COMPLEXITY AND VARIETIES FOR INFINITELY GENERATED MODULES

D. J. BENSON, JON F. CARLSON, AND J. RICKARD

1. Introduction

In the past fifteen years the theory of complexity and varieties of modules has become a standard tool in the modular representation theory of finite groups. Moreover the techniques have been used in the study of integral representations [8] and have been extended to the representation theories of objects such as groups of finite virtual cohomological dimension [1], infinitesimal subgroups of algebraic groups and restricted Lie algebras [14, 16]. In all cases some sort of finiteness condition on the module category has been required to make the theory work. Usually this comes in the form of stipulating that all modules under consideration be finitely generated. While the restrictions have been efficient for most applications to date, there are very good reasons for wanting to develop a theory that will accommodate infinitely generated modules. One reason might be the possibility of extending the techniques of representations to other classes of infinite groups. Another reason is that some recent work has revealed a few of the defects of the finiteness requirement. One such problem can be summarized as follows.

Let G be a finite group and k a field of characteristic p dividing |G|. In the paper of Carlson, Donovan and Wheeler [10], they consider the quotient category of finitely generated kG-modules by the subcategory of modules of less than maximal complexity. In this quotient the endomorphism ring of the trivial module is a direct product of local rings corresponding to the conjugacy classes of elementary abelian

2 D. J. BENSON, JON F. CARLSON, AND J. RICKARD

subgroups of maximal rank (i.e., the components of the maximal ideal spectrum of H^{∗}(G, k) of maximal dimension). There is no decomposition of the trivial module corresponding to the idempotents in this endomorphism ring, because idempotents fail to split in the quotient category. On the other hand, a direct sum of a suitable number of copies of the trivial module does have such a decomposition (see also [9]). Hence the quotient category suffers from a failure of the Krull–Schmidt property, and with the failure, the module theory becomes hard work. The problem here really is with the finite generation requirement. For if the category had infinite direct sums then the idempotents in the endomorphism ring could be made to split, and the Krull–Schmidt Theorem would be recovered. We therefore reach the somewhat startling conclusion that in the larger quotient category, in which we allow infinitely generated modules, the trivial module decomposes as a direct sum of infinitely generated modules, corresponding to the conjugacy classes of elementary abelian subgroups of maximal rank.

Our aim in this paper is to provide definitions of the complexity and the varieties of modules that do not depend on the modules being finitely generated. It is certainly natural to ask that the subcategory of all modules having a fixed complexity c or smaller be closed under direct sums and summands. Its intersection with the subcategory of finitely generated modules should be the finitely generated modules of complexity c or smaller. In addition, the corresponding stable category should be triangulated, meaning that if two terms of a short exact sequence are in the subcategory then so is the third. Finally the functor on the quotient categories, induced by the inclusions of the finite categories into the infinite, should be fully faithful.

We actually propose several definitions for complexity. All but one of them turn out to be the same. But even the odd one, the minimum category, coincides with

COMPLEXITY AND VARIETIES FOR INFINITELY GENERATED MODULES 3

the others if we restrict ourselves to countably generated modules. The most easily testable of the conditions defining complexity is the condition for membership of the subcategory \underline{E_{c}}, which says that, if we allow adding finite dimensional projective summands, any finite subset of a module of complexity c is contained in a finitely generated submodule of complexity at most c. Other definitions are given in terms of a condition which assures that the functors on the quotient categories are fully faithful, filtered colimits of finite dimensional modules and, of course, dimensions of varieties. Each of these seems to provide a unique insight into the nature of the subcategories.

As in the case of finitely generated modules, the varieties give the most information about modules and their complexities. However there is a big difference in the case of infinitely generated modules. It is that the variety of a module M is a collection V_{G}(M) of closed homogeneous subvarieties of V_{G}(k), rather than a single closed subvariety V_{G}(M) ⊆ V_{G}(k). In the finitely generated case, this collection is exactly the collection of closed homogeneous subvarieties of V_{G}(M), but for an infinitely generated module this collection need not have a unique maximal element. However, V_{G}(M) does have a dimension, namely the maximal dimension of an element. This turns out to be equal to the complexity of the module.

The paper is organized roughly as follows. As one motivation for the extension of complexity, we present in Section 3 an account of how to split idempotents when we are allowed infinite direct sums. The two main definitions of complexity and the proof of their equivalence is given next. Section 5 gives the definition in terms of filtered colimits, and shows how this implies the statement about the minimum subcategory. Next it is shown how the splitting of idempotents allows us to recover the Krull–Schmidt Theorem in the quotient categories. The final sections contain the definitions of the varieties and the analog for infinitely generated modules of the

4 D. J. BENSON, JON F. CARLSON, AND J. RICKARD

Alperin–Evens theorem.

In an appendix we have included several examples of what happens in the category of infinitely generated modules. The category has many more pitfalls than we had originally anticipated, and we have tried to illustrate some of the strange behavior with the examples. In addition, it is actually possible to see how the splitting of the idempotents works in some easy cases using diagrams for modules. As one example we show how, in the complexity quotient category, the trivial module for the dihedral group decomposes as the direct sum of two infinitely generated modules. The interested reader may wish to use the appendix as a companion to the text.

2. Definitions and preliminaries

Most of the concepts used in this paper involve standard constructions of categories, cohomology rings and representations. However, for the benefit of the reader we begin with a brief review of some of the definitions. For general reference, we refer the reader to the books by Benson [5] and Evens [13] for information on cohomology and representations. For more details on triangulated categories and quotient categories see Happel [15].

Throughout the paper we assume that G is a finite group and that k is an algebraically closed field of characteristic p > 0. The cohomology ring H^{∗}(G, k) is a finitely generated graded commutative k-algebra, and it has a maximal ideal spectrum V_{G}(k) which is a homogeneous affine variety. By a theorem of Quillen, V_{G}(k) has dimension equal to the p-rank of G and the components of V_{G}(k) correspond to the conjugacy classes of maximal elementary abelian p-subgroups of G. In fact, the minimal prime ideals of H^{∗}(G, k) are precisely the radicals of the kernels of the restriction maps

res_{G,E} : H^{∗}(G, k) → H^{∗}(E, k)

COMPLEXITY AND VARIETIES FOR INFINITELY GENERATED MODULES 5

for maximal elementary abelian p-subgroups E of G. These last facts do not play a large role in this paper, but they are crucial to the decompositions of the trivial module in the quotient categories that are a major motivation for our writing.

For a finitely generated kG-module M, let I_{G}(M) ⊆ H^{∗}(G, k) be the annihilator in H^{∗}(G, k) of the cohomology ring Ext_{kG}^{∗}(M, M). Notice that I_{G}(M) also annihilates Ext_{kG}^{∗}(M, N) for all kG-modules N. This is because Ext_{kG}^{∗}(M, N) is an Ext_{kG}^{∗}(M, M)-module. The variety of M is the closed subset V_{G}(M) ⊆ V_{G}(k) consisting of all maximal ideals (points of V_{G}(k)) which contain I_{G}(M). So V_{G}(M) is a closed homogeneous subvariety, and it has dimension at most the p-rank of G.

The complexity of M is, in fact, the dimension of V_{G}(M). The original definition of Alperin was in terms of the polynomial rate of growth of the cohomology of M, but this is equivalent to our definition. An important fact about varieties which we need is that if

0 ⟶ M′ ⟶ M ⟶ M′′ ⟶ 0

is an exact sequence, then the variety of any one of the modules, M, M′ or M′′, is a subset of the union of the varieties of the other two. Hence if two of the modules in the sequence have complexity at most c, then the third also has complexity at most c. It is a consequence of this fact that the stable category of all modules of complexity at most c is triangulated.

We write mod(kG) for the category of all finitely generated kG-modules, and stmod(kG) for the corresponding stable category. That is, stmod(kG) has the same objects as mod(kG), but for M and N in mod(kG)

Hom_{stmod(kG)}(M, M) = Hom_{kG}(M, N)/PHom_{kG}(M, N)

where PHom_{kG}(M, N) is the set of all homomorphisms from M to N which factor through a projective module. The module category is abelian while the stable category is triangulated. The basic properties of triangulated categories can be

6 D. J. BENSON, JON F. CARLSON, AND J. RICKARD

summarized as follows.

An additive category C is triangulated if it has triangles which satisfy certain axioms. First we need a translation functor T : C → C, which is an additive selfequivalence of the category C. Then a triangle in C is a sextuple (X, Y, Z, u, v, w) of objects X, Y, Z and morphisms

X ⟶^{u} Y ⟶^{v} Z ⟶^{w} T X.

