suppose that Λ and Γ are projective as k-modules. Then the following are equivalent.

(i) Λ and Γ are derived equivalent, that is, there exists a tilting complex T over Λ such that the endomorphism ring End_{D(Mod Λ)}(T) is isomorphic to Γ.

(ii) There exists an exact equivalence K(Inj Λ) → K(Inj Γ).

(iii) There exists an exact equivalence D^{b}(mod Λ) → D^{b}(mod Γ).

Proof. (i) ⇒ (ii): In Rickard [24], it is shown that Λ and Γ admit a standard derived equivalence. Thus there is a bounded complex P of Γ-Λ-bimodules which in each degree is finitely generated projective over Λ and over Γ. The functor Hom_{Λ}(P, −) sends complexes of injective Λ-modules to complexes of injective Γ-modules and semi-injective complexes to semi-injective complexes. The last assertion follows from the isomomrphism

Hom_{Γ}(A, Hom_{Λ}(P, X)) ≅ Hom_{Λ}(A ⊗_{Γ} P, X).

Thus Hom_{Λ}(P, −) induces the following commutative diagram of exact functors

\xymatrix{D(Mod Λ) \ar[d]^{(Q_{Λ})_{ρ}} \ar[r]^{Hom_{Λ}(P,−)} & D(Mod Γ) \ar[d]^{(Q_{Γ})_{ρ}} \\ K(Inj Λ) \ar[r]^{Hom_{Λ}(P,−)} & K(Inj Γ)}

because we know from Corollary 6.2 that the right adjoint functors (Q_{Λ})_{ρ} and (Q_{Γ})_{ρ} identify the derived categories with the full subcategories formed by all semi-injective complexes. By our assumption, the functor D(Mod Λ) → D(Mod Γ) is an equivalence which induces an equivalence D^{b}(mod Λ) → D^{b}(mod Γ). Now we apply Proposition 2.1 as follows. The commutativity of the diagram implies that Hom_{Λ}(P, −) induces an equivalence K^{c}(Inj Λ) → K^{c}(Inj Γ). Then a standard dévissage argument shows that Hom_{Λ}(P, −) induces an equivalence K(Inj Λ) → K(Inj Γ) since K(Inj Λ) is compactly generated and the functor preserves all coproducts.

(ii) ⇒ (iii): An exact equivalence K(Inj Λ) → K(Inj Γ) induces an exact equivalence K^{c}(Inj Λ) → K^{c}(Inj Γ) and therefore an exact equivalence D^{b}(mod Λ) → D^{b}(mod Γ), again by Proposition 2.1.

(iii) ⇒ (i): Let F: D^{b}(mod Γ) → D^{b}(mod Λ) be an exact equivalence. Then T = F Γ is a tilting complex with End_{D^{b}(mod Λ)}(T) ≅ Γ. □

9. Bousfield localization

We recall briefly some basic facts about Bousfield localization. Let T be a triangulated with arbitrary coproducts. We fix a full triangulated subcategory S of T which is localizing in the sense that S is closed under taking all coproducts. Then we have a sequence

S ⟶^{I} T ⟶^{Q} T/S

of canonical functors and observe that I has a right adjoint I_{ρ} if and only if Q has a right adjoint Q_{ρ}. In this case we call the sequence a localization sequence. We follow Rickard [25] and write E_{S} = I ∘ I_{ρ} and F_{S} = Q_{ρ} ∘ Q. Note that E_{S} and F_{S} are idempotent functors.

Let us collect the basic facts of such a localization sequence.

Lemma 9.1. A localization sequence S ⟶^{I} T ⟶^{Q} T/S has the following properties. (i) The functor Q_{ρ} is fully faithful and identifies T/S with the full subcategory

S^{⊥} = {Y ∈ T | Hom_{T}(X, Y) = 0 for all X ∈ S}.

(ii) We have

S = {X ∈ T | Hom_{T}(X, Y) = 0 for all Y ∈ S^{⊥}}.

(iii) For each object X of T, there exists up to isomorphism a unique exact

triangle

X′ ⟶ X ⟶ X′′ ⟶ X′[1]

with X′ ∈ S and X′′ ∈ S^{⊥}. (iv) For each object X of T, the adjunction morphisms E_{S}X → X and X →

F_{S}X fit into an exact triangle

E_{S}X ⟶ X ⟶ F_{S}X ⟶ E_{S}X[1].

