MODULE CATEGORIES FOR GROUP ALGEBRAS OVER COMMUTATIVE RINGS

DAVE BENSON, SRIKANTH B. IYENGAR, HENNING KRAUSE WITH AN APPENDIX BY GREG STEVENSON

Abstract. We develop a suitable version of the stable module category of a finite group G over an arbitrary commutative ring k. The purpose of the construction is to produce a compactly generated triangulated category whose compact objects are the finitely presented kG-modules. The main idea is to form a localisation of the usual version of the stable module category with respect to the filtered colimits of weakly injective modules. There is also an analogous version of the homotopy category of weakly injective kG-modules and a recollement relating the stable category, the homotopy category, and the derived category of kG-modules.

Contents

1. Introduction 1 2. Weakly projective and weakly injective modules 3 3. An example 5 4. Filtered colimits of weakly injective modules 6 5. Stable module categories 8 6. Base change 10 7. Thick subcategories 14 8. The homotopy category of weakly injective modules 16 Appendix A. Disconnecting spectra via localisation sequences 20 References 23

1. Introduction

The purpose of this paper is to develop a version of the stable module category of a finite group G over an arbitrary commutative ring k. We start with the category Mod(kG) whose objects are all kG-modules and whose morphisms are the module homomorphisms. If k were a field, the stable module category StMod(kG) would be formed by taking the same objects, but with morphisms

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

where PHom_{kG}(M, N) is the linear subspace consisting of those morphisms that factor through some projective kG-module. This is then a triangulated category that is compactly generated, with compact objects the finitely generated kG-modules; see [12]. The crucial property that makes this work is that the group algebra kG is self injective, so projectives modules are the same as the injective modules.

Version from 5th August 2012.

The authors are grateful to the Mathematische Forschungsinstitut at Oberwolfach for support through a “Research in Pairs” visit. The research of the second author was partly supported by NSF grant DMS 1201889.

Over an arbitrary commutative ring k however, projective modules are no longer injective, so one needs to adjust this construction. Instead, we use the fact that weakly projective modules are weakly injective, where “weakly” means “relative to the trivial subgroup”. Thus we define PHom_{kG}(M, N) to be the morphisms that factor through a weakly projective module, and construct StMod(kG) as above. This is again a triangulated category, and the compact objects in it are the finitely presented kG-modules; we write stmod(kG) for the full subcategory of modules isomorphic in StMod(kG) to a finitely presented module.

This is not quite good enough to make a decent stable module category, as there are several problems with this construction:

– The category StMod(kG) is usually not compactly generated by stmod(kG).

– The functor k_{(|G|)} ⊗_{k} − : StMod(kG) → StMod(k_{(|G|)}G) is not an equivalence of categories, where k_{(|G|)} denotes k localised by inverting all integers coprime to |G|.

The reason for both of these is that filtered colimits of weakly projective modules are not necessarily weakly projective; see Example 3.3.

To remedy these deficiencies, we form the Verdier quotient of StMod(kG) with respect to the localising subcategory of cw-injectives, namely filtered colimits of weakly injective modules. The principal features of this quotient category, which we denote StMod^{cw}(kG), are collected in the result below.

Theorem 1.1. The following statements hold:

(1) The category StMod^{cw}(kG) is a compactly generated triangulated category.

(2) The natural functor stmod(kG) → StMod^{cw}(kG) identifies stmod(kG) with the full subcategory consisting of the compact objects.

(3) The functor k_{(|G|)} ⊗ −: StMod^{cw}(kG) → StMod^{cw}(k_{(|G|)}G) is an equivalence of triangulated categories.

(4) An object in StMod^{cw}(kG) is zero if and only if its image in StMod^{cw}(k_{(p)}G) is zero for each prime p dividing |G|.

These results are proved in Sections 5 and 6, building on properties of weakly injective modules and their filtered colimits, presented in the preceding sections.

The equivalence in (3) above is deduced from general results on StMod^{cw}(kG) in Section 6, concerning base change along a homomorphism k → k′ of commutative rings, and leading to the following local-global principle:

Theorem 1.2. A kG-module M is cw-injective if and only if M_{𝔭} is a cw-injective k_{𝔭}G-module for every maximal ideal 𝔭 ⊆ k.

In Section 8 we develop a version of the homotopy category of weakly injective kG-modules, namely, we introduce a Verdier quotient K_{pur}(WInj kG) of the homotopy category of weakly injective kG-modules that has the following properties.

Theorem 1.3. The category K_{pur}(WInj kG) is a compactly generated triangulated category and the compact objects form a full triangulated category that is equivalent to the bounded derived category D^{b}(mod kG) of finitely presented kG-modules. Moreover, there is a recollement

\xymatrix{StMod^{cw}(kG) \ar[r] & K_{pur}(WInj kG) \ar[r] \ar[l] \ar[l] & D_{pur}(Mod kG) \ar[l] \ar[l]}

where D_{pur}(Mod kG) denotes the pure derived category of kG-modules.

When k is a field, the thick tensor ideals of stmod(kG), and the localising tensor ideals of StMod(kG), have been classified, in terms of appropriate subsets of the Zariski spectrum of the cohomology ring H^{∗}(G, k); see [4, 5]. One of our motivations for undertaking the research reported here is to investigate possible extensions of these results to a general commutative ring k. The tensor product of kG-modules again induces a tensor triangulated structure on the stable module category. However, in Section 7 we produce an infinite chain of thick tensor ideals of stmod(ℤG) for any group G with at least two elements, and explain why this implies that the classification in terms of cohomological support [4] does not carry over verbatim to this context.

The same example shows that the spectrum of the tensor triangulated category stmod(ℤG) in the sense of Balmer [1] is disconnected. Again, this is in contrast to the case when the ring of coefficients is a field. A detailed explanation of this phenomenon can be found in Appendix A.

2. Weakly projective and weakly injective modules

In this section we recall basic results on relative projectives and relative injectives for group algebras over commutative rings, obtained by Gaschütz [10] and D. G. Higman [13], building on the work of Maschke and Schur. See also [3, §3.6]. The novelty, if any, is a systematic use of Quillen’s [26] notion of exact categories.

Throughout this work G is a finite group, k a commutative ring of coefficients and kG the group algebra. The category of kG-modules is denoted Mod(kG), and mod(kG) is its full subcategory consisting of all finitely presented modules.

Restriction to the trivial subgroup. The functor Mod(kG) → Mod(k) that restricts a kG-module to the trivial subgroup has a left adjoint kG⊗_{k} − and a right adjoint Hom_{k}(kG, −). These functors are isomorphic, since G is finite; we identify them, and write M↑^{G} for kG ⊗_{k} M; the G action is given by h(g ⊗ m) = hg ⊗ m. Adjunction yields, for each kG-module M, natural morphisms of kG-modules

ι_{M}: M ⟶ M↑^{G}  and π_{M}: M↑^{G} ⟶ M where
ι_{M}(m) = ∑_{g∈G} g ⊗ g^{−1}m and π_{M}(g ⊗ m) = gm .

Note that ι_{M} is a monomorphism whilst π_{M} is an epimorphism.

Weakly projective and weakly injective modules. An exact category in the sense of Quillen [26] (see also Appendix A of Keller [16]) is an additive category with a class of short exact sequences satisfying certain axioms. For example, every additive category is an exact category with respect to the split exact structure.

We give Mod(kG) the structure of an exact category with respect to the ksplit short exact sequences. Said otherwise, we consider for Mod(k) the split exact structure, and declare that a sequence of kG-modules is k-split exact if it is split exact when restricted to the trivial subgroup.

For any kG-module M, the following sequences of kG-module are k-split exact:

0 ⟶ M ⟶^{ι_{M}} M↑^{G} ⟶ Coker ι_{M} ⟶ 0

0 ⟶ Ker π_{M} ⟶ M↑^{G} ⟶^{π_{M}} M ⟶ 0 .

Indeed, each element of M↑^{G} can be written uniquely as ∑_{g∈G} g ⊗_{k} m_{g}. The map assigning such an element to m_{1}, where 1 is the identity of G, is a k-splitting of ι_{M}, whilst the map M → M↑^{G} assigning m to 1 ⊗_{k} m is a k-splitting of π_{M}.

Definition 2.1. A kG-module P is said to be weakly projective if Hom_{kG}(P, −) preserves exactness of k-split exact sequences; equivalently, given a k-split exact sequence 0 → L → M → N → 0 of kG-modules, any morphism P → N lifts to M.

In the same vein, a kG-module P is weakly injective if Hom_{kG}(−, P) preserves exactness of k-split exact sequences; equivalently, given a k-split exact sequence 0 → L → M → N → 0 of kG-modules, any morphism L → P extends to M.

The trace map. We recall a characterisation of weakly projective and injective kG-modules in terms of the trace map.

