COMPLEXITY AND VARIETIES FOR INFINITE GROUPS, II

D. J. BENSON

Abstract. This is the second part of a two part paper developing the module theory, and a theory of complexity and varieties for modules, in the context of Kropholler’s class of groups LH𝔉. The numbering is a continuation of the numbering for the first part [4].

Contents (Part II): 10. Closed model categories. 11. Idempotent functors. 12. Generating stmod(kΓ) and StMod(kΓ). 13. Nilpotence. 14. Complexity. 15. Varieties for modules. 16. The thick subcategories C(V). 17. Connectedness. 18. Examples. 19. An application. 20. Some open problems.

10. Closed model categories

Quillen [13] introduced the notion of a closed model category as an axiomatization of homotopy theory. In [14] he reformulates the axioms, and we use the version given there for the sake of convenience. In this section, we describe a closed model category structure on Mod(kΓ) and mod(kΓ), in such a way that the homotopy category Ho Mod(kΓ) is equivalent to StMod(kΓ) and Ho mod(kΓ) is equivalent to stmod(kΓ).

We begin by describing the fibrations, cofibrations and weak equivalences in Mod(kΓ), and then we verify axioms CM1–5 of Quillen [14]. The definitions in mod(kΓ) are the same, and the verification is essentially the same.

Definition 10.1. Let Γ be a group and k a commutative ring of coefficients. A module homomorphism M → N in Mod(kΓ) is taken to be a fibration if it is surjective. It is taken to be a cofibration if it is injective, and the cokernel is cofibrant (see Definition 4.1). It is taken to be a weak equivalence if it passes down to an isomorphism in StMod(kΓ). See Proposition 8.2 for an explicit description of what this means in module theoretic terms. A module homomorphism is called a trivial cofibration if it is both a cofibration and a weak equivalence. It is called a trivial fibration if it is both a fibration and a weak equivalence.

Lemma 10.2. A module homomorphism in Mod(kΓ) is a trivial cofibration if and only if it is injective with projective cokernel.

The author is partly supported by a grant from the NSF.

Proof. This follows immediately from Proposition 5.4.

Lemma 10.3. A module homomorphism in Mod(kΓ) is a trivial fibration if and only if it is surjective and the kernel has finite projective dimension.

Proof. This follows by applying Proposition 8.2 to the kernel and the zero module.

Proposition 10.4. (Axiom CM4) Given a commutative diagram

[figure: M; α; i; M′; p; N; β; N′]

there is a map N → M′ making both triangles commute, provided either (i) i is a cofibration and p is a trivial fibration, or (ii) i is a trivial cofibration and p is a fibration.

Proof. (i) First we treat the special case where M = 0, so that N is cofibrant. Let X be the pullback of β and p. Then X → N is surjective. We write K for the kernel, which is isomorphic to the kernel of p, and hence has finite projective dimension by Lemma 10.3. The lifting property now follows by applying Lemma 4.5 to the short exact sequence

0 → K → X → M → 0.

For the general case, we perform the following construction. Let π : N → L be the cokernel of i and let X be the pushout of α and i. Then we have a short exact sequence

0 → M′ → X → L → 0.

Let K be the kernel of the map X → N′ defined by the square, and let K′ be the kernel of p. Then we have a commutative diagram

[figure: 0; 0; 0; K′; M′; p; N′; 0; 0; K; X; N′; 0; L; L; 0; 0]

COMPLEXITY AND VARIETIES FOR INFINITE GROUPS, II 31

Applying Lemma 4.5 to the sequence

0 → K′ → K → L → 0,

we see that it splits. Composing with K → X, we see that

0 → M′ → X → L → 0

splits. This gives us a map X → M′ and hence a map γ : N → M′ satisfying α = γ ∘ i, so that the upper triangle commutes. The lower triangle need not commute, but the difference β − p ∘ γ vanishes on composition with i. So it defines a map η : L → N′ with η ∘ π = β − p ∘ γ. Applying the case we proved first (with M = 0 and N = L), we obtain a map ζ : L → M′ with p ∘ ζ = η. Set γ′ = γ + ζ ∘ π : N → M′. Then

γ′ ∘ i = γ ∘ i + ζ ∘ π ∘ i = α
p ∘ γ′ = p ∘ γ + p ∘ ζ ∘ π = (β − η ∘ π) + η ∘ π = β.

(ii) This is similar but easier. This time, the cokernel of i is projective, by Lemma 10.2. Write π : N → P for this cokernel. Then π splits, so there are maps η : P → N, ζ : N → M with π ∘ η = Id_{P}, ζ ∘ i = Id_{M} and i ∘ ζ + η ∘ π = Id_{N}. Since P is projective and p is surjective, there exists a map θ : P → M′ such that p ∘ θ = β ∘ η. Now set γ = α ∘ ζ + θ ∘ π : N → M′. Then

γ ∘ i = α ∘ ζ ∘ i + θ ∘ π ∘ i = α
p ∘ γ = p ∘ α ∘ ζ + p ∘ θ ∘ π = β ∘ i ∘ ζ + β ∘ η ∘ π = β.

Proposition 10.5. (Axiom CM5) (i) Every map in Mod(kΓ) can be factored as a cofibration followed by a trivial fibration.

(ii) Every map in Mod(kΓ) can be factored as a trivial cofibration followed by a fibration.

Proof. (i) Let α : M → N be a module homomorphism in Mod(kΓ). Choose r large enough so that the map

IdB̄^{⊗r} ⊗ Ω^{r}α : B̄^{⊗r} ⊗_{k} Ω^{r}M → B̄^{⊗r} ⊗_{k} Ω^{r}N

can be taken to be injective with cofibrant cokernel. Also arrange that the maps

ρ_{M} : B̄^{⊗r} ⊗_{k} Ω^{r}M → M ρ_{N} : B̄^{⊗r} ⊗_{k} Ω^{r}N → N

(Remark 4.3) are surjective, so that by Proposition 8.2 the kernels K and K′ have finite projective dimension. Let X be the pushout of the maps IdB̄^{⊗r} ⊗ Ω^{r}α and ρ_{M}. Then M → X is a cofibration. Using Lemma 4.4, we obtain a map η : X → N as in the following diagram so that the composite M → X → N is equal to α.

[figure: 0; K; B̄^{⊗r} ⊗_{k} Ω^{r}M; ρ_{M}; M; α; 0; 0; K; B̄^{⊗r} ⊗_{k} Ω^{r}N; ρ_{N}; X; η; 0; N]

Since ρ_{N} is surjective it follows that η is surjective. Its kernel K′′ fits into a diagram

[figure: 0; 0; 0; K; K′; K′′; 0; 0; K; B̄^{⊗r} ⊗_{k} Ω^{r}N; X; 0; N; N; 0; 0]

Since K and K′ have finite projective dimension, so does K′′, and so X → N is a trivial fibration by Lemma 10.3.

(ii) Let α : M → N be a module homomorphism in Mod(kΓ). Choose a projective module P surjecting onto N, and factor α as M → M ⊕ P → N. Here, we have used the zero map from M to P, so that M → M ⊕ P is a trivial cofibration by Lemma 10.2, and M ⊕ P → N is surjective and hence a fibration.

Theorem 10.6. Let Γ be a group and k a commutative Noetherian ring of coefficients. With the above definitions for the fibrations, cofibrations and weak equivalences, Mod(kΓ) is a closed model category.

Proof. We check axioms CM1–5 of Quillen [14]. The difficult ones, axioms CM4 and CM5, are proved in Propositions 10.4 and 10.5. The remaining axioms CM1–3 are trivial to verify; for convenience we state them below. CM1 Mod(kΓ) has all finite limits and colimits.

CM2 If M_{1} →^{f} M_{2} →^{g} M_{3} are module homomorphisms in Mod(kΓ), and two of f, g, g ∘ f are weak equivalences, then so is the third.

A map f : M → N is said to be a retract of f′ : M′ → N′ if there are maps ϕ_{1} : M → M′, ϕ_{2} : N → N′, ψ_{1} : M′ → M and ψ_{2} : N′ → N such that ϕ_{2} ∘ f = f′ ∘ ϕ_{1}, ψ_{2} ∘ f′ = f ∘ ψ_{1}, ψ_{1} ∘ ϕ_{1} = Id_{M} and ψ_{2} ∘ ϕ_{2} = Id_{N}. CM3 If f : M → N is a retract of f′ : M′ → N′ and f′ is a fibration (resp. cofibration, weak equivalence) then f is also a fibration (resp. cofibration, weak equivalence).

Proposition 10.7. Let Γ be an LH𝔉-group and k a commutative Noetherian ring of coefficients. With the above definitions of fibration, cofibration and weak equivalence, mod(kΓ) is a closed model category.

Proof. All the constructions used in the proof keep us inside mod(kΓ), by Remark 8.4.

We now give one of the standard consequences of Theorem 10.6 and Proposition 10.7.

Proposition 10.8. In Mod(kΓ) (or mod(kΓ)), any two of the three families of fibrations, cofibrations and weak equivalences determine the third in the following way:

(i) A map is a cofibration (resp. trivial cofibration) if and only if it has the left lifting property with respect to all trivial fibrations (resp. fibrations).

(ii) A map is a fibration (resp. trivial fibration) if and only if it has the right lifting property with respect to all trivial cofibrations (resp. cofibrations).

(iii) A map is a weak equivalence if and only if it may be factored as f = p ∘ i with i a trivial cofibration and p a trivial fibration.