There is a finite variant of Bousfield localization for compactly generated triangulated categories which Rickard [25] introduced into representation theory. Here we use the tensor product ⊗_{k} which is defined on K(Inj kG).

Let S_{0} be a class of compact objects of K(Inj kG) and denote by S = Loc(S_{0}) the localizing subcategory of K(Inj kG) which is generated by S_{0}. Then the sequence

S ⟶^{I} K(Inj kG) ⟶^{Q} K(Inj kG)/S

of canonical functors is a localization sequence. Moreover, S is compactly generated and the subcategory S^{c} of compact objects equals the thick subcategory Thick(S_{0}) of K^{c}(Inj kG) which is generated by S_{0}.

Now suppose that S_{0} is a thick tensor ideal of K^{c}(Inj kG). Thus S_{0} is by definition a thick subcategory and a tensor ideal, that is, X ⊗_{k} Y belongs to S_{0} for all X in S_{0} and Y in K^{c}(Inj kG). Then S = Loc(S_{0}) is a localizing tensor ideal and therefore the exact triangle

E_{S}ik ⟶ ik ⟶ F_{S}ik ⟶ E_{S}ik[1]

induces for each X in K(Inj kG) an exact triangle

X ⊗_{k} E_{S}ik ⟶ X ⊗_{k} ik ⟶ X ⊗_{k} F_{S}ik ⟶ X ⊗_{k} E_{S}ik[1]

which is isomorphic to

E_{S}X ⟶ X ⟶ F_{S}X ⟶ E_{S}X[1].

10. Varieties

In this section, we indicate how the theory of support for kG-modules from [4] may be modified to work in K(Inj kG).

Let H^{∗}(G, k) be the cohomology ring of G, and denote by Spec^{∗} H^{∗}(G, k) the set of homogeneous prime ideals of H^{∗}(G, k). We consider the Zariski topology on Spec^{∗} H^{∗}(G, k), that is, a subset of Spec^{∗} H^{∗}(G, k) is Zariski closed if it is of the form

V(𝔞) = {𝔭 ∈ Spec^{∗} H^{∗}(G, k) | 𝔞 ⊆ 𝔭}

for some homogeneous ideal 𝔞 of H^{∗}(G, k). We write 𝔪 = H^{+}(G, k) for the unique maximal ideal of H^{∗}(G, k) and obtain the projective variety

Proj H^{∗}(G, k) = Spec^{∗} H^{∗}(G, k) \ V (H^{+}(G, k)) = Spec^{∗} H^{∗}(G, k) \ {𝔪}.

Now fix a specialization closed subset V ⊆ Spec^{∗} H^{∗}(G, k), that is, 𝔭 ⊆ 𝔮 and 𝔭 ∈ V imply 𝔮 ∈ V. We obtain the localizing tensor ideal

S_{V} = Loc({X ∈ K^{c}(Inj kG) | H^{∗}(G, X)_{𝔮} = 0 for all 𝔮 ∈ Spec^{∗} H^{∗}(G, k) \ V}), of K(Inj kG). To simplify our notation, we write

E_{V} = E_{S_{V}} and F_{V} = F_{S_{V}}.

Now fix 𝔭 ∈ Spec^{∗} H^{∗}(G, k) and let

V_{𝔭} = {𝔮 ∈ Spec^{∗} H^{∗}(G, k) | 𝔭 ⊆ 𝔮} and W_{𝔭} = {𝔮 ∈ Spec^{∗} H^{∗}(G, k) | 𝔮 ⊈ 𝔭}.

Note that W_{𝔭} \ V_{𝔭} = {𝔭}. We define

κ_{𝔭} = (F_{W_{𝔭}} ∘ E_{V_{𝔭}})ik ≅ (E_{V_{𝔭}} ∘ F_{W_{𝔭}})ik.

For example, one computes

κ_{𝔪} = (E_{V_{𝔪}} ∘ F_{W_{𝔪}})ik = E_{V_{𝔪}}ik = pk.

Given X in K(Inj kG), we have

X ⊗_{k} κ_{𝔭} ≅ (F_{W_{𝔭}} ∘ E_{V_{𝔭}})X ≅ (E_{V_{𝔭}} ∘ F_{W_{𝔭}})X

and the variety of X is by definition

V_{G}(X) = {𝔭 ∈ Spec^{∗} H^{∗}(G, k) | X ⊗_{k} κ_{𝔭} ≠ 0}.

