COMPLEXITY AND VARIETIES FOR INFINITE GROUPS, I

D. J. BENSON

Abstract. This two part paper generalizes the usual notion of complexity and varieties for modules over the group algebra of a finite group, to a large class of infinite groups. The context is modules of type F P_{∞} for groups in Kropholler’s class LH𝔉. One of the main results is that the category of such modules is generated in a suitable sense by modules induced from finite elementary abelian subgroups. This implies that an element of complete cohomology of such a module is nilpotent if and only if its restriction to every finite elementary abelian subgroup is nilpotent. It also implies that the complexity of a module of type F P_{∞} is finite, and that the variety is supported on some finite collection of finite elementary abelian subgroups. An example is given which shows that the complexity does not determine the rate of growth of the number of generators in a projective resolution, in the way it does for finite groups.

Contents (Part I): 1. Introduction. 2. LH𝔉-groups. 3. Complete cohomology. 4. Cofibrant modules. 5. Complete resolutions. 6. Complete cohomology for finite groups.

7. Induced modules. 8. Triangulated categories. 9. Derived categories. 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.

1. Introduction

Chouinard’s theorem [7] states that if k is a commutative ring of coefficients and G is a finite group then a kG-module is projective if and only if its restriction to every elementary abelian subgroup is projective. Dade’s lemma [11] states that if E is a finite elementary abelian p-group and k is an algebraically closed field of characteristic p then a finitely generated kE-module is projective if and only if its restriction to every cyclic shifted subgroup is projective. These facts lead to the notion of the variety of a module, which is zero if and only if the module is projective. The dimension of the variety measures the polynomial rate of growth of the minimal resolution of the module, and is called the complexity.

For infinite groups, there can be modules which have finite projective dimension without being projective. Moreover, the group algebra is seldom Noetherian, so rather than

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

2 D. J. BENSON

examining finitely generated modules, we look at modules of type F P_{∞}, namely modules which have a resolution by finitely generated projective modules. So the natural question is whether a module of type F P_{∞} has finite projective dimension if and only if its restriction to every finite elementary abelian subgroup does. Bieri [3], p. 186 suggests the question of whether the weaker statement holds, that if Γ has type F P_{∞} (i.e., ℤ has type F P_{∞} as a ℤΓ-module) then the center of Γ is finitely generated and the torsion-free subgroups of Γ have finite cohomological dimension. Even in this generality, the answer is no. Brown and Geoghegan [5] have shown that Richard Thompson’s group (see Section 2) is torsion-free of type F P_{∞} but does not have finite cohomological dimension.

Kropholler [14] has recently introduced a large class LH𝔉 of (discrete) groups, and proved that for a group Γ in this class, a ℤΓ-module of type F P_{∞} has finite projective dimension if and only if its restriction to every finite subgroup does. This class of groups includes, for example, all solvable by finite groups, all groups with finite virtual cohomological dimension, and all groups with a faithful representation as endomorphisms of a Noetherian module over a commutative ring. All groups known not to be in LH𝔉 are variants of the Brown–Geoghegan Beispiel.

The purpose of this paper is to develop the module theory and a theory of complexity and varieties for modules of type F P_{∞} over an LH𝔉-group. The paper begins with an extensive discussion of the module theory, and describes in detail the structure of various versions of the module category. Complete cohomology gives rise to a notion of stable module category, which may be viewed either as a quotient of the derived category (Section 9) or as the homotopy category for a suitable closed model structure on the module category (Section 10).

Our approach to complexity and varieties uses a generalization of Rickard’s idempotent functor approach [17] to this context. This allows us to prove the following remarkable fact about modules of type F P_{∞} for LH𝔉-groups. First, there are some obvious modules of type F P_{∞}, which may be obtained by inducing finitely generated modules from finite elementary abelian subgroups. Then there are some obvious operations which preserve type F P_{∞}: the kernel of a surjective morphism (or the cokernel of an injective morphism) between modules of type F P_{∞} has type F P_{∞}; a finite direct sum of modules of type F P_{∞} and any summand of such a module again has type F P_{∞}. The remarkable fact is that these modules and operations generate all modules of type F P_{∞}. This is proved in Section 12, where an equivalent formulation is given in terms of filtrations of modules.