Definition 2.2. If M and N are kG-modules, the trace map

Tr_{G}: Hom_{k}(M, N) → Hom_{kG}(M, N)

is defined by

Tr_{G}(θ)(m) = ∑_{g∈G} g(θ(g^{−1}m)) for m ∈ M.

A straightforward calculation justifies the following properties of the trace map.

Lemma 2.3. Fix a homomorphism θ ∈ Hom_{k}(M, N).

(1) If α: M′ → M is a morphism of kG-modules, then Tr_{G}(θα) = Tr_{G}(θ)α.

(2) If β: N → N′ is a morphism of kG-modules, then Tr_{G}(βθ) = β Tr_{G}(θ). □

For any kG-module M, let θ_{M} ∈ End_{k}(M↑^{G}) be the map given defined by

θ_{M}(g ⊗ m) = \begin{cases}1 ⊗ m & g = 1 \\ 0 & g ≠ 1. \end{cases}

Lemma 2.4. The map Tr_{G}(θ_{M}) is the identity morphism on M↑^{G}. □

Theorem 2.5. Let ρ: M → N be a morphism of kG-modules. The following conditions are equivalent:

(1) In Mod(kG), the map ρ factors through the surjection π_{N}: N↑^{G} → N.

(2) In Mod(kG), the map ρ factors through the injection ι_{M}: M → M↑^{G}.

(3) There exists θ ∈ Hom_{k}(M, N) such that Tr_{G}(θ) = ρ.

Proof. (1) ⇒ (3): If there exists a homomorphism ρ′: M → N↑^{G} of kG-modules with ρ = π_{N}ρ′ then ρ = Tr_{G}(π_{N}θ_{N}ρ′), by Lemmas 2.3 and 2.4.

(2) ⇒ (3) follows as above.

(3) ⇒ (1) and (2): Suppose there exists θ ∈ Hom_{k}(M, N) with Tr_{G}(θ) = ρ. The map ρ′ ∈ Hom_{kG}(M, N↑^{G}) with ρ′(m) = ∑_{g∈G} g ⊗ θ(g^{−1}m) satisfies π_{N}ρ′ = ρ, and ρ′′ ∈ Hom_{kG}(M↑^{G}, N) with ρ′′(g ⊗ m) = gθ(m) satisfies ρ′′ι_{M} = ρ. □

Criteria for weak projectivity and weak injectivity. The result below is due to D. G. Higman [13]. Part (5) is called Higman’s criterion for weak projectivity.

Theorem 2.6. The following conditions on a kG-module P are equivalent: (1) The module P is weakly projective.

(2) The natural morphism π_{P} : P ↑^{G} → P is a split surjection.

(3) The module P is weakly injective.

(4) The natural morphism ι_{P} : P → P ↑^{G} is a split injection.

(5) There exists θ ∈ End_{k}(P) such that Tr_{G}(θ) = id_{P}.

(6) There exists a kG-module M such that P is a direct summand of M↑^{G}. In particular, M↑^{G} is weakly projective and weakly injective for any kG-module M.

Proof. We first prove that conditions (1), (2), and (5) are equivalent; analogous arguments settle the equivalence of (3), (4) and (5).

(1) ⇒ (2): If P is weakly projective then the identity morphism of P lifts to a splitting of P ↑^{G} → P.

(2) ⇒ (5): If α: P → P ↑^{G} is a splitting then by Lemma 2.3 we have

Tr_{G}(π_{P} θ_{P ↑^{G}}α) = π_{P} Tr_{G}(θ_{P ↑^{G}})α = π_{P} α = id_{P} .

(5) ⇒ (1): Let θ ∈ End_{k}(P) with Tr_{G}(θ) = id_{P}. Given a k-split exact sequence 0 → L → M → N → 0 and a morphism α: P → N of kG-modules, choose a k-lift α̃: P → M of α; it follows from Lemma 2.3 that Tr_{G}(α̃θ) is a kG-lift of α.

(4) ⇒ (6): Take M = P.

(6) ⇒ (5): A module of the form M↑^{G} satisfies (5) by Lemma 2.4. It remains to observe that this property descends to direct summands, by Lemma 2.3. □

The theorem above shows that the exact category Mod(kG) is a Frobenius category. This means that there are enough projective objects and enough injective objects, and that both coincide; see Section I.2 of Happel [12] for details.

When k is a field, weak injectivity is equivalent to injectivity. The following corollary of Theorem 2.6 is thus a generalisation of Maschke’s theorem.

Corollary 2.7. Let k be a commutative ring and G a finite group. The following conditions are equivalent:

(1) Each kG-module is weakly injective.

(2) The trivial kG-module k is weakly injective.

(3) The order of G is invertible in k.

Proof. Both non-trivial implications follow from Higman’s criterion: (2) ⇒ (3) because Tr_{G}(θ) = |G|θ for any θ ∈ End_{k}(k) ≅ k, whilst (3) ⇒ (1) because Tr_{G}(\frac{1}{|G|} id_{P}) = id_{P} for any kG-module P. □

3. An example

In this section, we construct modules that are filtered colimits of weakly injective kG-modules but are not themselves weakly injective. In what follows, given an integer n, we write k_{(n)} for the localisation of k that inverts all integers r such that (r, n) = 1. For any kG-module M, we denote M_{(n)} the kG-module k_{(n)} ⊗_{k} M; this can also be described as a filtered colimit:

M_{(n)} = colim_{(r,n)=1}(M ⟶^{r} M) .

Definition 3.1. Let k be a commutative ring, G a finite group, and n an integer. For any kG-module M define Γ_{n}M by forming the following exact sequence

(3.1)  0 ⟶ M ⟶ M↑^{G} ⊕ M_{(n)} ⟶ Γ_{n}M ⟶ 0

where the monomorphism sends m ∈ M to (ι_{M}(m), m). This sequence is k-split exact, since ι_{M} is k-split.

Theorem 3.2. If |G| divides n, the kG-module Γ_{n}M is a filtered colimit of weakly injective kG-modules.

Proof. It is expedient to use the stable category, StMod(kG), defined in Section 5.

The kG-module Γ_{n}M is given by the following pushout

\xymatrix{M \ar[r]^{ι_{M}} \ar[d] & M↑^{G} \ar[d] \\ M_{(n)} \ar[r] & Γ_{n}M}

and this is a filtered colimit of pushouts of the form

\xymatrix{M \ar[r]^{ι_{M}} \ar[d]^{r} & M↑^{G} \ar[d] \\ M \ar[r] & M_{(r)}}

with (r, n) = 1. In particular, Γ_{n}M is a filtered colimit of the M_{(r)}. The hypothesis is that |G| divides n, so one gets

Tr_{G}(\frac{n}{|G|} id_{M}) = \frac{n}{|G|} Tr_{G}(id_{M}) = \frac{n}{|G|}(|G| id_{M}) = n id_{M} .

Therefore the multiplication morphism M ⟶^{n} M factors through a weakly injective module, by Theorem 2.5. For each integer r with (r, n) = 1, there exist integers a and b such that na + rb = 1, so the preceding computation implies that the morphism M ⟶^{r} M is an isomorphism in the stable category StMod(kG), with inverse the multiplication by a. It follows that the morphism on the right in the diagram is also an isomorphism. Therefore each M_{(r)} is weakly injective. □

The kG-module Γ_{n}M need not be itself weakly injective.

Example 3.3. Let G be a finite group and set k = ℤ. Given n > 1, the ℤG-module Γ_{n}ℤ constructed above is weakly injective if and only if |G| = 1.

To prove this, first observe that Hom_{ℤ}(ℤ_{(n)}, ℤ) = 0 so Hom_{ℤG}(ℤ_{(n)}, ℤ) = 0 as well. If Γ_{n}ℤ is weakly injective, then the sequence (3.1) splits and ι_{ℤ} is a split monomorphism, since the second component of the inverse of ℤ → ℤ↑^{G} ⊕ ℤ_{(n)} is zero. It follows that ℤ is weakly injective, by Theorem 2.6, and therefore |G| is invertible in ℤ, by Corollary 2.7. Thus |G| = 1. The other direction is clear.

4. Filtered colimits of weakly injective modules

Motivated by the example in the previous section we study filtered colimits of weakly injective modules.

Definition 4.1. We say that a kG-module is cw-injective if it is isomorphic in Mod(kG) to a filtered colimit of weakly injective modules.

Weakly injective modules are cw-injective, but not conversely; see Example 3.3. However, these notions coincide for finitely presented modules, as is proved in part (2) of the next result.

Lemma 4.2. Let M be a kG-module.

(1) The module M is cw-injective if and only if every morphism N → M with N finitely presented factors through a weakly injective module.

(2) If M is finitely presented and cw-injective, then it is weakly injective.

Proof. (1) If M = colim M_{α}, a filtered colimit of weakly injective modules M_{α}, then each morphism N → M with N finitely presented factors through one of the structure morphisms M_{α} → M.