Proof. (i) The axiom CM4(i) says that a cofibration has the left lifting property with respect to all trivial fibrations. Conversely, if α : M → N has the left lifting property with respect to all trivial fibrations, then using axiom CM5(i) we may write α = pi with i : M → X a cofibration and p : X → N a trivial fibration. Looking at the diagram

[figure: M; α; X; p; N; N]

we have a map β : N → X with βα = i and pβ = id_{N}. So α is a retract of i, and by axiom CM3 α is a cofibration.

The axiom CM4(ii) implies that a trivial cofibration has the left lifting property with respect to all fibrations. Conversely, if α : M → N has the left lifting property with respect to all fibrations, then we write α = pi with i : M → X a trivial cofibration and

34 D. J. BENSON

p : X → N a fibration. As before, we obtain a map β : N → X with βα = i and pβ = id_{N}. So f is a retract of i, and by axiom CM3, f is a trivial cofibration.

The proof of (ii) is dual to the proof of (i), and (iii) follows immediately from axioms CM2 and CM5.

For any closed model category C, one may form the homotopy category Ho C by formally inverting the weak equivalences. Quillen proves in [13] that this category is equivalent to the category πC_{cf} whose objects are the objects in C which are both cofibrant and fibrant, and where the morphisms are the “homotopy classes” of maps in a suitable sense.

In the case of Mod(kΓ) and mod(kΓ), every object is fibrant, so we obtain equivalences between Ho Mod(kΓ) and πMod_{c}(kΓ) and between Ho mod(kΓ) and πmod_{c}(kΓ).

Proposition 10.9. Morphisms in Mod_{c}(kΓ) are homotopic if and only if they differ by a map which factors through a projective module.

Proof. To prove this, we identify a path object N^{I} for a module N ([12] §I.1, Definition 4). If π : P_{N} → N is a projective kΓ-module mapping onto N, then we may factor the diagonal map N → N ⊕ N as

N \underrightarrow{( \begin{matrix}Id & N \\ 0 \end{matrix} )} N ⊕ P_{N} \underrightarrow{( \begin{matrix}Id_{N} & 0 \\ Id_{N} & π \end{matrix} )} N ⊕ N

where the first map is a trivial cofibration and the second is a fibration. Two maps f, f′ : M → N are homotopic if and only if f ⊕ f′ : M → N ⊕ N lifts to N ⊕ P_{N}. In other words, if and only if there exists a map f ⊕ ϕ : M → N ⊕ P_{N} such that f + π ∘ ϕ = f′, namely if and only if f − f′ factors through π.

Theorem 10.10. Let Γ be a group and k a commutative ring of coefficients. The obvious functor from Mod(kΓ) to StMod(kΓ) induces an equivalence of categories from Ho Mod(kΓ) to StMod(kΓ). If Γ is an LH𝔉-group, then the obvious functor from mod(kΓ) to stmod(kΓ) induces an equivalence of categories from Ho mod(kΓ) to stmod(kΓ).

Proof. This follows from Proposition 10.9 and Lemma 8.5.

11. Idempotent functors

Throughout this section, Γ is an LH𝔉-group and k is a commutative Noetherian ring of coefficients of finite global dimension.

We construct idempotent functors E_{C} and F_{C} on StMod(kΓ) for each thick subcategory C of stmod(kΓ) as described in Section 5 of Rickard [15]. This construction is based on a

COMPLEXITY AND VARIETIES FOR INFINITE GROUPS, II 35

well known construction in stable homotopy theory. For the convenience of the reader, we briefly review the construction and properties here.

Let C be a thick subcategory of stmod(kΓ). In other words, C is a triangulated subcategory which is closed under taking direct summands. We define C^{⊕} to be the smallest thick subcategory of StMod(kΓ) containing C and closed under countable direct sums. [Again, if k is a field of coefficients and StMod(kΓ) is defined without the countability hypothesis, we define C^{⊕} in the same way with respect to arbitrary direct sums.] We define C^{→} to be the full subcategory of StMod(kΓ) whose objects are filtered colimits in Mod(kΓ) of modules in C. It is easy to check that C^{→} is thick, and so C^{⊕} is contained in C^{→}. If M is in C^{→} then any map from a module in stmod(kΓ) to M factors in StMod(kΓ) through a module in C.

Lemma 11.1. C = C^{⊕} ∩ stmod(kΓ).

Proof. If N is a module in C^{⊕} ∩ stmod(kΓ) then certainly N is in C^{→}, and so the identity map on N factors through a module in C, say N → M → N. By Proposition 8.2, this gives us a module isomorphism for some r ≥ 0

Ω^{r}(M) ⊕ P ≅ Ω^{r}(N) ⊕ Q

where P is projective, but Q need not be. Since M and N are in stmod(kΓ), we may assume that Ω^{r}(M) and Ω^{r}(N) are finitely generated. So Ω^{r}(N) is isomorphic to a direct summand of Ω^{r}(M) ⊕ P′ for some finitely generated projective module P′. Finally, this implies that N is isomorphic in stmod(kΓ) to a direct summand of B̄^{⊗r} ⊗ Ω^{r}(M). Since C is closed under the translation τ and under taking direct summands, the lemma is proved.

Let M be a module in StMod(kΓ). We inductively construct a sequence of modules M = M_{0}, M_{1}, … as follows. Evaluation gives a homomorphism from the direct sum of the modules C_{i} ⊗_{k} Hom_{kΓ}(C_{i}, M_{n}) (where C_{i} runs over a complete set of representatives of isomorphism classes of modules in C) to M_{n} which we complete to a distinguished triangle in StMod_{c}(kΓ) to define M_{n+1}:

⨁_{i}(C_{i} ⊗_{k} Hom_{kΓ}(C_{i}, M_{n})) → M_{n} → M_{n+1} →

Using Lemma 8.5, we see that for a module C in C, every morphism C → M_{n} in StMod_{c}(kΓ) lifts to Mod(kΓ), and then factors through the left hand term in this triangle, and so the composite C → M_{n} → M_{n+1} is zero.

36 D. J. BENSON

Each M_{n} comes equipped with a map M → M_{n}, which we complete to a distinguished triangle A_{n} → M → M_{n} → in StMod(kΓ). The homotopy colimit of these triangles gives us a distinguished triangle

E_{C}(M) → M → F_{C}(M) →

which is used as the definition of E_{C} and F_{C}.

If C is any module in C then any map C → F_{C}(M) factors through some M_{n}, and then the composite with M_{n} → M_{n+1} is zero. So F_{C}(M) is C-local, in the sense that there are no nonzero maps to it from any module in C.

It is not hard to prove from the construction, that each A_{n}, and hence also the module E_{C}(M) is in C^{⊕}. The map E_{C}(M) → M is universal among maps to M from modules in C^{⊕}, because if C is in C^{⊕} then applying Êxt_{kΓ}^{0}(C, −) to the above triangle gives an isomorphism

Êxt_{kΓ}^{0}(C, E_{C}(M)) ≅ Êxt_{kΓ}^{0}(C, M).

Similarly, the map M → F_{C}(M) is universal among maps from M to a C-local object, because if L is C-local then applying Êxt_{kΓ}^{0}(−, L) to the above triangle gives an isomorphism

Êxt_{kΓ}^{0}(F_{C}(M), L) ≅ Êxt_{kΓ}^{0}(M, L).

Since these observations are formal consequences of the facts that E_{C}(M) is in C^{⊕} and F_{C}(M) is C-local, it follows that any other triangle with M as its middle term and whose left and right term have these properties, is isomorphic to the triangle above. In particular, the result of the construction does not depend (up to isomorphism) on the choices made. This implies that E_{C} and F_{C} can be regarded as functors on StMod(kΓ). We refer to this uniqueness statement as Rickard’s characterization of this triangle.

Proposition 11.2. Let C be a thick subcategory of stmod(kΓ) and M be a module in StMod(kΓ). Then

(i) The following are equivalent:

(a) M is in C^{⊕}.

(b) E_{C}(M) → M is an isomorphism in StMod(kΓ).

(c) F_{C}(M) ≅ 0 in StMod(kΓ).

(ii) The following are equivalent:

(a) M is C-local.

(b) E_{C}(M) ≅ 0 in StMod(kΓ).

(c) M → F_{C}(M) is an isomorphism in StMod(kΓ).

COMPLEXITY AND VARIETIES FOR INFINITE GROUPS, II 37

If C ≤ C′ are thick subcategories of stmod(kG), there are induced natural transformations E_{C} → E_{C′} and F_{C} → F_{C′} which induce a map of triangles

[figure: E_{C}(M); M; F_{C}(M); E_{C′}(M); M; F_{C′}(M)]

for any object M in StMod(kΓ).

Proposition 11.3. Let C ≤ C′ be thick subcategories of stmod(kΓ). Then the following are equivalent:

(i) E_{C}(M) → E_{C′}(M) is an isomorphism for all M in StMod(kΓ).

(ii) F_{C}(M) → F_{C′}(M) is an isomorphism for all M in StMod(kΓ).

(iii) C = C′.

(iv) C^{⊕} = (C′)^{⊕}.

(v) M is C-local if and only if M is C′-local.

Proof. Equivalence of (i) and (ii) follows from one of the axioms for a triangulated category (which is really the five lemma). Clearly (iii) implies (i) and (ii). By Proposition 11.2, (i) implies (iv), and (ii) is equivalent to (v). Since C = C^{⊕} ∩ stmod(kΓ), (iv) implies (iii).

12. Generating stmod(kΓ) and StMod(kΓ)

Throughout this section, Γ is an LH𝔉-group and k is a commutative Noetherian ring of coefficients of finite global dimension. We apply the idempotent functor approach and Theorem 7.6 to deduce information about generation of the categories stmod(kΓ) and StMod(kΓ).