COMPLEXITY AND VARIETIES FOR INFINITE GROUPS, I 3

It is worth pointing out some immediate consequences of this statement (see Sections 13 and 14). The first is expressed in terms of complete cohomology, which is explained in Section 3.

Theorem 1.1. 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_{∞}. If θ ∈ Êxt_{kΓ}^{∗}(M, M) restricts to a nilpotent element on each finite elementary abelian subgroup of Γ, then θ is nilpotent.

Theorem 1.2. Let Γ be an LH𝔉-group and k a field of coefficients, and let M be a kΓmodule of type F P_{∞}. Then there is an upper bound for the complexities of the restrictions of M to finite subgroups of Γ.

Since the restriction of a module of type F P_{∞} to a finite subgroup is not necessarily finitely generated, the version of complexity we must use for the finite subgroups is the recently developed theory of Benson, Carlson and Rickard [2].

Motivated by Kropholler’s theorem, we define the complexity of a module of type F P_{∞} for an LH𝔉-group to be the maximum complexity of a restriction to a finite subgroup, or equivalently to a finite elementary abelian p-subgroup (p = char(k)). In Section 16, we prove that the thick subcategory of stmod(kΓ) consisting of modules of complexity at most c is generated by modules induced from finitely generated modules of complexity at most c for finite elementary abelian p-subgroups of Γ. An example in Section 18 shows that the usual interpretation of complexity in the case of a finite group, in terms of the rate of growth of projective resolutions, does not generalize to modules of type F P_{∞}, even for finitely generated abelian groups.

Part of the difficulty involved in defining varieties for modules in the context of modules of type F P_{∞} for an LH𝔉-group is the question of what to use as the “ambient” variety. In the case of a finite group, one uses the maximal ideal spectrum of the cohomology ring. By a theorem of Quillen [16], this is equal (at the level of underlying topological space) to a colimit of affine spaces corresponding to the elementary abelian p-subgroups. The colimit is over the “Quillen category”, whose objects are the elementary abelian p-subgroups, and whose morphisms are generated by conjugations and inclusions. In our context, we use the rather ad hoc approach of defining the ambient variety as the corresponding colimit. It really isn’t a variety in the usual sense, but rather a possibly infinite dimensional object V_{Γ}^{r} obtained by patching together open subsets of varieties using a colimit over the Quillen category. It would be reassuring to know of a corresponding cohomological construction, but currently we know of no such thing.

4 D. J. BENSON

Once these preliminaries are out of the way, it is relatively easy to set up a theory of varieties for modules of type F P_{∞} for an LH𝔉-group. Associated to each such module M, there is a collection V_{Γ}^{r}(M) of closed homogeneous irreducible subvarieties of V_{Γ}^{r}. This collection is supported on some finite collection of finite elementary abelian p-subgroups, and may be thought of as determining whether tensor products M ⊗_{k} N have finite projective dimension. We prove analogs in this context of the tensor product theorem and Carlson’s connectedness theorem.

Finally, we mention a theorem which follows from the theory described in this paper (see Section 19).

Theorem 1.3. 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.

We remark that this theorem has been proved by H.-W. Henn under the assumption that Γ is an H_{1}𝔉-group of type F P_{∞}. Kropholler and Mislin have proved that an LH𝔉-group of type F P_{∞} is necessarily an H_{1}𝔉-group, so that in fact this theorem is not new. Our proof, on the other hand, is quite different from that of Henn, which uses a modification of the techniques of Quillen [16].