As to the converse, one has that M = colim_{N→M} N, with the colimit taken over the category of morphisms N → M where N runs through all finitely presented kGmodules. Suppose that each morphism N → M factors through a weakly injective module. Thus each morphism N → M factors through N↑^{G}, by Theorem 2.5. The module N↑^{G} is finitely presented and it follows that the morphisms N → M with N weakly injective form a cofinal subcategory. Thus M is a filtered colimit of weakly injective kG-modules.

(2) Let M be a finitely presented kG-module. If M is cw-injective, then (1) implies that the identity morphism id_{M} factors through a weakly injective module. Thus M is weakly injective, by Theorem 2.6. □

Next we present criteria for cw-injectivity in terms of the purity of certain canonical morphisms. For background on purity, see the books of Jensen and Lenzing [15] and Prest [24, 25].

Definition 4.3. An exact sequence

0 → M′ → M → M′′ → 0

of kG-modules is said to be pure exact if for every right kG-module N, the sequence

0 → N ⊗_{kG} M′ → N ⊗_{kG} M → N ⊗_{kG} M′′ → 0

is exact. This is equivalent to the statement that for any finitely presented kGmodule P, each morphism P → M′′ lifts to a morphism P → M, and to the statement that the sequence is a filtered colimit of split exact sequences.

A morphism M → M′′ appearing in a pure exact sequence as above is called a pure epimorphism, whilst M′ → M is called a pure monomorphism.

Criteria for cw-injectivity. The following tests for cw-injectivity are analogues of the criterion for weak injectivity formulated in Theorem 2.6.

Theorem 4.4. The following properties of a kG-module M are equivalent:

(1) The module M is cw-injective.

(2) The natural morphism π_{M}: M↑^{G} → M is a pure epimorphism.

(3) The natural morphism ι_{M}: M → M↑^{G} is a pure monomorphism.

Proof. (1) ⇒ (2) and (3): If M is a filtered colimit of weakly injective modules M_{α}, then M↑^{G} → M is a filtered colimit of split epimorphisms M_{α}↑^{G} → M_{α} and hence is a pure epimorphism. Similarly, M → M↑^{G} is a filtered colimit of split monomorphisms M_{α} → M_{α}↑^{G} and hence is a pure monomorphism.

(2) ⇒ (1): When M↑^{G} → M is a pure epimorphism any morphism N → M with N finitely presented lifts to M↑^{G}, so Lemma 4.2 yields that M is cw-injective.

(3) ⇒ (1): Let N → M be a morphism of kG-modules with N finitely presented. We obtain the following commutative diagram with exact rows.

\xymatrix{0 \ar[r] & N \ar[r] \ar[d] & N↑^{G} \ar[r] \ar[d] & N′ \ar[r] \ar[d] & 0 \\ 0 \ar[r] & M \ar[r] & M↑^{G} \ar[r] & M′ \ar[r] & 0}

When M → M↑^{G} is a pure monomorphism, M↑^{G} → M′ is a pure epimorphism, so the morphism N′ → M′ lifts to M↑^{G}; note that N′ is also finitely presented. Thus the morphism N → M extends to N↑^{G}, so M is cw-injective, by Lemma 4.2. □

Corollary 4.5. The class of cw-injective modules is closed under (1) filtered colimits,

(2) products, and

(3) pure submodules and pure quotient modules.

Proof. We use the characterisation from Lemma 4.2 to verify (1) and (2).

(1) Let M = colim M_{α} be a filtered colimit of cw-injective modules. Each morphism N → M with N finitely presented factors through one of the structure morphisms M_{α} → M. The morphism N → M_{α} then factors through a weakly injective module. Thus M is cw-injective.

(2) Let M = ∏_{α} M_{α} be a product of cw-injective modules and fix a finitely presented module N. Each morphism N → M_{α} factors through a weakly injective module and therefore through the natural morphism N → N↑^{G}. Thus each morphism N → M factors through N↑^{G}, and it follows that M is cw-injective.

(3) Let M → N be a pure monomorphism with N cw-injective. An application of Theorem 4.4 yields that the composite M → N → N↑^{G} is a pure monomorphism; it factors through the natural monomorphism M → M↑^{G}, so the latter is pure as well. Another application of Theorem 4.4 shows that M is cw-injective.

The argument for a pure quotient module is analogous □

5. Stable module categories

In this section we introduce the stable module category StMod(kG) and an appropriate variant which we denote by StMod^{cw}(kG). We begin with a brief discussion of compactly generated triangulated categories.

Compactly generated triangulated categories. Let T be a triangulated category with suspension Σ: T ⟶^{∼} T. For a class C of objects we define its perpendicular categories to be the full subcategories

C^{⊥} = {Y ∈ T | Hom_{T}(Σ^{n}X, Y) = 0 for all X ∈ C, n ∈ ℤ}
⊥C = {X ∈ T | Hom_{T}(X, Σ^{n}Y) = 0 for all Y ∈ C, n ∈ ℤ}.

We write Thick(C) and Loc(C) for the thick subcategory and the localising subcategory, respectively, of T generated by C.

Suppose that T has set-indexed coproducts. An object X ∈ T is compact if the representable functor Hom_{T}(X, −) preserves set-indexed coproducts. The category T is compactly generated if there is a set C of compact objects such that T admits no proper localising subcategory containing C.

The following proposition provides a useful method to construct compactly generated triangulated categories and to identify its subcategory of compact objects. As usual, T^{c} denotes the full subcategory of compact objects in T.

Proposition 5.1. Let T be a triangulated category with set-indexed coproducts, C a set of compact objects in T, and set V = C^{⊥}. Then the composite

Loc(C) ↣ T ↠ T/V

is an equivalence of triangulated categories. Moreover, the category T/V is compactly generated and the equivalence identifies Thick(C) with (T/V)^{c}.

Proof. Set U = Loc(C). Observe that since C consists of compact objects, the Verdier quotient T/U does exist, in that the morphisms between a given pair of objects form a set; see [23, Proposition 9.1.19]. It follows that the Verdier quotient T/V exists, since U^{⊥} = V. Moreover, the composite U ↣ T ↠ T/V is an equivalence, by [23, Theorem 9.1.16], with the warning that there U^{⊥} is denoted ⊥U. The category U is compactly generated by construction. The description of the compacts follows from standard properties of compact objects; see Lemma 2.2 in [22]. □

Corollary 5.2. If T′ ⊆ T is a localising subcategory containing Loc(C), then the natural functor

T′/(V ∩ T′) ⟶ T/V

is an equivalence of triangulated categories.

Proof. This follows from Lemma 5.3 below, since the functor induces an equivalence between the categories of compact objects. □

The following test for equivalence of triangulated categories is implicit in [17, §4.2]; for details see [5, Lemma 4.5].

Lemma 5.3. Let F: S → T be an exact functor between compactly generated triangulated categories and suppose F preserves set-indexed coproducts. If F restricts to an equivalence S^{c} ⟶^{∼} T^{c}, then F is an equivalence of categories. □

The stable module category. The following lemma describes the morphisms that are annihilated when one passes to the stable module category; it is an immediate consequence of Theorems 2.5 and 2.6.

Lemma 5.4. For a morphism M → N of kG-modules, the following conditions are equivalent:

(1) The morphism factors through some weakly projective module.

(2) The morphism factors through the natural epimorphism N↑^{G} → N.

(3) The morphism factors through some weakly injective module.

(4) The morphism factors through the natural monomorphism M → M↑^{G}. □

The stable module category StMod(kG) has the same objects as Mod(kG), but the morphisms are given by

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

where PHom_{kG}(M, N) is the k-submodule of Hom_{kG}(M, N) consisting of morphisms M → N satisfying the equivalent conditions of Lemma 5.4. As Mod(kG) is a Frobenius category, StMod(kG) is a triangulated category; see [12, Section I.2]. The exact triangles in it are induced by the k-split exact sequences of kG-modules.

Let stmod(kG) be the full subcategory of StMod(kG) consisting of objects isomorphic to finitely presented kG-modules. This is a thick subcategory consisting of compact objects. The result below is an immediate consequence of Lemma 4.2.

Lemma 5.5. In StMod(kG), the perpendicular category (stmod kG)^{⊥} coincides with the localising subcategory whose objects are the cw-injective kG-modules. □

Set V = (stmod kG)^{⊥} and modify the stable module category by defining

StMod^{cw}(kG) = StMod(kG)/V.

This Verdier quotient exists by Proposition 5.1, since stmod(kG) consists of compact objects. Given Lemma 5.5, the following result is a special case of Proposition 5.1.

Theorem 5.6. The composite functor

Loc(stmod kG) ↣ StMod(kG) ↠ StMod^{cw}(kG)

is an equivalence of triangulated categories. The category StMod^{cw}(kG) is compactly generated and the equivalence identifies stmod(kG) with the full subcategory consisting of all compacts objects of StMod^{cw}(kG). □

For later use, we record a remark about functors between exact categories.

