IDEMPOTENT MODULES IN THE STABLE CATEGORY

JEREMY RICKARD

November 1994, revised June 1995

Abstract. Let G be a finite group and k be an algebraically closed field of prime characteristic. Corresponding to each closed homogeneous subvariety W of the maximal ideal spectrum of H^{∗}(G, k) we construct (usually infinite-dimensional) kGmodules E(W) and F (W) which are idempotent in the sense that E(W) and F (W) are isomorphic (up to projective summands) to E(W) ⊗ E(W) and F (W) ⊗ F (W) respectively. We study the properties of these modules, and as an application we use them to describe natural direct sum decompositions of modules in quotient categories.

1. Introduction

In the study of modular representation theory of finite groups, the notions of complexity and varieties for finitely generated modules (see [2, Chapter 5] for a good exposition) have proved very fruitful over the last several years. More recently, it has been realized that it is desirable to extend the theory to infinitely generated modules. In Carlson, Donovan and Wheeler’s study [8] of complexity quotient categories they observed that although the trivial module was indecomposable in such a category, its endomorphism ring could have non-trivial idempotents. In [4] it was shown that the quotient category construction could be extended to arbitrary modules, and that the idempotents in the endomorphism ring of the trivial module corresponded to a direct sum decomposition into infinite dimensional modules.

One tool that was very useful in the development of the theory of varieties for modules was the kG-module L_{ζ} associated to a homogeneous element ζ ∈ H^{n}(G, k) in the mod p cohomology ring of a finite group G. This module, introduced by Carlson [7], is just the kernel of a map Ω^{n}k ⟶ k representing ζ. One important property of L_{ζ} is that its variety is precisely the hypersurface V_{G}⟨ζ⟩ defined by ζ in the maximal ideal spectrum V_{G} of H^{∗}(G, k). Of course, this hypersurface does not determine ζ uniquely: for example, we could replace ζ by ζ^{2} without changing the hypersurface, but the modules L_{ζ} and L_{ζ^{2}} are different.

In this paper we hope to convince the reader that once one accepts the idea of introducing infinitely generated modules, one can do even better. We shall construct (infinitely generated) modules E(W) and F (W) associated to each closed homogeneous subvariety W of V_{G}. The variety of E(W) is W, and in fact it is, in a sense that we will make precise, the ‘universal’ module with variety W.

Let us list some of the properties of these modules: these properties are best stated in the context of the stable module category StMod(kG) of (not necessarily finitely generated) kG-modules (see Section 3).

• There is a canonical distinguished triangle

E(W) ⟶^{η_{W}} k ⟶^{ϵ_{W}} F (W) ⟶

where k denotes the one-dimensional trivial module.

• For any module M, the map

M ⊗ η_{W} : M ⊗ E(W) ⟶ M

is the universal map (in the stable category) to M from a module whose variety is contained in W.

• In the stable category, E(W) is isomorphic to E(W) ⊗ E(W) and F (W) is isomorphic to F (W) ⊗ F (W) — they are ‘idempotent modules’.

• If W_{1} and W_{2} are any two closed homogeneous subvarieties of V_{G} then there are natural isomorphisms

E(W_{1} ∩ W_{2}) ≅ E(W_{1}) ⊗ E(W_{2})

and

F (W_{1} ∪ W_{2}) ≅ F (W_{1}) ⊗ F (W_{2})

in the stable category, and there are distinguished triangles (‘Mayer-Vietoris triangles’)

E(W_{1} ∩ W_{2}) ⟶ E(W_{1}) ⊕ E(W_{2}) ⟶ E(W_{1} ∪ W_{2}) ⟶

and

F (W_{1} ∩ W_{2}) ⟶ F (W_{1}) ⊕ F (W_{2}) ⟶ F (W_{1} ∪ W_{2}) ⟶ .

• The modules E(W) and F (W) behave well with respect to restriction. If H is a subgroup of G, then the restriction of E(W) to H is the kH-module E((res_{G,H}^{∗})^{−1}W ).

Let us end this introduction by saying something about the construction of the modules E(W) and F (W). In fact, we shall give two constructions. The first, given in Section 5, is a special case of something far more general, and involves nothing very original: it is the same construction that is involved in the proof of the Brown Representability Theorem from algebraic topology. The second construction, given in Section 6, is less general, but has the advantage that it gives more insight into the explicit structure of the modules. It is convenient to have both constructions available when we study the properties of the modules.

2. Notation and basic definitions

Throughout this paper, G will be a finite group and k will be an algebraically closed field of characteristic p > 0.

By Mod(kG) we will denote the category of (not necessarily finitely-generated) right kG-modules, and by mod (kG) the full subcategory consisting of finitely generated modules. By StMod(kG) and stmod(kG) we will denote the corresponding stable categories: i.e., the categories whose objects are the same as those of Mod(kG) and mod (kG) respectively, but whose morphisms are equivalence classes of module homomorphisms, where two homomorphisms

α, β : M ⟶ N

are equivalent if α − β factors through a projective module. If M and N are kGmodules, then we shall use the notation \underline{Hom}(M, N) for the k-vector space of maps from M to N in the category StMod(kG).

When we use the tensor product sign without any subscript, as in M ⊗N, this will mean the tensor product over k of the kG-modules M and N, itself regarded as a kG-module in the usual way, via the diagonal action of G. Since the tensor product of any kG-module with a projective module is itself projective, the tensor product of two objects of StMod(kG) makes sense.

3. The stable module category

The category stmod(kG) has been used extensively in representation theory, its ‘infinite dimensional’ analogue StMod(kG) rather less so. In this section we shall point out that most of the familiar properties of stmod(kG) are shared by StMod(kG).

First of all, StMod(kG) is a triangulated category; this follows from the fact that, since kG is a quasi-Frobenius algebra, a kG-module is projective if and only if it is injective—even if the module is infinite-dimensional [9]. We shall use the language of triangulated categories extensively in this paper, so let us remind the reader of the relevant structure possessed by StMod(kG).

Given a module M, we can define a new module ΩM to be the kernel of a surjective map from a projective module to M. In StMod(kG), this module is independent (up to canonical isomorphism) of the choice of surjection, and actually defines a selfequivalence

Ω : StMod(kG) ⟶ StMod(kG)

of the stable category, and so Ω^{n}M makes sense, even for negative n.

For any two modules X and Y, there is a natural isomorphism

\underline{Hom}(X, Ω^{−1}Y) ≅ Ext_{kG}^{1}(X, Y),

and so, given a short exact sequence

0 ⟶ L ⟶ M ⟶ N ⟶ 0

of modules, there is a canonical map (in StMod(kG)) from N to Ω^{−1}L. A diagram

L ⟶ M ⟶ N ⟶ Ω^{−1}L

isomorphic to one arising in this way is called a distinguished triangle. We shall often abbreviate the diagram to

L ⟶ M ⟶ N ⟶ .

With all this extra structure (the self-equivalence Ω and the class of distinguished triangles) StMod(kG) becomes a triangulated category in the sense of Verdier. For a recent exposition of the theory of triangulated categories, see Weibel’s book [13, Chapter 10].

The following lemma is well-known (and obvious, by the Krull-Schmidt theorem) for finitely generated modules.

Lemma 3.1. (a) Let M be a kG-module. Then M is the direct sum of a projective module and a module M_{pf} (the projective-free part of M) which has no non-zero projective summand.

(b) If X and Y are kG-modules with no non-zero projective summands, then a module homomorphism α : X ⟶ Y is an isomorphism in Mod(kG) if and only if it is an isomorphism in StMod(kG).

(c) The projective-free part M_{pf} is determined up to isomorphism by M.