I would like to take this opportunity to thank Peter Kropholler for introducing me to his LH𝔉-groups and explaining to me the key facts about the module theory for these groups. For the sake of clarity, some arguments from Kropholler [14] and from Cornick and Kropholler [8, 9, 10] are repeated here. I would also like to thank Jon Carlson, Jonathan Cornick and Hans-Werner Henn for taking an interest in this work and discussing various aspects of it with me.

2. LH𝔉-groups

For the convenience of the reader, we briefly recall from [14] the definition of Kropholler’s class of LH𝔉-groups. The reader should consult [8, 9, 10, 14, 15] for further information on this class of groups.

Given any class 𝔛 of groups, we define a class of groups H_{α}𝔛 for each ordinal α as follows. For α = 0 we define H_{0}𝔛 = 𝔛. For α > 0 we define a group Γ to be in H_{α}𝔛 if Γ acts cellularly on a finite dimensional contractible CW complex X, in such a way that the setwise stabilizer of each cell is equal to the pointwise stabilizer, and is in H_{β}𝔛 for some β < α. The class H𝔛 is then the union of all the H_{α}𝔛. The letter H stands for “hierarchy”. The algebraic consequence of this definition is that the augmented cellular chains on X

0 → C_{n}(X) → ⋯ → C_{0}(X) → ℤ → 0

form a finite resolution of ℤ as a ℤΓ-module, with each C_{j}(X) a direct sum of modules induced up from subgroups in H_{β}𝔛 for β < α.

We denote by 𝔉 the class of finite groups, and we form the class H𝔉 as above. Finally, a group Γ is said to be an LH𝔉-group if every finitely generated subgroup of Γ is an H𝔉-group.

To orient the reader, a few remarks are in order at this stage. First, LH𝔉 is subgroup closed, extension closed and closed under directed unions, amalgamated free products and HNN extensions. It contains every solvable by finite group and every group admitting a faithful representation as automorphisms of a Noetherian module over a commutative ring (“linear group”). Every one relator group is in LH𝔉 (cf. Bieri [3], Examples 2.6). By a construction of Serre (Brown [4] §VIII.3, Exercise), every group of finite virtual cohomological dimension is in H_{1}𝔉. The class LH𝔉 does not contain Richard Thompson’s group

Γ = ⟨x_{0}, x_{1}, … | x_{n}^{x_{i}} = x_{n+1} ∀i < n⟩
= ⟨a, b | [ab^{−1}, a^{−1}ba], [ab^{−1}, a^{−2}ba^{2}]⟩

(where a = x_{0} and b = x_{1}). Brown and Geoghegan [5] have shown that this group is torsion-free and has type F P_{∞} but not type F P. Kropholler [14] deduced from this (see Theorem 3.3 in the next section) that it is not in LH𝔉. Since the group of all permutations on any infinite set contains Richard Thompson’s group, this permutation group is also not in LH𝔉.

The group SL(3, ℤ) has finite virtual cohomological dimension, so it is in H_{1}𝔉, but any action on a tree has to fix a point (Margulis; see Serre [18], I.6.6). It follows that restricting ones attention to one dimensional complexes in the definition of H𝔉-groups does alter the class of groups obtained.

By constructing a suitable tree out of cosets, it can be shown that if Γ_{1} ⊆ Γ_{2} ⊆ ⋯ is an ascending sequence of groups in H_{α}𝔉 then Γ = ⋃_{n} Γ_{n} is in H_{α+1}𝔉. So both H𝔉 and LH𝔉 are closed under ascending unions in this sense. For example, a free abelian group of countable rank is in H_{2}𝔉 but not in H_{1}𝔉.

The additive group of the rationals is in H_{1}𝔉, as it has finite cohomological dimension. It is not known whether the additive group of the reals is in H𝔉, although it is certainly in LH𝔉. In fact, it is not known whether there are any groups in LH𝔉 which do not lie in H_{3}𝔉. According to Kropholler, an example of a group which is in H_{3}𝔉 but not in H_{2}𝔉 is the semidirect product ℚ(t)^{+}⋊ℚ(t)^{×} with the obvious action.

