DECOMPOSING THICK SUBCATEGORIES OF THE STABLE MODULE CATEGORY

HENNING KRAUSE

Abstract. Let \underline{mod} kG be the stable category of finitely generated modular representations of a finite group G over a field k. We prove a Krull-Schmidt theorem for thick subcategories of \underline{mod} kG. It is shown that every thick tensor-ideal C of \underline{mod} kG (i.e. a thick subcategory which is a tensor ideal) has a (usually infinite) unique decomposition C = ∐_{i∈I} C_{i} into indecomposable thick tensor-ideals. This decomposition follows from a decomposition of the corresponding idempotent kG-module E_{C} into indecomposable modules. If C = C_{W} is the thick tensor-ideal corresponding to a closed homogeneous subvariety W of the maximal ideal spectrum of the cohomology ring H^{∗}(G, k), then the decomposition of C reflects the decomposition W = ⋃_{i=1}^{n} W_{i} of W into connected components.

Introduction

In modular representation theory of finite groups, one frequently passes to the stable module category which is a triangulated category. Following ideas from stable homotopy theory, Benson, Carlson, and Rickard studied in a number of papers the lattice of thick subcategories of the stable module category. In [4], these authors obtain for p-groups a description of this lattice which is an analogue of a result for the stable homotopy category due to Devinatz, Hopkins, and Smith [8, 11]. In this paper we prove a Krull-Schmidt theorem for this lattice which we now explain.

Let Λ = kG be the group algebra of a finite group G over a field k. The stable category \underline{mod} Λ of finitely generated Λ-modules is a triangulated category and carries a tensor product which is induced from the tensor product on mod Λ. A full subcategory C of \underline{mod} Λ is called a thick tensor-ideal in \underline{mod} Λ if C is a thick subcategory and X ⊗ Y ∈ C for all X ∈ C and Y ∈ \underline{mod} Λ. Given a thick tensor-ideal C, we say that a family (C_{i})_{i∈I} of thick tensor-ideals in \underline{mod} Λ is a decomposition of C if

(1) the objects in C are the finite coproducts of objects from the C_{i};

(2) C_{i} ∩ C_{j} = 0 for all i ≠ j in I. A decomposition (C_{i})_{i∈I} of C is denoted by C = ∐_{i∈I} C_{i}, and we say that C is indecomposable if C ≠ 0 and any decomposition C = C_{1} ∐ C_{2} implies C_{1} = 0 or C_{2} = 0.

Theorem A. Every thick tensor-ideal C in \underline{mod} Λ has a unique decomposition C = ∐_{i∈I} C_{i} into indecomposable thick tensor-ideals. Conversely, given any family (C_{i})_{i∈I} of thick tensor-ideals in \underline{mod} Λ satisfying C_{i} ∩ C_{j} = 0 for all i ≠ j in I, there exists a thick tensor-ideal C such that C = ∐_{i∈I} C_{i}.

Before we discuss this result in more detail let us mention that there seems to be no analogous statement for the stable homotopy category. Our analysis of thick subcategories in \underline{mod} Λ is based on the analysis of certain objects in the stable category \underline{Mod} Λ of all Λ-modules. In fact, we are able to compute the centre Z[C∐] of the localizing subcategory C∐ of \underline{Mod} Λ which is generated by a thick tensor-ideal C in \underline{mod} Λ. Recall that the centre of an additive category is the endomorphism ring of the identity functor.

We prove that Z[C∐] is a product of commutative local rings, and the corresponding decomposition id_{C} = (e_{i})_{i∈I} of the identity functor into central idempotents gives the decomposition C = ∐_{i∈I} C_{i} if we define C_{i} = e_{i}(C) for every i. Note that the centre of the stable homotopy category is ℤ which is not a local ring. Therefore one cannot expect an analogue of Theorem A for the stable homotopy category because any Krull-Schmidt theorem requires local endomorphism rings for the indecomposable objects.

In [18], Rickard introduced for every thick subcategory C of \underline{mod} Λ a particular Λmodule E_{C}. It is obtained from the trivial representation k by applying a right adjoint e: \underline{Mod} Λ → C∐ of the inclusion C∐ → \underline{Mod} Λ, i.e. E_{C} = e(k). If C is a tensor-ideal, then the canonical map E_{C} → k induces an isomorphism E_{C} ⊗ E_{C} → E_{C} in \underline{Mod} Λ, and therefore E_{C} is called idempotent. In recent years, the study of such modules has become one of the most successful methodologies in modular representation theory of finite groups. We obtain Theorem A from the following result since Z[C∐] ≃ \underline{End}_{Λ}(E_{C}).