Theorem 12.1. Let C be the thick subcategory of stmod(kΓ) generated by the modules induced from finitely generated modules for finite elementary abelian subgroups of Γ. Then C is equal to stmod(kΓ) and C^{⊕} is equal to StMod(kΓ).

Proof. We have C ≤ stmod(kΓ). If M is in StMod(kΓ) and is C-local, then by Theorem 7.6, M ↓_{G} has finite projective dimension for every finite subgroup G ≤ Γ. So by Proposition 3.2, M is stmod(kΓ)-local. Now applying Proposition 11.3, we deduce that C = stmod(kΓ).

If M is in StMod(kΓ) then F_{C}(M) is C-local, and so F_{C}(M) ↓_{G} has finite projective dimension for every finite subgroup G ≤ Γ. Applying Theorem 5.7, we deduce that F_{C}(M) has finite projective dimension, so that it is isomorphic to the zero object in StMod(kΓ). We now apply Proposition 11.2 to deduce that M is in C^{⊕}, so that C^{⊕} = StMod(kΓ).

38 D. J. BENSON

Remarks 12.2. (i) As elsewhere in this paper, in the case where k is a field, we may apply this theorem with the alternative definition of StMod(kΓ) using all (not necessarily countably presented) modules M such that B ⊗_{k} M has finite projective dimension. In this case, in the definition of C^{⊕} we use arbitrary (not necessarily countable) direct sums.

(ii) If k is a field of characteristic p and E is a finite elementary abelian p-group then the only simple kE-module is the trivial module k. It follows that k generates the category of all finitely generated kE-modules, and so the only module we need induce in the above theorem is the trivial module. It follows that in this case, stmod(kΓ) is generated by permutation modules on the cosets of finite elementary abelian p-subgroups.

It is worth separating out the various parts of this theorem and stripping them of their categorical trimmings.

Theorem 12.3. Let E be the smallest collection of kΓ-modules satisfying:

(i) If E is a finite elementary abelian subgroup of Γ and N is a finitely generated kEmodule then N ↑^{E} is in E.

(ii) If M_{1} and M_{2} are in E and f : M_{1} → M_{2} is a surjective kΓ-module homomorphism then Ker(f) is in E.

(iii) If M_{1} and M_{2} are in E and f : M_{1} → M_{2} is an injective kΓ-module homomorphism then Coker(f) is in E.

(iv) M_{1} ⊕ M_{2} is in E if and only if M_{1} and M_{2} are in E.

Then E consists precisely of the kΓ-modules of type F P_{∞}.

Proof. Since the modules described in (i) have type F P_{∞}, and the operations described in (ii), (iii) and (iv) preserve the F P_{∞} property, every module in E has type F P_{∞}.

Condition (i) implies that every finitely generated free module is in E, so by Condition (iv), every finitely generated projective module is in E. Therefore by (ii), if M is in E then so is Ω(M). Since k has finite global dimension, we may apply Corollary 5.3 to see that for M in E, some Ω^{r}(M) has a complete resolution which splits on tensoring with B.

Let C be the full subcategory of stmod(kΓ) consisting of modules which are isomorphic in stmod(kΓ) to modules in the collection E. Then the above remarks show that C is thick, so it must be equal to the thick subcategory of stmod(kΓ) generated by the modules induced from finitely generated modules for finite elementary abelian subgroups of Γ. So by Theorem 12.1, C = stmod(kΓ).

Finally, we apply Proposition 8.2 to deduce that for any module M of type F P_{∞}, some Ω^{r}(M) is in E and so M is in E.

COMPLEXITY AND VARIETIES FOR INFINITE GROUPS, II 39

Following an unpublished idea of Carlson for the finite case, we can reword this as follows.

Corollary 12.4. Let M be a kΓ-module of type F P_{∞}. Then there exists an integer r ≥ 0 and a kΓ-module N of type F P_{∞} such that Ω^{r}M ⊕ N (in mod(kΓ)) has a finite filtration

0 = M_{0} ⊆ M_{1} ⊆ ⋯ ⊆ M_{n} = Ω^{r}M ⊕ N

with each M_{i}/M_{i−1} isomorphic (again in mod(kΓ)) to a module induced from a finitely generated module for a finite elementary abelian subgroup of Γ.

Proof. By the theorem, it suffices to show that the full subcategory C of stmod(kΓ) whose objects are the modules satisfying the above condition is thick.

First, by Proposition 8.2, every object isomorphic in stmod(kΓ) to an object in C is itself in C. Second, if M is in C then so are B̄ ⊗_{k} M and Ω(k) ⊗_{k} M, so C is closed under the translation functor τ and its inverse. If 0 → M → M′ → M′′ → 0 is a short exact sequence with M and M′′ in C then M′ is obviously in C. By rotating triangles and using closure under τ and τ^{−1}, this shows that C is triangle closed. It is clearly closed under taking direct summands, so it is thick.

Theorem 12.5. Let E^{⊕} be the smallest collection of kΓ-modules satisfying:

(i) If E is a finite elementary abelian subgroup of Γ and N is a finitely generated kEmodule then N ↑^{Γ} is in E^{⊕}.

(ii) If M_{1} and M_{2} are in E^{⊕} and f : M_{1} → M_{2} is a surjective kΓ-module homomorphism then Ker(f) is in E^{⊕}.

(iii) If M_{1} and M_{2} are in E^{⊕} and f : M_{1} → M_{2} is an injective kΓ-module homomorphism then Coker(f) is in E^{⊕}.

(iv) A finite or countable direct sum ⨁_{i} M_{i} is in E^{⊕} if and only if each M_{i} is in E^{⊕}. Then E^{⊕} consists precisely of the countably presented kΓ-modules M such that B ⊗_{k} M

has finite projective dimension.

Theorem 12.6. Let Γ be an LH𝔉-group and k be a field of coefficients. Let E^{⊕} be the smallest collection of kΓ-modules satisfying:

(i) If E is a finite elementary abelian subgroup of Γ and N is a finitely generated kEmodule then N ↑^{Γ} is in E^{⊕}.

(ii) If M_{1} and M_{2} are in E and f : M_{1} → M_{2} is a surjective kΓ-module homomorphism then Ker(f) is in E^{⊕}.

(iii) If M_{1} and M_{2} are in E^{⊕} and f : M_{1} → M_{2} is an injective kΓ-module homomorphism then Coker(f) is in E^{⊕}.

40 D. J. BENSON

(iv) An arbitrary direct sum ⨁_{i} M_{i} is in E^{⊕} if and only if each M_{i} is in E^{⊕}.

Then E^{⊕} consists precisely of the kΓ-modules M such that B ⊗_{k} M has finite projective dimension.

13. Nilpotence

In this section, we give an application of the theorems of the last section to composites of homomorphisms. This section is based on an unpublished idea of Carlson in the case where Γ is finite. We begin by defining the notion of the level of a module.

Let Γ be an LH𝔉-group and k be a commutative Noetherian ring of coefficients of finite global dimension. We say that a module M in StMod(kΓ) has level zero if it has finite projective dimension. We say that it has level one if it does not have level zero, but it is isomorphic in StMod(kΓ) to a direct summand of a module induced from a finite elementary abelian subgroup of Γ. Inductively, if M fits into a distinguished triangle

M′ → M → M′′ →

where M′ has level one and M′′ has level n − 1, and M does not have level strictly less than n, then we say that it has level n. If M does not have level n for any integer n ≥ 0, we say that M has level ∞.

Corollary 12.4 may be interpreted as giving sufficient conditions for a kΓ-module to have some finite level.

Lemma 13.1. Let Γ ≤ Γ′ be groups, and k be a commutative ring of coefficients. Let M be a kΓ-module and N be a kΓ′-module. If a kΓ-module homomorphism α : N ↑^{Γ}→ M on restriction to Γ′ factors through a projective kΓ′-module then α factors through a projective kΓ-module.

Proof. Denote by i : N → N ↑^{Γ}↓_{Γ′} the natural homomorphism of kΓ′-modules given by i(x) = 1 ⊗ x. Then the image of α under the isomorphism

Hom_{kΓ}(N ↑^{Γ}, M) ≅ Hom_{kΓ′}(N, M ↓_{Γ′})

is the composite

α′ : N ⟶^{i} N ↑^{Γ}↓_{Γ′} ⟶^{α|_{Γ′}} M ↓_{Γ′} .

So if α|_{Γ′} factors through a projective kΓ′-module P then so does α′, and so α factors through P ↑^{Γ}.

Definition 13.2. An element θ ∈ Êxt_{kΓ}^{r}(M, M) is said to be nilpotent if some composite θ ∘ ⋯ ∘ θ ∈ Êxt_{kΓ}^{nr}(M, M) is zero.

COMPLEXITY AND VARIETIES FOR INFINITE GROUPS, II 41

An element ϕ ∈ Êxt_{kΓ}^{r}(M, N) is said to be tensor nilpotent if some tensor power

⨂^{n}(ϕ) ∈ Êxt_{kΓ}^{nr}(⨂^{n} M, ⨂^{n} N)

is zero.

Theorem 13.3. Let Γ be an LH𝔉-group and k a commutative Noetherian ring of coefficients of finite global dimension. Let M be a module in StMod(kΓ) of level n.