3. Complete cohomology

Our methods depend on some categorical constructions on a suitable version of the stable module category. We begin by introducing complete cohomology, which is used to define the morphisms in this category. Most of the material in this and the next section is due to Kropholler [14] and to Kropholler and Cornick [8, 9, 10].

If R is any ring and M is an R-module, we define Ω(M) to be the kernel of a surjective homomorphism from a projective module onto M. By Schanuel’s lemma, this is well defined up to adding and removing projective direct summands. If M and N are Rmodules, we define \underline{Hom}_{R}(M, N) to be the quotient of Hom_{R}(M, N) by the additive subgroup consisting of homomorphisms which factor through a projective module. There is a natural homomorphism

\underline{Hom}_{R}(M, N) → \underline{Hom}_{R}(Ω(M), Ω(N)),

and we define the complete cohomology via

Êxt_{R}^{r}(M, N) = lim_{\overrightarrow{i}} \underline{Hom}_{R}(Ω^{r+i}(M), Ω^{i}(N)).

This definition makes sense for any integer r. Since lim_{⟶} is exact, there are long exact sequences in each variable, which extend infinitely far to the left and to the right. The usefulness of this definition depends upon the following lemma.

Lemma 3.1. A module M for a ring R has finite projective dimension if and only if Êxt_{R}^{0}(M, M) = 0.

Proof. This is statement (4.2) of [14]. One direction is clear; in the other direction, if Êxt_{R}^{0}(M, M) = 0 then the identity element of Hom_{R}(M, M) goes to zero in the limit, and hence in \underline{Hom}_{R}(Ω^{r}M, Ω^{r}M) for some r ≥ 0. But then Ω^{r}M is projective.

In the case of modules over an LH𝔉-group, this may be applied using the following proposition.

Proposition 3.2. Suppose that M and N are kΓ-modules, where Γ is an LH𝔉-group and k is a commutative ring of coefficients. Suppose that M has type F P_{∞}, and that the restriction of N to every finite subgroup of Γ has finite projective dimension. Then Êxt_{kΓ}^{0}(M, N) = 0.

Proof. This is proved in Theorem A of Cornick and Kropholler [8]. The proof consists of applying statement (3.3) of Kropholler [14] to the functors Êxt_{kΓ}^{ℓ}(M, −).

Combining Lemma 3.1 and Proposition 3.2 gives the main theorem of Kropholler [14]:

Theorem 3.3. Let Γ be an LH𝔉-group and k a commutative ring of coefficients. If M is a kΓ-module of type F P_{∞} whose restriction to every finite subgroup has finite projective dimension, then M has finite projective dimension.

Now for any set S, we denote by B(S, k) the set of functions from S to k which take only finitely many different values in k. We make this into a commutative k-algebra by pointwise multiplication. If a group Γ acts on S then it acts on B(S, k) in the obvious way: g(ϕ)(s) = ϕ(g^{−1}s). In case S = Γ, with action given by multiplication, we write B = B(Γ, k).

We shall be considering kΓ-modules M such that B⊗_{k}M has finite projective dimension, and kΓ-modules M such that B ⊗_{k} M is projective. We shall discuss in Remark 5.9 to what extent these conditions depend on the exact definition of the module B.

Lemma 3.4. (i) For any set S, and any coefficient ring k, the module B(S, k) is free as a k-module.

(ii) For every finite subgroup G of Γ, B = B(Γ, k) restricts to a free kG-module.

Proof. (i) If k = ℤ, it follows from a result of Bergman (see Lemma 97.2 of Fuchs [13]) that B(S, ℤ) is free as an abelian group. For more general coefficient rings k, we now use that fact that B(S, k) = k ⊗_{ℤ} B(S, ℤ).