Theorem B. The Λ-module E_{C} has, up to isomorphism, a unique decomposition E_{C} = ∐_{i∈I} E_{i} into indecomposable Λ-modules with local endomorphism ring. Moreover, the endomorphism ring of E_{C} has the following properties:

(1) ⋂_{n∈ℕ} 𝔯^{n} = 0 for the Jacobson radical 𝔯 of \underline{End}_{Λ}(E_{C});

(2) if C is a tensor-ideal, then \underline{End}_{Λ}(E_{C}) ≃ ∏_{i∈I} \underline{End}_{Λ}(E_{i})

The decomposition of the module E_{C} is the consequence of a more general result. In fact, we introduce the concept of an endofinite object in a compactly generated triangulated category and prove that any endofinite object decomposes uniquely into indecomposable objects. Theorem B then follows since E_{C} is an endofinite object in the localizing subcategory of \underline{Mod} Λ which is generated by C.

Interesting examples of endofinite objects also arise in stable homotopy theory. For instance, the classifying space BG of G is endofinite in the category of spectra [15]. Therefore we obtain a new proof for the existence of a stable splitting of BG which does not involve Segal’s conjecture; see Benson’s survey [2] for more details.

Let us end this introduction with a geometric interpretation of Theorem A in terms of the projective prime ideal spectrum of the cohomology ring H^{∗}(G, k). More precisely, we denote by V_{G}(k) the set of closed homogeneous irreducible non-zero subvarieties of the maximal ideal spectrum V_{G}(k) of H^{∗}(G, k). Generalizing the definition of a closed homogeneous subvariety of V_{G}(k), Benson, Carlson, and Rickard consider in [3] subsets W of V_{G}(k) which are closed under specialization, i.e. W ∈ W and V ⊆ W implies V ∈ W. For instance, any closed homogeneous subvariety W of V_{G}(k) is determined by W = {V ∈ V_{G}(k) | V ⊆ W }. Given any subset W of V_{G}(k) closed under specialization, we say that W is projectively connected if any decomposition W = W_{1} ∪ W_{2} into subsets of V_{G}(k) closed under specialization such that W_{1}∩W_{2} = ∅ implies W = W_{1} or W = W_{2}. In [4], Benson, Carlson, and Rickard establish a bijection W ↦ C_{W} between the subsets of V_{G}(k) which are closed under specialization and the thick tensor-ideals in \underline{mod} Λ. Under this bijection the decomposition C = ∐_{i∈I} C_{i} of C = C_{W} into indecomposable thick tensor-ideals corresponds to the decomposition W = ⋃_{i∈I} W_{i} of W into projectively connected non-empty components, i.e. W_{i} ∩ W_{j} = ∅ and C_{i} = C_{W_{i}} for all i ≠ j in I. We have therefore the following consequence.

Theorem C. Let C be a thick tensor-ideal in \underline{mod} Λ, and suppose that W is the corresponding subset W of V_{G}(k) closed under specialization such that C = C_{W}. Then the following conditions are equivalent:

(1) W is non-empty and projectively connected; (2) C is an indecomposable thick tensor-ideal in \underline{mod} Λ;

(3) E_{C} is an indecomposable object in \underline{Mod} Λ.

1. Endofiniteness

Let T be a triangulated category [19] and suppose that arbitrary coproducts exist in T . An object X in T is called compact if for every family (Y_{i})_{i∈I} in T the canonical map ∐_{i} Hom(X, Y_{i}) → Hom(X, ∐_{i} Y_{i}) is an isomorphism. We denote by T_{0} the full subcategory of compact objects in T and observe that T_{0} is a triangulated subcategory of T . Following [17], the category T is called compactly generated provided that the isomorphism classes of objects in T_{0} form a set, and Hom(C, X) = 0 for all C in T_{0} implies X = 0 for every object X in T .

In [7], Crawley-Boevey introduced the concept of endofiniteness for locally finitely presented categories. We make the following analogous definition for a compactly generated triangulated category T .

Definition 1.1. An object X in T is called endofinite if for every compact object C in T the End(X)-module Hom(C, X) has finite composition length.

The following result collects some useful properties of endofinite objects.

Theorem 1.2. An endofinite object X has the following properties:

(1) there exists, up to isomorphism, a unique decomposition X = ∐_{i∈I} X_{i} into indecomposable objects with local endomorphism ring;

(2) ⋂_{n∈ℕ} 𝔯^{n} = 0 for the Jacobson radical 𝔯 of End(X);

(3) if ϕ: Y → X is a phantom map, i.e. the induced map Hom(C, Y) → Hom(C, X) is zero for all compact C, then ϕ = 0.

Proof. The proof uses the fact that the category T can be embedded (modulo the phantom maps in T ) into an abelian Grothendieck category. We recall briefly this construction and refer to [14] for more details. Let Mod T_{0} be the category of additive functors T_{0}^{op} → Ab into the category of abelian groups, and denote by mod T_{0} the full subcategory of finitely presented functors. We shall use the Yoneda functor

h: T ⟶ Mod T_{0}, X ↦ H_{X} = Hom(−, X)|_{T_{0}}.