Proof. (a) Since any filtered colimit of projective kG-modules is projective [1], the union of any chain of projective submodules of M is projective. Thus, by Zorn’s Lemma, M has a maximal projective submodule. Since projective kG-modules are also injective, this submodule is a summand of M. By maximality, a complement cannot have a non-zero projective submodule.

(b) Suppose α is an isomorphism in StMod(kG). If α is not injective, then let S be a simple submodule of the kernel. Since the composition

S ⟶ X ⟶^{α} Y

is zero, the inclusion S ⟶ X must be zero in StMod(kG), and so must factor through I(S), the injective hull of S. But then the map I(S) ⟶ X must be injective, as S is the unique minimal submodule of I(S), contradicting the assumption that X has no projective summand. Similarly, by considering a simple quotient of the cokernel, α must be surjective, and so α is a module isomorphism.

(c) This follows from (b).

Thus isomorphism classes of objects of StMod(kG) correspond naturally to isomorphism classes of projective-free modules. This correspondence respects direct sums.

Lemma 3.2. (a) Let M and N be kG-modules. Then

(M ⊕ N)_{pf} ≅ M_{pf} ⊕ N_{pf}.

(b) A kG-module M is indecomposable in StMod(kG) if and only if its projectivefree part is indecomposable as a module.

Proof. (a) We just have to show that M_{pf} ⊕ N_{pf} is itself projective-free. If P were an indecomposable projective submodule, then since

soc(P) ⟶ P ⟶ M_{pf} ⊕ N_{pf}

is non-zero, its composition with the projections M_{pf} ⊕ N_{pf} onto at least one of the summands (M_{pf} say) must be non-zero. But then the map from P to M_{pf} must be injective.

(b) This follows from (a).

We shall occasionally use the following criterion to prove that \underline{Hom}(M, N) = 0. The proof is the same as in the finite-dimensional case.

Lemma 3.3. Let M, N be kG-modules such that Hom_{k}(M, N) is a projective kGmodule. Then \underline{Hom}(M, N) = 0.

Proof. Let α : M ⟶ N be a module homomorphism. Then α is the composition of the maps

M ⟶ M ⊗ Hom_{k}(M, N) : m ↦ m ⊗ α

and

M ⊗ Hom_{k}(M, N) ⟶ N : m′ ⊗ β ↦ β(m′).

Thus α factors through a projective module.

Let us end this section by recalling some of the ideas of complexity and varieties for infinite dimensional modules, as defined in [4]. We shall assume that the reader is familiar with these concepts for finite-dimensional modules — Chapter 5 of Benson’s book [2] is a good reference.

If M is a (possibly infinitely generated) kG-module, then the complexity of M is defined to be the smallest natural number c such that every map X ⟶ M from a finitely generated module X factors through a finitely generated module of complexity c. This is the same [4, Theorem 5.3] as the smallest natural number c such that M is a filtered colimit of finitely generated modules of complexity at most c.

In [4, Definition 7.2], a tentative definition is given of the variety of a (possibly infinitely generated) module M. As pointed out in that paper, this definition has certain drawbacks, and might be better thought of as describing the closure of the variety of the module. However, these subtleties will not concern us here, as all we shall need is a description of which modules have variety contained in a given closed subvariety of V_{G}, the maximal ideal spectrum of H^{∗}(G, k), and this is uncontroversial.

Definition. (cf. [4, Definition 7.2]) Let M be a kG-module and let W be a closed homogeneous subvariety of V_{G}. We say that ‘the variety of M is contained in W ’ if W is defined by a graded ideal of H^{∗}(G, k) that is contained in the radical of the annihilator of every finitely generated graded H^{∗}(G, k)-submodule of Ext_{kG}^{∗}(N, M) for every finitely generated kG-module N.

By [4, Theorem 7.5] this is equivalent to demanding that M be a filtered colimit of finitely generated modules whose varieties are contained in W.

4. Homotopy colimits in the stable module category

It is quite rare for a sequence

X_{1} ⟶^{α_{1}} X_{2} ⟶^{α_{2}} X_{3} ⟶^{α_{3}} …

of maps in a triangulated category to have a colimit in the sense of category theory. This is because such a colimit would, so long as the category had countable direct sums, be the cokernel of the map

⨁_{i=0}^{∞} X_{i} ⟶^{1−α} ⨁_{i=0}^{∞} X_{i}, (*)

where α is the map induced by the α_{i}, and the only maps in a triangulated category that have cokernels are compositions of split epimorphisms followed by split monomorphisms. However, there is a substitute for the colimit that does exist whenever the category has countable direct sums. This is the mapping telescope construction of algebraic topology, also called [5] the homotopy colimit, hocolim(X_{i}), of the sequence, and it is constructed by completing the map (*) to a distinguished triangle

⨁_{i=0}^{∞} X_{i} ⟶^{1−α} ⨁_{i=0}^{∞} X_{i} ⟶ hocolim(X_{i}) ⟶ . (**)

Note that, although the homotopy colimit is well-defined up to isomorphism, it is not in general well-defined up to unique isomorphism in the way that a genuine colimit would be (and so perhaps we should pedantically refer to a rather than the homotopy colimit). This is because the completion of a map to a distinguished triangle in a triangulated category is in general only unique up to non-unique isomorphism.

The construction described above is very well-known and widely used in topology; Bökstedt and Neeman’s paper [5], which gives several applications of a more algebraic nature, provides a good introduction for ignorant algebraists.

Before we go on to study homotopy colimits in the special case of the stable module category, there are two general facts that we shall need.

Lemma 4.1. Let

X_{1} ⟶^{α_{1}} X_{2} ⟶^{α_{2}} X_{3} ⟶^{α_{3}} …

be a sequence of maps in a triangulated category with countable direct sums. If, for each i > 0, there is k ≥ i such that the composition

α_{i}α_{i+1} … α_{k}

is zero, then

hocolim(X_{i}) ≅ 0.

Proof. The conditions ensure that the map

1 + ∑_{i=1}^{∞} α^{i} : ⨁_{i=0}^{∞} X_{i} ⟶ ⨁_{i=0}^{∞} X_{i}

makes sense, since on each X_{j} only finitely many of the α^{i} are non-zero. This map is an inverse to the map 1 − α of (*), which is therefore an isomorphism. Hence hocolim(X_{i}), the third corner of the distinguished triangle (**) containing this map, is zero.

Lemma 4.2. Let

X_{1} ⟶ Y_{1} ⟶ Z_{1} ⟶
↓  ↓  ↓
X_{2} ⟶ Y_{2} ⟶ Z_{2} ⟶
↓  ↓  ↓
⋮  ⋮  ⋮
↓  ↓  ↓
X_{n} ⟶ Y_{n} ⟶ Z_{n} ⟶
↓  ↓  ↓
⋮  ⋮  ⋮

be a sequence of maps between distinguished triangles in a triangulated category (e.g., StMod(kG)) with countable direct sums and with the property that a countable direct sum of distinguished triangles is itself a distinguished triangle. Then there is a distinguished triangle

hocolim(X_{i}) ⟶ hocolim(Y_{i}) ⟶ hocolim(Z_{i}) ⟶ .

Proof. By a general fact about triangulated categories due to Verdier (see, for example, [13, Exercise 10.2.6]), there is a diagram

⨁ X_{i}  ⟶  ⨁ Y_{i}  ⟶  ⨁ Z_{i}  ⟶  ⨁ X_{i}[1]
↓  ↓  ↓  ↓
⨁ X_{i}  ⟶  ⨁ Y_{i}  ⟶  ⨁ Z_{i}  ⟶  ⨁ X_{i}[1]
↓  ↓  ↓  ↓
hocolim(X_{i}) ⟶ hocolim(Y_{i}) ⟶ hocolim(Z_{i}) ⟶ hocolim(X_{i})[1]
↓  ↓  ↓  ∗  ↓
⨁ X_{i}[1] ⟶  ⨁ Y_{i}[1] ⟶  ⨁ Z_{i}[1] ⟶  ⨁ X_{i}[2]

which commutes, except for the square marked ∗, which anti-commutes, and where all the rows and columns are distinguished triangles.