(i) Suppose we are given a sequence of kΓ-modules

M = M_{0}, M_{1}, …, M_{n} = N

and elements θ_{i} ∈ Êxt_{kΓ}^{t_{i}}(M_{i}, M_{i+1}) (0 ≤ i < n, t_{i} ∈ ℤ) such that res_{Γ,E}(θ_{i}) is equal to the zero element of Êxt_{kE}^{t_{i}}(M_{i} ↓_{E}, M_{i+1} ↓_{E}) for all finite elementary abelian subgroups E ≤ Γ. Then the Yoneda composite

θ_{n−1} ∘ ⋯ ∘ θ_{1} ∘ θ_{0} ∈ Êxt_{kΓ}^{t_{0}+···+t_{n−1}}(M, N)

is zero.

(ii) If θ ∈ Êxt_{kΓ}^{t}(M, M) restricts to a nilpotent element on each finite elementary abelian subgroup of Γ, then θ is nilpotent.

(iii) If ϕ ∈ Êxt_{kΓ}^{r}(M, N) restricts to a tensor nilpotent element of Êxt_{kE}^{r}(M ↓_{E}, N ↓_{E}) for every finite elementary abelian subgroup E ≤ Γ then ϕ is tensor nilpotent.

Proof. (i) Choose a distinguished triangle

M′ ⟶^{α} M ⟶^{β} M′′ →

such that M′ has level one and M′′ has level n − 1. Choose r ≥ 0 so that Ω^{r}M′′ is a direct summand of a module induced from an elementary abelian subgroup E ≤ Γ, and also so that θ_{0} is represented by a homomorphism θ̂_{0} : Ω^{t_{0}+r}M_{0} → Ω^{r}M_{1} whose restriction to E factors through a projective module. Then by Lemma 13.1, θ̂_{0} ∘ Ω^{t_{0}+r}(α) factors through a projective kΓ-module, and hence θ_{0} ∘ α = 0 in Êxt_{kΓ}^{t_{0}}(M′, M_{1}). It follows from the long exact sequence in the first variable, that there exists an element θ′_{0} ∈ Êxt_{kΓ}^{t_{0}}(M′′, M_{1}) such that θ′_{0} ∘ β = θ_{0}. The map θ_{1} ∘ θ′_{0} restricts to zero in Êxt_{kE}^{t_{0}+t_{1}}(M′′ ↓_{E}, M_{2} ↓_{E}) for every elementary abelian subgroup E ≤ Γ. So by induction on n, the composite

θ_{n−1} ∘ ⋯ ∘ θ_{2} ∘ (θ_{1} ∘ θ′_{0}) ∈ Êxt_{kΓ}^{t_{0}+···+t_{n−1}}(M′′, N)

is zero. Composing with β, we see that θ_{n−1} ∘ ⋯ ∘ θ_{0} is zero in Êxt_{kΓ}^{∗}(M, N).

(ii) In the above analysis, it was only necessary to assume that θ_{i} restricted to zero on the finite elementary abelian subgroups from which modules were induced to build up M. There are only a finite number of such subgroups. So if θ ∈ Êxt_{kΓ}^{t}(M, M) is nilpotent on

42 D. J. BENSON

restriction to each of these subgroups, then we can replace θ by a power of θ to make it zero on each such subgroup. So it follows from (i) that θ is nilpotent.

(iii) Replace ϕ by a tensor power of ϕ to make it zero on restriction to each of the finite elementary abelian subgroups from which modules were induced to build up M. Then ⨂^{n}(ϕ) can be written as a composite

⨂^{n}(ϕ) = (ϕ ⊗ ⨂^{n−1}(Id_{M})) ∘ (Id_{N} ⊗ ϕ ⊗ ⨂^{n−2}(Id_{M})) ∘ ⋯ ∘ (⨂^{n−1}(Id_{N}) ⊗ ϕ).

Applying (i), we find that ⨂^{n}(ϕ) = 0.

Corollary 13.4. Let Γ be an LH𝔉-group, k be a commutative Noetherian ring of coefficients of finite global dimension, and M be a kΓ-module of type F P_{∞}. Then there exists an integer n such that the conclusion of Theorem 13.3 holds.

Proof. This follows from Theorem 13.3 and Corollary 12.4.

14. Complexity

The following theorem, together with Kropholler’s theorem (3.3), provide the motivation for our definition of complexity for a module of type F P_{∞} for an LH𝔉-group. We begin by remarking that we use the definition of complexity for an infinitely generated module for a finite group given by Benson, Carlson and Rickard [5].

Theorem 14.1. Let Γ be an LH𝔉-group and k be a field of coefficients, and let M be a kΓmodule of type F P_{∞}. Then there is an upper bound on the complexities of the restrictions of M to finite subgroups of Γ.

Proof. Let C_{n} be the full subcategory of stmod(kΓ) consisting of the modules M such that the restriction to each finite subgroup has complexity at most n. Let C_{∞} be the full subcategory consisting of the modules which are in C_{n} for some n ≥ 0. Then C_{n} and C_{∞} are thick subcategories of stmod(kΓ). By Theorem 12.1, stmod(kΓ) is generated by modules which are in some C_{n}. So we have C_{∞} = stmod(kΓ).

Definition 14.2. Let Γ be an LH𝔉-group and k be a field of coefficients, and let M be a kΓ-module of type F P_{∞}. We define the complexity c_{Γ}(M) to be the maximum complexity of a restriction M ↓_{G} to a finite subgroup G of Γ.

Proposition 14.3. Let Γ be an LH𝔉-group and k be a field of coefficients. Then a kΓmodule M has complexity zero if and only if it has finite projective dimension.

Proof. This follows from Kropholler’s theorem 3.3, and Corollary 5.6 of [5].

The following is the analog of the Alperin–Evens theorem [1] in this context.

Theorem 14.4. Let Γ be an LH𝔉-group and k be a field of coefficients, and let M be a kΓ-module of type F P_{∞}. Then

c_{Γ}(M) = max_{E≤Γ} c_{E}(M ↓_{E}).

Here, E runs over the finite elementary abelian p-subgroups of Γ.

Proof. This follows immediately from Theorem 9.2 of [5].

15. Varieties for modules

If G is a finite group and k is an algebraically closed field of characteristic p, then a theorem of Quillen [12] describes the maximal ideal spectrum of the cohomology ring. Writing V_{G} for this variety, Quillen’s theorem states that at the level of topological spaces, V_{G} is the colimit of the varieties V_{E} as E runs over the category A_{G} whose objects are the elementary abelian p-subgroups of G and whose morphisms are generated by the conjugations and inclusions. Furthermore, V_{E} is related to the rank variety V_{E}^{r} = J(kE)/J^{2}(kE) by a Frobenius twist, which is again an isomorphism at the level of topological spaces.

For a finitely generated kG-module M, there is an associated closed homogeneous subvariety V_{G}(M) whose dimension is equal to the complexity of M. Expositions of this theory can be found in Benson [3] and Evens [10]. For infinitely generated modules, a corresponding theory was recently developed by Benson, Carlson and Rickard [6]. Instead of a single variety, associated to an infinitely generated module is a collection V_{G}(M) of closed homogeneous irreducible subvarieties of V_{G}. Again, the maximum dimension of an element of V_{G}(M) is equal to the complexity of M.

In this section, we develop the corresponding theory for an LH𝔉-group Γ. Our definition of the ambient rank variety V_{Γ}^{r} is somewhat ad hoc. We define it as a colimit over the Quillen category. We associate to each module M in Mod(kΓ) a collection V_{Γ}^{r}(M) of closed homogeneous irreducible subvarieties of V_{Γ}. If M is in mod(kΓ), there is an upper bound to the dimensions of the varieties in V_{Γ}^{r}(M). In fact, in this case V_{Γ}^{r}(M) is supported on some finite set of finite elementary abelian p-subgroups. The largest dimension of an element of V_{Γ}^{r}(M) is equal to the complexity of M.

Let Γ be an LH𝔉-group and k be an algebraically closed field of characteristic p. For the purpose of this section, we use the version of Mod(kΓ) in which we take all (not necessarily countably presented) modules M such that B ⊗_{k} M has finite projective dimension. The

44 D. J. BENSON

advantage of this is that if M is in Mod(kΓ) and N is any kΓ-module then M ⊗_{k} N is in Mod(kΓ).

We define the Quillen category A_{p}(Γ) to have as its objects the finite elementary abelian p-subgroups E ≤ Γ. The arrows from E to E′ are the (injective) group homomorphisms induced by conjugations in Γ.

If E is a finite elementary abelian p-group, we write J(kE) for the radical of the group algebra, and we set V_{E}^{r} = J(kE)/J^{2}(kE). This is an affine space whose dimension is equal to the p-rank of E, topologized with the Zariski topology. An injective group homomorphism ρ : E → E′ induces an injective map of affine spaces ρ_{∗} : V_{E}^{r} → V_{E′}^{r}.

We define

V_{Γ}^{r} = lim_{\overrightarrow{E∈A_{p}(Γ)}} V_{E}^{r},

as a topological space. In other words, a point in V_{Γ}^{r} is an equivalence class of pairs consisting of a finite elementary abelian p-subgroup E and a point x ∈ V_{E}^{r}. The equivalence relation is generated by setting (E, x) ∼ (E′, x′) when there is a morphism ρ : E → E′ in A_{p}(Γ) with ρ_{∗}(x) = x′.