If X is endofinite, then Hom(F, H_{X}) is a End(H_{X})-module of finite length for all F in mod T_{0}, and therefore H_{X} is an endofinite object of Mod T_{0} in the sense of [7]. We claim that H_{X} is an injective object. To this end let ε: 0 → H_{X} → M → N → 0 be an exact sequence. Every map F → N with F in mod T_{0} factors through M → N since H_{X} is fp-injective, i.e. Ext^{1}(−, H_{X}) vanishes on finitely presented objects, by [14, Lemma 1.6]. Thus ε is a pure-exact sequence which is split exact since every endofinite object is pure-injective [7, 3.6]. The injectivity of H_{X} implies that h induces an isomorphism End(X) → End(H_{X}) by [14, Lemma 1.7]. Any endofinite object in Mod T_{0} has a decomposition into indecomposable objects with local endomorphism ring by [7, Theorem 3.5.2], and this gives the desired decomposition of X in T .

Let us give an alternative proof for (1). To this end consider the localizing subcategory L of Mod T_{0} which is generated by all F in mod T_{0} with Hom(F, H_{X}) = 0. We can form the quotient category Mod T_{0}/L in the sense of [9], and obtain a locally finite Grothendieck category since the objects in mod T_{0} form a generating set of finite length objects. Moreover, the quotient functor q: Mod T_{0} → Mod T_{0}/L identifies H_{X} with an injective object in Mod T_{0}/L. More precisely, q(H_{X}) is fp-injective in Mod T_{0}/L since H_{X} is fp-injective in Mod T_{0} by [14, Lemma 1.6]. However, any fpinjective object in a locally noetherian category is injective by an analogue of Baer’s criterion for Grothendieck categories. It follows from [12, Corollary 2.11] that H_{X} is L-closed, i.e. Hom(L, H_{X}) = 0 = Ext^{1}(L, H_{X}), since L is generated by finitely presented objects. Therefore H_{X} is an injective object in Mod T_{0} since q(H_{X}) is injective, and End(H_{X}) ≃ End(q(H_{X})); see [9]. Any injective object in a locally finite category decomposes uniquely into indecomposables with local endomorphism ring [9, IV.2], and this gives an alternative proof for (1) since

End(X) ≃ End(H_{X}) ≃ End(q(H_{X})).

Part (2) follows from the general fact that ⋂_{n∈ℕ} rad^{n} End(Q) = 0 for every injective object Q in a locally finite category [9, IV.4]. To prove (3) consider a phantom map ϕ: Y → X. The functor h induces an isomorphism Hom(Y, X) → Hom(H_{Y} , H_{X}), by [14, Lemma 1.7], since H_{X} is injective, and therefore ϕ = 0. This finishes the proof.

We include for later reference the following lemma.

Lemma 1.3. Let S be a localizing subcategory of T which is generated by compact objects from T , and denote by e: T → S a right adjoint of the inclusion S → T .

(1) S is a compactly generated triangulated category.

(2) If X is an endofinite object in T , the e(X) is endofinite in S.

Proof. Suppose that S is generated by a class D of objects in T_{0} and denote by C the thick subcategory of T_{0} which is generated by D. Clearly, C ⊆ S_{0}. Suppose now that Hom(S_{0}, X) = 0 for some X in S, and denote by R the localizing subcategory of all Y in T such that Hom(Y, X[n]) = 0 for all n ∈ ℤ. Clearly, R contains S and therefore X = 0. It is not hard to check that S_{0} = C (e.g., see [16, Lemma 2.2]), and we conclude that S is compactly generated.

Now let X be an endofinite object in T and fix C in S_{0}. The functorial isomorphism Hom(C, X) ≃ Hom(C, e(X)) shows that Hom(C, e(X)) has finite length over End(e(X)) since C is also compact in T . Therefore e(X) is endofinite in S.

Example 1.4. (1) A spectrum X is endofinite in the stable homotopy category if and only if the stable homotopy groups π_{n}^{s}(X) have finite length over the endomorphism ring [X, X] of X for all n ∈ ℤ; see [15].

(2) A Λ-module X is an endofinite object in the stable module category \underline{Mod} Λ if and only if the endolength of X (i.e. the length of X as module over End_{Λ}(X)) is finite; see [13, Corollary 3.3].

2. Decomposing thick subcategories

Let Λ = kG be the group algebra of a finite group G over a field k. We consider the category Mod Λ of (right) Λ-modules, and mod Λ denotes the full subcategory of all finitely generated Λ-modules. The algebra Λ is quasi-Frobenius, i.e. projective and injective Λ-modules coincide. Therefore the stable category \underline{Mod} Λ is triangulated; e.g. see [10]. Recall that the objects in \underline{Mod} Λ are those of Mod Λ, and for two Λ-modules X, Y one defines \underline{Hom}_{Λ}(X, Y) to be Hom_{Λ}(X, Y) modulo the subgroup of maps which factor through a projective Λ-module. Note that the projection functor Mod Λ → \underline{Mod} Λ preserves coproducts. Thus \underline{Mod} Λ has arbitrary coproducts, and it is not difficult to check that an object X in \underline{Mod} Λ is compact if and only if X ≃ Y in \underline{Mod} Λ for some finitely generated Λ-module Y. Therefore we shall not distinguish between (\underline{Mod} Λ)_{0} and \underline{mod} Λ. The description of the compact objects shows that \underline{Mod} Λ is compactly generated.