Lemma 10.1. Let 𝔭 ∈ Spec^{∗} H^{∗}(G, k). Then V_{G}(κ_{𝔭}) = {𝔭}.

Proof. The proof is essentially the same as the proof of Lemma 10.4 of [4]. □

Lemma 10.2. A complex X in K(Inj kG) is acyclic if and only if V_{G}(X) is contained in Proj H^{∗}(G, k).

Proof. A complex X is acyclic if and only if X ⊗_{k} pk = 0 if and only if V_{G}(X) ⊆ Proj H^{∗}(G, k). □

It follows that κ_{𝔭} is in K_{ac}(Inj kG) ≅ StMod kG unless 𝔭 = 𝔪, and that these modules agree with the modules κ_{V} introduced in [4].

11. Objects with injective cohomology

Modules over kG with injective cohomology were studied in [6]. In this section, we indicate how this theory may be modified to work in K(Inj kG). The theory is actually easier than in StMod kG, because it does not involve a discussion of injective modules over the non-Noetherian Tate cohomology ring.

Let I be an injective H^{∗}(G, k)-module. Then the functor from K(Inj kG) to the category of abelian groups which takes an object X to

Hom_{H^{∗}(G,k)}(H^{∗}(G, X), I)

takes triangles to exact sequences and coproducts to products. So by Brown representability (see Neeman [22]) there is an object T (I) in K(Inj kG) satisfying

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

The assignment I ↦ T (I) extends via Yoneda’s lemma to a functor

T: Inj H^{∗}(G, k) → K(Inj kG).

A dimension shifting argument (see [6, §3]) shows that we obtain an isomorphism of graded H^{∗}(G, k)-modules

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

In particular, setting X = ik, we see that H^{∗}(G, T (I)) ≅ I for all I in Inj H^{∗}(G, k), and setting X = T (I′) we see that

Hom_{K(Inj kG)}(T (I′), T (I)) ≅ Hom_{H^{∗}(G,k)}(I′, I),

so that the functor T is fully faithful. In particular, if 𝔭 ∈ Spec^{∗} H^{∗}(G, k) and I_{𝔭} is the injective hull of H^{∗}(G, k)/𝔭, then using Matlis [20], we have

End_{K(Inj kG)}^{∗}(T (I_{𝔭})) ≅ H^{∗}(G, k)𝔭^{∧} = lim_{n^{⟵}} H^{∗}(G, k)_{𝔭}/𝔭_{𝔭}^{n}.

Proposition 11.1. Let I_{𝔪} = H_{∗}(G, k), the graded dual of H^{∗}(G, k). This is the injective hull of the trivial H^{∗}(G, k)-module k = H^{∗}(G, k)/𝔪 where 𝔪 = H^{+}(G, k) is the maximal ideal generated by the positive degree elements. Then T (I_{𝔪}) ≅ pk, the projective resolution of k.

Proof. The proof is essentially the same as the proof of Lemma 3.1 of [6]. □

Proposition 11.2. Let H be a subgroup of G, and write T_{G} and T_{H} for the functor T with respect to kG and kH respectively. If I is an injective H^{∗}(G, k)-module, we have

T_{G}(I)↓_{H} ≅ T_{H}(Hom_{H^{∗}(G,k)}^{∗}(H^{∗}(H, k), I)).

Proof. The proof is essentially the same as the proof of Proposition 7.1 of [6]. □

Proposition 11.3. Let 𝔭 ∈ Spec^{∗} H^{∗}(G, k). Then V_{G}(T (I_{𝔭})) = {𝔭}.

Proof. The proof is essentially the same as the proof of Theorem 7.3 of [6]. □

It follows that T (I_{𝔭}) is in K_{ac}(Inj kG) ≅ StMod kG unless 𝔭 = 𝔪, and that these objects agree with the objects of the same name constructed in [6].

Theorem 11.4. Let 𝔭 be a homogeneous prime ideal in H^{∗}(G, k), and let d be the Krull dimension of H^{∗}(G, k)/𝔭. Then

κ_{𝔭} ≅ T (I_{𝔭}[d]).