We topologize V_{Γ}^{r} as follows. For each finite elementary abelian p-subgroup E ≤ Γ, we have a map V_{E}^{r} → V_{Γ}^{r}. A subset of V_{Γ}^{r} is open (respectively closed) if and only if its preimage in each V_{E}^{r} is open (respectively closed). In general, there need be no upper bound to the ranks of the finite elementary abelian p-subgroups of Γ, and so V_{Γ}^{r} is not necessarily a variety in the usual sense. But since the image of each V_{E}^{r} → V_{Γ}^{r} can be viewed as an affine variety, we can view V_{Γ}^{r} as an ind-object (this concept is dual to proobjects as described in the Appendix to Artin–Mazur [2]) in the category of affine varieties over k.

If M is a module in Mod(kΓ), then for each finite elementary abelian p-subgroup E of Γ we have a collection V_{E}^{r}(M ↓_{E}) of closed subvarieties of V_{E}^{r} as defined in [6]. If ρ : E → E′ is a morphism in A_{Γ} then the image of V_{E}^{r}(M ↓_{E}) is equal to the set of those elements of V_{E′}^{r}(M ↓_{E′}) which lie in the image of ρ_{∗} : V_{E}^{r} → V_{E′}^{r}. So we define V_{Γ}^{r}(M) to be the union over all finite elementary abelian p-subgroups E ≤ Γ of the collection of closed homogeneous subsets obtained by taking the image of V_{E}^{r}(M ↓_{E}) under the map V_{E}^{r} → V_{Γ}^{r}.

For a finite elementary abelian p-group E and a kE-module N, we have V_{E}^{r}(N) = V_{E}^{r}(Ω(N)). So it follows from Proposition 8.2 that if M and M′ are isomorphic in StMod(kΓ) then V_{Γ}^{r}(M) = V_{Γ}^{r}(M′).

COMPLEXITY AND VARIETIES FOR INFINITE GROUPS, II 45

Theorem 15.1. Let Γ be an LH𝔉-group and k be an algebraically closed field of characteristic p. For a module M in Mod(kΓ), V_{Γ}^{r}(M) = ∅ if and only if M has finite projective dimension.

Proof. If M has finite projective dimension then M is isomorphic to the zero module in StMod(kΓ), and so V_{Γ}^{r}(M) = ∅. Conversely, if V_{Γ}^{r}(M) = ∅ then the restriction of M to every finite elementary abelian p-subgroup E ≤ Γ is projective by Corollary 5.6 of [6]. By Chouinard’s theorem [9], the restriction of M to every finite subgroup G ≤ Γ is projective. So by Theorem 5.7, M has finite projective dimension.

Theorem 15.2. Let Γ be an LH𝔉-group and k be an algebraically closed field of characteristic p. For a module M in mod(kΓ), V_{Γ}^{r}(M) = ∅ if and only if M has finite projective dimension.

Proof. This follows in the same way using Theorem 3.3 instead of Theorem 5.7.

Theorem 15.3. Let Γ be an LH𝔉-group and k be an algebraically closed field of characteristic p. For a kΓ-module M of type F P_{∞}, the maximum dimension of an element of V_{Γ}^{r}(M) is equal to c_{Γ}(M).

Proof. Since V_{Γ}^{r}(M) is defined in terms of restrictions to finite elementary abelian psubgroups, by Theorem 14.4 it suffices to prove this statement for Γ = E a finite elementary abelian p-group. In this context, the result belongs to [6] but does not in fact appear there. Corollary 7.7 of [5] does not suffice in this context, because this uses the definition of variety of a module made obsolete by [6]. However, it is not hard to deduce the statement we want from Proposition 9.1 of [5].

Lemma 15.4. Let Γ be an LH𝔉-group and k be an algebraically closed field of characteristic p. If ρ : E → Γ is the inclusion of a finite elementary abelian p-subgroup E ≤ Γ, then for any kE-module M we have

V_{Γ}^{r}(M ↑^{Γ}) = ρ_{∗}(V_{E}^{r}(M)).

Proof. This follows easily from the definitions and the Mackey decomposition theorem.

Lemma 15.5. Let Γ be an LH𝔉-group and k be an algebraically closed field of characteristic p. If M_{α}, α ∈ I is a collection of kΓ-modules then

V_{Γ}^{r}(⨁_{α∈I} M_{α}) = ⋃_{α∈I} V_{Γ}^{r}(M_{α}).

46 D. J. BENSON

Proof. Restricting to a finite elementary abelian p-subgroup E ≤ Γ, then extending the field and restricting to a cyclic shifted subgroup, the direct sum is projective if and only if all the summands are projective.

Proposition 15.6. Let Γ be an LH𝔉-group and k be an algebraically closed field of characteristic p. Given any collection V of nonempty closed homogeneous irreducible subvarieties V ⊆ V_{Γ}^{r} (each contained in the image of some ρ_{∗} : V_{E}^{r} → V_{Γ}^{r} where ρ : E → Γ is the inclusion of a finite elementary abelian p-subgroup E ≤ Γ), there exists a module M in Mod(kΓ) with

V_{Γ}^{r}(M) = V.

Proof. First suppose that V = {V } with V in the image of ρ_{∗} : E → Γ. Let V = ρ_{∗}(V_{0}) with V_{0} a nonempty closed homogeneous irreducible subvariety of V_{E}^{r}. Let V_{E} be the maximal ideal spectrum of H^{∗}(E, k), and let β^{∗} : V_{E} → V_{E}^{r} be the semilinear isogeny of affine varieties induced by the Bockstein homomorphism (cf. section 6 of [6]). Then V_{1} = (β^{∗})^{−1}(V_{0}) is a nonempty closed homogeneous irreducible subvariety of V_{E}, and we set M = κ(V_{1}) ↑^{Γ}, where κ(V_{1}) is the module defined in section 8 of [6]. By corollary 9.2 of [6], we have V_{E}^{r}(κ(V_{1})) = {V_{0}}, and so by Lemma 15.4 we have V_{Γ}^{r}(M) = {V }.

For an arbitrary collection V, we take the direct sum of the corresponding collection of modules produced as above, and use Lemma 15.5.

Lemma 15.7. Let Γ be an LH𝔉-group and k be an algebraically closed field of characteristic p. If

M_{1} → M_{2} → M_{3} →

is a distinguished triangle in StMod(kΓ), then for {i, j, k} = {1, 2, 3}, we have

V_{Γ}^{r}(M_{i}) ⊆ V_{Γ}^{r}(M_{j}) ∪ V_{Γ}^{r}(M_{k}).

In particular, the two largest of c_{Γ}(M_{1}), c_{Γ}(M_{2}) and c_{Γ}(M_{3}) are equal.

Proof. Restricting the triangle to a finite elementary abelian p-subgroup E ≤ Γ, then extending the field and restricting to a cyclic shifted subgroup, either all three modules are projective or at most one of them is.

Theorem 15.8. Let Γ be an LH𝔉-group and k be an algebraically closed field of characteristic p. If M is a kΓ-module of type F P_{∞} then there is a finite collection A of finite elementary abelian p-subgroups E ≤ Γ such that each element of V_{Γ}^{r}(M) is in the image of V_{E}^{r} → V_{Γ}^{r} for some E ∈ A.

COMPLEXITY AND VARIETIES FOR INFINITE GROUPS, II 47

Proof. For any collection A of finite elementary abelian p-subgroups E ≤ Γ, we denote by C_{A} the thick subcategory of stmod(kΓ) generated by modules induced from finitely generated modules for subgroups E ∈ A. It follows from Theorem 12.1 that stmod(kΓ) is generated by modules each of which lies in C_{A} with A = {E} for some finite elementary abelian p-subgroup E ≤ Γ. So every module of type F P_{∞} lies in C_{A} for some finite collection A. The theorem now follows by applying Lemmas 15.4 and 15.7.

Theorem 15.9. Let Γ be an LH𝔉-group and k be an algebraically closed field of characteristic p. If M and N are kΓ-modules in Mod(kΓ) then

V_{Γ}^{r}(M ⊗_{k} N) = V_{Γ}^{r}(M) ∩ V_{Γ}^{r}(N).

Proof. First, we note that Mod(kΓ) is closed under taking tensor products (in fact, it is even closed under tensor products with arbitrary kΓ-modules), and the theorem makes sense. Since V_{Γ}^{r}(M) is defined in terms of restrictions to finite elementary abelian psubgroups E of Γ, the theorem follows from Corollary 7.5 of [6].

16. The thick subcategories C(V)

Throughout this section, let Γ be an LH𝔉-group and k an algebraically closed field of characteristic p. Let V be a collection of nonempty closed homogeneous irreducible subvarieties of V_{Γ}^{r}. Then we write C(V) for the full subcategory of stmod(kΓ) whose objects are the kΓ-modules M satisfying V_{Γ}^{r}(M) ⊆ V. It is easy to see, using Lemma 15.7, that C(V) is a thick subcategory of stmod(kΓ).

For example, we denote by V_{c} the collection of all nonempty closed homogeneous irreducible subvarieties of V_{Γ}^{r} of dimension at most c. Then by Theorem 15.3, C(V_{c}) is exactly the thick subcategory of kΓ-modules in stmod(kΓ) which have complexity at most c.

Theorem 16.1. Suppose that V is a collection of nonempty closed homogeneous irreducible subvarieties of V_{Γ}^{r}, and that V is closed under specialization (i.e., if V ∈ V and W ⊆ V then W ∈ V).

(i) If ρ : E → Γ is the inclusion of a finite elementary abelian p-subgroup E ≤ Γ, and N is a finitely generated kE-module with

ρ_{∗}(V_{E}^{r}(N)) ∈ V

then N ↑^{Γ} is in C(V).

(ii) The thick subcategory of stmod(kΓ) generated by these induced modules is equal to C(V).

(iii) The following are equivalent for a module M in StMod(kΓ):

48 D. J. BENSON

(a) M is C(V)-local.