2.1. Thick subcategories. Let C be a thick subcategory of \underline{mod} Λ, i.e. C is a full triangulated subcategory of \underline{mod} Λ which is closed under taking direct summands. We denote by C∐ the localizing subcategory of \underline{Mod} Λ which is generated by C. Recall that a full triangulated subcategory of \underline{Mod} Λ is localizing if it is closed under taking coproducts. The inclusion C∐ → \underline{Mod} Λ has a right adjoint e: \underline{Mod} Λ → C∐ (e.g., see [16, 18]), and we denote by ε: E_{C} = e(k) → k the corresponding map for the trivial representation k. Using the results about endofinite objects we can now prove the first portion of Theorem B.

Theorem 2.1. The Λ-module E_{C} has, up to isomorphism, a unique decomposition E_{C} = ∐_{i∈I} E_{i} into indecomposable Λ-modules with local endomorphism ring. Moreover, ⋂_{n∈ℕ} 𝔯^{n} = 0 for the Jacobson radical 𝔯 of \underline{End}_{Λ}(E_{C}).

Proof. The trivial representation k is certainly an endofinite object in \underline{Mod} kG, and therefore E_{C} is endofinite in C∐ by Lemma 1.3. We can apply Theorem 1.2 and obtain a decomposition E_{C} = ∐_{i∈I} E_{i} into indecomposables in \underline{Mod} Λ plus the assertion about the Jacobson radical of \underline{End}_{Λ}(E_{C}). It is well-known that any Λ-module X has a decomposition X = X_{P} ∐ P_{X} such that X_{P} has no non-zero projective direct sumand and P_{X} is projective [13, Proposition 5.10]. Clearly, any projective Λ-module is a coproduct of indecomposable modules since Λ is artinian. On the other hand, the endomorphisms of X_{P} which factor through projectives belong to the Jacobson radical of End_{Λ}(X_{P}), and therefore the decomposition of E_{C} in \underline{Mod} Λ gives the desired decomposition in Mod Λ.

Remark 2.2. Let 𝔰 be the Jacobson radical of End_{Λ}(E_{C}). Then (⋂_{n∈ℕ} 𝔰^{n})^{l} = 0 where l denotes the Loewy length of Λ.

2.2. Thick tensor-ideals. Let C be a thick subcategory of \underline{mod} Λ and assume in addition that X ⊗ Y ∈ C for all X ∈ C and Y ∈ \underline{mod} Λ. We call a subcategory C with these properties a thick tensor-ideal in \underline{mod} Λ. We shall use the following description of the objects in C∐ which is due to Rickard.

Proposition 2.3. Suppose that C is a thick tensor-ideal in \underline{mod} Λ. Then the following are equivalent for an object X in \underline{Mod} Λ:

(1) X belongs to C∐;

(2) ε ⊗ X: E_{C} ⊗ X → k ⊗ X ≃ X is an isomorphism.

Proof. Combine Proposition 5.6 and Proposition 5.13 from [18].

The following lemma will be useful for decomposing a thick tensor-ideal.

Lemma 2.4. Let C and D be thick tensor-ideals in \underline{mod} Λ. Then the following are equivalent:

(1) C ∩ D = 0;

(2) C ⊗ D = 0, i.e. X ⊗ Y = 0 for all X ∈ C and Y ∈ D;

(3) \underline{Hom}_{Λ}(C, D) = 0, i.e. \underline{Hom}_{Λ}(X, Y) = 0 for all X ∈ C and Y ∈ D;

(4) \underline{Hom}_{Λ}(C∐, D∐) = 0, i.e. \underline{Hom}_{Λ}(X, Y) = 0 for all X ∈ C∐ and Y ∈ D∐;

(5) C∐ ⊗ D∐ = 0, i.e. X ⊗ Y = 0 for all X ∈ C∐ and Y ∈ D∐;

(6) E_{C} ⊗ E_{D} = 0.

(7) C∐ ∩ D∐ = 0; Proof. (1) ⇒ (2) Clear, since C ⊗ D ⊆ C ∩ D.

(2) ⇒ (3) Let X ∈ C and Y ∈ D. Then the k-dual X^{∗} is a direct summand of X^{∗} ⊗ X ⊗X^{∗} and belongs therefore to C . It follows that Hom_{k}(X, Y) ≃ X^{∗}⊗Y is a projective Λ-module since C ⊗ D = 0. Any Λ-map X → Y factors through X ⊗ Hom_{k}(X, Y), and therefore \underline{Hom}_{Λ}(X, Y) = 0.

(3) ⇒ (4) Let D′ be the localizing subcategory of all X in \underline{Mod} Λ such that \underline{Hom}_{Λ}(C, X) = 0. Clearly, D∐ ⊆ D′ since D ⊆ D′. Now let C′ be the localizing subcategory of all X in \underline{Mod} Λ such that \underline{Hom}_{Λ}(X, D′) = 0. We have C∐ ⊆ C′ since C ⊆ C′, and therefore \underline{Hom}_{Λ}(C∐, D∐) = 0.