We should emphasize that not every sextuple is a triangle. The triangles are a distinguished set of sextuples with some special properties. A morphism of triangles is a triple f, g, h such that the diagram

[figure: X; u; f; Y; v; g; Z; w; h; T X; T f; X′; u′; Y′; v′; Z′; w′; T X′]

commutes. The axioms for triangles assert that every sextuple isomorphic to a triangle is a triangle. Every morphism u : X → Y is in a unique triangle, up to isomorphism. If Id_{X} : X → X is the identity then the third object in the triangle of Id_{X} is the zero object. A sextuple (X, Y, Z, u, v, w) is a triangle if and only if (Y, Z, T X, v, w, −T u) is also a triangle. Given two triangles (X, Y, Z, u, v, w) and (X′, Y′, Z′, u′, v′, w′), and maps f : X → X′, g : Y → Y′, there must exist h : Z → Z′ such that (f, g, h) is a morphism of triangles. Finally we have the “octahedral axiom”, which we do not explicitly state. In the case of the stable category, it is basically the third isomorphism theorem.

It is actually not very difficult to prove that the stable category of kG-modules modulo projectives is triangulated. The key observation is that short exact sequences in the module category correspond to triangles in the stable category. That is, if

0 ⟶ A ⟶^{α} B ⟶^{β} C ⟶ 0

is exact then there is a triangle (A, B, C, α, β, γ) in the stable category. Indeed

COMPLEXITY AND VARIETIES FOR INFINITELY GENERATED MODULES 7

if α : M → N is a morphism in the stable category then we can find projective modules P and Q and representatives α′ : M → N ⊕ P, α′′ : M ⊕ Q → P such that α′ is injective and α′′ is surjective. That is, we have exact sequences

0 ⟶ M ⟶^{α′} N ⊕ P ⟶^{β} V ⟶ 0

and

0 ⟶ U ⟶^{γ} M ⊕ Q ⟶^{α′′} N ⟶ 0.

Finally, it can be shown that V ≅ Ω^{−1}(U) ⊕ P′ for some projective module P′. Here Ω^{−1}(U) is defined to be the cokernel of the map θ : U → Q′ where Q′ is an injective hull for U. Thus Ω^{−1} is the translation functor for the stable category and

U ⟶^{γ} M ⟶^{α} N ⟶^{β} Ω^{−1}(U)

is a triangle.

Finally we should mention the quotient construction. Suppose that C is a triangulated and M ⊆ C is a triangulated full subcategory. Then we can form the quotient category C/M = Q by a localization process. The objects in Q are the same as those in C. For morphisms in Q we invert any morphism in C if the third object of the triangle of that morphism is in M. So if M and N are objects in Q, then a morphism from M to N is represented by a diagram M ⟵^{s} U ⟶^{t} N of objects and morphisms such that the third object in the triangle of the morphism s : U → M is in M. Thus the morphism has the form ts^{−1}. Now if f : V → U is also invertible then we must allow that

ts^{−1} = (tf)(sf)^{−1}.

So a morphism is actually an equivalence class of pairs (t, s). The equivalence relation is generated by (t, s) ∼ (tf, sf) as above. The quotient category is universal for functors taking the morphisms s described above to isomorphisms.

8 D. J. BENSON, JON F. CARLSON, AND J. RICKARD

In fact, the localization process may be performed using either left or right fractions. In other words, we can represent morphisms in C/M by diagrams M ⟶^{t′} U′ ⟵^{s′} N where the third object in the triangle of the morphism s′ : U′ → N is in M. To see this, observe that the opposite category of a triangulated category is triangulated, and use the universal property of the quotient.

3. Splitting of idempotents

If M is an object in an additive category, and e ∈ End(M) is idempotent, we say that e splits if M ≅ M_{1} ⊕ M_{2} in such a way that

e = ( \begin{matrix}Id & 0 \\ 0 & 0 \end{matrix} ) .

Here, we are viewing End(M) as the ring of 2 × 2 matrices of the form

( \begin{matrix}α & β \\ γ & δ \end{matrix} )

with α ∈ End(M_{1}), β ∈ Hom(M_{2}, M_{1}), γ ∈ Hom(M_{1}, M_{2}) and δ ∈ End(M_{2}).

Lemma 3.1. Let C be a triangulated category with countable direct sums, and with the property that a countable direct sum of triangles is a triangle. If e is an idempotent in the endomorphism ring of an object M in C, then e splits.

Proof. This is essentially Proposition 3.2 of Bökstedt and Neeman [6]. For the convenience of the reader, we recall their proof here. Suppose that e ∈ End(M) is idempotent. Let M̂ = M ⊕ M ⊕ ⋯ be a countable direct sum of copies of M, and consider the endomorphism

ϕ = ( \begin{matrix}e & 1 − e & 0 & ⋯ \\ 0 & e & 1 − e \\ 0 & 0 & e \\ ⋮ & & & ... \end{matrix} ) ∈ End(M̂).

We may view this endomorphism schematically as

[figure: M; e; M; ⊕; ⊕; M; e; 1−e; M; ⊕; ⊕; ⋮; 1−e; ⋮]

Complete this to a triangle

M_{1} → M̂ ⟶^{ϕ} M̂ → T (M_{1})

(where T is the translation functor) to define the object M_{1}. Similarly, swapping the roles of e and 1 − e, we get an endomorphism ψ ∈ End(M̂), which we complete to a triangle

M_{2} → M̂ ⟶^{ψ} M̂ → T (M_{2})

to define the object M_{2}.

Because

( \begin{matrix}1 − e & e \\ −e & 1 − e \end{matrix} ) ( \begin{matrix}1 − e & 0 \\ 0 & e \end{matrix} ) ( \begin{matrix}1 − e & −e \\ e & 1 − e \end{matrix} ) = ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) ,

and

( \begin{matrix}1 − e & e \\ −e & 1 − e \end{matrix} ) ( \begin{matrix}e & 0 \\ 0 & 1 − e \end{matrix} ) ( \begin{matrix}1 − e & −e \\ e & 1 − e \end{matrix} ) = ( \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} ) ,

we may rechoose a decomposition of M̂ ⊕ M̂ as a direct sum of copies of M in such a way that ϕ ⊕ ψ is the endomorphism

[figure: M ⊕ M; a; M ⊕ M; ⊕; ⊕; M ⊕ M; a; b; M ⊕ M; ⊕; ⊕; ⋮; b; ⋮]

10 D. J. BENSON, JON F. CARLSON, AND J. RICKARD

where a = ( \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} ) and b = ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ). It follows that there is a triangle

M ⟶ M̂ ⊕ M̂ ⟶^{ϕ⊕ψ} M̂ ⊕ M̂ → T (M)

and so M_{1}⊕M_{2} ≅ M. It is clear that, with respect to this direct sum decomposition, e has the desired form. □

Let M be an object in a quotient category C/C′, where C′ is a full triangulated subcategory of a triangulated category C. Suppose that C and C′ are closed under countable direct sums, and that a countable direct sum of triangles is a triangle. Then an idempotent e ∈ End_{C/C′}(M) may be represented by maps M ←^{u} M′ →^{v} M, where u and v are arrows in C, and u has the property that when completed to a triangle, the third vertex lies in C′. We may arrange this in such a way that the same map u is used to represent 1 − e by maps M ←^{u} M′ →^{w} M. Then the diagram used to split M in the proof of the above lemma is as follows.

[figure: M; M′; u; v; M; ⊕; ⊕; ⊕; M; M′; u; v; w; M; ⊕; ⊕; ⊕; ⋮; w; ⋮]

Since the map u is an isomorphism in C/C′, we may as well leave out these maps and just complete the following diagram to a triangle:

[figure: M′; v; M; ⊕; ⊕; M′; v; w; M; ⊕; ⊕; ⋮; w; ⋮]

COMPLEXITY AND VARIETIES FOR INFINITELY GENERATED MODULES 11

4. Complexity

We define M_{c} to be the subcategory of stmod(kG) consisting of the modules of complexity at most c. This is a triangulated subcategory, and so we may form quotient categories stmod(kG)/M_{c} and M_{c}/M_{c′} for c′ < c.

Let Mod(kG) be the category of all kG-modules (both finitely and infinitely generated), and StMod(kG) be the corresponding stable category. This is triangulated in the same way as stmod(kG) (see Section 2), using the following lemma.