(b) V_{Γ}^{r}(M) ∩ V = ∅.

(c) Êxt_{kΓ}^{0}(N, M ↓_{E}) = 0 for each module N described in (i).

(d) Êxt_{kΓ}^{0}(N ↑^{Γ}, M) = 0 for each module N ↑^{Γ} described in (i).

(iv) The subcategory C(V)^{⊕} consists precisely of those modules M in StMod(kΓ) satisfying V_{Γ}^{r}(M) ⊆ V.

Proof. Since V is closed under specialization, part (i) follows from Lemma 15.4 and the fact that for a finitely generated kE-module M, V_{E}^{r}(M) is equal to the collection of nonempty closed homogeneous irreducible subvarieties of V_{E}^{r}(M). We prove parts (ii) and (iii) simultaneously as follows. Let C′(V) be the thick subcategory of stmod(kΓ) generated by these induced modules. By (i), we have C′(V) ⊆ C(V). We add a fifth condition to (iii):

(e) M is C′(V)-local. If we can prove that (a)–(e) are equivalent, then by Proposition 11.3 the equivalence of (a) and (e) proves (ii).

The implication (a) ⇒ (e) follows from the fact that C′(V) ≤ C(V).

The implication (e) ⇒ (d) follows from the definition of C′(V)-local.

The implication (d) ⇒ (c) follows from Corollary 5.11.

To prove the implication (c) ⇒ (b), suppose that V ∈ V_{Γ}^{r}(M)∩V. Since V is irreducible, it is in the image of ρ_{∗} : V_{E}^{r} → V_{Γ}^{r} for some finite elementary abelian p-subgroup E ≤ Γ. So we can write V = ρ_{∗}(V_{0}) for some nonempty closed homogeneous irreducible subvariety V_{0} ∈ V_{E}^{r}(M ↓_{E}). Write V_{0} = β^{∗}(V_{1}) for V_{1} ⊆ V_{E}(k) (see Section 6 of [6]), and let E(V_{1}) and F (V_{1}) be the idempotent modules described in Section 6 of [15] and Section 8 of [6]. Then there is a distinguished triangle

E(V_{1}) ⊗_{k} M ↓_{E}→ M ↓_{E}→ F (V_{1}) ⊗_{k} M ↓_{E}→ .

Since V_{E}^{r}(F (V_{1})) = {V ∈ V_{E}^{r} | V ⊈ V_{0}} (Corollary 9.2 of [6]), we have (using Corollary 7.5 of [6])

V_{E}^{r}(M ↓_{E}) ⊈ V_{E}^{r}(F (V_{1}) ⊗_{k} M ↓_{E}).

It follows that the map E(V_{1}) ⊗_{k} M → M does not factor through a projective (since otherwise M ↓_{E} would be isomorphic to a direct summand of F (V_{1}) ⊗_{k} M ↓_{E}), and so Êxt_{kE}^{0}(E(V_{1}) ⊗_{k} M ↓_{E}, M ↓_{E}) ≠ 0. Furthermore, E(V_{1}) ⊗_{k} M ↓_{E} is a filtered colimit of finitely generated modules whose rank variety is contained in V_{0}. So some finitely generated module N in that filtered system satisfies Êxt_{kE}^{0}(N, M ↓_{E}) ≠ 0.

Finally, we prove the implication (b) ⇒ (a). Since E_{C(V)}(M) is in C(V)^{⊕}, we have V_{Γ}^{r}(E_{C(V)}(M)) ⊆ V. Since we already know that (a) ⇒ (b), and F_{C(V)}(M) is C(V)-local,

COMPLEXITY AND VARIETIES FOR INFINITE GROUPS, II 49

we have V_{Γ}^{r}(F_{C(V)}(M)) ∩ V = ∅. Thus by Theorem 15.1, E_{C(V)}(M) is isomorphic to the zero object in StMod(kΓ). So by Proposition 11.2 (ii), M is C(V)-local.

To prove (iv), if M is in C(V)^{⊕} then clearly V_{Γ}^{r}(M) ⊆ V. Conversely, suppose that V_{Γ}^{r}(M) ⊆ V. Then by construction, F_{C(V)}(M) is a filtered colimit of modules M_{α} with V_{Γ}^{r}(M_{α}) ⊆ V, and so V_{Γ}^{r}(F_{C(V)}(M)) ⊆ V. On the other hand, F_{C(V)}(M) is C(V)-local, so by (iii) we have V_{Γ}^{r}(F_{C(V)}(M)) ∩ V = ∅. Thus V_{Γ}^{r}(F_{C(V)}(M)) = ∅, and so by Theorem 15.1, F_{C(V)}(M) has finite projective dimension, or in other words it is isomorphic to the zero module in StMod(kΓ). Applying Proposition 11.2 (i), we deduce that M is in C(V)^{⊕}.

Theorem 16.2. Suppose that V is a collection of nonempty closed homogeneous irreducible subvarieties of V_{Γ}^{r}, and that V is closed under specialization. Let E be the smallest collection of kΓ-modules satisfying:

(i) If E is a finite elementary abelian p-subgroup of Γ, V is a closed homogeneous irreducible subvariety of V_{E}^{r} such that ρ_{∗}(V) ∈ V (where ρ : E → Γ is the inclusion) and N is a finitely generated kE-module such that V_{E}^{r}(N) = V, then N ↑^{Γ} is in E.

(ii) If M_{1} and M_{2} are in E and f : M_{1} → M_{2} is a surjective kΓ-module homomorphism then Ker(f) is in E.

(iii) If M_{1} and M_{2} are in E and f : M_{1} → M_{2} is an injective kΓ-module homomorphism then Coker(f) is in E.

(iv) M_{1} ⊕ M_{2} is in E if and only if both M_{1} and M_{2} are in E.

Then E consists precisely of the kΓ-modules M of type F P_{∞} such that V_{Γ}^{r}(M) ⊆ V.

Proof. This follows from Theorem 16.1 in the same way as Theorem 12.3 follows from Theorem 12.1.

This allows us to formulate the following strengthening of Corollary 12.4.

Corollary 16.3. Let M be a kΓ-module of type F P_{∞}. Then there exists an integer r ≥ 0 and a kΓ-module N of type F P_{∞} such that Ω^{r}M ⊕ N (in mod(kΓ)) has a finite filtration

0 = M_{0} ⊆ M_{1} ⊆ ⋯ ⊆ M_{n} = Ω^{r}M ⊕ N

with each M_{i}/M_{i−1} isomorphic (again in mod(kΓ)) to a module N ↑^{Γ} induced from a finitely generated module N for a finite elementary abelian p-subgroup of Γ, such that ρ_{∗}(V_{E}^{r}(N)) ∈ V_{E}^{r}(M) (where ρ : E → Γ is the inclusion).

Theorem 16.4. Suppose that V is a collection of nonempty closed homogeneous irreducible subvarieties of V_{Γ}^{r}, and that V is closed under specialization. Let E^{⊕} be the smallest collection of kΓ-modules satisfying:

50 D. J. BENSON

(i) If E is a finite elementary abelian subgroup of Γ, W is a collection of closed homogeneous irreducible subvarieties of V_{E}^{r} such that ρ_{∗}(W) ⊆ V (where ρ : E → Γ is the inclusion), and N is a finitely generated kE-module such that V_{E}^{r}(N) = W, then N ↑^{E} is in E^{⊕}.

(ii) If M_{1} and M_{2} are in E^{⊕} and f : M_{1} → M_{2} is a surjective kΓ-module homomorphism then Ker(f) is in E^{⊕}.

(iii) If M_{1} and M_{2} are in E^{⊕} and f : M_{1} → M_{2} is an injective kΓ-module homomorphism then Coker(f) is in E^{⊕}.

(iv) An arbitrary direct sum ⨁_{i} M_{i} is in E^{⊕} if and only if each M_{i} is in E^{⊕}.

Then E^{⊕} consists precisely of the kΓ-modules M such that B ⊗_{k} M has finite projective dimension, and such that V_{Γ}^{r}(M) ⊆ V.

Corollary 16.5. The category C(V_{c}) of modules of complexity at most c is equal to the thick subcategory of stmod(kΓ) generated by the modules of the form N ↑^{Γ}, as N runs over the finitely generated kE-modules of complexity at most c and E runs over the elementary abelian p-subgroups of Γ.

Proof. This follows by applying part (i) of Theorem 16.1 to the collection V = V_{c}.

An interesting consequence of this corollary concerns a possible generalization of the original definition of complexity for finitely generated modules for finite groups. If M is a kΓ-module of type F P_{∞}, write α_{n}(M) for the minimum number of generators for any candidate for Ω^{n}(M) as a kΓ-module. It follows from Corollary 16.5 that α_{n}(M) is bounded by a polynomial in n of degree one less than the complexity c_{Γ}(M). For finite groups this is known to be the best possible bound. However, even for finitely generated abelian groups, this turns out to be false. An example can be found in the next section.

There remains the question of whether the Poincaré series

p_{M}(t) = ∑_{n=0}^{∞} α_{n}(M)t^{n}

is the power series expansion of a rational function of t, as is the case when Γ is finite. The best we can say is that if M does not have finite projective dimension then α_{n}(M) is sandwiched between 1 and some positive multiple of n^{c−1}, and so

lim sup_{n→∞} α_{n}(M)^{1/n} = 1

so that p_{M}(t) defines an analytic function of t around t = 0 with radius of convergence equal to one.