(4) ⇒ (5) If \underline{Hom}_{Λ}(C∐, D∐) = 0, then C∐ ⊗ D∐ = 0 since C∐ ⊗ D∐ ⊆ C∐ ∩ D∐ by Proposition 2.3.

(5) ⇒ (6) Clear.

(6) ⇒ (7) Clear, since E_{C} ⊗ E_{D} ⊗ X ≃ X for every X ∈ C∐ ∩ D∐ by Proposition 2.3.

(7) ⇒ (1) Clear.

We continue with two results which establish the strong connection between decompositions of C and E_{C}. To this end the following definition is needed. Let ε: E → k be a map in \underline{Mod} Λ. We denote by L_{ε} the localizing subcategory of all X in \underline{Mod} Λ such that ε ⊗ X is an isomorphism. The intersection C_{ε} = L_{ε} ∩ \underline{mod} Λ gives a thick subcategory of \underline{mod} Λ, and we observe that C_{ε} is a thick tensor-ideal if ε ⊗ E is an isomorphism.

Proposition 2.5. Let C be a thick tensor-ideal in \underline{mod} Λ. Every decomposition E_{C} = ∐_{i} E_{i} satisfying E_{i} ⊗ E_{j} = 0 for all i ≠ j, induces a decomposition C = ∐_{i} C_{i} of C such that E_{i} ≃ E_{C_{i}} for all i.

Proof. Let (ε_{i}): ∐_{i} E_{i} → k be the decomposition of ε: E_{C} → k. The assumption E_{i} ⊗ E_{j} = 0 for all i ≠ j implies that ε_{i} ⊗ E_{i} is an isomorphism for all i. Therefore we get a decomposition C = ∐_{i} C_{i} of C if we define C_{i} = C_{ε_{i}} for all i. It remains to show that E_{i} ≃ E_{C_{i}} for all i. Observe first that E_{i} ⊗ E_{C_{j}} = 0 for all i ≠ j since E_{i} ⊗ C_{j} = 0. Now consider the canonical map ϕ: ∐_{j} E_{C_{j}} → k. We have E_{C_{i}} ⊗ E_{C_{j}} = 0 for all i ≠ j by Lemma 2.4. Therefore C ⊆ L_{ϕ} and consequently C∐ ⊆ L_{ϕ}. It follows that

E_{i} ≃ E_{i} ⊗ (∐_{j} E_{C_{j}}) ≃ ∐_{j}(E_{i} ⊗ E_{C_{j}}) ≃ E_{i} ⊗ E_{C_{i}}.

On the other hand, E_{i} ⊗ E_{C_{i}} ≃ E_{C_{i}} since C_{i}^{∐} ⊆ L_{ε_{i}}. Thus E_{i} ≃ E_{C_{i}} for all i.

Proposition 2.6. Let C be a thick tensor-ideal in \underline{mod} Λ. Every decomposition C = ∐_{i} C_{i} of C induces a decomposition E_{C} = ∐_{i} E_{i} such that E_{i} ⊗ E_{j} = 0 for all i ≠ j and E_{C_{i}} ≃ E_{i} for all i.

Proof. We define E_{i} = E_{C_{i}} for all i and observe that E_{i} ⊗ E_{j} = 0 for all i ≠ j by Lemma 2.4. We claim that E_{C} ≃ ∐_{i} E_{i}. To this end consider the canonical map ϕ: ∐_{i} E_{i} → k. Observe that C∐ ⊆ L_{ϕ} since C ⊆ L_{ϕ}. Therefore E_{C} ≃ (∐_{i} E_{i}) ⊗ E_{C}. On the other hand,

(∐_{i} E_{i}) ⊗ E_{C} ≃ ∐_{i}(E_{i} ⊗ E_{C}) ≃ ∐_{i} E_{i}

since C_{i} ⊆ C for all i. Thus E_{C} ≃ ∐_{i} E_{i}.

We are now in a position to prove the decomposition theorem for thick tensor-ideals.

Proof of Theorem A. Let C be a thick tensor-ideal in \underline{mod} Λ and fix the decomposition E_{C} = ∐_{i∈I} E_{i} in \underline{Mod} Λ into indecomposable objects which exists by Theorem 2.1. We claim that E_{i}⊗E_{j} = 0 for all i ≠ j in I. In fact, for every i ∈ I we obtain a decomposition

E_{i} ≃ E_{i} ⊗ (∐_{s∈I} E_{s}) ≃ ∐_{s∈I}(E_{i} ⊗ E_{s}),

and we have E_{i} ≃ E_{i} ⊗ E_{t} for precisely one t ∈ I since E_{i} is indecomposable. The same argument gives E_{t} ≃ E_{t} ⊗ E_{i}, and therefore i = t. Thus E_{i} ⊗ E_{j} = 0 for all i ≠ j in I. We obtain a decomposition C = ∐_{i∈I} C_{i} from Proposition 2.5, and it follows from Proposition 2.6 that every C_{i} is indecomposable. Moreover, the uniqueness of the decomposition C = ∐_{i∈I} C_{i} (up to permutation of the C_{i}) follows from the corresponding uniqueness of the decomposition E_{C} = ∐_{i∈I} E_{i}.