(ii) If G is a finite subgroup of Γ then choose a set S = {g_{α}} of left coset representatives of G in Γ. Then as a kG-module, B is isomorphic to the induced module kG ⊗_{k} B(S, k), so freeness follows from freeness of B(S, k) as a k-module.

Denote by i : k → B the inclusion of the constant functions, and write B̄ for the cokernel. Denote by μ : B ⊗_{k} B → B the multiplication map. These are homomorphisms of kΓ-modules.

Proposition 3.5. Suppose that Γ is a group, and k is a commutative ring of coefficients. If M is a kΓ-module satisfying Êxt_{kΓ}^{0}(M, B ⊗_{k} M) = 0 then B ⊗_{k} M has finite projective dimension as a kΓ-module.

8 D. J. BENSON

Proof. This is proved in Section 2 of Cornick–Kropholler [8], by observing that the image of i ⊗ Id_{M} under the composite map

Hom_{kΓ}(M, B ⊗_{k} M) → Êxt_{kΓ}^{0}(M, B ⊗_{k} M) ⟶^{Id_{B}∪−}

Êxt_{kΓ}^{0}(B ⊗_{k} M, B ⊗_{k} B ⊗_{k} M) ⟶^{(μ⊗Id_{M})_{∗}} Êxt_{kΓ}^{0}(B ⊗_{k} M, B ⊗_{k} M)

is equal to the identity map. The second term here is the zero group, so we can apply Lemma 3.1 to deduce that B ⊗_{k} M has finite projective dimension.

Corollary 3.6. Suppose that Γ is an LH𝔉-group, and k is a commutative ring of coefficients. If M is a kΓ-module of type F P_{∞}, which has finite projective dimension as a k-module, then B ⊗_{k} M has finite projective dimension as a kΓ-module. In particular, for some r ≥ 0, B ⊗_{k} Ω^{r}(M) is projective.

Proof. The restriction of B ⊗_{k} M to any finite subgroup has finite projective dimension, so by Proposition 3.2, Êxt_{kΓ}^{0}(M, B ⊗_{k} M) = 0. Now apply Proposition 3.5.

The last statement of the corollary follows from the identity

B ⊗_{k} Ω^{r}(M) ⊕ (projective) ≅ Ω^{r}(B ⊗_{k} M) ⊕ (projective),

which in turn follows from Schanuel’s lemma.

4. Cofibrant modules

Definition 4.1. Let Γ be a group and k be a commutative ring of coefficients. A kΓmodule M is said to be cofibrant if B ⊗_{k} M is projective.

This terminology will be justified in Section 10.

Lemma 4.2. Suppose that Γ is a group, and k is a commutative ring of coefficients. Let M be a cofibrant kΓ-module. Then there are projective modules P_{1}, P_{2} and P_{3} such that

M ⊕ P_{1} ≅ B̄ ⊗_{k} ΩM ⊕ P_{2} ≅ Ω(B̄ ⊗_{k} M) ⊕ P_{3}.

Proof. Let 0 → ΩM → P_{M} → M → 0 be a short exact sequence with P_{M} projective. Then we have short exact sequences

0 → B̄ ⊗_{k} ΩM → B̄ ⊗_{k} P_{M} → B̄ ⊗_{k} M → 0

and

0 → M → B ⊗_{k} M → B̄ ⊗_{k} M → 0.

Also let

0 → Ω(B̄ ⊗_{k} M) → P_{B̄⊗_{k}M} → B̄ ⊗_{k} M → 0

be a short exact sequence with P_{B̄⊗_{k}M} projective. The lemma now follows by applying Schanuel’s lemma to the above sequences.