COMPLEXITY AND VARIETIES FOR INFINITE GROUPS, II 51

Theorem 16.6. Suppose that V is a collection of nonempty closed homogeneous irreducible subvarieties of V_{Γ}^{r}, and that V is closed under specialization. Suppose that M and N are kΓ-modules such that B ⊗_{k} M and B ⊗_{k} N have finite projective dimension, and that V_{Γ}^{r}(M) ⊆ V and V_{Γ}^{r}(N) ∩ V = ∅. Then Êxt_{kΓ}^{i}(M, N) = 0 for all i ∈ ℤ.

Proof. If M is induced from a finitely generated module M_{0} for a finite elementary abelian p-subgroup E ≤ Γ with ρ_{∗}(V_{E}^{r}(M_{0})) ∈ V (ρ : E → Γ the inclusion) then using Corollary 5.11 (ii) we have

Êxt_{kΓ}^{i}(M, N) ≅ Êxt_{kΓ}^{i}(M_{0} ↑^{Γ}, N)
≅ Êxt_{kE}^{i}(M_{0}, N ↓_{E})
≅ Ĥ^{i}(E, Hom_{k}(M_{0}, N ↓_{E}))
≅ Ĥ^{i}(E, Hom_{k}(M_{0}, k) ⊗_{k} N ↓_{E}).

Since

V_{E}^{r}(Hom_{k}(M_{0}, k) ⊗_{k} N ↓_{E}) = V_{E}^{r}(Hom_{k}(M_{0}, k)) ∩ V_{E}^{r}(N ↓_{E})
= V_{E}^{r}(M_{0}) ∩ V_{E}^{r}(N ↓_{E}) = ∅,

Hom_{k}(M_{0}, k) ⊗_{k} N ↓_{E} is projective by Corollary 7.5 of [6] and so Êxt_{kΓ}^{i}(M, N) = 0.

17. Connectedness

In this section, we generalize to the context of LH𝔉-groups a theorem of Carlson [8] for finite groups. Namely, he proves in that paper that the variety V_{G}(M) of a finitely generated indecomposable module M for a finite group G over an algebraically closed field is (projectively) connected. For an LH𝔉-group Γ, it is not enough for the module M just to be indecomposable. It must be stably indecomposable in the sense that it does not decompose nontrivially in StMod(kΓ). By Proposition 8.2, this is the same as saying that no Ω^{n}(M) decomposes in Mod(kΓ) as a direct sum of two nonprojective modules. A more precise statement is given in the following theorem.

Theorem 17.1. Let Γ be an LH𝔉-group, and let k be an algebraically closed field of coefficients of characteristic p. Let V_{1} and V_{2} be closed homogeneous subvarieties of V_{Γ}^{r} satisfying V_{1} ∩ V_{2} = {0}. Suppose that M is a kΓ-module such that B ⊗_{k} M has finite projective dimension, and such that every element of V_{Γ}^{r}(M) is in either V_{1} or V_{2}. Then M is isomorphic in StMod(kΓ) to a direct sum M_{1} ⊕ M_{2} in such a way that every element of V_{Γ}^{r}(M_{1}) lies in V_{1} and every element of V_{Γ}^{r}(M_{2}) lies in V_{2}. Equivalently, for some n ≥ 0, Ω^{n}(M) plus a projective module has such a direct sum decomposition in Mod(kΓ).

52 D. J. BENSON

Proof. Let V_{1} be the collection of closed homogeneous irreducible subvarieties of V_{1}, V_{2} that of V_{2}, and V = V_{1} ∪ V_{2}. Let E_{1}^{⊕}, E_{2}^{⊕} and E^{⊕} be the corresponding collections of kΓ-modules, as described in Theorem 16.4. Let Ẽ^{⊕} be the collection of modules of the form M_{1} ⊕ M_{2} with M_{1} in E_{1}^{⊕} and M_{2} in E_{2}^{⊕}. It is clear that Ẽ^{⊕} ⊆ E^{⊕}, and we must prove equality. This amounts to showing that Ẽ^{⊕} satisfies conditions (i)–(iv) of Theorem 16.4 with respect to the collection V.

To prove that condition (i) is satisfied by Ẽ^{⊕} amounts to proving the theorem in the case of a finite elementary abelian p-group. In this case, the theorem follows from Corollary 8.2 of Rickard [15].

To prove that condition (ii) holds, let f : M_{1} ⊕ M_{2} → M′_{1} ⊕ M′_{2} be a surjective homomorphism, where V_{Γ}^{r}(M_{i}) ⊆ V_{i} and V_{Γ}^{r}(M′_{i}) ⊆ V_{i} for i = 1, 2. Then Ker(f) fits into a short exact sequence of the form

0 → Ω(M′_{1}) ⊕ Ω(M′_{2}) → Ker(f) ⊕ (projective) → M_{1} ⊕ M_{2} → 0.

By Theorem 16.6, we have

Êxt_{kΓ}^{1}(M_{1}, Ω(M′_{2})) = Êxt_{kΓ}^{1}(M_{2}, Ω(M′_{1})) = 0.

So for some n ≥ 0 we have

Ext_{kΓ}^{1}(Ω^{n}(M_{1}), Ω^{n+1}(M′_{2})) = Ext_{kΓ}^{1}(Ω^{n}(M_{2}), Ω^{n+1}(M′_{1})) = 0.

It follows that the corresponding translate of the sequence

0 → Ω^{n+1}(M′_{1}) ⊕ Ω^{n+1}(M′_{2}) → Ω^{n}(Ker(f)) ⊕ (projective) → Ω^{n}(M_{1}) ⊕ Ω^{n}(M_{2}) → 0

splits as a direct sum of an extension of Ω^{n}(M_{1}) by Ω^{n+1}(M′_{1}) and an extension of Ω^{n}(M_{2}) by Ω^{n+1}(M′_{2}). Thus Ω^{n}(Ker(f)) plus a projective module splits as a direct sum in the desired manner.

The proof that conditions (iii) and (iv) hold are similar.

18. Examples

We begin with an example which serves two purposes. First, it illustrates the necessity for the full force of the definition of varieties for infinitely generated modules for finite groups given in [6], when defining V_{Γ}^{r}(M) in Section 15. Second, it shows that a periodic module of type F P_{∞} for an LH𝔉-group can have complexity two, a phenomenon which does not occur in the finite case.

Let Γ = ℤ/2×ℤ/2×ℤ, generated by commuting elements g_{1} and g_{2} of order two and g_{3} of infinite order. Then there is exactly one maximal finite elementary abelian subgroup, namely E = ⟨g_{1}, g_{2}⟩. Let k be an algebraically closed field of characteristic two, so that

COMPLEXITY AND VARIETIES FOR INFINITE GROUPS, II 53

V_{Γ}^{r} ≅ V_{E}^{r} ≅ 𝔸^{2}(k). Let M be the kΓ-module with k-basis elements x_{i} and y_{i} (i ∈ ℤ), and with Γ-action given by

g_{1}(x_{i}) = x_{i} + y_{i}  g_{1}(y_{i}) = y_{i}
g_{2}(x_{i}) = x_{i} + y_{i+1}  g_{2}(y_{i}) = y_{i}
g_{3}(x_{i}) = x_{i+1}  g_{3}(y_{i}) = y_{i+1}.

Then M is generated as a kΓ-module by the basis element x_{0}. Since Γ is polycyclic by finite, it follows that kΓ is Noetherian, and M has type F P_{∞} as a kΓ-module.

Let K be an algebraically closed extension of k of transcendence degree at least two. For λ_{1}, λ_{2} ∈ K (not both zero), set α = λ_{1}(g_{1} − 1) + λ_{2}(g_{2} − 1) ∈ KE. Then α(x_{i}) = λ_{1}y_{i} + λ_{2}y_{i+1} while α(y_{i}) = 0. If λ_{1} and λ_{2} are both nonzero, then the restriction of M to the shifted subgroup ⟨1 + α⟩ ⊆ KE ⊆ KΓ has a free submodule of codimension one, spanned by the x_{i} and the elements ∑_{i} γ_{i}y_{i} with ∑_{i}(λ_{1}/λ_{2})^{i}γ_{i} = 0. Furthermore, (K ⊗_{k} M) ↓_{⟨1+α⟩} is a direct sum of this submodule with the one dimensional submodule spanned by y_{0}. So (K ⊗_{k} M) ↓_{⟨1+α⟩} is not free. On the other hand, if either λ_{1} or λ_{2} is zero then (K ⊗_{k} M)↓_{⟨1+α⟩} is free.

It follows that V_{Γ}^{r}(M) is equal to the collection of all closed homogeneous irreducible subvarieties of V_{Γ}^{r} ≅ V_{E}^{r} ≅ 𝔸^{2}(k) except for the two lines through the origin corresponding to the two coordinate axes. This is not equal to V_{E}^{r}(N) for any finitely generated kEmodule N.

To see that M is periodic of period one, we notice that using x_{0} as a generator for M, we have

M ≅ kΓ/(g_{3}(g_{1} − 1) − (g_{2} − 1)).

The ideal generated by g_{3}(g_{1} − 1) − (g_{2} − 1) in kΓ has a k-basis consisting of the elements x′_{i} = g_{3}^{i+1}(g_{1} − 1) − g_{3}^{i}(g_{2} − 1) and y′_{i} = g_{3}^{i}(g_{1} − 1)(g_{2} − 1). These elements are acted on by Γ in exactly the same way as the basis elements x_{i} and y_{i} of M. So we have a short exact sequence

0 → M → kΓ → M → 0.

It follows that M has a free resolution of the following form:

⋯ → kΓ → kΓ → kΓ → M → 0.