Lemma 5.7. Let F: C → D be a functor between exact categories. If F admits a right adjoint that is exact, then F preserves projectivity; if it admits a left adjoint that is exact, then F preserves injectivity.

Proof. Suppose F has a right adjoint, say F′, that is exact. For any projective object X of C, it follows from the isomorphism

Hom_{D}(F (X), −) ≅ Hom_{C}(X, F′(−))

that Hom_{D}(F (X), −) maps an epimorphism in D to a surjective map, that is to say, that F (X) is a projective object of D.

The argument for the other case is exactly analogous. □

The tensor product. The category of kG-modules carries a tensor product; it is the tensor product over k with diagonal G-action. Note that this tensor product is exact in each variable with respect to the k-split exact structure.

Lemma 5.8. Let M, N be kG-modules.

(1) If M is weakly injective, then M ⊗_{k} N is weakly injective.

(2) If M is cw-injective, then M ⊗_{k} N is cw-injective.

Proof. The functor − ⊗_{k} N has Hom_{k}(N, −) as a right adjoint that is exact with respect to the k-split exact structure. Thus, Lemma 5.7 implies that − ⊗_{k} N preserves weak projectivity, which coincides with weak injectivity. The functor − ⊗_{k} N preserves filtered colimits, and therefore cw-injectivity as well. □

It follows from the lemma that the tensor product on Mod(kG) passes to a tensor product on StMod(kG); this is compatible with the triangulated structure, so that stmod(kG) and StMod(kG) are tensor triangulated categories. The cw-injective kG-modules form a tensor ideal in StMod(kG) and hence StMod^{cw}(kG) inherits a tensor triangulated structure.

6. Base change

In this section we prove a number of results that track the behaviour of the stable module category StMod^{cw}(kG) under changes of the coefficient ring k. We remind the reader that Mod(kG) is the category of kG-modules with exact structure defined by the k-split short exact sequences.

Let ϕ: k → k′ be a homomorphism of commutative rings. Assigning the element ∑ α_{g}g to ∑ ϕ(α_{g})g is then a homomorphism of rings kG → k′G. We consider the corresponding restriction functor Mod(k′G) → Mod(kG) and the base change functor k′ ⊗_{k} −: Mod(kG) → Mod(k′G).

Exact structure. Base change and restriction are compatible with the exact structure for modules over kG and k′G respectively.

Lemma 6.1. Let ϕ: k → k′ be a ring homomorphism.

(1) Restriction takes k′-split exact sequences of k′G-modules to k-split exact sequences of kG-modules.

(2) Base change takes k-split exact sequences of kG-modules to k′-split exact sequences of k′G-modules.

(3) Restriction and base change preserve weak injectivity and cw-injectivity.

Proof. Parts (1) and (2) are clear. As to (3), it follows from Lemma 5.7 that restriction and base change preserve weak injectivity, because the former is right adjoint to the latter, and they are both exact functors; one uses again the property that weak injectivity and weak projectivity coincide. Since both functors also preserve filtered colimits, they preserve cw-injectivity as well. □

Localisation at |G|. Recall that for any kG-module M and integer n we write M_{(n)} for the localisation of M inverting all integers coprime to n.

Theorem 6.2. For any integer n that is divisible by |G| the localisation functor

(−)_{(n)} : StMod^{cw}(kG) ⟶ StMod^{cw}(k_{(n)}G)

is an equivalence of triangulated categories, and induces an equivalence

(−)_{(n)} : stmod(kG) ⟶ stmod(k_{(n)}G).

Proof. To begin with note that, for any n, localisation does induce an exact functor of triangulated categories, by Lemma 6.1. It suffices to prove that this is an equivalence on StMod^{cw}(kG); the second equivalence follows by restriction to compact objects; see Theorem 5.6.

Consider the restriction functor

F: StMod^{cw}(k_{(n)}G) ⟶ StMod^{cw}(kG).

The composite F (−)_{(n)} is the identity. As to the other composite, for each kGmodule M, the morphism in Definition 3.1 induces an exact triangle

M ⟶ F (M_{(n)}) ⟶ Γ_{n}M ⟶

in StMod(kG), as M↑^{G} is zero there. By Theorem 3.2, the kG-module Γ_{n}M is cw-injective, so the first morphism in the triangle defines a natural isomorphism in StMod^{cw}(kG) from the identity functor to F ((−)_{(n)}). Thus (−)_{(n)} is an equivalence of categories with inverse F. □

Remark 6.3. As Example 3.3 shows, Γ_{n}M need not be weakly injective, so the localisation functor

(−)_{(n)} : StMod(kG) ⟶ StMod(k_{(n)}G)

is not necessarily an equivalence of categories.

Concerning Lemma 6.1, it is easy to construct examples of kG-modules that are weakly injective after base change along a homomorphism k → k′, but are not themselves weakly injective. We now focus on certain classes of homomorphisms where such a phenomenon cannot occur.

Pure monomorphisms. The notion of a pure monomorphism has been recalled in Definition 4.3.

Lemma 6.4. Let ϕ: k → k′ be a homomorphism of rings that is a pure monomorphism of k-modules, and let M be a kG-module. If k′ ⊗_{k} M is cw-injective as a kG-module, then so is M.

Proof. The canonical morphism ϕ ⊗_{k} M: M → k′ ⊗_{k} M is a pure monomorphism of kG-modules, since for any right kG-module N we have

N ⊗_{kG} (ϕ ⊗_{k} M) ≅ ϕ ⊗_{k} (N ⊗_{kG} M)

where the second is a monomorphism, by the hypothesis on ϕ. Thus the desired statement follows from the fact that a pure submodule of a cw-injective module is cw-injective, by Corollary 4.5. □

Proposition 6.5. Let {k → k_{i}}_{i∈I} be homomorphisms of rings such that the induced map X → ∏_{i}(k_{i} ⊗_{k} X) is a monomorphism for any k-module X. If M is a kG-module with the property that k_{i} ⊗_{k} M is cw-injective as kG-module for each i ∈ I, then M is cw-injective.

Proof. We claim that for any kG-module M, the natural morphism

η_{M}: M ⟶ ∏_{i∈I}(k_{i} ⊗_{k} M)

is a pure monomorphism of kG-modules. Indeed, for any right kG-module N the morphism η_{(N⊗_{kG}M)} factors as

N ⊗_{kG} M ⟶^{N⊗_{kG}η_{M}} N ⊗_{kG} ∏_{i∈I}(k_{i} ⊗_{k} M) ⟶ ∏_{i∈I} (k_{i} ⊗_{k} (N ⊗_{kG} M))

where the morphism on the right is the canonical one. Since η_{(N⊗_{kG}M)} is a monomorphism, by hypothesis, so is N ⊗_{kG} η_{M}. This settles the claim.

Now if each k_{i} ⊗_{k} M is cw-injective as a kG-module, then so is their product and hence also its pure submodule M, by Corollary 4.5. □

A local-global principle. From Proposition 6.5 we obtain the following localto-global type results for detecting cw-injectivity. As usual, the localisation of a k-module M at a prime ideal 𝔭 of k is denoted M_{𝔭}.

Corollary 6.6. A kG-module M is cw-injective if and only if M_{𝔭} is a cw-injective k_{𝔭}G-module for every maximal ideal 𝔭 ⊆ k.

Proof. If M is cw-injective, so is each M_{𝔭} by Lemma 6.1. The converse is a special case of Proposition 6.5, since the canonical map X → ∏_{𝔭} X_{𝔭} is a monomorphism for each k-module X; see [20, Theorem 4.6]. □

The next result adds to Theorem 6.2.

Corollary 6.7. Fix an integer n divisible by |G|, and let M be a kG-module. The following conditions are equivalent:

(1) The kG-module M is cw-injective.

(2) The k_{(n)}G-module M_{(n)} is cw-injective.

(3) The k_{(p)}G-module M_{(p)} is cw-injective for every prime p dividing n.

Proof. (1) ⇔ (2): This follows from Theorem 6.2.

(2) ⇒ (3): This is by Lemma 6.1, applied to the homomorphism k_{(n)} → k_{(p)}.

(3) ⇒ (2): We apply Proposition 6.5 to the family k_{(n)} → k_{(p)}, as p ranges over the primes dividing n. To this end, let X be a k_{(n)}-module, and note that there is a natural isomorphism

k_{(p)} ⊗_{k_{(n)}} X ⟶^{∼} ℤ_{(p)} ⊗_{ℤ_{(n)}} X

with X viewed as a ℤ_{(n)}-module via restriction. Since the ideals (p) in ℤ_{(n)}, where p divides n, is a complete list of maximal ideals of ℤ_{(n)}, it follows from [20, Theorem 4.6] that the induced map X → ∏_{p} X_{(p)} is a monomorphism, as desired. □

Epimorphisms. Recall that a homomorphism R → S of rings is an epimorphism (in the category of rings) if and only if the canonical morphism S ⊗_{R} M → M is an isomorphism for each S-module M; equivalently, if the multiplication map S ⊗_{R} S → S is an isomorphism. This means that the restriction functor Mod(S) → Mod(R) is fully faithful. For example, each localisation R → R[Σ^{−1}] inverting the elements of a subset Σ ⊆ R is an epimorphism.