Proof. If d > 0 then both objects are in K_{ac}(Inj kG) ≅ StMod kG and the theorem is proved in [3, 5]. If d = 0 then 𝔭 = 𝔪 and both sides are isomorphic to the projective resolution pk. □

12. Chouinard and Dade

In this section we describe the analogues in K(Inj kG) of the theorem of Chouinard [12] and of Benson, Carlson and Rickard’s version [4] of Dade’s lemma [13].

Theorem 12.1. Let G be finite group and k a field of characteristic p. An object in K(Inj kG) is semi-injective, respectively semi-projective, respectively zero, if and only if its restriction to every elementary abelian p-subgroup of G is semi-injective, respectively semi-projective, respectively zero.

Proof. It follows from the recollement (6.3) that an object X in K(Inj kG) is semiinjective, resp. semi-projective, if and only if Hom_{k}(tk, X) = 0, resp. X ⊗_{k} tk = 0. By Chouinard’s theorem [12] in StMod kG, this is true if and only if the restriction of Hom_{k}(tk, X), resp. X ⊗_{k} tk to each elementary abelian p-subgroup E of G is zero. This is equivalent to the statement that the restriction of X to each such E is semi-injective, resp. semi-projective.

If an object X in K(Inj kG) restricts to zero on every elementary abelian psubgroup then it is acyclic, so it is in K_{ac}(Inj kG) ≃ StMod(kG). So we can apply Chouinard’s theorem in StMod(kG) to deduce that X ≅ 0. □

Now if E = ⟨g_{1}, …, g_{r}⟩ is an elementary abelian group of rank r, we write X_{i} for the element g_{i} − 1 ∈ J(kE), the radical of the group algebra. If K is an extension field of k, and λ = (λ_{1}, …, λ_{r}) is a non-zero point in affine space 𝔸^{r}(K), then

X_{λ} = λ_{1}X_{1} + ⋯ + λ_{r}X_{r}

is an element of J(KE) satisfying X_{λ}^{p} = 0, and ⟨1 + X_{λ}⟩ is a cyclic subgroup of order p in the group algebra KE. It is called a cyclic shifted subgroup of E over K.

Theorem 12.2. An object in K(Inj kE) is semi-injective, respectively semi-pro-jective, respectively zero if and only if for all extension fields K of k and all cyclic shifted subgroups of E over K the restriction is semi-injective, respectively semiprojective, respectively zero.

Proof. The proof follows the same lines as the proof of Theorem 12.1, using the version of Dade’s lemma in [4, Theorem 5.2] instead of Chouinard’s theorem. We also need to observe that

K ⊗_{k} Hom_{k}(tk, X) ≅ Hom_{K}(tK, K ⊗_{k} X)

and

K ⊗_{k} (X ⊗_{k} tk) ≅ (K ⊗_{k} X) ⊗_{K} tK.  □

Remark 12.3. As in [4], it suffices to check the hypothesis for K the algebraic closure of an extension of k of transcendence degree r − 1.

13. Homotopy colimits and localizing subcategories

The goal of this section is to show that in the stable module category

StMod kG ≃ K_{ac}(Inj kG),

the homotopy category of complexes of injectives K(Inj kG) and the derived category D(Mod kG), localizing subcategories are closed under taking filtered colimits in the corresponding category of chain complexes and chain homomorphisms. This amounts to filling in the details of arguments of Bousfield and Kan [9] and Bökstedt and Neeman [8] for the sake of easy access.

Let C denote one of the categories K_{ac}(Inj kG), D(Mod kG), K(Inj kG) (the arguments work in other situations, but it seems difficult to make precise the conditions on C). Let I be a small category, and let ϕ: I → C be a covariant functor. Then we call ϕ an I-diagram in C. We define the homotopy colimit of the diagram ϕ to be the total complex of the double complex formed from finite chains of maps in I in the following manner.

(13.1)  ⋯ ⟶^{d_{3}} ⨁_{i→j→k} ϕ(i) ⟶^{d_{2}} ⨁_{i→j} ϕ(i) ⟶^{d_{1}} ⨁_{i} ϕ(i)

We regard this as a complex of objects in C, where the differentials are alternating sums over deleted objects in the chain in the usual way. So for example d_{1} takes the copy of ϕ(i) indexed by i ⟶^{α} j via ϕ(α) to ϕ(j) minus the identity to ϕ(i); while d_{2} takes the copy of ϕ(i) indexed by i ⟶^{α} j ⟶^{β} k via ϕ(α) to the copy of ϕ(j) indexed by j ⟶^{β} k minus the identity to the copy of ϕ(i) indexed by i ⟶^{β∘α} k plus the identity to the copy of ϕ(i) indexed by i ⟶^{α} j. It is easy to see that d_{j} ∘ d_{j+1} = 0. Note that the cokernel of d_{1} is \underrightarrow{colim}_{I} ϕ. We write \underrightarrow{hocolim}_{I} ϕ or

\underrightarrow{hocolim}_{i∈I} ϕ(i) for the homotopy colimit.