Our next lemma shows that homotopy colimits in the stable module category can be interpreted in terms of genuine colimits in the module category.

Lemma 4.3. Let

X_{1} ⟶^{α_{1}} X_{2} ⟶^{α_{2}} X_{3} ⟶^{α_{3}} …

be a sequence of module homomorphisms in Mod(kG). The colimit colim(X_{i}) in Mod(kG) is isomorphic, in StMod(kG), to the homotopy colimit hocolim(X_{i}).

Proof. The module homomorphism

⨁_{i=0}^{∞} X_{i} ⟶^{1−α} ⨁_{i=0}^{∞} X_{i}

is injective and its cokernel is colim(X_{i}), so there is a short exact sequence

0 ⟶ ⨁_{i=0}^{∞} X_{i} ⟶^{1−α} ⨁_{i=0}^{∞} X_{i} ⟶ colim(X_{i}) ⟶ 0.

Hence there is a distinguished triangle

⨁_{i=0}^{∞} X_{i} ⟶^{1−α} ⨁_{i=0}^{∞} X_{i} ⟶ colim(X_{i}) ⟶

in StMod(kG), and so, by the definition of the homotopy colimit, colim(X_{i}) is isomorphic to hocolim(X_{i}).

5. A general construction

In this section, we shall describe a very general construction in StMod(kG). The idea of the construction will be familiar to many topologists, being more or less identical to that involved in the Brown Representability Theorem [6]. For the convenience of those readers who are not familiar with this area of algebraic topology, we shall give a complete presentation of the properties of the construction, and we shall use special properties of StMod(kG) whenever this simplifies matters, even where it is not strictly necessary. Readers interested in Brown representability and its generalization, Bousfield localization, in the context of a general triangulated category are advised to look at recent papers of Neeman [10, 11, 12], in which he persuasively argues that these ideas are much underused outside algebraic topology.

Throughout this section, C will be an épaisse subcategory of stmod(kG), i.e., a full triangulated subcategory that is closed under taking direct summands; later we shall impose extra properties on C. Associated to C there are two full triangulated subcategories of StMod(kG): C^{⊕} will be the smallest full triangulated subcategory of StMod(kG) that contains C and is closed under arbitrary direct sums, and C⃗ will be the full subcategory of StMod(kG) whose objects are modules which are filtered colimits (in Mod(kG)) of modules in C. For example, C might be the stable category of finitely generated kG-modules of complexity at most c, in which case the objects of C⃗ are what are defined in [4] to be the (not necessarily finitely generated) modules of complexity at most c.

The proofs of [4, Theorem 4.2, Theorem 5.3] work in this more general situation to show that an object M of StMod(kG) is in C⃗ if and only if it has the property that any map from a finitely generated module to M factors, in StMod(kG), through an object of C, and hence that C⃗ is indeed a triangulated category. Also, this alternative description makes it clear that C⃗ is closed under arbitrary direct sums, and so C^{⊕} is contained in C⃗.

The alternative description of C⃗ also makes it easy to see that the finitely generated modules in C⃗ (or C^{⊕}) are precisely the objects of C – this uses the fact that C is closed under taking direct summands. In [4, Theorem 5.3] it is shown that, for C the stable category of modules of complexity at most c, the smallest full triangulated subcategory of StMod(kG) that is closed under countable direct sums coincides with the full subcategory of countably generated modules in C⃗. We shall see later that, for the same C, the categories C⃗ and C^{⊕} coincide.

Let us start by giving some trivial properties of C^{⊕}.

Lemma 5.1. If

C_{1} ⟶ C_{2} ⟶ C_{3} ⟶ …

is a sequence of maps between objects of C^{⊕}, then hocolim(C_{i}) is in C^{⊕}.

Proof. By the definition of hocolim(C_{i}), it is in the smallest triangulated subcategory of StMod(kG) that contains all the C_{i} and is closed under countable direct sums.

Definition. Let L be a subcategory of a triangulated category S. An object Y of S is called L-local if Hom_{S}(X, Y) = 0 for all objects X of L.

Lemma 5.2. An object of StMod(kG) is C-local if and only if it is C^{⊕}-local.

Proof. Suppose Y is C-local. Consider objects X such that \underline{Hom}(X, Ω^{i}Y) = 0 for all i. The class of such objects forms a full triangulated subcategory of StMod(kG), since a distinguished triangle in StMod(kG) gives rise to a long exact sequence upon applying the functor \underline{Hom}(−, Y). Also, this class of objects is closed under taking arbitrary direct sums and contains C. It therefore contains C^{⊕}. The other implication is trivial.

Now we shall describe the construction that we are going to study.

Let X be an object of StMod(kG). Let us set X_{0} = X: we shall inductively construct a sequence of objects X_{0}, X_{1},… as follows.

There is a natural module homomorphism

⨁_{i}(C_{i} ⊗_{k} Hom_{kG}(C_{i}, X_{n})) ⟶ X_{n},

where C_{i} runs over a complete set of representatives of isomorphism classes of modules in C. By construction this map has the property that every map C ⟶ X_{n}, where C is in C, factors through it. We can now define X_{n+1} up to (non-unique) stable isomorphism by completing this map to a distinguished triangle

⨁_{i}(C_{i} ⊗_{k} Hom_{kG}(C_{i}, X_{n})) ⟶ X_{n} ⟶ X_{n+1} ⟶ .

Each X_{n} comes equipped with a map from X, namely the composition

X = X_{0} ⟶ X_{1} ⟶ … ⟶ X_{n},

and so we can define objects A_{n} by the distinguished triangles

A_{n} ⟶ X ⟶ X_{n} ⟶ .

There is a sequence of maps of distinguished triangles

A_{0} ⟶ X ⟶ X_{0} ⟶
↓ ∥ ↓
A_{1} ⟶ X ⟶ X_{1} ⟶
↓ ∥ ↓
A_{2} ⟶ X ⟶ X_{2} ⟶
↓  ∥  ↓
⋮  ⋮  ⋮

and taking homotopy colimits, we get, by Lemma 4.2, a distinguished triangle

T_{C}(X) : E_{C}(X) ⟶ X ⟶ F_{C}(X) ⟶ .

The completion of maps to distinguished triangles is a construction that is only unique up to non-unique isomorphism. Since the construction of the triangle T_{C}(X) involved many such completions, it is not immediately clear that the triangle we end up with is independent of the choices made, but we shall soon see that this is so.

We shall now study the properties of this construction in a series of easy steps.

Lemma 5.3. For all i, A_{i} is in C^{⊕}.

Proof. This is clearly true for A_{0} = 0. Suppose inductively that it is true for A_{n}. By the octahedral axiom, there is a commutative diagram

X ⟶ X_{n} ⟶ Ω^{−1}A_{n} ⟶
∥ ↓ ↓
X ⟶ X_{n+1} ⟶ Ω^{−1}A_{n+1} ⟶
↓  ↓
C = C
↓  ↓

where C is a direct sum of objects of C and all the rows and columns are distinguished triangles. Hence A_{n+1} is in C^{⊕}, since both A_{n} and C are.

Proposition 5.4. E_{C}(X) is in C^{⊕}.

Proof. Since E_{C}(X) is hocolim(A_{i}), this follows from Lemmas 5.1 and 5.3.

Proposition 5.5. F_{C}(X) is C-local, and hence C^{⊕}-local.