Lemma 6.8. If ϕ: k → k′ is a ring epimorphism, then so is the induced homomorphism kG → k′G.

Proof. If ϕ is an epimorphism, then the induced morphism k′ ⊗_{k} k′ → k′ is an isomorphism. It follows that the induced morphism k′G ⊗_{kG} k′G → k′G is an isomorphism. Thus kG → k′G is a ring epimorphism. □

Proposition 6.9. Let k → k′ be a ring epimorphism and set

U = {M ∈ StMod^{cw}(kG) | k′ ⊗_{k} M = 0}
V = {M ∈ StMod^{cw}(kG) | M ⟶^{∼} k′ ⊗_{k} M .}

Then base change induces an equivalence of triangulated categories

StMod^{cw}(kG)/U ⟶^{∼} StMod^{cw}(k′G)

while restriction induces an equivalence of triangulated categories

StMod^{cw}(k′G) ⟶^{∼} V.

Proof. Restriction followed by base change is naturally isomorphic to the identity. By general principles, this implies that restriction is fully faithful, while base change is up to an equivalence a quotient functor; see Proposition 1.3 of Chap. I in [9]. □

Remark 6.10. Let k → k′ be an epimorphism such that k′ is finitely presented as a k-module, and set

C = {M ∈ stmod(kG) | k′ ⊗_{k} M = 0}
D = {M ∈ stmod(kG) | M ⟶^{∼} k′ ⊗_{k} M}.

Then C and D form thick tensor ideals of stmod(kG) such that the inclusion of C admits a right adjoint while the inclusion of D admits a left adjoint. Moreover, C^{⊥} = D and C = ^{⊥}D. Thus the inclusions of C and D and their adjoints form a localisation sequence

[figure: C; stmod(kG); D.]

In this situation the Balmer spectrum of stmod(kG) is disconnected, provided that C ≠ 0 ≠ D. Recall from [1] that for any essentially small tensor triangulated category T there is an associated spectral topological space Spc T which provides a universal notion of tensor-compatible supports for objects of T. The above localisation sequence yields a decomposition

Spc(stmod kG) = Spc C ∐ Spc D

by Theorem A.5. We refer to Appendix A for further explanations.

In the next section we describe an example that illustrates the preceding remark.

7. Thick subcategories

Let G be a finite group and fix a prime p ∈ ℤ.

For each positive integer n, the canonical homomorphism ϕ_{n}: ℤ → ℤ/p^{n} is an epimorphism of rings, and ℤ/p^{n} is finitely presented over ℤ. Restriction via ϕ_{n} identifies the ℤ/p^{n}G-modules with all ℤG-modules that are annihilated by p^{n}.

Let D_{n} be the full subcategory of stmod(ℤG) whose objects are the ℤG-modules isomorphic to modules annihilated by p^{n}; this is a tensor ideal thick subcategory and restriction via ϕ_{n} identifies it with stmod(ℤ/p^{n}G). Note that any kG-module M is in D_{n} when M ⊗_{ℤ} M is in D_{n}. This is easy to verify and means that D_{n} is radical in the sense of Balmer [1].

When p does not divide |G|, one has D_{n} = 0 for each n, by Theorem 6.2.

Proposition 7.1. When p divides |G| one has D_{n} ≠ D_{n+1} for each n ≥ 1, so these tensor ideal thick subcategories form a strictly increasing chain:

D_{1} ⊊ D_{2} ⊊ D_{3} ⊊ … ⊆ stmod(ℤG) .

Moreover, ℤ ∉ D_{n} for any n, so ⋃_{n⩾1} D_{n} is a proper thick subcategory of stmod(ℤG).

Proof. Fix an n ≥ 1. Evidently ℤ/p^{n+1} is in D_{n+1}; we claim that it is not in D_{n}. Indeed, the inclusion D_{n} → stmod(ℤG) admits ℤ/p^{n} ⊗_{ℤ} − as a left adjoint; see Remark 6.10. It thus suffices to show that the natural morphism ε: ℤ/p^{n+1} → ℤ/p^{n} is not an isomorphism in stmod(ℤG). Assume to the contrary that it admits an inverse, say a morphism σ: ℤ/p^{n} → ℤ/p^{n+1}. Observe that σ(1) = ap, for some integer a, since σ is to be ℤ-linear.

In stmod(ℤG) the morphism εσ is the identity map of ℤ/p^{n}, so that in mod(kG) the endomorphism id −εσ of ℤ/p^{n} admits a factorisation

\xymatrix{ℤ/p^{n} \ar@{-->}[dr]_{κ} \ar[rr]^{id −εσ} &  & ℤ/p^{n} \\  & (ℤ/p^{n})G \ar@{-->}[ur]_{π}}

where π is the natural morphism; see Theorem 2.5. Write κ(1) = ∑_{g∈G} g ⊗ c_{g} with c_{g} ∈ ℤ/p^{n}. Note that c_{g} = c_{1} for each g, since κ is ℤG-linear. Thus the commutativity of the diagram above yields

1 − ap = (id −εσ)(1) = π(∑_{g∈G} g ⊗ c_{1}) = |G|c_{1} in ℤ/p^{n}.

This is a contradiction, for p divides |G|. Thus, D_{n} ≠ D_{n+1}, as claimed.

It remains to verify that ℤ is not in D_{n} for any n ≥ 1. As above, it suffices to verify that the natural morphism ℤ → ℤ/p^{n} is not an isomorphism in stmod(ℤG), but this is clear because Hom_{ℤ}(ℤ/p^{n}, ℤ) = 0. □

In order to connect the preceding computation to the problem of classifying thick subcategories of the stable module category, we recall some basics concerning relative cohomology; see [3, §3.9].

Relative cohomology. Let k be a commutative ring and G a finite group. A relative projective resolution of a kG-module M is a complex P of kG-modules with each P^{i} weakly projective, P^{i} = 0 when i > 0, and equipped with a morphism of complexes ε: P → M of kG-modules such that the augmented complex

⋯ ⟶ P^{−1} ⟶ P^{0} ⟶^{ε} M ⟶ 0

is k-split exact. Relative projective resolutions exists and are unique up to homotopy; they can be constructed starting with the morphism π_{M}: M↑^{G} → M. For any such P, and kG-module N, we set

(7.1)  Ext_{G,1}^{∗}(M, N) = H^{∗}(Hom_{kG}(P, N)) .

These k-modules are independent of the choice of P. Note that

Ext_{G,1}^{∗}(M, N) ≅ Ext_{kG}^{∗}(M, N)

when M is projective over k, since any kG-projective resolution of M is a relative projective resolution. In particular, Ext_{G,1}^{∗}(k, k) ≅ Ext_{kG}^{∗}(k, k).

As usual, Ext_{G,1}^{∗}(M, M) is a graded k-algebra, and Ext_{G,1}^{∗}(M, N) is a graded bimodule with Ext_{G,1}^{∗}(N, N) acting on the left and Ext_{G,1}^{∗}(M, M) on the right. The functor − ⊗_{k} M, with diagonal G-action, induces a homomorphism

ζ_{M}: Ext_{G,1}^{∗}(k, k) → Ext_{G,1}^{∗}(M, M)

of graded k-algebras, and the Ext_{G,1}^{∗}(k, k) action on Ext_{G,1}^{∗}(M, N) through ζ_{M} and through ζ_{N} coincide, up to the usual sign.

For a kG-module M we set

H^{∗}(G, M) = Ext_{G,1}^{∗}(k, M) .

The k-algebra H^{∗}(G, k) is graded-commutative and H^{∗}(G, M) is a graded module over it. Observe that for each kG-module N, there is a natural isomorphism of graded H^{∗}(G, k) modules:

(7.2)  Ext_{G,1}^{∗}(M, N) ≅ H^{∗}(G, Hom_{k}(M, N)) .

Indeed, let P be a kG-projective resolution of k, so that P ⊗_{k} M with diagonal G-action is a relative projective resolution of M. The isomorphism above is induced by the adjunction isomorphism

Hom_{kG}(P ⊗_{k} M, N) ≅ Hom_{kG}(P, Hom_{k}(M, N)) .

Supports. Let now k be a noetherian commutative ring and M a kG-module. The ring H^{∗}(G, k) is then finitely generated as a k-algebra, hence a noetherian ring, and the H^{∗}(G, k)-module H^{∗}(G, M) is finitely generated whenever M is finitely generated; see Evens [8]. From (7.2) one then deduces that Ext_{G,1}^{∗}(M, N) is finitely generated over H^{∗}(G, k) for any pair of finitely generated kG-modules M and N.