We say that I is a right filter if it is a small category satisfying:

(i) given objects x and y in I there exists an object z in I and arrows x → z and y → z, and

(ii) given objects x and y in I and arrows f, g: x → y, there exists an object z in I and an arrow α: y → z such that α ∘ f = α ∘ g.

For example, I could be a poset in which every pair of elements has an upper bound. If I is a right filter, then an I-diagram ϕ: I → C is called a filtered system in C. We assume that every filtered system in C has a colimit, which we write as \underrightarrow{colim}_{I} ϕ or as \underrightarrow{colim}_{i∈I} ϕ(i).

Whether I is a filtered system or a more general small category, there is an obvious map \underrightarrow{hocolim}_{I} ϕ → \underrightarrow{colim}_{I} ϕ.

Lemma 13.2 (Bousfield and Kan [9]). Let ϕ be an I-diagram in C. Then

\underrightarrow{hocolim}_{I} ϕ → \underrightarrow{colim}_{I} ϕ

is an equivalence.

Proof. In the case where I has a terminal object, say ℓ, there is a homotopy on the complex (13.1) sending the copy of ϕ(i) indexed by i → ⋯ → j to the copy in one degree higher indexed by i → ⋯ → j → ℓ. This is a homotopy from the identity to the projection onto the subcomplex consisting of the single copy of ϕ(ℓ) in degree zero. This proves that the map from the homotopy colimit to the colimit is an equivalence (i.e., passes down to an isomorphism in the corresponding homotopy category) in this case.

The homotopy colimit can be written as a colimit of homotopy colimits over smaller diagrams, so we have

\underrightarrow{hocolim}_{I} ϕ = \underrightarrow{colim}_{ℓ∈I} \underrightarrow{hocolim}_{I/ℓ} ϕ → \underrightarrow{colim}_{ℓ∈I} \underrightarrow{colim}_{I/ℓ} ϕ = \underrightarrow{colim}_{I} ϕ.

Since I/ℓ has a terminal object,

\underrightarrow{hocolim}_{I/ℓ} ϕ → \underrightarrow{colim}_{I/ℓ} ϕ

is an equivalence, and it remains to prove that a colimit of equivalences is an equivalence. This is where the mild assumptions on the category C come in. Bousfield and Kan were working in the homotopy category of simplicial sets, where equivalences are detected by maps from spheres, and any such map to the filtered colimit factors through some term in the filtered system.

For a countable filtered system, we can argue as follows. If there is no terminal object, then we may choose a cofinal subsystem consisting of a countable sequence of objects and maps

ϕ(0) ⟶^{α_{0}} ϕ(1) ⟶^{α_{1}} ϕ(2) ⟶^{α_{2}} · · · .

Then the colimit fits into a triangle

⨁_{n} ϕ(n) ⟶^{1−α} ⨁_{n} ϕ(n) → \underrightarrow{colim}_{n} ϕ(n).

It follows that a colimit of equivalences is an equivalence in this case. So it is only for uncountable filtered systems that there is any problem.

In the category StMod kG ≃ K_{ac}(Inj kG), equivalences are detected by maps from the modules Ω^{n}S for n ∈ ℤ and S simple, in the sense that for a map f: M → N, if for all n ∈ ℤ and S simple

f_{∗}: \underline{Hom}_{kG}(Ω^{n}S, M) → \underline{Hom}_{kG}(Ω^{n}S, N)

is an isomorphism, then f is an equivalence. So the argument of Bousfield and Kan works here: any map from Ω^{n}S to a filtered colimit factors through some term in the filtered system.

The same argument works in D(Mod kG), where equivalences are detected by maps from perfect complexes, and any map from a perfect complex to a filtered colimit factors through some object in the system.