Lemma 4.1. Let S be a Sylow p-subgroup of G. For a (possibly infinitely generated) kG-module M, the following conditions are equivalent:

(i) The module M is projective.

(ii) The module M is injective.

(iii) The restriction M ↓_{S} is projective.

(iv) The restriction M ↓_{S} is injective.

(v) Every S-fixed element m ∈ M can be written as ∑_{g∈S} g(m′) for some m′ ∈ M.

Proof. One proves that (i) ⇔ (iii) ⇔ (v) ⇔ (iv) ⇔ (ii). □

Actually, projective modules are the same as injective modules for an arbitrary finite dimensional self-injective algebra, but the proof is a little more messy.

We define various subcategories of StMod(kG) which are possible contenders for the category of modules of complexity at most c, and prove that these subcategories almost coincide.

(i) We define \underline{Min_{c}} to be the smallest full triangulated subcategory of the category StMod(kG) containing M_{c} and closed under taking direct summands and countable direct sums. The intersection of any collection of such subcategories is also one, and so there is a unique smallest one.

12 D. J. BENSON, JON F. CARLSON, AND J. RICKARD

(ii) We define \underline{U_{c}} to be the full subcategory of StMod(kG) consisting of those modules M with the property that any map ϕ : X → M from a finitely generated module X factors through a finitely generated module of complexity at most c. Equivalently, given any such map ϕ, there exists a map ψ : Y → X in stmod(kG) such that ϕ ∘ ψ = 0 in StMod(kG) and ψ passes down to an isomorphism in stmod(kG)/M_{c}.

The property defining \underline{U_{c}} is exactly the condition which will be needed for the results of Section 6.

(iii) We define \underline{E_{c}} to be the full subcategory of StMod(kG) consisting of those modules M with the property that given any finite subset S of M, there is a finitely generated projective module P and a finitely generated submodule M′ of M ⊕ P such that M′ is in M_{c} (i.e., M′ has complexity at most c), and M′ contains the elements (s, 0) ∈ M ⊕ P with s ∈ S.

Theorem 4.2. We have

\underline{Min_{c}} ⊆ \underline{U_{c}} = \underline{E_{c}}.

Proof. (i) \underline{Min_{c}} ⊆ \underline{U_{c}}: The subcategory \underline{U_{c}} is clearly closed under taking direct summands and countable direct sums. Given a triangle in StMod(kG) with two corners in \underline{U_{c}}, represent it in Mod(kG) by a short exact sequence

0 → M_{1} → M_{2} → M_{3} → 0

with M_{2} and M_{3} in \underline{U_{c}}. We claim that M_{1} is also in \underline{U_{c}}, so that \underline{U_{c}} is triangulated and hence contains \underline{Min_{c}}. Given a map X → M_{1} with X finitely generated, we may factor the composite X → M_{1} → M_{2} as a composite X → Z → M_{2} with Z in M_{c}. By adding a suitable projective to Z, we may assume X → Z is injective. Then there is an induced map Z/X → M_{3}, which we may factor as a composite

COMPLEXITY AND VARIETIES FOR INFINITELY GENERATED MODULES 13

Z/X → Z′ → M_{3} with Z′ in M_{c}.

0 → X → Z → Z′
↓ ↓ ↓
0 → M_{1} → M_{2} → M_{3} → 0

Add a suitable projective module P to Z and make Z ⊕ P → Z′ surjective with kernel Z′′ in M_{c}. Then X → M_{1} factors through Z′′.

(ii) \underline{E_{c}} ⊆ \underline{U_{c}}: If M is a kG-module representing an object in \underline{E_{c}}, then given a map ϕ : X → M with X finitely generated, say by a set S, then we may find a projective module P and a finitely generated submodule M′ of M ⊕ P such that M′ is in M_{c}, and the elements of M ⊕ P of the form (ϕ(S), 0) are in M′. This means that ϕ factors as X → M′ ↪ M ⊕ P → M, and so M lies in \underline{U_{c}}.

(iii) \underline{U_{c}} ⊆ \underline{E_{c}}: If M is a kG-module representing an object in \underline{U_{c}} and S is a finite subset of M, let X be the submodule of M generated by S. Then the inclusion X ⟶^{i} M factors as X ⟶^{α} Z ⟶^{β} M, where Z is a finitely generated module of complexity at most c. Let Z ⟶^{σ} Z′ be the cokernel of α, and let Z′ ⟶^{γ} P be the injective hull of Z′. This is a projective module since kG is self-injective. The map (β, γσ) : Z → M ⊕ P is injective, and (β, γσ) ∘ α = (i, 0), so the image of Z is a submodule of M ⊕ P containing the elements (s, 0) with s ∈ S. □

Definition 4.3. We define the complexity of a (possibly infinitely generated) kGmodule M to be c if M is in \underline{U_{c}} but not in \underline{U_{c−1}}.

5. Filtered colimits

In this section, we characterize the modules of complexity at most c as the modules which may be written as filtered colimits of finitely generated modules of complexity at most c. The examples in the Appendix show that we may not assume that the

14 D. J. BENSON, JON F. CARLSON, AND J. RICKARD

maps in the filtered colimit are inclusions. We begin by recalling some formalities on filtered colimits.

If I is a small indexing category (“small” means that the class of objects is a set, and “indexing” has the empty meaning), then a diagram of type I in a category C is a covariant functor F : I → C. A colimit of F is an object X ∈ C together with a natural transformation λ_{X} from F to the constant functor c_{X} : I → C with value X, and universal in the sense that given a natural transformation λ to another constant functor c_{Y}, there is a unique map X → Y whose composite with λ_{X} gives λ. If the colimit of F exists, we write lim_{\overrightarrow{I}} F for it. If all such colimits exist, they form a functor lim_{\overrightarrow{I}} : C^{I} → C, where C^{I} denotes the category whose objects are the functors from I to C and whose arrows are the natural transformations. In this case, we have an adjunction

Hom_{C}(lim_{\overrightarrow{I}} F, Y) ≅ Hom_{C^{I}}(F, c_{Y}).

If σ : J → I is a covariant functor between indexing categories, then composition gives a functor C^{I} → C^{J}, and hence a natural transformation

lim_{\overrightarrow{J}}(− ∘ σ) → lim_{\overrightarrow{I}}(−).

We recall (Artin–Mazur [3], Appendix) that a small category I is right filtering if

(a) given any pair of objects i, i′ in I, there is an object i′′ and morphisms i → i′′ and i′ → i′′, and

(b) if i\overrightarrow{→}i′ are two morphisms in I then there is a morphism i′ → i′′ such that the composites i → i′′ are equal.

A filtered colimit is a colimit of a functor F : I → C from a small right filtering category I to a category C.

COMPLEXITY AND VARIETIES FOR INFINITELY GENERATED MODULES 15

If C is a category of sets or modules, then the elements of lim_{\overrightarrow{I}} F are equivalence classes of elements in the disjoint union of the F (i), as i runs over the objects in I. The equivalence relation is given as follows. If α ∈ F (i) and α′ ∈ F (i′), then α ∼ α′ if there are arrows ρ : i → i′′, ρ′ : i′ → i′′ such that F (ρ)(α) = F (ρ′)(α′). The module structure is defined similarly: for example, if α ∈ F (i) and α′ ∈ F (i′) then we choose arrows ρ : i → i′′ and ρ′ : i′ → i′′, and instead of adding α and α′, we add F (ρ)(α) and F (ρ′)(α′) in F (i′′).

Filtered colimits of modules possess the following property not shared by all colimits: an element α ∈ F (i) goes to zero in lim_{\overrightarrow{I}} F if and only if there is an arrow ρ : i → i′ in I such that F (ρ)(α) = 0 in F (i′).

A functor σ : J → I between indexing categories is right cofinal if

(c) for every object i in I there is an object j in J and a morphism i → σj in I, and

(d) if i and j are objects in I and J, and i\overrightarrow{→}σj are morphisms in I, then there is a morphism j → j′ in J such that the composites i\overrightarrow{→}σj → σj′ are equal.

Lemma 5.1. Composing with a right cofinal functor preserves filtered colimits. In other words, if I and J are small right filtering categories, σ : J → I is right cofinal, and F : I → C is a diagram of type I in C, then the natural map

lim_{\overrightarrow{J}}(F ∘ σ) → lim_{\overrightarrow{I}} F

is an isomorphism.