We write V_{G} for the homogeneous prime ideals of H^{∗}(G, k) not containing the ideal of elements of positive degree, H^{⩾1}(G, k), and for each finitely generated kG-module M, set

V_{G}(M) = {𝔭 ∈ V_{G} | Ext_{G,1}^{∗}(M, M)_{𝔭} ≠ 0} .

This is a closed subset of V_{G} that we call the support of M. It coincides with

{𝔭 ∈ V_{G} | Ext_{G,1}^{∗}(M, N)_{𝔭} ≠ 0 for some N ∈ mod kG}

since the action of H^{∗}(G, k) on Ext_{G,1}^{∗}(M, N) factors through that of Ext_{G,1}^{∗}(M, M).

To formulate the next result, we recall that k is regular if it is noetherian and each finitely generated k-module has a finite projective resolution. For any k-module M, we write Supp_{k} M for the subset {𝔭 ∈ Spec k | M_{𝔭} ≠ 0} of Spec k.

Proposition 7.2. Let k be a regular ring. Given finitely generated k-modules M, N with trivial G-action, if Supp_{k} M ⊆ Supp_{k} N, then V_{G}(M) ⊆ V_{G}(N).

Proof. It is not hard to verify that

Supp_{k} M = Supp_{k} End_{k}(M) .

Indeed, as M is finitely generated localisation at a prime 𝔭 induces an isomorphism

End_{k}(M)_{𝔭} ≅ End_{k_{𝔭}}(M_{𝔭}) .

Thus, after localisation, we have to verify M = 0 if and only if End_{k}(M) = 0, and this is clear.

Consequently, if Supp_{k} M ⊆ Supp_{k} N, then Supp_{k} End_{k}(M) ⊆ Supp_{k} End_{k}(N). Then Hopkins’ theorem [11, Theorem 11] (see also [21, Theorem 1.5]) implies that End_{k}(M) is in the thick subcategory generated by End_{k}(N), in the derived category of k-modules, and hence in the derived category of kG-modules; this is where we need the trivial action. It follows from (7.2) that V_{G}(M) ⊆ V_{G}(N). □

Remark 7.3. When k is a field, assigning a subcategory C of stmod(kG) to its support, ⋃_{M∈C} V_{G}(M), induces a bijection between the tensor ideal thick subcategories and the specialisation closed subsets of V_{G}. This result is due to Benson, Carlson, and Rickard [4, Theorem 3.4] for k algebraically closed, and proved in [5, Theorem 11.4] in general. It follows from Proposition 7.1 that such a classification result cannot hold when k = ℤ: Pick a prime p dividing |G|; then ℤ/p^{n} is not in the tensor ideal thick subcategory of stmod(ℤG) generated by ℤ/p when n > 1. However, we claim that there are equalities

V_{G}(ℤ/p^{n}) = V_{G}(ℤ/p) for all n ≥ 1.

This follows from Proposition 7.2, since Supp_{ℤ}(ℤ/p^{n}) = {(p)} for each n ≥ 1.

8. The homotopy category of weakly injective modules

In this section we study the homotopy category of weakly injective modules and a pure variant. As before, k will be a commutative ring and G a finite group. An important ingredient of this discussion is an additional exact structure for the category of kG-modules.

Let WInj kG denote the full subcategory of weakly injective (equivalently, weakly projective) kG-modules, and winj kG is the full subcategory of finitely presented modules in WInj kG.

Exact structures. Recall that a sequence 0 → M′ → M → M′′ → 0 in Mod(kG) is k-split exact if it is split exact when restricted to the trivial subgroup. The sequence is k-pure exact if it is pure exact when restricted to the trivial subgroup.

Lemma 8.1. For an exact sequence η: 0 → M′ → M → M′′ → 0 of kG-modules, the following conditions are equivalent:

(1) The sequence is k-pure exact.

(2) For every finitely presented weakly injective kG-module P, each morphism P → M′′ lifts to a morphism P → M.

Proof. Recall from Definition 4.3 that η is a pure exact sequence of kG-modules if and only if for every finitely presented kG-module Q, each morphism Q → M′′ lifts to a morphism Q → M. Next observe that each finitely presented weakly injective kG-module is a direct summand of Q↑^{G} = kG ⊗_{k} Q for some finitely presented k-module Q. Applying

Hom_{kG}(kG ⊗_{k} Q, −) ≅ Hom_{k}(Q, −)

to the morphism M → M′′ yields the assertion. □

For any additive category C we write K(C) for the homotopy category of cochain complexes in C. As usual, K^{b}(C) is the full subcategory of bounded complexes.

For a specified exact structure on C, a complex X = (X^{n}, d^{n}) is said to be acyclic if there are exact sequences

0 → Z^{n} ⟶^{u^{n}} X^{n} ⟶^{v^{n}} Z^{n+1} → 0

in C such that d^{n} = u^{n+1}v^{n} for all n ∈ ℤ. We say X is acyclic in almost all degrees if such exact sequences exist for almost all n.

The bounded derived category. We consider the category of finitely presented kG-modules with the k-split exact structure and define the corresponding bounded derived category as the Verdier quotient

D^{b}(mod kG) = K^{b}(mod kG)/K_{ac}^{b}(mod kG),

where K_{ac}^{b}(mod kG) is the full subcategory of acyclic complexes, with respect to the k-split structure, in K^{b}(mod kG). The natural functor

(8.1)  K^{+,b}(winj kG) ⟶ D^{b}(mod kG)

is an equivalence of triangulated categories; its inverse identifies a complex with its weakly injective resolution. Here K^{+,b}(winj kG) is the full subcategory of K(winj kG) of complexes X with X^{n} = 0 for n ≪ 0 and X acyclic in almost all degrees.

Proposition 8.2. The composite functor

mod(kG) ↣ D^{b}(mod kG) ↠ D^{b}(mod kG)/K^{b}(winj kG)

induces an equivalence of triangulated categories

stmod(kG) ⟶^{∼} D^{b}(mod kG)/K^{b}(winj kG).

Proof. Adapt the proof for group algebras over a field; see Theorem 2.1 in [27]. □

Given bounded complexes X, Y of finitely presented kG-modules and n ∈ ℤ, set

Ext_{G,1}^{n}(X, Y) = Hom_{D^{b}(mod kG)}(X, Y [n]).

This notation is consistent with our previous definition (7.1).

The pure derived category. The pure derived category of Mod(kG) is by definition the following Verdier quotient

D_{pur}(Mod kG) = K(Mod kG)/K_{k-pac}(Mod kG),

where K_{k-pac}(Mod kG) denotes the full subcategory consisting of complexes that are acyclic with respect to the k-pure exact structure.

Proposition 8.3. The derived category D_{pur}(Mod kG) is a compactly generated triangulated category. The inclusion winj kG → Mod(kG) induces an equivalence of triangulated categories between K^{b}(winj kG) and the full subcategory of compact objects in D_{pur}(Mod kG).

Proof. Consider the triangulated category K(Mod kG) and identify each kG-module with the corresponding complex concentrated in degree zero. Then each object in winj kG is compact in K(Mod kG) and

(winj kG)^{⊥} = K_{k-pac}(Mod kG).

Now apply Proposition 5.1. □

Lemma 8.4. The composite K(WInj kG) ↣ K(Mod kG) ↠ D_{pur}(Mod kG) induces an equivalence of triangulated categories

K(WInj kG)/K_{k-pac}(WInj kG) ⟶^{∼} D_{pur}(Mod kG).

Proof. This is an immediate consequence of Corollary 5.2. □

Remark 8.5. When k is a field, every exact sequence of k-modules is pure exact, so D_{pur}(Mod kG) is the usual derived category of kG-modules. On the other extreme, if G is trivial, D_{pur}(Mod kG) is the pure derived category of Mod(k) studied in [18].

The pure homotopy category. The pure homotopy category of WInj kG is by definition the following Verdier quotient

K_{pur}(WInj kG) = K(WInj kG)/K_{pac}(WInj kG)

where K_{pac}(WInj kG) denotes the full subcategory consisting of complexes that are acyclic with respect to the pure exact structure.

Proposition 8.6. The homotopy category K_{pur}(WInj kG) is compactly generated and the natural functor (8.1) induces an equivalence of triangulated categories between the full subcategory of compact objects in K_{pur}(WInj kG) and D^{b}(mod kG).

Proof. Choose for each M ∈ mod(kG) a weakly injective resolution iM. Arguing as in the proof of [19, Lemma 2.1], one can verify that the morphism M → iM induces, for each X ∈ K(WInj kG), an isomorphism

Hom_{K(Mod kG)}(iM, X) ⟶^{∼} Hom_{K(Mod kG)}(M, X) .