Suppose now that (C_{i})_{i∈I} is a family of thick tensor ideals in \underline{mod} Λ with C_{i} ∩ C_{j} = 0 for all i ≠ j in I. We define C to be the full subcategory of all finite coproducts of objects from the C_{i}. Clearly, C is a tensor-ideal. By Lemma 2.4, \underline{Hom}_{Λ}(C_{i}, C_{j}) = 0 for all i ≠ j. Therefore C is a thick subcategory of \underline{mod} Λ since any map ϕ: X → Y in C has a decomposition (ϕ_{i}): ∐_{i} X_{i} → ∐_{i} Y_{i} with ϕ_{i} ∈ C_{i} for all i. It follows that C is a thick tensor-ideal with C = ∐_{i∈I} C_{i}. This finishes the proof of Theorem A.

The following corollary summarizes the basic properties of a decomposition C = ∐_{i∈I} C_{i}.

Corollary 2.7. Let C = ∐_{i∈I} C_{i} be a decomposition of a thick tensor-ideal in \underline{mod} Λ. Then E_{C} ≃ ∐_{i∈I} E_{C_{i}} and C is indecomposable if and only if E_{C} is indecomposable. Moreover, \underline{End}_{Λ}(E_{C}) ≃ ∏_{i∈I} \underline{End}_{Λ}(E_{C_{i}}).

2.3. The centre of a localizing subcategory. Given an additive category A, we denote by Z[A] the centre of A which is the endomorphism ring of the identity functor id_{A}. More precisely, Z[A] is the ring of all natural transformation id_{A} → id_{A}. Note that Z[A] is a commutative ring. Suppose now that C is a thick tensor-ideal in \underline{mod} Λ and let C∐ be the localizing subcategory which is generated by C. Then we have the following description of Z[C∐].

Proposition 2.8. The map Z[C∐] → \underline{End}_{Λ}(E_{C}), ϕ ↦ ϕ_{E_{C}}, is a ring isomorphism. In particular, \underline{End}_{Λ}(E_{C}) is a commutative ring.

Proof. We construct an inverse for Z[C∐] → \underline{End}_{Λ}(E_{C}). To this end consider the canonical map ε: E_{C} → k and recall from Proposition 2.3 that for every X in C∐ the map ε_{X} = ε ⊗ X: E ⊗ X → X is an isomorphism. If ψ is an endomorphism of E_{C}, then we obtain for every X in C∐ an endomorphism ε_{X} ^{∘}(ψ ⊗ X)^{∘} ε_{X}^{−1} of X. In fact, the map

\underline{End}_{Λ}(E_{C}) ⟶ Z[C∐], ψ ↦ (ε_{X} ^{∘}(ψ ⊗ X)^{∘} ε_{X}^{−1})_{X}

gives a ring homomorphism which is an inverse for Z[C∐] → \underline{End}_{Λ}(E_{C}).

Corollary 2.9. Z[C∐] is a product of commutative local rings.

3. Decomposing Varieties

3.1. Varieties for finitely generated modules. Let H^{∗}(G, k) = Ext_{Λ}^{∗}(k, k) be the cohomology ring of G over the field k, and we assume for simplicity that k is algebraically closed. We denote by V_{G}(k) the maximal ideal spectrum of H^{∗}(G, k) and for every finitely generated Λ-module X we denote by V (X) the corresponding closed homogeneous subvariety of V_{G}(k) (see [1, Chapter 5] for precise definitions). Given a closed homogeneous subvariety W of V_{G}(k), the finitely generated Λ-modules whose variety is contained in W form a thick tensor-ideal C_{W} in \underline{mod} Λ. We shall see that the decomposition C_{W} = ∐_{i} C_{i} into indecomposable thick tensor-ideals corresponds to the decomposition W = ⋃_{i} W_{i} of W into connected components, i.e. C_{i} = C_{W_{i}} for all i. We proceed in two steps.

Proposition 3.1. Let W = ⋃_{i=1}^{n} W_{i} be a union of closed homogeneous subvarieties of V_{G}(k) such that W_{i} ∩ W_{j} = {0} for all i ≠ j. Then C_{W} = ∐_{i=1}^{n} C_{W_{i}}.

Proof. Clearly, C_{W_{i}} ⊆ C_{W} and C_{W_{i}} ∩ C_{W_{j}} = 0 for all i ≠ j. Suppose now that X ∈ C_{W} and we may assume that X is indecomposable. Then V (X) = ⋃_{i=1}^{n}(V (X) ∩ W_{i}) and therefore V (X) = V (X) ∩ W_{i} for some i since the variety of an indecomposable module is connected [6]. Thus X ∈ C_{W_{i}} and therefore C_{W} = ∐_{i=1}^{n} C_{W_{i}}.

Proposition 3.2. Let W be a closed homogeneous subvariety of V_{G}(k). Every decomposition C_{W} = ∐_{i} C_{i} into thick tensor-ideals induces a decomposition W = ⋃_{i} W_{i} into closed homogeneous subvarieties such that W_{i} ∩ W_{j} = {0} for all i ≠ j and C_{W_{i}} = C_{i} for all i.