Remark 4.3. Even if M is not cofibrant, the diagram

[figure: 0; B̄ ⊗_{k} ΩM; B̄ ⊗_{k} P_{M}; B̄ ⊗_{k} M; 0; 0; M; B ⊗_{k} M; B̄ ⊗_{k} M; 0]

(where P_{M} is a projective module surjecting onto M) constructs for us a map

B̄ ⊗_{k} ΩM → M

whose image in \underline{Hom}_{kΓ}(B̄ ⊗_{k} ΩM, M) is well defined, because B̄ ⊗_{k} P_{M} is projective.

Lemma 4.4. Let Γ be a group and k be a commutative ring of coefficients. Given a kΓ-module homomorphism M → M′, the square

\xymatrix{B̄ ⊗_{k} ΩM \ar[r] \ar[d] & M \ar[d] \\ B̄ ⊗_{k} ΩM′ \ar[r] & M′}

may be made to commute. If M → M′ is surjective, this may be achieved just by altering the top horizontal map.

Proof. First we treat the case where M → M′ is surjective. In this case, we form the pullback X of B̄ ⊗_{k} M → B̄ ⊗_{k} M′ and B ⊗_{k} M′ → B̄ ⊗_{k} M′ and notice that the map from B ⊗_{k} M to X is surjective. This allows us to lift the obvious map B̄ ⊗ P_{M} → X to B̄ ⊗_{k} P_{M} → B ⊗_{k} M. The induced map B̄ ⊗_{k} ΩM → M then makes the diagram commute.

Next, if M → M′ is injective, we may assume that P_{M′} = P_{M} ⊕ P_{M′/M}. We then lift separately on P_{M} and P_{M′/M}.

Finally, a general map M → M′ may be factored as a surjective map followed by an injective map, and we proceed as above.

Lemma 4.5. Let Γ be a group, k be a commutative ring of coefficients, and M be a kΓ-module.

(i) If M is cofibrant then every extension of M by a kΓ-module of finite projective dimension splits.

(ii) If B ⊗_{k} M has finite projective dimension and every extension of M by a kΓ-module of finite projective dimension splits, then M is cofibrant.

10 D. J. BENSON

Proof. (i) Suppose that

0 → M′ ⟶^{α} M′′ ⟶^{β} M → 0

is a short exact sequence with M cofibrant, and that M′ has finite projective dimension. We argue by induction on the projective dimension of M′.

First suppose that M′ is projective. Consider the diagram

[figure: 0; M′; α; M′′; β; M; 0; 0; B ⊗_{k} M′; B ⊗_{k} M′′; B ⊗_{k} M; 0]

The left hand vertical map splits because B̄ ⊗_{k} M′ is projective, and the bottom row splits because B ⊗_{k} M is projective. It follows that α splits.

For the general case, let P and P′ be projective modules mapping onto M and M′. Then we obtain a surjective homomorphism from P ⊕ P′ to M′′ and a commutative diagram as follows.

[figure: 0; 0; 0; 0; ΩM′; Ωα; P′; M′; α; 0; 0; ΩM′′; Ωβ; P ⊕ P′; M′′; β; 0; 0; ΩM; P; M; 0; 0; 0; 0]

By the inductive hypothesis, the sequence

0 → ΩM′ → ΩM′′ → ΩM → 0

splits.

Since M is cofibrant, we may assume that the module P has been taken large enough so that the map B̄ ⊗_{k} ΩM → M is split surjective (see Lemma 4.2 and Remark 4.3). By Lemma 4.4, the diagram

[figure: B̄ ⊗_{k} M′′; IdB̄⊗Ωβ; M′′; β; B̄ ⊗_{k} ΩM; M]

COMPLEXITY AND VARIETIES FOR INFINITE GROUPS, I 11

may be made to commute without disrupting the lower horizontal map. Both the lower horizontal map and the left hand vertical map are split surjections, and hence β is also a split surjection.

(ii) Choosing ΩM large enough so that B̄⊗_{k}ΩM → M can be taken to be surjective, the kernel has finite projective dimension and hence the map splits. If M were not cofibrant then B ⊗_{k} B̄ ⊗_{k} ΩM would have smaller projective dimension than B ⊗_{k} M, which is absurd because the latter is isomorphic to a direct summand of the former.