For the category K(Inj kG), we pass to K(Mod kG) and use the fact that for each simple kG-module S the injective resolution S → iS induces an isomorphism

Hom_{K(Mod Λ)}(iS, X) ≅ Hom_{K(Mod Λ)}(S, X)

for all X in K(Inj kG) by [18, Lemma 2.1]. In K(Mod kG) any map from S to a filtered colimit factors through some object in the system since S is finitely presented. Thus equivalences in K(Inj kG) are detected by maps from the injective resolutions iS of simple kG-modules S. □

Theorem 13.3. Let L be a localizing subcategory of C. Then L is closed under taking filtered colimits in the underlying category of chain complexes.

Proof. According to Lemma 13.2, it suffices to show that the homotopy colimit is in L.

For n ≥ 0, write X(n) for the total complex of the truncation of the sequence (13.1) consisting of just the last n + 1 objects and the maps d_{n}, …, d_{1}. Since each ϕ(i) is in L and L is closed under direct sums, each of the terms in (13.1) is in L, and so by induction on n, X(n) is in L.

There are inclusions α_{n}: X(n) → X(n + 1), and we have a short exact sequence of complexes

0 → ⨁_{n} X(n) ⟶^{1−α} ⨁_{n} X(n) → \underrightarrow{colim}_{n} X(n) → 0.

The corresponding triangle shows that

\underrightarrow{hocolim}_{I} ϕ = \underrightarrow{colim}_{n} X(n)

is in L. □

14. K(Inj kE) for an elementary abelian 2-group E

Let

E = ⟨g_{1}, …, g_{r}⟩ ≅ (ℤ/2)^{r}

be an elementary abelian 2-group of rank r, and let k be a field of characteristic two. Let

H^{∗}(E, k) = k[x_{1}, …, x_{r}].

where the polynomial generators x_{1}, …, x_{r} have degree one. The purpose of this section is to give an equivalence of triangulated categories

K(Inj kE) ≃ D_{dg}(k[x_{1}, …, x_{r}]).

This can be viewed as a version of Bernstein–Gel′fand–Gel′fand duality [7], and is also related to a construction of Carlsson [11].

First we discuss the cyclic group of order two. The discussion begins with the observation that the reduced bar construction on a cyclic group of order two is the minimal resolution. The Alexander–Whitney map on the reduced bar construction is strictly associative, and so it follows that the minimal resolution supports a strictly associative comultiplication. Applying Hom_{k(ℤ/2)}(−, k) to the reduced bar construction gives a differential graded algebra quasi-isomorphic to cochains on B(ℤ/2). From this, it follows that we have a quasi-isomorphism of differential graded algebras

C^{∗}(B(ℤ/2); k) ≃ H^{∗}(ℤ/2, k)

where the right hand side is regarded as a differential graded algebra with zero differential. A differential graded algebra is said to be formal if it is quasi-isomorphic to its cohomology. The statement above says that C^{∗}(B(ℤ/2); k) is formal.

Using the Künneth theorem and the Eilenberg–Zilber theorem, it follows that C^{∗}(BE; k) is also formal, since we have quasi-isomorphisms

C^{∗}(BE; k) ≃ C^{∗}(B(ℤ/2); k) ⊗_{k} ⋯ ⊗_{k} C^{∗}(B(ℤ/2); k)
≃ H^{∗}(ℤ/2, k) ⊗_{k} ⋯ ⊗_{k} H^{∗}(ℤ/2, k)
≅ H^{∗}(E, k) = k[x_{1}, …, x_{r}].

Thus we have equivalences of categories

(14.1)  D_{dg}(C^{∗}(BE; k)) ≃ D_{dg}(H^{∗}(E, k)) = D_{dg}(k[x_{1}, …, x_{r}]).

Theorem 14.2. Let E be an elementary abelian 2-group and k a field of characteristic two. Then there is an equivalence of triangulated categories

K(Inj kE) ≃ D_{dg}(H^{∗}(E, k)) = D_{dg}(k[x_{1}, …, x_{r}]).

Proof. This follows by combining the equivalences

K(Inj kE) ≃ D_{dg}(End_{kE}(ik)) ≃ D_{dg}(C^{∗}(BE; k))
≃ D_{dg}(H^{∗}(E, k)) = D_{dg}(k[x_{1}, …, x_{r}])

coming from Proposition 3.1, Theorem 4.1 and equation (14.1). □

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

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Institut für Mathematik, Universität Paderborn, 33095 Paderborn, Germany

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