Proof. Assume first that C_{i} is indecomposable for all i. We can choose a finitely generated Λ-module X such that W = V (X) (e.g., see [1, Corollary 5.9.2]) and we assume in addition that the number of indecomposable direct summands of X is minimal. The decomposition C_{W} = ∐_{i} C_{i} gives a decomposition W = ⋃_{i} W_{i} if we define W_{i} = V (X_{i}) for all i where X = ∐_{i} X_{i} denotes the decomposition of X such that X_{i} ∈ C_{i} for all i. In fact, W = V (∐_{i} X_{i}) = ⋃_{i} V (X_{i}) (e.g., see [1, Proposition 5.7.5]), and W_{i} ∩ W_{j} = {0} for all i ≠ j since V (X_{i}) ∩ V (X_{j}) = V (X_{i} ⊗ X_{j}) (e.g., see [1, Theorem 5.7.11]) and V (P) = {0} for any projective Λ-module P (e.g., see [1, Proposition 5.7.2]). Note that C_{W_{i}} ∩ C_{i} ≠ 0 for all i, since 0 ≠ X_{i} ∈ C_{W_{i}} ∩ C_{i}. We obtain from Proposition 3.1 a decomposition C_{W} = ∐_{i} C_{W_{i}}, and the uniqueness of the decomposition C_{W} = ∐_{i} C_{i} into indecomposable thick tensor-ideals from Theorem A implies C_{W_{i}} = C_{i} for all i. If C_{W} = ∐_{i} C_{i} is an arbitrary decomposition, then we can pass to a refinement C_{W} = ∐_{i}(∐_{j∈I_{i}} C_{ij}) with C_{i} = ∐_{j∈I_{i}} C_{ij} and C_{ij} indecomposable for all i and j, by Theorem A. Using the assertion for indecomposable decompositions, one obtains a decomposition W = ⋃_{i}(⋃_{j∈I_{i}} W_{ij}) such that C_{W_{ij}} = C_{ij} for all i, j. If we define W_{i} = ⋃_{j∈I_{i}} W_{ij} for every i, then Proposition 3.1 gives

C_{W_{i}} = ∐_{j∈I_{i}} C_{W_{ij}} = ∐_{j∈I_{i}} C_{ij} = C_{i}.

This finishes the proof.

Recall that a closed homogeneous subvariety W of V_{G}(k) is projectively connected if any decomposition W = W_{1} ∪ W_{2} into closed homogeneous subvarieties with W_{1} ∩ W_{2} = {0} implies W = W_{1} or W = W_{2}. We obtain the following immediate consequences of Proposition 3.1 and Proposition 3.2.

Corollary 3.3. For a closed homogeneous subvariety W of V_{G}(k) the following conditions are equivalent:

(1) W is non-zero and projectively connected;

(2) C_{W} is an indecomposable thick tensor-ideal in \underline{mod} Λ;

(3) E_{C_{W}} is an indecomposable object in \underline{Mod} Λ.

Corollary 3.4. Let W be a closed homogeneous subvariety of V_{G}(k), and let W = ⋃_{i=1}^{n} W_{i} be the decomposition of W into projectively connected non-zero components. Then C_{W} = ∐_{i=1}^{n} C_{W_{i}} is the decomposition of C_{W} into indecomposable thick tensor ideals. Moreover, the corresponding idempotent Λ-module E_{C_{W}} has precisely n non-projective indecomposable direct summands.

We give now an example of a thick tensor-ideal C such that the decomposition C = ∐_{i∈I} C_{i} into indecomposable thick tensor-ideals is infinite. However, it is clear that card I ≤ card S whenever there is a set S of indecomposable Λ-modules such that C is the smallest thick tensor-ideal containing S.

Example 3.5. Let G = ℤ_{2} × ℤ_{2} be the Klein four group and char k = 2. The finitely generated Λ-modules of complexity at most 1 form a thick tensor-ideal C, and C = ∐_{λ∈ℙ^{1}(k)} C_{λ} where {C_{λ} | λ ∈ ℙ^{1}(k)} is the set of indecomposable thick tensor-ideal of the form C_{V_{G}⟨ζ⟩} for some 0 ≠ ζ ∈ H^{1}(G, k), and V_{G}⟨ζ⟩ denotes the closed homogenous subvariety of maximal ideals containing ζ.

In [5], Benson and Gnacadja conjecture that for every closed homogeneous subvariety W of V_{G}(k) the module E_{C_{W}} is pure-injective. This conjecture becomes true provided one works in the appropriate category. Recall from [14] the following concept of purity for a compactly generated triangulated category T . A map X → Y in T is a pure monomorphism if the induced map Hom(C, X) → Hom(C, Y) is a monomorphism for every compact object C in T , and X is pure-injective if every pure monomorphism X → Y splits. For example, a Λ-module X is pure-injective in Mod Λ if and only if X is a pure-injective object in \underline{Mod} Λ. The statement in part (3) of Theorem 1.2 shows that every endofinite object X in T is pure-injective. We conclude that for every thick subcategory C of \underline{mod} Λ the corresponding Λ-module E_{C} is a pure-injective object in the localizing subcategory of \underline{Mod} Λ which is generated by C.