Proof. See Proposition 1.8 of the Appendix of Artin–Mazur [3]. □

Lemma 5.2. Suppose that C is a triangulated subcategory of StMod(kG) which is

16 D. J. BENSON, JON F. CARLSON, AND J. RICKARD

closed under taking countable direct sums. Suppose that we are given a diagram

M_{0} ⟶^{ρ_{0}} M_{1} ⟶^{ρ_{1}} ⋯

in Mod(kG) with the property that each M_{i}, regarded as an object in StMod(kG), lies in C. Then the colimit of the M_{i} in Mod(kG), regarded as an object in StMod(kG), lies in C.

Proof. The argument for this is given in §2 of Bökstedt and Neeman [6], and goes as follows. Let M = ⨁_{i} M_{i}, and consider the endomorphism

ϕ = ( \begin{matrix}1 & 0 & 0 & ⋯ \\ −ρ_{0} & 1 & 0 \\ 0 & −ρ_{1} & 1 \\ ⋮ & & & ... \end{matrix} )

of M. Then ϕ is injective, and its cokernel is isomorphic to the required colimit. It follows that if we complete ϕ to a triangle in C

M → M → M′ → T (M),

then M′ is isomorphic, in StMod(kG), to the required colimit. □

We define \underline{C_{c}} to be the full subcategory of StMod(kG) consisting of filtered colimits of modules in M_{c}.

Theorem 5.3. We have

\underline{Min_{c}} ⊆ \underline{U_{c}} = \underline{C_{c}} = \underline{E_{c}}.

Furthermore, \underline{Min_{c}} consists of the countably generated modules in \underline{U_{c}} = \underline{E_{c}}.

Proof. We have already proved much of this in Theorem 4.2. The statement that \underline{U_{c}} = C_{c} follows from Proposition 2.1 of Lenzing [17] with a little work, but for the benefit of the reader we include a self contained proof here. The following statements complete the proof of the Theorem:

COMPLEXITY AND VARIETIES FOR INFINITELY GENERATED MODULES 17

(i) \underline{U_{c}} ⊆ \underline{C_{c}}: Given an object in \underline{U_{c}}, represent it by a module M in Mod(kG). Let I be the category whose objects are the morphisms X → M with X finitely generated, and whose morphisms are the commutative triangles. (Actually, to avoid set theoretic problems, namely to ensure that I is small, we should really replace I by a small skeletal subcategory.) Let J be the full subcategory of I whose objects are the morphisms X → M with X finitely generated of complexity at most c. Then M is the filtered colimit of the functor I → Mod(kG) which sends X → M to X. The functor J → I is right cofinal, by definition of \underline{U_{c}}, and so Lemma 5.1, M is also the filtered colimit of the same functor from J to Mod(kG).

(ii) \underline{C_{c}} ⊆ \underline{E_{c}}: If S is a finite subset of a filtered colimit M = lim_{\overrightarrow{I}} F with F (i) in M_{c} for all i in I, then S lies in the image of some F (i) → M. The kernel of F (i) → M is equal to the kernel of F (ρ) for some arrow ρ : i → j in I. Let K be the kernel of F (j) → M, and K → P be an injective map with P a projective module. Then Im(F (ρ)) ∩ K = 0, so K → P lifts to a map F (j) → P in such a way that the composite F (i) → F (j) → P is zero. It follows that the map F (j) → M ⊕ P is injective, and the elements (s, 0) ∈ M ⊕ P with s ∈ S lie in the image.

(iii) Every countably generated module in \underline{C_{c}} lies in \underline{Min_{c}}: If M is a countably generated module in Mod(kG) representing an object in \underline{C_{c}}, then we may express M as a colimit of a diagram M_{0} → M_{1} → ⋯ of modules which, when regarded as objects in StMod(kG), lie in M_{c}. By Lemma 5.2, the colimit M, regarded as an object in StMod(kG), lies in \underline{Min_{c}}. □

We end this section with some further properties of filtered colimits. If I is a right filtering category, and F_{1}, F_{2}, F_{3} : I → Mod(kG) are functors, then a short exact sequence

0 → F_{1} ⟶^{α} F_{2} ⟶^{β} F_{3} → 0

18 D. J. BENSON, JON F. CARLSON, AND J. RICKARD

consists of natural transformations α : F_{1} → F_{2} and β : F_{2} → F_{3} with the property that for all objects i in I,

0 → F_{1}(i) ⟶^{α(i)} F_{2}(i) ⟶^{β(i)} F_{3}(i) → 0

is a short exact sequence of modules.

Proposition 5.4. If 0 → F_{1} ⟶^{α} F_{2} ⟶^{β} F_{3} → 0 is a short exact sequence of functors I → Mod(kG) as above, then

0 → lim_{\overrightarrow{I}} F_{1} ⟶^{α_{∗}} lim_{\overrightarrow{I}} F_{2} ⟶^{β_{∗}} lim_{\overrightarrow{I}} F_{3} → 0

is also a short exact sequence.

Proof. This is Grothendieck’s axiom AB5, which is well known to be satisfied by module categories. □

Proposition 5.5. A filtered colimit M = lim_{\overrightarrow{I}} F of projective kG-modules is a projective kG-module.

Proof. In general, a filtered colimit of flat modules is flat. For a group algebra, or more generally for a (left) perfect ring, flat (left) modules are the same as projective modules (Bass [4]). □

Corollary 5.6. A (possibly infinitely generated) kG-module is projective if and only if it has complexity zero. □

6. The Krull–Schmidt Theorem

In this section, we prove that if c′ < c, then the functor M_{c}/M_{c′} → \underline{U_{c}}/\underline{U_{c′}}

COMPLEXITY AND VARIETIES FOR INFINITELY GENERATED MODULES 19

induced by the inclusion functor M_{c} → \underline{U_{c}} is faithful. Our object is to define a subcategory M_{c,c′} of \underline{U_{c}}/\underline{U_{c′}} by taking direct summands of objects in M_{c}/M_{c′}. Then we prove the Krull–Schmidt theorem in M_{c,c−1}, and deduce a cancellation theorem for M_{c}/M_{c−1}.

Lemma 6.1. Let c′ < c. Suppose that U → M is a morphism in \underline{U_{c}} representing an isomorphism in \underline{U_{c}}/\underline{U_{c′}}, and with M in M_{c}. Then there is a morphism M′ → U in \underline{U_{c}} representing an isomorphism in \underline{U_{c}}/\underline{U_{c′}}, and with M′ in M_{c}.

Proof. Begin by completing U → M to a triangle

U → M → W → Ω^{−1}U

in \underline{U_{c}}, so that W is in \underline{U_{c′}}. Then M → W factors as M → Y → W with Y in M_{c′}. Complete M → Y to a triangle

M′ → M → Y → Ω^{−1}M′.

Then M′ is in M_{c} and M′ → M represents an isomorphism in M_{c}/M_{c′}. The composite M′ → M → W is zero, so M′ → M factors as M′ → U → M, and M′ → U represents an isomorphism in \underline{U_{c}}/\underline{U_{c′}}. □

Proposition 6.2. If c′ < c, then the functor

M_{c}/M_{c′} → \underline{U_{c}}/\underline{U_{c′}}

induced by the inclusion functor M_{c} → \underline{U_{c}} is fully faithful.

Proof. To say that a functor is fully faithful means that it induces isomorphisms on hom sets.

Suppose that M ←^{s} U →^{t} N is a morphism in \underline{U_{c}}/\underline{U_{c′}}, between objects M and N in M_{c} (i.e., U is in \underline{U_{c}}, and the third vertex of the triangle containing U → M is in

20 D. J. BENSON, JON F. CARLSON, AND J. RICKARD

\underline{U_{c′}}). By the lemma, there is a homomorphism M′ →^{ϕ} U in \underline{U_{c}} which represents an isomorphism in \underline{U_{c}}/\underline{U_{c′}}, and with M′ in M_{c}. So M ←^{sϕ} M′ →^{tϕ} N represents the same morphism in \underline{U_{c}}/\underline{U_{c′}}.

If M ← M′ → N and M ← M′′ → N represent the same morphism in \underline{U_{c}}/\underline{U_{c′}}, with both M′ and M′′ in M_{c}, then another application of the lemma shows that they represent the same morphism in M_{c}/M_{c′}. □

Definition 6.3. We define M_{c,c′} to be the smallest subcategory of \underline{U_{c}}/\underline{U_{c′}} which contains M_{c}/M_{c′} and which is closed under taking direct summands.