Proof. Let C be an object of C. Since C is finitely generated, every map

C ⟶^{α} F_{C}(X) = hocolim(X_{i})

must factor through X_{n} ⟶ F_{C}(X) for some n. But by the construction of X_{n+1}, the composition

C ⟶ X_{n} ⟶ X_{n+1}

is zero, since C is in C. Hence α must be zero, and so F_{C}(X) is C-local. By Lemma 5.2 it is also C^{⊕}-local.

Proposition 5.6. The map E_{C}(X) ⟶ X that occurs in the distinguished triangle T_{C}(X) is the universal map in StMod(kG) from an object of C^{⊕} to X.

Proof. Let C be an object of C^{⊕}. By Proposition 5.5, the long exact sequence obtained by applying \underline{Hom}(C, −) to the triangle T_{C}(X) gives an isomorphism

\underline{Hom}(C, E_{C}(X)) ≅ \underline{Hom}(C, X).

Hence any map to X from an object of C^{⊕} factors uniquely through E_{C}(X).

Proposition 5.7. The map X ⟶ F_{C}(X) that occurs in the distinguished triangle T_{C}(X) is the universal map in StMod(kG) from X to a C-local object.

Proof. If L is C-local, then the functor \underline{Hom}(−, L) applied to T_{C}(X) gives, by Proposition 5.4, an isomorphism

\underline{Hom}(F_{C}(X), L) ≅ \underline{Hom}(X, L).

Thus any map from X to L factors uniquely through F_{C}(X).

Remark. Propositions 5.6 and 5.7 are formal consequences of the fact that E_{C}(X) is in C^{⊕} and that F_{C}(X) is C-local, and hence C^{⊕}-local. Thus any other distinguished triangle whose middle object is X and whose other objects have these properties will be canonically isomorphic to T_{C}(X). Also, these facts show that not only is the distinguished triangle T_{C}(X) determined uniquely up to isomorphism, it is even functorial in X.

In many of the examples that we are interested in, we can give an even better description of E_{C}(X) and F_{C}(X). To do this, we shall need to impose some extra conditions on C; for convenience of reference we shall make the following definition.

Definition. An épaisse subcategory C of stmod(kG) is called a tensor-ideal subcategory if it satisfies the following two conditions:

(i) If M is in C, then its dual M^{∨} is also in C.

(ii) If M is in C and N is in stmod(kG), then M ⊗ N is in C.

For example, the stable category of finitely generated kG-modules of complexity at most c is a tensor-ideal subcategory of stmod(kG). Another example is the stable category of finitely generated modules whose variety is contained in some fixed closed subvariety W of V_{G}. Actually, condition (i) follows from condition (ii), since M^{∨} is a summand of M^{∨} ⊗ M ⊗ M^{∨}, and C is closed under taking direct summands.

Lemma 5.8. Let C be a tensor-ideal subcategory of stmod(kG). If C is in C^{⊕} and X is a finitely generated kG-module, then C ⊗ X is in C^{⊕}.

Proof. Since C is tensor-ideal, C is contained in the full subcategory of StMod(kG) consisting of those modules Y such that Y ⊗X is in C^{⊕}. Also it is clearly triangulated and closed under arbitrary direct sums, and so it contains C^{⊕}.

Proposition 5.9. Let C be a tensor-ideal subcategory of stmod(kG). If C is in C^{⊕} and X is any kG-module, then C ⊗ X is in C^{⊕}.

Proof. By Lemma 5.8, the class of modules X for which this is true contains all finitely generated modules. It is closed under arbitrary direct sums, and so contains all semi-simple modules. Finally, it is closed under extensions, and so contains all modules, by induction on the length of the radical filtration.

Lemma 5.10. Let C be a tensor-ideal subcategory of stmod(kG). If L is C-local and X is finitely generated, then L ⊗ X is C-local.

Proof. Let C be in C. Then, since X is finitely generated, there is an isomorphism

\underline{Hom}(C, L ⊗ X) ≅ \underline{Hom}(C ⊗ X^{∨}, L).

Since C is tensor-ideal, C ⊗ X^{∨} is in C, and so \underline{Hom}(C ⊗ X^{∨}, L) is zero.

Proposition 5.11. Let C be a tensor-ideal subcategory of stmod(kG). If L is C-local and X is any kG-module, then L ⊗ X is C-local.

Proof. The class of modules X for which this is true contains all finitely generated modules, by Lemma 5.10. It is closed under arbitrary direct sums, since for a finitely generated module C the functor \underline{Hom}(C, −) commutes with direct sums, and so it contains all semisimple modules. It is closed under extensions, and so it contains all modules, as in the proof of Proposition 5.9.

Lemma 5.12. Let C be a tensor-ideal subcategory of stmod(kG). For any kG-module X, X ⊗ E_{C}(k) is in C^{⊕} and X ⊗ F_{C}(k) is C-local, and hence C^{⊕}-local, where k is the trivial kG-module.

Proof. This follows immediately from Propositions 5.4, 5.5, 5.9 and 5.11.

Proposition 5.13. Let C be a tensor-ideal subcategory of stmod(kG). The distinguished triangle

X ⊗ E_{C}(k) ⟶ X ⟶ X ⊗ F_{C}(k) ⟶,

obtained by taking the tensor product of X with T_{C}(k), is isomorphic to T_{C}(X).

Proof. It follows from Lemma 5.12, just as in the proofs of Propositions 5.6 and 5.7, that the maps in the distinguished triangle X ⊗ T_{C}(k) share the universal properties of the maps in T_{C}(X).

Lemma 5.14. If C is in a tensor-ideal subcategory C of stmod(kG), then C ⊗ F_{C}(X) ≅ 0 in StMod(kG).

Proof. By Propositions 5.6 and 5.13, the first map in the distinguished triangle

C ⊗ T_{C}(k) : C ⊗ E_{C}(k) ⟶ C ⟶ C ⊗ F_{C}(k) ⟶

is the universal map from an object of C^{⊕} to C. Since C is already in C^{⊕}, this must be an isomorphism, and so C ⊗ F_{C}(k) must be zero. Hence, for any module X,

C ⊗ F_{C}(X) ≅ C ⊗ F_{C}(k) ⊗ X ≅ 0

in StMod(kG).

Proposition 5.15. If C is a tensor-ideal subcategory of stmod(kG) and C is in C⃗, then C ⊗ F_{C}(X) ≅ 0 in StMod(kG).

Proof. Suppose C is the filtered colimit colim(C_{i}) of modules in C. Then, in Mod(kG),

C ⊗ F_{C}(X) ≅ colim(C_{i} ⊗ F_{C}(X)),

which is projective, since a filtered colimit of projective kG-modules is projective.

Lemma 5.16. If C is a tensor-ideal subcategory of stmod(kG) and C is in C⃗, then F_{C}(C) ≅ 0 in StMod(kG).

Proof. By Proposition 5.13, F_{C}(C) is stably isomorphic to C ⊗ F_{C}(k), which is projective, by Proposition 5.15.

Theorem 5.17. If C is a tensor-ideal subcategory of stmod(kG), then the categories C^{⊕} and C⃗ are the same.

Proof. We saw at the beginning of this section that C^{⊕} is contained in C⃗. To prove the other inclusion, suppose that C is in C⃗. Then C ≅ E_{C}(C), by Lemma 5.16, which is in C^{⊕} by Proposition 5.4.

Lemma 5.18. If C is a tensor-ideal subcategory of stmod(kG), then the tensor product E_{C}(k) ⊗ F_{C}(k) is projective.

Proof. This follows immediately from Propositions 5.4 and 5.15 and Theorem 5.17.