3.2. Varieties for infinitely generated modules. In [3], Benson, Carlson, and Rickard generalize the definition of the variety V (X) of a finitely generated Λ-module X. To this end they consider the set V_{G}(k) of closed homogeneous irreducible non-zero subvarieties of the maximal ideal spectrum V_{G}(k) of H^{∗}(G, k). Note that V_{G}(k) can be identified with the projective prime ideal spectrum Proj H^{∗}(G, k). Under this bijection a union of Zariski-closed subsets of Proj H^{∗}(G, k) corresponds to a subset of V_{G}(k) which is closed under specialization. Recall that a subset W of V_{G}(k) is closed under specialization if W ∈ W and V ⊆ W implies V ∈ W. It is clear that the subsets of V_{G}(k) which are closed under specialization form the closed subsets of a topology on V_{G}(k). Given any Λ-module X, there is associated a subset V(X) of V_{G}(k) which is closed under specialization [3, Definition 10.2]. For example, V(X) = {V ∈ V_{G}(k) | V ⊆ V (X)} if X is finitely generated. The following result is due to Benson, Carlson, and Rickard [4]. It is an analogue of Hopkins’ classification of thick subcategories of the homotopy category of unbounded complexes of finitely generated projective modules over a commutative noetherian ring [11].

Proposition 3.6. The map W ↦ C_{W} = \{X ∈ \underline{mod} Λ | V(X) ⊆ W\} defines a bijective correspondence between the subsets of V_{G}(k) which are closed under specialization and the thick tensor-ideals in \underline{mod} Λ.

Proof. See Theorem 3.4 in [4].

Clearly, the map W ↦ C_{W} is inclusion preserving and identifies the projectively connected non-empty subsets of V_{G}(k) with the indecomposable thick tensor-ideals in \underline{mod} Λ. Therefore Theorem C is a direct consequence of Corollary 2.7 and the preceding proposition.

References

[1] D.J. Benson, Representations and cohomology II, Cambridge University Press (1991).

[2] D.J. Benson, Stably splitting BG, Bull. Amer. Math. Soc. 33 (1996), 189–198.

[3] D.J. Benson, J.F. Carlson, and J. Rickard, Complexity and varieties for infinitely generated modules, II, Math. Proc. Cambridge Philos. Soc. 120 (1996), 597–615.

[4] D.J. Benson, J.F. Carlson, and J. Rickard, Thick subcategories of the stable module category, Fund. Math. 153 (1979), 59–80.

[5] D.J. Benson and G.P. Gnacadja, Phantom maps and purity in modular representation theory I, Preprint (1997).

[6] J.F. Carlson, The variety of an indecomposable module is connected, Invent. Math. 77 (1984), 291–299.

[7] W.W. Crawley-Boevey, Locally finitely presented additive categories, Comm. Algebra 22 (1994), 1644–1674.

[8] E.S. Devinatz, M.J. Hopkins, and J.H. Smith, Nilpotence and stable homotopy theory, I, Ann. Math. (2) 128 (1988), 207–241.

[9] P. Gabriel, Des catégories abéliennes, Bull. Soc. Math. France 90 (1962), 323–448.

[10] D. Happel, Triangulated categories in the representation theory of finite dimensional algebras, London Math. Soc. Lecture Note Ser. 119 (1988).

[11] M.J. Hopkins, Global methods in homotopy theory, in: Homotopy theory (Durham, 1985), London Math. Soc. Lecture Note Ser. 117 (1987), 73–96.

[12] H. Krause, The spectrum of a locally coherent category, J. Pure Appl. Algebra 114 (1997), 259– 271.

[13] H. Krause, Stable equivalence preserves representation type, Comment. Math. Helv. 72 (1997), 266–284.

[14] H. Krause, Smashing subcategories and the telescope conjecture – an algebraic approach, Preprint, Universität Bielefeld (1998).

[15] H. Krause and U. Reichenbach, Endofiniteness in stable homotopy theory, in preparation.

[16] A. Neeman, The Connection between the K-theory localization theorem of Thomason, Trobaugh and Yao and the smashing subcategories of Bousfield and Ravenel, Ann. Sci. École Norm. Sup. (4) 25 (1992), 547–566.

[17] A. Neeman, The Grothendieck duality theorem via Bousfield’s techniques and Brown representability, J. Amer. Math. Soc. 9 (1996), 205–236.

[18] J. Rickard, Idempotent modules in the stable category, J. London Math. Soc. 56 (1997), 149–170.

[19] J.L. Verdier, Catégories derivées, état 0, Springer Lec. Notes 569 (1977), 262–311.

Fakultät für Mathematik, Universität Bielefeld, 33501 Bielefeld, Germany

E-mail address: henning@mathematik.uni-bielefeld.de