Thus each iM is compact in K(WInj kG) and

{iM | M ∈ mod(kG)}^{⊥} = K_{pac}(WInj kG).

Now apply Proposition 5.1. □

Recollements. A recollement is by definition a diagram of exact functors

\xymatrix{T′ \ar[r]|{I} & T \ar[r]|{Q} \ar[l]^{I_{λ}} \ar[l]_{I_{ρ}} & T′′ \ar[l]^{Q_{λ}} \ar[l]_{Q_{ρ}}}

satisfying the following conditions, where S denotes the full subcategory consisting of objects X ∈ T satisfying Q(X) = 0.

(1) The functors (I_{λ}, I), (I, I_{ρ}), (Q_{λ}, Q), (Q, Q_{ρ}) form adjoint pairs.

(2) The functor I induces an equivalence T′ ⟶^{∼} S.

(3) The functor Q induces an equivalence T/S ⟶^{∼} T′′.

Proposition 8.7. Let T be a triangulated category with set-indexed products and coproducts. Fix a pair of essentially small thick subcategories C and D consisting of compact objects such that D ⊆ C. Then there is an induced recollement of compactly generated triangulated categories

\xymatrix{(D^{⊥})/(C^{⊥}) \ar[r]|{I} & T/(C^{⊥}) \ar[r]|{Q} \ar[l] \ar[l] & T/(D^{⊥}) \ar[l] \ar[l]} .

Restricting the left adjoints of I and Q to the full subcategories of compact objects yields up to equivalence and idempotent completion an sequence of exact functors:

[figure: C/D; C; D] .

Proof. We have a pair of localising subcategories C^{⊥} ⊆ D^{⊥} which induce a quotient functor T/(C^{⊥}) ↠ T/(D^{⊥}) with kernel (D^{⊥})/(C^{⊥}); see Proposition 2.3.1 of Chap. II in [29]. The categories C^{⊥} and D^{⊥} form subcategories of T that are closed under set-indexed products and coproducts. Thus I and Q preserve set-indexed products and coproducts. It follows from Brown representability that I and Q both have left and right adjoints.

A left adjoint of a coproduct preserving functor preserves compactness. The description of the categories of compact objects follows from Proposition 5.1. □

A recollement. We apply Proposition 8.7: Let T = K(WInj kG) and set

C = K^{+,b}(winj kG) and D = K^{b}(winj kG) .

Note that

C^{⊥} = K_{pac}(WInj kG) and D^{⊥} = K_{k-pac}(WInj kG) .

The inclusion K_{pac}(WInj kG) ↣ K_{k-pac}(WInj kG) induces the following commutative diagram of exact functors.

[figure: K_{pac}(WInj kG); K_{pac}(WInj kG); K_{k-pac}(WInj kG); K(WInj kG); D_{pur}(Mod kG); StMod^{cw}(kG); I; K_{pur}(WInj kG); Q; D_{pur}(Mod kG)]

The occurrence of the stable module category in this diagram is explained by the following lemma.

Lemma 8.8. Taking a complex of kG-modules to its cycles in degree zero induces an equivalence of triangulated categories

K_{k-pac}(WInj kG)/K_{pac}(WInj kG) ⟶^{∼} StMod^{cw}(kG).

Proof. Denote by K_{ac}(WInj kG) the category of complexes that are acyclic with respect to the k-split exact structure. Taking cycles in degree zero induces an equivalence of triangulated categories

K_{ac}(WInj kG) ⟶^{∼} StMod(kG).

This functor identifies K_{pac}(WInj kG) ∩ K_{ac}(WInj kG) with the category of cwinjective kG-modules by Lemma 5.5, and induces therefore an equivalence

K_{ac}(WInj kG)/(K_{pac}(WInj kG) ∩ K_{ac}(WInj kG)) ⟶^{∼} StMod^{cw}(kG).

Now observe that

K_{ac}(WInj kG) ⊆ K_{k-pac}(WInj kG)

and apply Corollary 5.2. □

The following result establishes the analogue of Proposition 6.1 in [6] for general rings of coefficients.

Theorem 8.9. The functors I and Q induce a recollement

\xymatrix{StMod^{cw}(kG) \ar[r]|{I} & K_{pur}(WInj kG) \ar[r]|{Q} \ar[l] \ar[l] & D_{pur}(Mod kG) \ar[l] \ar[l]} .

Restricting the left adjoints of I and Q to the full subcategories of compact objects yields up to equivalence the following sequence

[figure: stmod(kG); D^{b}(mod kG); K^{b}(winj kG).]

Proof. We apply Proposition 8.7. The functor Q is a quotient functor by Lemma 8.4, and its kernel identifies with K_{k-pac}(WInj kG)/K_{pac}(WInj kG); see Proposition 2.3.1 of Chap. II in [29]. The description of the kernel as stable module category follows from Lemma 8.8. The description of the compact objects follows from Propositions 8.2, 8.3, and 8.6. □

Remark 8.10. The adjoints of I and Q in this recollement admit explicit descriptions. More precisely, I_{λ} = tk ⊗_{k} −, I_{ρ} = Hom_{k}(tk, −), Q_{λ} = pk ⊗_{k} −, Q_{ρ} = Hom_{k}(pk, −). Here, we write pk for a weakly projective resolution, and tk for a Tate resolution of the trivial kG-module k; see [6, §6].

Appendix A. Disconnecting spectra via localisation sequences

by Greg Stevenson

Let S be an essentially small tensor triangulated category, for instance the stable module category stmod(kG) of a group algebra kG. In [1] Balmer has associated to S a spectral topological space Spc S which provides a universal notion of tensorcompatible supports for objects of S. We briefly recall the definition of Spc S. A proper tensor ideal (our tensor ideals are always assumed to be thick) P of S is prime if s ⊗ s′ ∈ P implies s ∈ P or s′ ∈ P for all s, s′ ∈ S. The spectrum of S is then defined to be the set Spc S of prime tensor ideals of S endowed with the Zariski topology which is given by the basis of closed subsets

{supp_{S} s = {P ∈ Spc S | s ∉ P} | s ∈ S}.

The subset supp_{S} s is called the support of the object s. More generally one can define the support of any subcategory C ⊆ S by

supp_{S} C = ⋃_{c∈C} supp_{S} c.

The spectrum is contravariantly functorial with respect to strong monoidal exact functors [1, Proposition 3.6] i.e., given a strong monoidal exact functor i_{∗}: R → S there is a spectral map Spc(i_{∗}): Spc S → Spc R.

We show that if a tensor triangulated category S admits a semiorthogonal decomposition given by a pair of tensor ideals then the spectrum of S is disconnected. In the remainder of this section, S is an essentially small tensor triangulated category, with unit object 1, and

[figure: R; i_{∗}; i^{!}; S; j^{∗}; j_{∗}; T]

is a localisation sequence such that both R and T are tensor ideals of S, when viewed as subcategories via the embeddings i_{∗} and j_{∗}. We recall that this diagram being a localisation sequence, or semiorthogonal decomposition, means that i_{∗} and j_{∗} are fully faithful exact functors such that i^{!} is right adjoint to i_{∗} and j^{∗} is left adjoint to j_{∗}. Moreover, (i_{∗}R)^{⊥} = j_{∗}T and i_{∗}R = ^{⊥}(j_{∗}T).

In general one must ask that both i_{∗}R and j_{∗}T are tensor ideals; i_{∗}R being an ideal does not necessarily imply j_{∗}T is an ideal and vice versa. However, there is a special case in which these implications hold and we feel it is worth making explicit.

We say the tensor triangulated category S is rigid if the symmetric monoidal structure on S is closed, with internal hom Hom(−, −) and, setting s^{∨} = Hom(s, 1) for s ∈ S, the natural morphism

s^{∨} ⊗ s′ → Hom(s, s′)

is an isomorphism for all s, s′ ∈ S. For further details the interested reader can consult Section 1 of [7] in the general case and Appendix A of [14] for further details in the triangulated case.

Lemma A.1. Let S be an essentially small rigid tensor triangulated category. If R is a tensor ideal of S, then so are the thick subcategories R^{⊥} and ^{⊥}R.

Proof. It is clear from the adjunction between the tensor product and internal hom that R^{⊥} is closed under the functors Hom(s, −) for s ∈ S. As Hom(s, −) ≅ s^{∨}⊗(−) and the duality (−)^{∨} is a contravariant equivalence it is thus immediate that R^{⊥} is an ideal as claimed. The proof that ^{⊥}R is an ideal is similarly trivial. □

We now return to the setting where S is only assumed to be tensor triangulated and commence the proof of the promised result on disconnectedness of the spectrum. Recall that an object s ∈ S is said to be ⊗-idempotent if s ⊗ s ≅ s.