Theorem 5.19. Let C be a tensor-ideal subcategory of stmod(kG). There are stable isomorphisms

E_{C}(k) ≅ E_{C}(k) ⊗ E_{C}(k)

and

F_{C}(k) ≅ F_{C}(k) ⊗ F_{C}(k).

Proof. These isomorphisms follow from Lemma 5.18 by taking the tensor product of the distinguished triangle T_{C}(k) with E_{C}(k) and F_{C}(k) respectively.

Remark. The property of the modules E_{C}(k) and F_{C}(k) described in Theorem 5.19 explains the term idempotent module used in the title of this paper.

6. Another construction of idempotent modules

If W is a closed homogeneous subvariety of V_{G}, and if we take C(W) to be the full subcategory of stmod(kG) consisting of finitely generated modules whose variety is contained in W, then the general theory of Section 5 applies to give a distinguished triangle

E_{C(W)}(k) ⟶ k ⟶ F_{C(W)}(k) ⟶ .

It will be useful to have another construction of this triangle, and that is what we shall provide in this section.

By [4, Theorem 7.5], C⃗(W) is the full subcategory of StMod(kG) consisting of those modules which, by the definition of [4], have variety contained in W. By Theorem 5.17, this also coincides with the category C(W)^{⊕}.

Let us first deal with the case where W is a hypersurface in V_{G}, defined by a non-zero homogeneous element ζ ∈ H^{n}(G, k), where n > 0.

Since there is a natural isomorphism

H^{n}(G, k) ≅ \underline{Hom}(Ω^{i}k, Ω^{i−n}k)

for each i, we can consider the sequence of maps

k ⟶ Ω^{−n}k ⟶ Ω^{−2n}k ⟶ … ⟶ Ω^{−rn}k ⟶ …

in StMod(kG), each of which represents ζ, and we can define F (ζ) to be the homotopy colimit of this sequence. Note that F (ζ) comes equipped with a map from k, which we shall call ϵ_{ζ}.

We can complete the map ϵ_{ζ} to a distinguished triangle

T (ζ) : E(ζ) ⟶^{η_{ζ}} k ⟶^{ϵ_{ζ}} F (ζ) ⟶,

which defines, up to stable isomorphism, another module E(ζ).

Theorem 6.1. With ζ as above, let C(ζ) be the full subcategory of stmod(kG) consisting of modules with variety contained in the hypersurface V_{G}⟨ζ⟩ of V_{G} defined by ζ.

(a) The variety of E(ζ) is contained in V_{G}⟨ζ⟩. In other words, E(ζ) is in C⃗(ζ) = C(ζ)^{⊕}.

(b) F (ζ) is C(ζ)-local.

(c) The distinguished triangle T (ζ) is isomorphic to the triangle T_{C(ζ)}(k) of Section 5.

Proof. (a) For each r, we can choose a module homomorphism Ω^{−rn}k ⟶ Ω^{−(r+1)n}k that represents ζ, and by taking compositions of these maps we get an injective module homomorphism k ⟶ Ω^{−rn}k representing ζ^{r} for each r. These maps can be completed to short exact sequences

0 ⟶ k ⟶ Ω^{−rn}k ⟶ X_{r} ⟶ 0,

where X_{r} is stably isomorphic to Ω^{−nr−1}L_{ζ}, and so has variety contained in V_{G}⟨ζ⟩.

Consider now the following sequence of maps of short exact sequences.

0  0  0  0
↓  ↓  ↓  ↓
k =  k  =  k  = … =  k  = …
∥  ↓  ↓  ↓
k ⟶ Ω^{−n}k ⟶ Ω^{−2n}k ⟶ … ⟶ Ω^{−rn}k ⟶ …
↓ ↓ ↓ ↓
0 ⟶ X_{1} ⟶ X_{2} ⟶ … ⟶ X_{r} ⟶ …
↓  ↓  ↓  ↓
0  0  0  0

The colimit of the middle row is stably isomorphic to F (ζ), by Lemma 4.3, and the map from k, regarded as the colimit of the first row, is just ϵ_{ζ}, so we get a distinguished triangle

k ⟶^{ϵ_{ζ}} F (ζ) ⟶ colim(X_{r}) ⟶

in StMod(kG). Hence E(ζ) is stably isomorphic to Ω(colim(X_{r})), which is in C⃗(ζ). (b) Let C be in C(ζ). Then, since C is finitely generated,

Hom_{k}(C, F (ζ)) ≅ C^{∨} ⊗ F (ζ) ≅ colim(C^{∨} ⊗ Ω^{−nr}k).

Since the variety of C^{∨} is contained in V_{G}⟨ζ⟩, Ext^{∗}(C^{∨}, C^{∨}) is annihilated by some power ζ^{t} of ζ, and so the composition of any t maps in the sequence

C^{∨} ⟶ C^{∨} ⊗ Ω^{−n}k ⟶ C^{∨} ⊗ Ω^{−2n}k ⟶ …

is zero in StMod(kG), and so the colimit of this sequence is projective, by Lemma 4.1. Hence \underline{Hom}(C, F (ζ)) = 0 by Lemma 3.3.

(c) As in the proofs of Propositions 5.6 and 5.7, it follows from (b) and (c) that the maps η_{ζ} and ϵ_{ζ} share the universal properties of the maps in the distinguished triangle T_{C(ζ)}(k).

Now let W be a general closed homogeneous subvariety of V_{G}, and let us simplify notation by defining the distinguished triangle

T (W) : E(W) ⟶^{η_{W}} k ⟶^{ϵ_{W}} F (W) ⟶

to be T_{C(W)}(k).

We shall call E(W) the idempotent module associated with the variety W. It is also helpful to think of F (W) as being the idempotent module associated with the open complement of W: the distinguished triangle T (W) is then analogous to the short exact sequence of sheaves

0 ⟶ k_{X\V} ⟶ k ⟶ k_{V} ⟶ 0

associated with a closed subspace V of a space X and its open complement X\V, where k_{V} and k_{X\V} are the sheaves on X obtained by extending by zero the constant sheaves k on V and X\V respectively. We shall carry this analogy further when we consider Mayer-Vietoris triangles in Section 8.

If W = V_{G}, and so C(W) = stmod(kG), it is clear that T (W) is isomorphic to the triangle

k ⟶^{∼} k ⟶ 0 ⟶ .

If W ≠ V_{G}, then we can express W as an intersection of homogeneous hypersurfaces. We shall now see that we can describe T (W) in terms of the corresponding triangles for these hypersurfaces, which are themselves described by Theorem 6.1.

Proposition 6.2. (a) If U and V are closed homogeneous subvarieties of V_{G} then E(U) ⊗ E(V) is canonically isomorphic to E(U ∩ V).

(b) If W = V_{G}⟨ζ_{1}⟩ ∩ ⋯ ∩ V_{G}⟨ζ_{s}⟩ is the intersection of homogeneous hypersurfaces in V_{G}, then T (W) is isomorphic to the distinguished triangle whose first map is

E(ζ_{1}) ⊗ ⋯ ⊗ E(ζ_{s}) ⟶^{η_{ζ_{1}}⊗···⊗η_{ζs}} k.

Proof. (a) First of all, E(U) ⊗ E(V) has variety contained in E(U ∩ V), by [4, Proposition 8.2]. Consider the map

η_{U} ⊗ η_{V} : E(U) ⊗ E(V) ⟶ k ⊗ k ≅ k.

If α : M ⟶ k is any map in StMod(kG) from a module M with variety contained in U ∩ V, then α factors uniquely through η_{U}, by the universal property of this map, giving a map β : M ⟶ E(U). Now β in turn factors uniquely through E(U) ⊗ η_{V}, by the universal property of that map, and so α factors uniquely through η_{U} ⊗ η_{V}, and so that map shares the universal property of η_{U∩V}. Hence the modules E(U ∩V) and E(U) ⊗ E(V) are canonically isomorphic.