Lemma 6.4. The category M_{c,c′} is triangulated.

Proof. Given a morphism in M_{c,c′}, represent it by a morphism ϕ : M → N in \underline{U_{c}}. Complete it to a triangle

L → M ⟶^{ϕ} N → Ω^{−1}(L)

in \underline{U_{c}}. Choose modules M′ and N′ in \underline{U_{c}} so that M ⊕ M′ and N ⊕ N′ are isomorphic in \underline{U_{c}}/\underline{U_{c′}} to objects in M_{c}. We set

ϕ̃ = ( \begin{matrix}ϕ & 0 \\ 0 & 0 \end{matrix} ) : M ⊕ M′ → N ⊕ N′.

If we complete ϕ̃ to a triangle in \underline{U_{c}}, then the third corner is isomorphic in \underline{U_{c}} to L ⊕ M′ ⊕ Ω(N′). Thus L is isomorphic in \underline{U_{c}}/\underline{U_{c′}} to a direct summand of an object in M_{c}, and so L is in M_{c,c′}. □

Proposition 6.5. The endomorphism ring of a module in M_{c}/M_{c−1} is an Artin algebra.

COMPLEXITY AND VARIETIES FOR INFINITELY GENERATED MODULES 21

Proof. Let M be a finitely generated module of complexity at most c, and let I_{G}(M) be the annihilator in H^{∗}(G, k) of Ext_{kG}^{∗}(M, M). Then Ext_{kG}^{∗}(M, M) is a finitely generated module over the image of

H^{∗}(G, k) = Ext_{kG}^{∗}(k, k) ⟶^{⊗M} Ext_{kG}^{∗}(M, M).

This is a graded central subring isomorphic to H^{∗}(G, k)/I_{G}(M), and has Krull dimension at most c. Let S be the multiplicative subset of H^{∗}(G, k)/I_{G}(M) consisting of the homogeneous elements which do not belong to any prime ideal of dimension c. It is proved in §3 of Carlson, Donovan and Wheeler [10] that the endomorphism ring of M in M_{c}/M_{c−1} is equal to the ring of degree zero elements in the homogeneous localization [Ext_{kG}^{∗}(M, M)S^{−1}]_{0}. This is finitely generated as a module over [(H^{∗}(G, k)/I_{G}(M))S^{−1}]_{0}. Every prime ideal in the latter ring is maximal, so it is a finite direct sum of Artinian local rings. It follows that [Ext_{kG}^{∗}(M, M)S^{−1}]_{0} is an Artin algebra. □

Corollary 6.6. The endomorphism ring of an indecomposable module in M_{c,c−1} is a local Artin algebra.

Proof. By the above proposition, the endomorphism ring of any module in M_{c,c−1} is an Artin algebra. It follows from Lemma 3.1 that idempotents in M_{c,c−1} split, because \underline{U_{c}}/\underline{U_{c−1}} has countable direct sums. □

Theorem 6.7. The Krull–Schmidt theorem holds in the category M_{c,c−1}.

Proof. The Krull–Schmidt theorem holds in any additive category with the following properties:

(i) the endomorphism ring of an indecomposable object is a local ring, and (ii) every object decomposes as a finite direct sum of indecomposables. □

Corollary 6.8. If M_{1} ⊕ M_{2} ≅ M_{1} ⊕ M_{3} in M_{c}/M_{c−1} then M_{2} ≅ M_{3}.

Proof. If M_{1} ⊕ M_{2} ≅ M_{1} ⊕ M_{3} in M_{c}/M_{c−1} then the same holds in M_{c,c−1}, so by the theorem, M_{2} ≅ M_{3} in M_{c,c−1}. It follows from Proposition 6.2 that the functor M_{c}/M_{c−1} → M_{c,c−1} is fully faithful, and so M_{2} ≅ M_{3} in M_{c}/M_{c−1}. □

7. Varieties

Suppose that M is a (possibly infinitely generated) kG-module, and N is a finitely generated kG-module. Then Ext_{kG}^{∗}(N, M) is an H^{∗}(G, k)-module which is finitely generated whenever M is finitely generated as a kG-module.

Lemma 7.1. Suppose that U ≠ 0 is a finitely generated graded H^{∗}(G, k)-submodule of Ext_{kG}^{∗}(N, M). Then there exists a map ϕ : X → M with X a finitely generated kG-module, such that

U ⊆ Im(ϕ_{∗} : Ext_{kG}^{∗}(N, X) → Ext_{kG}^{∗}(N, M))

with equality in all large enough degrees. In particular \sqrt{Ann(U)} = \sqrt{Ann(Im(ϕ_{∗})}).

Proof. Choose non-zero homogeneous elements ζ_{1}, …, ζ_{t} generating U, with ζ_{i} ∈ Ext_{kG}^{n_{i}}(N, M). Represent each ζ_{i} by a cocycle ζ̂_{i} : Ω^{n_{i}}(N) → M. Then for j > 0 the map

Êxt_{kG}^{j−n_{i}}(N, N) ≅ Ext_{kG}^{j}(N, Ω^{n_{i}}(N)) ⟶^{(ζ̂_{i})_{∗}} Ext_{kG}^{j}(N, M)

is multiplication by ζ_{i}. So let X = ⨁_{i=1}^{t} Ω^{n_{i}}(N) and ϕ = ∑_{i=1}^{t} ζ̂_{i}. Then the image of ϕ_{∗} contains U, and in degrees which are at least as big as the maximum of the n_{i}, the image is equal to U. In smaller degrees, there can be superfluous elements coming from elements of negative degree in Êxt_{kG}^{∗}(N, N).

COMPLEXITY AND VARIETIES FOR INFINITELY GENERATED MODULES 23

The statement about radicals of annihilators is proved as follows. If 0 ≠ U ⊆ U′ is an inclusion of finitely generated graded H^{∗}(G, k)-modules with one dimensional quotient, then Ann(U′/U) consists of all elements of positive degrees, and therefore contains Ann(U). Since

Ann(U)Ann(U′/U) ⊆ Ann(U′) ⊆ Ann(U),

this means that \sqrt{Ann(U)} = \sqrt{Ann(U′)}. Applying this argument inductively, we see that this holds whenever U has finite codimension in U′. □

Definition 7.2. We define V_{G}(M) to be the set of closed homogeneous subvarieties of V_{G}(k) which are contained in subvarieties of the form V_{G}(Ann(U)), where U is a finitely generated graded submodule of Ext_{kG}^{∗}(N, M) for some finitely generated kG-module N.

We remark that if M is itself finitely generated, then V_{G}(M) consists of the closed homogeneous subvarieties of V_{G}(M), and hence has a unique maximal element. In general, V_{G}(M) need not have a unique maximal element (for example, take an infinite direct sum of finitely generated modules with different varieties of the same dimension), but it is closed under finite unions:

Lemma 7.3. The set V_{G}(M) has the following properties:

(i) if V ∈ V_{G}(M) and W ⊆ V is closed and homogeneous then W ∈ V_{G}(M). (ii) V_{1}, …, V_{t} ∈ V_{G}(M) then V_{1} ∪ ⋯ ∪ V_{t} ∈ V_{G}(M).

Proof. (i) This is clear from the definition.

(ii) Choose finitely generated kG-modules N_{i}, and choose finitely generated graded H^{∗}(G, k)-submodules U_{i} ⊆ Ext_{kG}^{∗}(N_{i}, M), so that V_{i} ⊆ W_{i} = V_{G}(Ann(U_{i})). Then

24 D. J. BENSON, JON F. CARLSON, AND J. RICKARD

U = U_{1}⊕⋯⊕U_{t} is a finitely generated graded H^{∗}(G, k)-submodule of Ext_{kG}^{∗}(⨁_{i} N_{i}, M), and Ann(U) = Ann(U_{1}) ∩ ⋯ ∩ Ann(U_{t}). Thus V_{1} ∪ ⋯ ∪ V_{t} ⊆ W_{1} ∪ ⋯ ∪ W_{t} = V_{G}(Ann(U)). □

If ζ is a non-zero element of H^{n}(G, k), we represent ζ by a cocycle ζ̂: Ω^{n}(k) → k, and write L_{ζ} for the kernel of ζ̂. Then L_{ζ} is a finitely generated kG-module with the property that V_{G}(L_{ζ}) is the closed homogeneous hypersurface V_{G}⟨ζ⟩ determined by regarding ζ as a homogeneous element of the coordinate ring of V_{G}(k). There is a triangle

L_{ζ} → Ω^{n}(k) ⟶^{ζ̂} k → Ω^{−1}(L_{ζ})

in stmod(kG).