In fact, if N is any module in Mod(kΓ) with V_{Γ}^{r}(N) equal to the collection of all closed homogeneous subvarieties of V_{Γ}^{r} except for some nonempty collection of lines through the origin then N is eventually periodic of period one. To see this, let 0 ≠ ζ ∈ H^{1}(Γ, k) be an

54 D. J. BENSON

element such that the line V_{Γ}^{r}(ζ) ⊆ V_{Γ}^{r} is not in V_{Γ}^{r}(N). Then ζ is represented by a short exact sequence

0 → k → L → k → 0,

and we have V_{Γ}^{r}(L) = {V_{Γ}^{r}(ζ)}. Therefore, using Theorem 15.9, we have V_{Γ}^{r}(L ⊗_{k} N) = ∅ and so by Theorem 15.2 L ⊗_{k} N has finite projective dimension. So for some r ≥ 0, L ⊗_{k} Ω^{r}N is projective. The short exact sequence

0 → Ω^{r}N → L ⊗_{k} Ω^{r}N → Ω^{r}N → 0

then shows that Ω^{r+1}N ≅ Ω^{r}N.

Our second example illustrates that Theorem 15.8 has some content. Let Γ be a semidirect product E⋊ℤ where E is an infinite elementary abelian group (i.e., an 𝔽_{p}-vector space) with basis elements g_{i}, i ∈ ℤ. The complementary copy of ℤ is generated by an element h which acts via hg_{i}h^{−1} = g_{i+1}. Let k be an algebraically closed field of characteristic p.

Since Γ is finitely generated and solvable, it is in H𝔉. Every finite elementary abelian p-subgroup of Γ lies in E, and so V_{Γ}^{r} ≅ V_{E}^{r}/⟨h⟩. Here, V_{E}^{r} is an infinite dimensional affine space over k, and ⟨h⟩ ≅ ℤ permutes the basis elements simply transitively.

Given any finite collection A of finite elementary abelian p-subgroups of Γ, there exists a finite subgroup E_{0} ≤ E containing every element of A. Therefore, it follows from Theorem 15.8 that given any kΓ-module of type F P_{∞}, there exists a finite elementary abelian subgroup E_{0} ≤ E such that every element of V_{Γ}^{r}(M) is contained in the image of V_{E_{0}}^{r} → V_{Γ}^{r}. It seems unclear whether every kΓ-module of type F P_{∞} is isomorphic to a module induced from some finite subgroup of Γ.

19. An application

The following theorem was proved by H.-W. Henn (unpublished) under the assumption that Γ is in H_{1}𝔉, using a modification of the techniques of Quillen [12]. Kropholler and Mislin have proved that an LH𝔉-group of type F P_{∞} is necessarily in H_{1}𝔉, so in fact the theorem is not new, but the proof certainly is.

Theorem 19.1. Let Γ be an LH𝔉-group of type F P_{∞}. Then there are only finitely many conjugacy classes in Γ of finite elementary abelian subgroups.

Proof. For any collection A of finite elementary abelian subgroups of Γ, we denote by C_{A} the thick subcategory of stmod(ℤΓ) generated by modules induced from finitely generated modules for subgroups in A. It follows from Theorem 12.1 that every module of type F P_{∞} is in C_{A} for some finite collection A. Choose a finite collection A_{0} such that the trivial

COMPLEXITY AND VARIETIES FOR INFINITE GROUPS, II 55

ℤΓ-module ℤ is in C_{A_{0}}. We claim that every finite elementary abelian subgroup must be contained in some conjugate of some element of A_{0}. Suppose to the contrary that E is a finite elementary abelian p-subgroup which is not contained in any element of A_{0}. It follows from Proposition 4.3 of [7] that there is a finitely generated ℤ-free ℤE-module U such that U ↓_{E′} is projective for all proper subgroups E′ < E, but U is not projective (so Ĥ^{∗}(E, U) ≠ 0). If k is an algebraically closed field of characteristic p, this is achieved by taking a module U such that k ⊗_{ℤ} U has as its rank variety a line through the origin not contained in any hyperplane defined over 𝔽_{p}.

If N is a ℤE_{0} module with E_{0} ∈ A_{0} then by Corollary 5.11 we have

Êxt_{ℤΓ}^{∗}(N ↑^{Γ}, Hom_{ℤE}(ℤΓ, U)) ≅ Êxt_{ℤΓ}^{∗}(N, Hom_{ℤE}(ℤΓ, U)↓_{E_{0}}).

Now by the Mackey decomposition theorem, Hom_{ℤE}(ℤΓ, U) ↓_{E_{0}} is a direct product of modules obtained by restricting U to a proper subgroup E′ of E, then conjugating and inducing to E_{0}. Since U_{E′} is projective, and an arbitrary product of projective ℤEmodules has projective dimension at most one, we see that the above Êxt group is zero. Since ℤ is in the thick subcategory generated by such modules N ↑^{Γ}, it follows that Êxt_{ℤΓ}^{∗}(ℤ, Hom_{ℤE}(ℤΓ, U)) = 0. Again applying Corollary 5.11, we see that this implies that Ĥ^{∗}(E, U) = 0, contradicting the construction of U.

Remark 19.2. (Kropholler) If Γ is an LH𝔉-group of type F P_{∞} then using Lemma 3.1 and Proposition 3.2 we have

ℚ ⊗_{ℤ} Êxt_{ℤΓ}^{0}(ℤ, ℤ) ≅ Êxt_{ℤΓ}^{0}(ℤ, ℚ) ≅ Êxt_{ℚΓ}^{0}(ℚ, ℚ) = 0.

It follows that the identity element of Êxt_{ℤΓ}^{0}(ℤ, ℤ) has finite additive order, and this gives an upper bound for the orders of the finite subgroups of Γ. I do not know whether there can be infinitely many conjugacy classes of finite subgroups.

20. Some open problems

1. Does Proposition 6.5 hold for non-countably presented modules? This does not seem to have any impact on the rest of this paper, but seems to be an interesting question in its own right.

2. Does Theorem 6.6 hold for uncountably generated modules? If so, then we can lift the countability hypothesis in Corollary 6.7, the definitions of StMod(kΓ) and E^{⊕} in Section 8 and Theorem 12.5.

3. It is tempting to speculate more generally that if G is a finite group, k is a commutative ring of coefficients, and M is a flat kG-module, then M is projective relative to the trivial subgroup. In other words, does M flat imply that the natural map kG ⊗_{k} M → M splits?

4. How generally does the second remark following Theorem 12.1 hold? In other words, are there coefficient rings other than a field, for which the permutation modules on cosets of finite elementary abelian subgroups always generate stmod(kΓ)?

5. Is there a good theory of complexity in the context where the coefficient ring is a commutative Noetherian ring of finite global dimension rather than a field?

6. Is there an analog V_{Γ} for LH𝔉-groups of the maximal ideal spectrum V_{G} of the cohomology ring of a finite group H^{∗}(G, k) over a field k of characteristic p? This should have as its underlying topological space the space V_{Γ}^{r} constructed in Section 15.

7. Let Γ be an LH𝔉-group and k be a field of characteristic p, and let M be a kΓ-module of type F P_{∞}. Is the Poincaré series p_{M}(t) described at the end of Section 16 equal to the power series expansion of a rational function of t?

8. What collections V_{Γ}^{r}(M) occur for modules M of type F P_{∞}?

9. In the second example discussed in Section 18, is every ℤΓ-module of type F P_{∞} induced from some finite elementary abelian p-subgroup of Γ?

10. Can an LH𝔉-group of type F P_{∞} have infinitely many conjugacy classes of finite subgroups?

References

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

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

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

[4] D. J. Benson. Complexity and varieties for infinite groups, I. Preprint, 1996.

ftp://byrd.math.uga.edu/pub/papers/benson/hf1.dvi

[5] D. J. Benson, J. F. Carlson and J. Rickard. Complexity and varieties for infinitely generated modules, I. Math. Proc. Camb. Phil. Soc. 118 (1995), 223–243. ftp://byrd.math.uga.edu/pub/papers/benson-carlson-rickard/bcr1.dvi

[6] D. J. Benson, J. F. Carlson and J. Rickard. Complexity and varieties for infinitely generated modules, II. To appear in Math. Proc. Camb. Phil. Soc. ftp://byrd.math.uga.edu/pub/papers/benson-carlson-rickard/bcr2.dvi

[7] D. J. Benson and N. Habegger. Varieties for modules and a problem of Steenrod. J. Pure & Applied Algebra 44 (1987), 13–34.

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

[9] L. Chouinard. Projectivity and relative projectivity over group rings. J. Pure & Applied Algebra 7 (1976), 287–302.

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

[11] L. Fuchs. Infinite abelian groups, Volume II. Academic Press, New York/London 1973.

[12] D. G. Quillen. Spectrum of an equivariant cohomology ring I, II. Ann. of Math. 94 (1971), 549–572, 573–602.

[13] D. G. Quillen. Homotopical algebra. Springer Lecture Notes in Mathematics 43, Springer-Verlag, Berlin/New York, 1967.

[14] D. G. Quillen. Rational homotopy theory. Ann. of Math. 90 (1969), 205–295.

[15] J. Rickard. Idempotent modules in the stable category. To appear in J. London Math. Soc. ftp://byrd.math.uga.edu/pub/papers/rickard/idemp.dvi (Papers archived on ftp://byrd.math.uga.edu/pub/papers can also be accessed through the web at http://www.math.uga.edu/∼djb/archive.html)

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

E-mail address: djb@byrd.math.uga.edu