(b) This follows from (a) by induction on s.

Remark. One property of E(W) that is clear from this alternative construction, which is not clear from the more general construction of Section 5, is that it is independent of the field k, in the sense that if k ⊂ K is a field extension of k and W (K) is the K-variety obtained from W by extending scalars, then

E(W (K)) ≅ E(W) ⊗_{k} K.

The corresponding statement is false for E_{C}(k) if C is, for example, the category of modules with complexity at most c.

7. Some properties of the idempotent modules: restrictions and tensor products

Let us start by considering what happens when we restrict the idempotent modules E(W) and F (W) to subgroups. Recall that if H ≤ G, then the inclusion map induces an algebra homomorphism

res_{G,H} : H^{∗}(G, k) ⟶ H^{∗}(H, k),

and hence a map of varieties

res_{G,H}^{∗} : V_{H} ⟶ V_{G}.

The following proposition follows easily from the second construction of the idempotent modules.

Proposition 7.1. If W is a closed homogeneous subvariety of V_{G}, then the restriction to H of the distinguished triangle T (W) is isomorphic to the distinguished triangle

T ((res_{G,H}^{∗})^{−1}W )

in StMod(kH).

Proof. If 0 ≠ ζ ∈ H^{∗}(G, k) is a homogeneous element of degree n, then F (ζ) is isomorphic to the homotopy colimit of the sequence of maps

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

in StMod(kG). Restricting to H, this sequence becomes

k ⟶^{ζ′} Ω^{−n}k ⟶^{ζ′} Ω^{−2n}k ⟶^{ζ′} …,

where ζ′ = res_{G,H}(ζ). Hence the restriction of F (ζ) to H is stably isomorphic to F (ζ′), and the restriction of ϵ_{ζ} is ϵ_{ζ′}, up to stable isomorphism.

The proposition now follows easily for a general subvariety W from Proposition 6.2.

The following lemma is an easy consequence of the universal properties of the idempotent modules.

Lemma 7.2. Let V ⊆ W be closed homogeneous subvarieties of V_{G}.

(a) The map η_{V} factors uniquely through η_{W}, and ϵ_{V} factors uniquely through ϵ_{W}.

(b) The maps

E(V) ⊗ η_{W} : E(V) ⊗ E(W) ⟶ E(V)

and

ϵ_{V} ⊗ F (W) : F (W) ⟶ F (V) ⊗ F (W)

are isomorphisms in StMod(kG).

Proof. (a) This follows from the universal properties of η_{W} and ϵ_{V}, since the variety of E(V) is contained in W, and F (W) is C(W)-local, and hence C(V)-local.

(b) Since E(V) ⊗ η_{W} is the universal map to E(V) from a module with variety in W, and since the variety of E(V) is itself contained in W, the map must be an isomorphism. Similarly ϵ_{V} ⊗ F (W) is an isomorphism.

Let us introduce some notation for the maps guaranteed by the first part of the lemma.

Definition. Let V ⊆ W be closed homogeneous subvarieties of V_{G}. We shall denote by η_{V,W} and ϵ_{V,W} the unique maps that make the diagrams

E(V) ⟶^{η_{V}} k
↓η^{V,W}  ∥
E(W) ⟶^{η_{W}} k

and

k ⟶^{ϵ_{V}} F (V)
∥  ↓ϵ^{V,W}
k ⟶^{ϵ_{W}} F (W)

commute.

The next result that we want to prove is that, for general V and W, F (V ∪ W) is stably isomorphic to the tensor product F (V) ⊗ F (W). This will follow by general nonsense, in the same way as Proposition 6.2, once we have shown that F (V)⊗F (W) is C(V ∪ W)-local. This fact is a consequence of the following proposition.

Proposition 7.3. Let V and W be closed homogeneous subvarieties of V_{G}. The épaisse subcategory of stmod(kG) generated by C(V) and C(W) is C(V ∪ W).

Proof. (cf. Carlson’s paper [7].) Since C(V ∪ W) contains both C(V) and C(W), it must contain the épaisse subcategory that they generate, so we just have to prove the other inclusion. Since this is trivially true for V = W = {0}, and since the poset of closed homogeneous subvarieties of V_{G} satisfies the descending chain condition, we may assume inductively that the proposition is true for all pairs (V′, W′) ≠ (V, W) of subvarieties such that V′ ⊆ V, W′ ⊆ W.

If one of V and W contains the other, then the proposition is trivially true, so suppose this is not so. Then there are homogeneous elements ζ, ξ of H^{∗}(G, k), of degrees m and n, say, such that V ⊆ V_{G}⟨ζ⟩, W ⊆ V_{G}⟨ξ⟩, but V ⊈ V_{G}⟨ξ⟩ and W ⊈ V_{G}⟨ζ⟩.

Let M be a finitely generated module whose variety is contained in V ∪ W. Since V ∪ W ⊆ V_{G}⟨ζξ⟩, some power (ζξ)^{s} of ζξ annihilates Ext^{∗}(M, M). But then we have

M ⊗ L_{(ζξ)^{s}} ≅ ΩM ⊕ Ω^{s(n+m)}M

in stmod(kG) [2, Proposition 5.9.5], and so M is in any épaisse subcategory of stmod(kG) that contains M ⊗ L_{(ζξ)^{s}}.

There is a distinguished triangle

Ω^{sn}L_{ζ^{s}} ⟶ L_{(ζξ)^{s}} ⟶ L_{ξ^{s}} ⟶

[2, Lemma 5.9.3], and so M ⊗ L_{(ζξ)^{s}} is in the triangulated subcategory of stmod(kG) generated by M ⊗ L_{ζ^{s}} and M ⊗ L_{ξ^{s}}. But the varieties of these modules are contained in

V ∪ (W ∩ V_{G}⟨ζ⟩)

and

(V ∩ V_{G}⟨ξ⟩) ∪ W

respectively, and so by the inductive hypothesis both modules are in the épaisse subcategory generated by C(V) and C(W).

Corollary 7.4. If V and W are closed homogeneous subvarieties of V_{G}, then the tensor product F (V) ⊗ F (W) is C(V ∪ W)-local.

Proof. The full subcategory of stmod(kG) consisting of objects X such that

\underline{Hom}(X, F (V) ⊗ F (W)) = 0

is an épaisse subcategory of stmod(kG), and must contain C(V) and C(W), because F (V) ⊗ F (W) is both C(V)-local and C(W)-local by Proposition 5.5. Therefore, by Proposition 7.3, this subcategory contains C(V ∪ W), and so F (V) ⊗ F (W) is C(V ∪ W)-local.

Theorem 7.5. If V and W are any closed homogeneous subvarieties of V_{G}, then F (V) ⊗ F (W) is stably isomorphic to F (V ∪ W).

Proof. By Corollary 7.4, F (V)⊗F (W) is C(V ∪W)-local. Let L be another C(V ∪W)- local module, and let α : k ⟶ L be a map in StMod(kG). Since L is C(V)-local, α factors uniquely through ϵ_{V}, giving a map α′ : F (V) ⟶ L. Since L is also C(W)- local, α′ factors uniquely through F (V) ⊗ ϵ_{W}, by the universal property of this map. Hence α factors uniquely through

ϵ_{V} ⊗ ϵ_{W} : k ⟶ F (V) ⊗ F (W),

and so this map shares the universal property of ϵ_{V ∪W}.

8. Mayer-Vietoris triangles

The following theorem should remind the reader of Mayer-Vietoris sequences in algebraic topology.