Lemma 7.4. Suppose that θ : X → M is a homomorphism of kG-modules, with X finitely generated, and that U is the image of

θ_{∗} : Ext_{kG}^{∗}(X, X) → Ext_{kG}^{∗}(X, M).

Let ζ ∈ H^{∗}(G, k) be an element which annihilates U. Then θ factors as a composition X → X ⊗ Ω^{−1}(L_{ζ}) → M.

Proof. This follows from the fact that the composite

Ω^{n}(k) ⊗ X ⟶^{ζ̂⊗1} X → M

is zero in StMod(kG). □

Theorem 7.5. Let V be a collection of closed homogeneous subvarieties of V_{G}(k) satisfying conditions (i) and (ii) of Lemma 7.3. Then for a kG-module M the following conditions are equivalent:

(i) V_{G}(M) ⊆ V.

COMPLEXITY AND VARIETIES FOR INFINITELY GENERATED MODULES 25

(ii) Every map ϕ : X → M from a finitely generated module X factors through a finitely generated module M′ with V_{G}(M′) ∈ V.

(iii) M is a filtered colimit lim_{\overrightarrow{I}} F of finitely generated kG-modules F (i) with V_{G}(F (i)) ∈ V.

(iv) Given any finite subset S of M, there is a finitely generated projective module P and a finitely generated submodule M′ of M ⊕ P such that V_{G}(M′) ∈ V, and M′ contains the elements (s, 0) ∈ M ⊕ P with s ∈ S.

Proof. The equivalence of (ii), (iii) and (iv) is exactly the same as the proof of Theorems 4.2 and 5.3. So we shall prove that (iii) ⇒ (i) ⇒ (ii).

First suppose that M = lim_{\overrightarrow{I}} F with V_{G}(F (i)) ∈ V for all i in I. If U is a finitely generated graded H^{∗}(G, k)-submodule of Ext_{kG}^{∗}(N, M) with N finitely generated, then by Lemma 7.1, we may replace U by Im(ϕ_{∗}), where ϕ : X → M with X finitely generated. This does not change \sqrt{Ann(U)}, and hence does not change V_{G}(Ann(U)). Since U is Noetherian, we may choose a finitely generated submodule M_{0} of M, containing Im(ϕ), in such a way that the kernel of Ext_{kG}^{∗}(N, X) → Ext_{kG}^{∗}(N, M) is the same as the kernel of Ext_{kG}^{∗}(N, X) → Ext_{kG}^{∗}(N, M_{0}). The inclusion M_{0} ↪ M factors as M_{0} → F (i) → M for some i in I, and so

Im(ϕ_{∗}) = Im(Ext_{kG}^{∗}(N, X) → Ext_{kG}^{∗}(N, F (i))).

Since V_{G}(F (i)) ∈ V, we have V_{G}(Ann(Im(ϕ_{∗}))) ∈ V, and so V_{G}(Ann(U)) ∈ V. Since this is true for all such U, we have V_{G}(M) ⊆ V.

Conversely, suppose that V_{G}(M) ⊆ V. We wish to show that every map X → M, with X finitely generated, factors through a finitely generated module M′ with V_{G}(M′) ∈ V. Let U be the image of

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

26 D. J. BENSON, JON F. CARLSON, AND J. RICKARD

Then V_{G}(Ann(U)) ∈ V_{G}(M) ⊆ V. So we may choose homogeneous elements

ζ_{1}, …, ζ_{t} ∈ Ann(U) ⊆ H^{∗}(G, k)

so that

V_{G}⟨ζ_{1}⟩ ∩ ⋯ ∩ V_{G}⟨ζ_{t}⟩ ∈ V.

For each i, we represent ζ_{i} by a cocycle ζ̂_{i} : Ω^{n_{i}}(k) → k. Tensoring the triangle

Ω^{n_{i}}(k) ⟶^{ζ̂_{i}} k → Ω^{−1}(L_{ζ_{i}}) → Ω^{n_{i}−1}(k)

with X, we obtain a triangle

Ω^{n_{i}}(X) → X → X ⊗ Ω^{−1}(L_{ζ_{i}}) → Ω^{n_{i}−1}(X)

in StMod(kG). Since ζ_{i} ∈ Ann(U), we may apply the lemma to see that X → M factors as X → X ⊗ Ω^{−1}(L_{ζ_{i}}) → M. Each ζ_{j} still annihilates the image of

Ext_{kG}^{∗}(X ⊗ Ω^{−1}(L_{ζ_{i}}), X ⊗ Ω^{−1}(L_{ζ_{i}})) → Ext_{kG}^{∗}(X ⊗ Ω^{−1}(L_{ζ_{i}}), M).

So we may use the lemma repeatedly to see that X → M factors as

X → X ⊗ Ω^{−1}(L_{ζ_{1}}) ⊗ ⋯ ⊗ Ω^{−1}(L_{ζ_{t}}) → M.

Furthermore, V_{G}(X ⊗ Ω^{−1}(L_{ζ_{1}}) ⊗ ⋯ ⊗ Ω^{−1}(L_{ζ_{t}})) ⊆ V_{G}⟨ζ_{1}⟩ ∩ ⋯ ∩ V_{G}⟨ζ_{t}⟩ ∈ V. □

Definition 7.6. We define the dimension of V_{G}(M) to be the maximal dimension of an element V ∈ V_{G}(M).

Corollary 7.7. The complexity of a (possibly infinitely generated) kG-module M is equal to the dimension of V_{G}(M).

Proof. This follows from the theorem together with Theorem 5.3. □

COMPLEXITY AND VARIETIES FOR INFINITELY GENERATED MODULES 27

8. Further properties of the varieties

In contrast with the finitely generated case, it is not true in general that if H is a subgroup of G then

V_{H}(M ↓_{H}) = (res_{G,H}^{∗})^{−1}V_{G}(M).

An example of this is given in the appendix. However, one inclusion still holds:

Proposition 8.1. If H is a subgroup of G then

V_{H}(M ↓_{H}) ⊆ (res_{G,H}^{∗})^{−1}V_{G}(M).

Proof. If U ⊆ Ext_{kH}^{∗}(N, M ↓_{H}) ≅ Ext_{kG}^{∗}(N ↑^{G}, M) is finitely generated as an H^{∗}(H, k)-module, then it is also finitely generated as an H^{∗}(G, k)-module, since H^{∗}(H, k) is finitely generated as a module over the image of the restriction map. We have

Ann_{H^{∗}(G,k)}(U) ⊆ res_{G,H}^{−1}Ann_{H^{∗}(H,k)}(U),

so that

V_{G}(Ann_{H^{∗}(G,k)}(U)) ⊇ res_{G,H}^{∗}V_{H}(Ann_{H^{∗}(H,k)}(U)).

This is true for all such U, which proves the proposition. □

It is also false in general, in contrast with the finitely generated case, that

V_{G}(M_{1} ⊗ M_{2}) = V_{G}(M_{1}) ∩ V_{G}(M_{2}).

An example of this is given in the appendix. Again, one inclusion still holds:

Proposition 8.2. If M_{1} and M_{2} are kG-modules then

V_{G}(M_{1} ⊗ M_{2}) ⊆ V_{G}(M_{1}) ∩ V_{G}(M_{2}).

28 D. J. BENSON, JON F. CARLSON, AND J. RICKARD

Proof. For the exterior tensor product we have

V_{G×G}(M_{1} ⊗ M_{2}) = {V_{1} × V_{2} | V_{1} ∈ V_{G}(M_{1}), V_{2} ∈ V_{G}(M_{2})}.

Now apply the previous proposition to the diagonal inclusion of G in G × G. □

Proposition 8.3. If {M_{i}, i ∈ I} is a collection of kG-modules and M = ⨁_{i∈I} M_{i}, then V_{G}(M) is the closure under finite unions of ⋃_{i∈I} V_{G}(M_{i}).