Lemma A.2. There are ⊗-idempotent objects Γ_{R}1 and L_{R}1 such that

Γ_{R}1 ⊗ (−) ≅ i_{∗}i^{!} and L_{R}1 ⊗ (−) ≅ j_{∗}j^{∗}.

In particular, Γ_{R}1 ⊗ L_{R}1 ≅ 0.

Proof. This is standard, see for instance [2, Theorem 2.13] □

Lemma A.3. Given objects r ∈ R, s ∈ S, and t ∈ T there are isomorphisms

s ⊗ i_{∗}r ≅ i_{∗}(i^{!}s ⊗ r) and s ⊗ j_{∗}t ≅ j_{∗}(j^{∗}s ⊗ t).

Proof. The proof of the second isomorphism is almost exactly as in [28, Lemma 8.2] and the proof of the first is similar. □

Lemma A.4. The map Spc(j^{∗}): Spc T → Spc S gives a homeomorphism onto supp_{S} j_{∗}T and Spc(i^{!}): Spc R → Spc S is a homeomorphism onto supp_{S} i_{∗}R. Furthermore, there are equalities

supp_{S} i_{∗}R = supp_{S} Γ_{R}1 and supp_{S} j_{∗}T = supp_{S} L_{R}1,

so that both subsets are closed in Spc S.

Proof. The arguments for R and T are similar so we only prove the result for T. We already know from [1, Proposition 3.11] that the localisation j^{∗} induces a homeomorphism

Spc T →^{∼} {P ∈ Spc S | R ⊆ P},

and it remains to identify this subset with the support of R^{⊥} = j_{∗}T.

First observe that since Γ_{R}1⊗L_{R}1 ≅ 0 every prime tensor ideal must contain one of these idempotents. Furthermore, no proper ideal can contain both idempotents as they build 1 via the localisation triangle

Γ_{R}1 → 1 → L_{R}1 → ΣΓ_{R}1.

So for P ∈ Spc S we have R ⊆ P if and only if L_{R}1 ∉ P if and only if j_{∗}T ⊈ P. This completes the argument via the following trivial fact

supp_{S} j_{∗}T = {P ∈ Spc S | j_{∗}T ⊈ P}.

The final statement now follows as

supp_{S} L_{R}1 = {P ∈ Spc S | L_{R}1 ∉ P}
= {P ∈ Spc S | Γ_{R}1 ∈ P}
= {P ∈ Spc S | R ⊆ P}
= Spc(j^{∗})(Spc T)
= supp_{S} j_{∗}T.  □

From this point onward let us be lazy and identify R and T with thick subcategories of S and view their spectra as subsets of Spc S.

Theorem A.5. The subsets Spc R = supp_{S} R and Spc T = supp_{S} T of Spc S are open and closed, and there is a decomposition

Spc S = Spc R ∐ Spc T.

In particular, if neither R nor T is the zero ideal the space Spc S is disconnected.

Proof. By [1, Proposition 3.11] and the last lemma respectively we can describe Spc R as both

{P ∈ Spc S | T ⊆ P} and supp_{S} R.

It is clear from the first description that Spc R is generalisation closed and from the second that Spc R is specialisation closed. By the last lemma this subset is constructible (since it is the support of an object) and hence it is both closed and open. The argument for T is the same. The existence of the claimed decomposition is clear: by Lemma A.2 we have ⊗-orthogonal Rickard idempotents, precisely one of which must be contained in any non-zero proper prime ideal (this can also be used to argue that both subsets in question are open). □

For completeness we give an easier proof of a stronger result in the rigid case.

Proposition A.6. Suppose S is as above but is furthermore assumed to be rigid. Then S is equivalent to R ⊕ T. In particular the spectrum decomposes as in the above theorem.

Proof. Using rigidity of S we have, for i ∈ ℤ,

Hom(L_{R}1, Σ^{i}Γ_{R}1) ≅ Hom(Σ^{−i}1, (L_{R}1)^{∨} ⊗ Γ_{R}1) ≅ 0,

the final isomorphism since thick tensor ideals are closed under (−)^{∨} and R∩T = 0. Hence the localisation triangle

Γ_{R}1 → 1 → L_{R}1 → ΣΓ_{R}1

splits yielding 1 ≅ Γ_{R}1 ⊕ L_{R}1. It is now clear that S splits. □

Corollary A.7. If S has a decomposition, as in the assumptions of this section, given by non-zero tensor ideals and 1 is indecomposable then S is not rigid. □

References

1. P. Balmer, The spectrum of prime ideals in tensor triangulated categories, J. Reine Angew. Math. 588 (2005), 149–168.

2. P. Balmer, G. Favi, Generalized tensor idempotents and the telescope conjecture, Proc. Lond. Math. Soc. (3) 102 (2011), 1161–1185.

3. D. J. Benson, Representations and Cohomology I: Basic representation theory of finite groups and associative algebras, Cambridge Studies in Advanced Mathematics, vol. 30, Cambridge University Press, 1991, reprinted in paperback, 1998.

4. D. J. Benson, J. F. Carlson, and J. Rickard, Thick subcategories of the stable module category, Fundamenta Mathematicae 153 (1997), 59–80.

5. D. J. Benson, S. B. Iyengar, and H. Krause, Stratifying modular representations of finite groups, Ann. of Math. 174 (2011), 1643–1684.

6. D. J. Benson and H. Krause, Complexes of injective kG-modules, Algebra & Number Theory 2 (2008), 1–30.

7. P. Deligne and J. S. Milne, Tannakian categories, in Hodge cycles, motives, and Shimura varieties, Lecture Notes in Mathematics, vol. 900, Springer-Verlag, Berlin/New York, 1982, pp. 101–228.

8. L. Evens, The cohomology ring of a finite group, Trans. Amer. Math. Soc. 101 (1961), 224– 239.

9. P. Gabriel and M. Zisman, Calculus of fractions and homotopy theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 35, Springer-Verlag, Berlin/New York, 1967.

10. W. Gaschütz, Über den Fundamentalsatz von Maschke zur Darstellungstheorie der endlichen Gruppen, Math. Z. 56 (1952), 376–387.

11. M. J. Hopkins, Global methods in homotopy theory, Homotopy Theory, Durham 1985, Lecture Notes in Mathematics, vol. 117, Cambridge University Press, 1987.

12. D. Happel, Triangulated categories in the representation theory of finite dimensional algebras, London Math. Soc. Lecture Note Series, vol. 119, Cambridge University Press, 1988.

13. D. G. Higman, Modules with a group of operators, Duke Math. J. 21 (1954), 369–376.

14. M. Hovey, J. H. Palmieri, and N. P. Strickland, Axiomatic stable homotopy theory, Mem. Amer. Math. Soc., vol. 128, no. 610, 1997.

15. C. U. Jensen and H. Lenzing, Model theoretic algebra, Gordon and Breach, 1989.

16. B. Keller, Chain complexes and stable categories, Manuscripta Math. 67 (1990), 379–417.

17. B. Keller, Deriving DG categories, Ann. Scient. Éc. Norm. Sup. (4) 27 (1994), 63–102.

18. H. Krause, Approximations and adjoints in homotopy categories, Math. Ann. 353 (2012), 765–781.

19. H. Krause, The stable derived category of a noetherian scheme, Compositio Math. 141 (2005), 1128–1162.

20. H. Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, 1986.

21. A. Neeman, The chromatic tower for D(R), Topology 31 (1992), 519–532.

22. A. Neeman, The connection between the K-theory localization theorem of Thomason, Trobaugh and Yao and the smashing subcategories of Bousfield and Ravenel, Ann. Scient. Éc. Norm. Sup. (4) 25 (1992), 547–566.

23. A. Neeman, Triangulated categories, Annals of Math. Studies, vol. 148, Princeton Univ. Press, 2001.

24. M. Prest, Model theory and modules, London Math. Society Lecture Note Series, vol. 130, Cambridge University Press, 1988.

25. M. Prest, Purity, spectra and localisation, Cambridge University Press, 2009.

26. D. G. Quillen, Higher algebraic K-theory I, Algebraic K-theory I: Higher K-theories, Lecture Notes in Mathematics, vol. 341, Springer-Verlag, Berlin/New York, 1973, pp. 85–147.

27. J. Rickard, Derived categories and stable equivalence, J. Pure & Applied Algebra 61 (1989), 303–317.

28. G. Stevenson, Support theory via actions of tensor triangulated categories, J. Reine Angew.

    Math., to appear, arxiv:1105.4692

29. J.-L. Verdier, Des catégories dérivées des catégories abéliennes, Astérisque, vol. 239, Société Math. de France, 1996.

Dave Benson, Institute of Mathematics, University of Aberdeen, King’s College, Aberdeen AB24 3UE, Scotland U.K.

Srikanth B. Iyengar, Department of Mathematics, University of Nebraska, Lincoln, NE 68588, U.S.A.

Henning Krause, Fakultät für Mathematik, Universität Bielefeld, 33501 Bielefeld, Germany.