Theorem 8.1. Let V and W be closed homogeneous subvarieties of V_{G}. Then there are distinguished triangles, which we shall call Mayer-Vietoris triangles,

E(V ∩ W) ⟶^{(η_{V ∩W,V} ,−η_{V ∩W,W})} E(V) ⊕ E(W) \underrightarrow{( \begin{matrix}η_{V,V ∪W} \\ η_{W,V ∪W} \end{matrix} )} E(V ∪ W) ⟶

and

F (V ∩ W) ⟶^{(ϵ_{V ∩W,V} ,−ϵ_{V ∩W,W})} F (V) ⊕ F (W) \underrightarrow{( \begin{matrix}ϵ_{V,V ∪W} \\ ϵ_{W,V ∪W} \end{matrix} )} F (V ∪ W) ⟶

in StMod(kG).

Proof. If we form a distinguished triangle

E′ ⟶ E(V) ⊕ E(W) \underrightarrow{( \begin{matrix}η_{V,V ∪W} \\ η_{W,V ∪W} \end{matrix} )} E(V ∪ W) ⟶,

then the map

E(V ∩ W) ⟶^{(η_{V ∩W,V} ,−η_{V ∩W,W})} E(V) ⊕ E(W)

factors through E′, since its composition with ( \begin{matrix}η_{V,V ∪W} \\ η_{W,V ∪W} \end{matrix} ) is zero. If we denote by α the factoring map, we can form a distinguished triangle

E(V ∩ W) ⟶^{α} E′ ⟶ C ⟶ .

Upon taking the tensor product with E(V), the diagram

E(V ∩ W) ⟶ E(V) ⊕ E(W) ⟶ E(V ∪ W)

becomes a split short exact sequence, since η_{V,V ∪W} ⊗ E(V) and η_{V ∩W,W} ⊗ E(V) are isomorphisms. Therefore α ⊗ E(V) becomes an isomorphism, and so C ⊗ E(V) ≅ 0 in StMod(kG), which implies that C ⊗ F (V) ≅ C.

Similarly, C ⊗F (W) ≅ C, and so C is also stably isomorphic to C ⊗F (V)⊗F (W). But, by Theorem 7.5, this is stably isomorphic to C ⊗ F (V ∪ W), which is projective, since the variety of C is contained in V ∪ W. Hence α is a stable isomorphism, and so we get the first of the Mayer-Vietoris triangles.

The proof for the second triangle starts similarly. We consider a distinguished triangle

F′ ⟶ F (V) ⊕ F (W) ⟶ F (V ∪ W) ⟶,

and note that the map from F (V ∩ W) to F (V) ⊕ F (W) factors through F′, giving a distinguished triangle

F (V ∩ W) ⟶^{β} F′ ⟶ D ⟶ .

The diagram

F (V ∩ W) ⟶ F (V) ⊕ F (W) ⟶ F (V ∪ W)

becomes a split short exact sequence upon tensoring with F (V) or F (W), and so map β becomes an isomorphism upon tensoring with these modules. Thus

D ⊗ E(V ∩ W) ≅ D ⊗ E(V) ⊗ E(W) ≅ D.

However, since all of the modules F (V ∩ W), F (V), F (W) and F (V ∪ W) become projective upon taking the tensor product with E(V ∩ W), it follows that the tensor product D ⊗ E(V ∩ W) is projective, and so β is a stable isomorphism.

We can easily recover from a simple special case of the Mayer-Vietoris triangles the theorem of Carlson [7] that the variety of an indecomposable module is projectively connected.

Corollary 8.2. Let V and W be closed homogeneous subvarieties of V_{G} such that V ∩ W = {0}.

(a) E(V ∪ W) is stably isomorphic to E(V) ⊕ E(W).

(b) If M is a kG-module whose variety is contained in V ∪ W but not in either of V or W, then M is decomposable.

Proof. (a) This follows immediately from the Mayer-Vietoris triangle, since E({0}) is projective.

(b) The conditions on M imply that, in StMod(kG),

M ≅ M ⊗ E(V ∪ W) ≅ (M ⊗ E(V)) ⊕ (M ⊗ E(W)),

and neither of these two summands can be stably isomorphic to M, since their varieties are contained in V and W respectively. Hence M is decomposable in StMod(kG), and so must be decomposable as a module, by Lemma 3.2.

9. Decompositions of quotient categories

Let M_{c} be the full subcategory of stmod(kG) consisting of the modules with complexity at most c. Thus, in the notation of Section 5, M⃗_{c} is the full subcategory of StMod(kG) consisting of the not necessarily finitely generated modules with complexity at most c. We are going to use the Mayer-Vietoris triangles of Section 8 to produce a block decomposition of the quotient category M⃗_{c}/M⃗_{c−1} that was studied in [4]. Let us first take this opportunity to make some general comments about such quotient categories.

I and the other authors of [4] have been taken to task for ignoring possible settheoretic difficulties in forming the quotient category M⃗_{c}/M⃗_{c−1}. Of course, such quotients can always be formed at the expense of having to work in an enlarged settheoretic universe, so from certain points of view this is not an important problem.

Let us point out that in the cases we are interested in, the construction can be carried out within the existing universe.

Actually, in this case it is not too hard to prove directly that the multiplicative system of morphisms with respect to which we have to localize in order to form the quotient is locally small in the sense of Weibel’s book [13, 10.3.6], and so the quotient category exists without the need to enlarge the universe. Alternatively, the machinery of Section 5 shows that the quotient can be formed in the existing universe, since there is a natural isomorphism

Hom_{M⃗_{c}/M⃗_{c−1}}(X, Y) ≅ \underline{Hom}(F_{M_{c−1}}(X), Y )

for any objects X and Y of M⃗_{c}.

Of course, this argument works more generally to show that the quotient T /T_{0} can be formed without any set-theoretic difficulties whenever T_{0} is any subcategory of StMod(kG) of the form C^{⊕} for some épaisse subcategory C of stmod(kG) and T is any full triangulated subcategory of StMod(kG) that contains T_{0}.

In Corollary 8.2, we noted that if V and W are closed homogeneous subvarieties of V_{G} whose intersection is {0}, then E(V ∪ W) ≅ E(V) ⊕ E(W). Of course, if the intersection is not {0} then this is no longer true; however, in a quotient category we do have such isomorphisms more generally.

Lemma 9.1. Let W_{1},…, W_{n} be closed homogeneous subvarieties of V_{G}, all with dimension at most c, such that, for i ≠ j, W_{i} ∩ W_{j} has dimension less than c, and let U = W_{1} ∪ ⋯ ∪ W_{n}. The map

E(W_{1}) ⊕ ⋯ ⊕ E(W_{n}) \underrightarrow{( ⋮_{η_{Wn,U}}^{η_{W_{1},U}} )} E(U)

is an isomorphism in the quotient category M⃗_{c}/M⃗_{c−1}.

Proof. Consider the distinguished triangle

E(W_{1} ∩ W_{2}) ⟶ E(W_{1}) ⊕ E(W_{2}) ⟶ E(W_{1} ∪ W_{2}) ⟶

of Theorem 8.1. Since the dimension of W_{1} ∩ W_{2} is less than c, E(W_{1} ∩ W_{2}) ≅ 0 in M⃗_{c}/M⃗_{c−1}. Hence the map E(W_{1}) ⊕ E(W_{2}) ⟶ E(W_{1} ∪ W_{2}) is an isomorphism in the quotient category. The lemma follows easily by induction on n.

Note that the lemma implies that if M is a finite-dimensional kG-module of complexity c, and if W_{1},…, W_{n} are the distinct c-dimensional components of the variety of M, then there is a natural direct sum decomposition of M in the quotient category M⃗_{c}/M⃗_{c−1}:

M ≅ M ⊗ E(W_{1} ∪ ⋯ ∪ W_{n}) ≅ (M ⊗ E(W_{1})) ⊕ ⋯ ⊕ (M ⊗ E(W_{n})).