Proof. The main point here is that if θ : X → M is a homomorphism with X finitely generated, then θ(X) ⊆ ⨁_{i∈J} M_{i} for some finite subset J ⊆ I. Hence, by Lemma 7.1, any finitely generated submodule of Ext_{kG}^{∗}(N, M), for N finitely generated, is contained in

Ext_{kG}^{∗}(N, ⨁_{i∈J} M_{i}) = ⨁_{i∈J} Ext_{kG}^{∗}(N, M_{i})

for J a finite subset of I. □

Proposition 8.4. Suppose that 0 → M_{1} ⟶^{α} M_{2} ⟶^{β} M_{3} → 0 is a short exact sequence of kG-modules. Then for {i, j, k} = {1, 2, 3} we have

V_{G}(M_{i}) ⊆ {V_{j} ∪ V_{k} | V_{j} ∈ V_{G}(M_{j}), V_{k} ∈ V_{G}(M_{k})}.

Proof. The proof is based on the long exact sequence for cohomology. We shall treat the case i = 2; the other two cases are similar. Suppose that U_{2} is a finitely generated H^{∗}(G, k)-submodule of Ext_{kG}^{∗}(N, M_{2}), and I_{2} is its annihilator in H^{∗}(G, k). We wish to show that V_{G}(I_{2}) ⊆ V_{1} ∪ V_{3} for some V_{1} ∈ V_{G}(M_{1}) and V_{3} ∈ V_{G}(M_{3}). Let I_{3} be the annihilator in H^{∗}(G, k) of the submodule U_{3} = β_{∗}(U_{2}) ⊆ Ext_{kG}^{∗}(N, M_{3}). Let U′_{2} = U_{2} ∩ Ker(β_{∗}). Then U′_{2} is finitely generated, and so U′_{2} = α_{∗}(U_{1}) for some finitely generated H^{∗}(G, k)-submodule U_{1} ⊆ Ext_{kG}^{∗}(N, M_{1}). Namely, we can take

COMPLEXITY AND VARIETIES FOR INFINITELY GENERATED MODULES 29

U_{1} to be the H^{∗}(G, k)-submodule generated by inverse images under α_{∗} of a set of generators for U′_{2}. Let I_{1} be the annihilator in H^{∗}(G, k) of U_{1}. Then we have an exact sequence U_{1} → U_{2} → U_{3} → 0 of H^{∗}(G, k)-modules, and so I_{2} ⊇ I_{1}.I_{3}. So setting V_{1} = V_{G}(I_{1}) and V_{3} = V_{G}(I_{3}), we have V_{G}(I_{2}) ⊆ V_{1} ∪ V_{3} as required. □

It follows from Proposition 8.4 that for an object in the quotient category \underline{U_{c}}/\underline{U_{c−1}}, the collection of elements of V_{G}(M) with the property that all irreducible components have dimension c is well defined. We write V_{G}^{c}(M) for this collection. For an object in M_{c,c−1}, V_{G}^{c}(M) is a finite set with a unique maximal element, which we write as V_{G}^{c}(M).

Proposition 8.5. If M is an object in M_{c,c−1} and V_{G}^{c}(M) = {V_{1}, …, V_{t}} is a finite set, then M is isomorphic to M_{1} ⊕ ⋯ ⊕ M_{t} in M_{c,c−1} with V_{G}^{(c)}(M_{i}) = {V_{i}} for 1 ≤ i ≤ t.

Proof. This follows directly from Theorem 3.2 of Carlson and Wheeler [11] and the Krull–Schmidt Theorem 6.7 for the category M_{c,c−1}. □

9. The Alperin–Evens Theorem

In [2], Alperin and Evens proved that the complexity of a finitely generated module is equal to the maximal complexity of a restriction to an elementary abelian p-subgroup of G. In this section, we prove this without the finite generation hypothesis.

Let K be an algebraically closed field of infinite transcendence degree over k. It follows from the Noether normalization lemma that we may choose a finite set of elements ζ_{1}, …, ζ_{s} in the same degree in H^{∗}(G, k), generating a subring over which the whole of H^{∗}(G, k) is finitely generated as a module. Choose independent

30 D. J. BENSON, JON F. CARLSON, AND J. RICKARD

transcendentals X_{ij} ∈ K, with 1 ≤ i, j ≤ s. Set

ξ_{i} = ∑_{j=1}^{s} X_{ij}ζ_{j}.

Proposition 9.1. Let M be a (not necessarily finitely generated) kG-module. Then the following are equivalent:

(i) M has complexity at most c, and

(ii) L_{ξ_{1}} ⊗_{K} ⋯ ⊗_{K} L_{ξ_{c}} ⊗_{K} (K ⊗_{k} M) is projective.

Proof. If V is a closed homogeneous subvariety of V_{G}(k), denote by KV the subvariety of V_{G}(K) defined by the same equations. Then for any such V ⊆ V_{G}(k) of dimension at most c, we have

V_{G}⟨ξ_{1}⟩ ∩ ⋯ ∩ V_{G}⟨ξ_{c}⟩ ∩ KV = {0}.

The proposition therefore follows from Corollary 5.6 and Corollary 7.7. □

Theorem 9.2. The complexity of a (not necessarily finitely generated) kG-module M is equal to the maximal complexity of a restriction of M to an elementary abelian p-subgroup E of M.

Proof. This follows from Proposition 9.1 and Chouinard’s theorem [12]. Chouinard proves, without any finite generation hypothesis, that a kG-module is projective if and only if its restriction to every elementary abelian p-subgroup is projective. □

Appendix: Examples

We begin with an example of the construction described in Section 3, to show how idempotents in the endomorphism ring of the trivial module in a quotient category give rise to a direct sum decomposition when we pass to infinitely generated modules.

COMPLEXITY AND VARIETIES FOR INFINITELY GENERATED MODULES 31

Let G = D_{8}, the dihedral group of order eight, and k be an algebraically closed field of characteristic two. The finitely generated indecomposable modules were classified by Ringel [18], and we shall use his string-type and band-type schemata for describing modules. Thus Ω(k) is the string-type module M(a^{−1}b^{−1}a^{−1}bab):

• •
| ||
• •
|| |
• •
\ //
•

Here, we have used single and double lines to distinguish the actions of the two chosen elements of the group algebra squaring to zero. We look at the quotient category Q of the stable category of kG-modules by the subcategory of modules of complexity one. Note in particular that all band-type modules are in this subcategory.

The cohomology ring is H^{∗}(G, k) = k[x, y, z]/(xy), where deg(x) = deg(y) = 1 and deg(z) = 2. The element x (resp. y) is represented by the cocycle x̂ (resp. ŷ) : Ω(k) → k sending the left-hand top basis element to 1 (resp. 0) and the right hand one to 0 (resp. 1). So ζ = x + y is represented by a cocycle ζ̂ : Ω(k) → k whose kernel L_{ζ} is a band-type module, and is hence is periodic, and so ζ̂ is an isomorphism in Q. The endomorphism ring of the trivial module in Q is the ring of degree zero elements in the homogeneous localization of H^{∗}(G, k) given by inverting all elements outside the ideals (x) and (y) (Carlson, Donovan and Wheeler [10], Theorem 3.5). The idempotents in this ring are ζ^{−1}x and ζ^{−1}y. So to construct the summand of the trivial module k corresponding to ζ^{−1}x, we form the map

[figure: Ω(k); x; k; ⊕; ⊕; Ω(k); x; y; k; ⊕; ⊕; ⋮; y; ⋮]

This map is surjective, so when we complete it to a triangle, the third vertex is just the kernel. This is the infinitely generated module

M(… ba^{−1}b^{−1}a^{−1}baba^{−1}b^{−1}a^{−1}bab)
•  •  •
// \  // \  ||
• • • • •
⋯  || | || |
• • • •
\ //  \ //
•  •

Thus the trivial module decomposes as a direct sum

M(… ba^{−1}b^{−1}a^{−1}baba^{−1}b^{−1}a^{−1}bab) ⊕ M(a^{−1}b^{−1}a^{−1}baba^{−1}b^{−1}a^{−1}baba^{−1} …)

\begin{matrix}• & • & • \\ // & \ & // & \ & || \\ • & • & • & • & • \\ ⋯ & || & | & || & | \\ • & • & • & • \\ \ & // & \ & // \\ • & • \end{matrix}  ⊕  \begin{matrix}• & • & • \\ | & // & \ & // & \ \\ • & • & • & • & • \\ || & | & || & | & ⋯ \\ • & • & • & • \\ \ & // & \ & // \\ • & • \end{matrix}

The map from this to the trivial module is the obvious one, which sends the righthand end vertex of the left summand and the left-hand end vertex of the right summand to 1 and the rest of the basis elements to zero. The kernel is an infinite string module M_{0}, which has complexity one. In fact, every finitely generated submodule of M_{0} is contained in a finitely generated periodic submodule.