This also works for infinite dimensional M, so long as M has variety contained in a finite union of irreducible varieties of dimension at most c. This decomposition has also been derived, using different methods, by Benson [3], who also studies its properties further.

To deal with arbitrary M, where we may need to consider an infinite number of c-dimensional varieties, we have to do some more work. Let us first simplify the notation a little.

Definition. For a natural number c, we define

T_{c} : E_{c} ⟶^{η_{c}} k ⟶^{ϵ_{c}} F_{c} ⟶

to be the distinguished triangle T_{M_{c}}. Thus we abbreviate the notation η_{M_{c}} to η_{c}, and so on.

Theorem 9.2. In the quotient category M⃗_{c}/M⃗_{c−1}, there is an isomorphism

E_{c} ≅ ⨁_{W} E(W),

where W runs over all irreducible c-dimensional closed homogeneous subvarieties of V_{G}.

Before we prove this, let us make a brief comment about the proof. The idea is quite simple: we would like to be able to say that E_{c} is the colimit of E(V), as V runs over all closed homogeneous subvarieties of V_{G} of dimension at most c, and then that, as a limiting case of Lemma 9.1, this is isomorphic in the quotient category to the direct sum ⨁ E(W). The problem is that, since we want to take a colimit of a more complicated filtered system than a simple sequence of maps, the homotopy colimit construction in StMod(kG) is not available, and so we need to take the colimit in the module category. (Actually, if we worked with a countable field k, we could avoid this.) This complicates the proof a little, since we have to lift natural commutative diagrams in the stable category to rather less natural commutative diagrams in the module category.

Proof of Theorem 9.2. Let {W_{i} : i ∈ I} be the set of all irreducible closed homogeneous subvarieties (defined over k) of V_{G} of dimension at most c. For each i ∈ I, let ϵ_{i} : k ⟶ F_{i} be a module homomorphism which is isomorphic to ϵ_{W_{i}} in the stable category. Thus there are short exact sequences

0 ⟶ k ⟶ F_{i} ⟶ F_{i}/k ⟶ 0,

where F_{i}/k is stably isomorphic to Ω^{−1}E(W_{i}).

For each non-empty finite subset J of I, define W_{J} to be the union ⋃_{j∈J} W_{j} and F_{J} to be the tensor product

F_{J} = ⨂_{j∈J} F_{j},

so that, by Theorem 7.5, F_{J} is stably isomorphic to F (W_{J}). If J′ ⊆ J, then the maps ϵ_{j} for j ∈ J\J′ induce a module homomorphism F_{J′} ⟶ F_{J}. Thus we get a filtered system of modules F_{J} indexed by the finite subsets of I.

Consider the colimit colim(F_{J}). If M is any finitely generated kG-module then

Hom_{k}(M, colim(F_{J})) ≅ colim(Hom_{k}(M, F_{J})).

If M has complexity at most c, then the variety of M is a finite union ⋃_{j∈J(M)} W_{j}. Since the set of finite subsets of I containing J(M) is cofinal in the set of all finite subsets of I, and since Hom_{k}(M, F_{J}) ≅ M^{∨} ⊗ F_{J} is projective for J(M) ⊆ J, by Lemma 5.14, it follows that Hom_{k}(M, colim(F_{J})) is projective, being a filtered colimit of projective modules. Hence colim(F_{J}) is M_{c}-local in StMod(kG), by Lemma 3.3.

If we fix a non-zero homomorphism π : kG ⟶ k, then we can define E_{J} to be the kernel of the (surjective) map

k ⊕ (F_{J} ⊗ kG) ⟶ F_{J}

induced by ϵ_{J} and F_{J} ⊗ π, and so we get a filtered system of short exact sequences

S_{J} : 0 ⟶ E_{J} ⟶ k ⊕ (F_{J} ⊗ kG) ⟶ F_{J} ⟶ 0.

Since filtered colimits of modules preserve exactness, there is a short exact sequence

0 ⟶ colim(E_{J}) ⟶ k ⊕ (colim(F_{J}) ⊗ kG) ⟶ colim(F_{J}) ⟶ 0.

Since E_{J} is stably isomorphic to E(W_{J}), it has complexity at most c, and so colim(E_{J}) has complexity at most c. Thus, as in the proofs of Propositions 5.6 and 5.7, the distinguished triangle

colim(E_{J}) ⟶ k ⟶ colim(F_{J}) ⟶

must be isomorphic to T_{c}. In particular, colim(E_{J}) is stably isomorphic to E_{c}.

From the short exact sequences S_{J}, we can extract module homomorphisms η_{J} : E_{J} ⟶ k which are compatible with the maps E_{J′} ⟶ E_{J} for J′ ⊆ J. When J = {j} is a singleton, we shall write E_{j} and η_{j} instead of E_{{j}} and η_{{j}}. Each map η_{J} is non-zero, and so surjective, since in the stable category it is isomorphic to η_{W_{J}}.

The direct sums

D_{J} = ⨁_{j∈J} E_{j},

together with the natural inclusion maps, form a filtered system of modules, indexed by the finite subsets of I, whose colimit is ⨁_{i∈I} E_{i}.

The maps η_{j} for j ∈ J induce maps σ_{J} : D_{J} ⟶ k that are compatible with the inclusion maps S_{J′} ⟶ D_{J} for J′ ⊆ J, and so we have surjective maps of filtered systems

D_{J} ⟵^{D_{J} ⊗η_{J}} D_{J} ⊗ E_{J} ⟶^{σ_{J} ⊗E_{J}} E_{J}.

In the stable category, D_{J} ⊗ η_{J} is an isomorphism, and so the kernel is projective. Since filtered colimits of kG-modules preserve exactness and projectivity, the natural map

colim(D_{J} ⊗ E_{J}) ⟶ colim(D_{J}) ≅ ⨁_{i∈I} E_{i}

is a stable isomorphism.

The map σ_{J} ⊗ E_{J} is isomorphic in the stable category to the natural map

⨁_{j∈J} E(W_{j}) ⟶ E( ⋃_{j∈J} W_{j}),

which is an isomorphism in the quotient category M⃗_{c}/M⃗_{c−1} by Lemma 9.1. Thus the complexity of the kernel of σ_{J} ⊗E_{J} is less than c, and so the kernel of the natural map

colim(D_{J} ⊗ E_{J}) ⟶ colim(E_{J})

has complexity less than c.

But we have already seen that colim(E_{J}) is stably isomorphic to E_{c}, and so E_{c} is isomorphic to ⨁_{i∈I} E(W_{i}) in M⃗_{c}/M⃗_{c−1}. In this quotient category all the terms E(W_{i}) corresponding to varieties W_{i} of dimension less than c are isomorphic to zero, and so the theorem follows.

It follows from Theorem 9.2 that any object M of M⃗_{c}/M⃗_{c−1} has a natural direct sum decomposition

M ≅ M ⊗ E_{c} ≅ ⨁_{W} M ⊗ E(W),

where W runs over all the irreducible subvarieties of V_{G} of dimension c.

Acknowledgements

I would like to thank the following people: Dave Benson and Jon Carlson for many interesting and useful discussions related to the results in this paper; everybody in the D.M.I. at the Ecole Normale Supérieure, where I proved the first results of this paper, for their hospitality, and especially Raphaël Rouquier, who listened patiently to disjointed accounts of the results as I discovered them; Nick Kuhn for pointing out the reference [6]; Amnon Neeman for answering my questions on Bousfield localization.

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University of Bristol, School of Mathematics, University Walk, Bristol BS8 1TW, ENGLAND

E-mail address: J.Rickard@bristol.ac.uk