COMPLEXITY AND VARIETIES FOR INFINITELY GENERATED MODULES 33

The syzygy of the above kernel, Ω(M_{0}), is the module M(… ba^{−1}ba^{−1} …):

• •
⋯ // \ // \ ⋯
• • •

This module has no finitely generated periodic submodules, despite the fact that it is the colimit of the following sequence of finitely generated periodic modules:

M(aba^{−1}b^{−1}, id) → M(a(ba^{−1})^{3}b^{−1}, id) → M(a(ba^{−1})^{5}b^{−1}, id) → ⋯

\begin{matrix}• \\ // & \ \\ • & • \\ | & || \\ • & · & · & · & • \end{matrix} → \begin{matrix}• & • & • \\ // & \ & // & \ & // & \ \\ • & • & • & • \\ | & || \\ • & · & · & · & · & · & · & · & · & · & · & · & • \end{matrix} → \begin{matrix}• & • & • & • & • \\ // & \ & // & \ & // & \ & // & \ & // & \ \\ • & • & • & • & • & • \\ | & || \\ • & · & · & · & · & · & · & · & · & · & · & · & · & · & · & · & · & · & · & · & • \end{matrix} → ⋯

The modules here are modules of band type, obtained by identifying the two end vertices joined by a dotted line. Each homomorphism has a one dimensional kernel spanned by the basis element obtained by identifying these two end vertices. The quotient by this one dimensional kernel is embedded into the middle of the diagram for the next module in the sequence. Each of the modules in this sequence is periodic, but the colimit has no finitely generated periodic submodules.

The same phenomenon does not appear to happen for the Klein four group in characteristic two, but does happen for all other non-cyclic abelian p-groups in characteristic p. Let

G = ⟨x, y | [x, y] = x^{p^{α}} = y^{p^{β}} = 1⟩

with p^{α} ≥ 3. Consider the directed system of modules

M_{2} ⟶^{ϕ_{2}} M_{4} ⟶^{ϕ_{4}} M_{6} ⟶^{ϕ_{6}} ⋯

where M_{n} is generated by a_{n,1}, …, a_{n,n}, b_{n} with relations

(y − 1)a_{n,i} = (x − 1)^{p^{α}−1}a_{n,i+1},  1 ≤ i ≤ n − 1
(y − 1)a_{n,n} = (x − 1)^{p^{α}−2}b_{n}
(y − 1)b_{n} = 0.

34 D. J. BENSON, JON F. CARLSON, AND J. RICKARD

Then M_{n} is periodic, because M_{n} ↓_{⟨x⟩} is free. The map ϕ_{n} is defined by

ϕ_{n}(a_{n,i}) = a_{n+2,i},  1 ≤ i ≤ n
ϕ_{n}(b_{n}) = (x − 1)a_{n+2,n+1}.

Let M be the direct limit of this system of modules. Then M has no periodic submodules.

Similarly, if

G = ⟨x, y, z⟩ ≅ (ℤ/2)^{3}

and p = 2, we consider the directed system

M_{2} ⟶^{ϕ_{2}} M_{4} ⟶^{ϕ_{4}} M_{6} ⟶^{ϕ_{6}} ⋯

where M_{n} is generated by a_{n,1}, …, a_{n,n} with relations

(z + 1)a_{n,i} = (x + 1)(y + 1)a_{n,i+1},  1 ≤ i ≤ n − 2
(z + 1)a_{n,n−1} = (x + 1)a_{n,n}
(z + 1)a_{n,n} = 0.

Then M_{n} is periodic, because M_{n} ↓_{⟨x,y⟩} is free. The map ϕ_{n} is defined by

ϕ_{n}(a_{n,i}) = a_{n+2,i},  1 ≤ i ≤ n − 1
ϕ_{n}(a_{n,n}) = (y + 1)a_{n+2,n}.

Let M be the direct limit of this system of modules. Then again M has no periodic submodules.

We now turn to some examples which demonstrate the problems with the restriction and tensor product theorems (see Section 8). Let G be the Klein four group and k be an algebraically closed field of characteristic two. Again we use the notation of Ringel [18]. Let M_{1} and M_{2} be the infinite string modules

M_{1} = M(ab^{−1}ab^{−1}ab^{−1} …) ; \begin{matrix}• & • & • \\ // & \ & // & \ & // & \ & ⋯ \\ • & • & • & • \end{matrix}  M_{2} = M(b^{−1}ab^{−1}ab^{−1} …) ; \begin{matrix}• & • & • \\ \ & // & \ & // & \ & ⋯ \\ • & • & • \end{matrix}

The module M_{2} has complexity one; indeed it is the union of finitely generated

periodic submodules. The module M_{1}, despite the fact that it is periodic (M_{1} ≅

Ω(M_{1})), has complexity two. The set V_{G}(M_{1}) consists of all subsets of V_{G}(k).

However, there is a subgroup H of order two in G with the property that M_{1} ↓_{H} is

free. So

V_{H}(M_{1} ↓_{H}) ≠ (res_{G,H}^{∗})^{−1}V_{G}(M_{1}).

See Proposition 8.1.

Let L be the module M(b). Then L is periodic, and is induced from a one

dimensional module for H. Since M_{1} ↓_{H} is free, so is M_{1} ⊗ L. So we have

V_{G}(M_{1} ⊗ L) ≠ V_{G}(M_{1}) ∩ V_{G}(L).

See Proposition 8.2.

In some sense, this example shows that the definition of V_{G}(M) is not quite right

as it stands. The variety V_{G}(M_{1}) in the above example should really be the collection

of homogeneous subvarieties of the open subset of V_{G}(k) given by removing the line

res_{G,H}^{∗}V_{H}(k).

References

[1] A. Adem. On the exponent of cohomology of discrete groups. Bull. London Math. Soc. 21 (1989), 585–590.

[2] J. L. Alperin and L. Evens. Representations, resolutions, and Quillen’s dimension theorem. J. Pure Appl. Algebra 22 (1981), 1–9.

[3] M. Artin and B. Mazur. Etale homotopy. Springer Lecture Notes in Mathematics 100, Springer-Verlag, Berlin/New York 1969.

[4] H. Bass. Finitistic dimension and a homological generalization of semi-primary rings. Trans. Amer. Math. Soc. 95 (1960), 466–488.

[5] D. J. Benson. Representations and cohomology II: cohomology of groups and modules. Cambridge studies in advanced mathematics, vol. 31, CUP, 1991.

[6] M. Bökstedt and A. Neeman. Homotopy limits in triangulated categories. Compositio Mathematica 86 (1993), 209–234.

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

[8] J. F. Carlson. Exponents of modules and maps. Invent. Math. 95 (1989), 13–24.

[9] J. F. Carlson. The decomposition of the trivial module in the complexity quotient category. Preprint, 1993.

[10] J. F. Carlson, P. W. Donovan and W. W. Wheeler. Complexity and quotient categories for group algebras. Preprint, 1993.

[11] J. F. Carlson and W. W. Wheeler. Varieties and localizations of module categories. Preprint, 1993.

[12] L. Chouinard. Projectivity and relative projectivity over group rings. J. Pure Appl. Algebra 7 (1976), 278–302.

[13] L. Evens. The cohomology of groups. Oxford Science Publications, OUP, 1991.

[14] E. M. Friedlander and B. J. Parshall. Support varieties for restricted Lie algebras. Invent. Math. 86 (1986), 553–562.

[15] D. Happel. Triangulated categories in the representation theory of finite dimensional algebras. London Math. Soc. Lecture Notes 119, CUP, 1988.

[16] J. C. Jantzen. Kohomologie von p-Lie-Algebren und nilpotente Elemente. Abh. Mat. Sem. Univ. Hamburg 76 (1986), 191–219.

[17] H. Lenzing. Homological transfer from finitely presented to infinite modules. In “Abelian group theory”, Springer Lecture Notes in Math. 1006, 734–761.

[18] C. M. Ringel. The indecomposable representations of the dihedral 2-groups. Math. Ann. 214 (1975), 19–34.

[19] J.-P. Serre. Sur la dimension cohomologique des groupes profinis. Topology 3 (1965), 413–420.

Department of Mathematics, University of Georgia, Athens, GA 30602, USA

Department of Mathematics, University of Georgia, Athens, GA 30602, USA

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