5. Complete resolutions

Definition 5.1. A complete resolution of a module M is defined to be an exact sequence of projective kΓ-modules indexed by the integers

⋯ → P_{2} → P_{1} → P_{0} → P_{−1} → P_{−2} → ⋯

such that the image of P_{0} → P_{−1} is isomorphic to M.

The module B may be used to manufacture complete resolutions of cofibrant modules as follows. Suppose that M is a cofibrant kΓ-module. Then the inclusion i : k → B induces an inclusion i ⊗ Id_{M} from M into the projective module B ⊗_{k} M, with cokernel B̄ ⊗_{k} M. The sequence

0 → M → B ⊗_{k} M → B̄ ⊗_{k} M → 0

splits on tensoring with B, since μ ⊗ Id_{M} is a splitting. It follows that M has a complete resolution which splits on tensoring with B (for i ≥ 0, B ⊗_{k} Ω^{i}M is projective so there’s no problem with the splitting there).

If M is finitely (resp. countably) generated, then the image of M → B ⊗_{k} M lies in a finitely (resp. countably) generated direct summand. The inclusion of M into this summand still splits on tensoring with B.

After dimension shifting, these remarks are summarized in the following proposition and corollary.

Proposition 5.2. Suppose that Γ is a group, and k is a commutative ring of coefficients. If M is a kΓ-module such that B⊗_{k} M has finite projective dimension then for some r ≥ 0, Ω^{r}(M) has a complete resolution which splits on tensoring with B.

Corollary 5.3. Suppose that Γ is a group, and k is a commutative ring of coefficients. If M is a kΓ-module of type F P_{∞}, which has finite projective dimension as a k-module, then for some r ≥ 0, Ω^{r}(M) has a complete resolution by a complex of finitely generated projective modules, which splits on tensoring with B.

12 D. J. BENSON

Proof. This follows from Corollary 3.6 and Proposition 5.2.

Another interesting consequence of this construction is the following.

Proposition 5.4. Suppose that Γ is a group, and k is a commutative ring of coefficients. Suppose that M is a cofibrant kΓ-module. If M has finite projective dimension then M is projective.

Proof. If M has projective dimension at most r (i.e., Ω^{r}(M) is projective) then so does B̄ ⊗_{k} ⋯ ⊗_{k} B̄ ⊗_{k} M, and hence (using the extended version of Schanuel’s lemma) M is projective.

Corollary 5.5. Suppose that G is a finite group and k is a commutative ring of coefficients. If M is projective as a k-module and has finite projective dimension as a kG-module then it is projective as a kG-module.

Proof. In the case of a finite group G, the kG-module B is isomorphic to kG. So if M is projective as a k-module then it is cofibrant as a kG-module. Now apply the proposition.

The following is an unpublished theorem of Cornick and Kropholler in the case of H𝔉groups, generalized by Kropholler to LH𝔉-groups. We begin with a lemma.

Lemma 5.6. Suppose that Γ_{α} (α < γ) is an ascending chain of groups, for some ordinal γ, with union ⋃_{α<γ} Γ_{α} = Γ, and k is a commutative ring of coefficients. If M is a kΓ-module

whose restriction to each subgroup Γ_{α} is projective, then M has projective dimension at most one.

Proof. Since the restriction of M to the trivial subgroup is projective, the natural map from P = kΓ ⊗_{k} M to M is a surjective homomorphism from a projective module onto M. For each of the subgroups Γ_{α}, let Q_{α} = kΓ ⊗_{kΓ_{α}} M, and let P_{α} be the kernel of the natural surjective map from P to Q_{α}.

If α < β < γ, then P_{α} ≤ P_{β} and Q_{β} is a quotient of Q_{α}. The module M ↓_{kΓ_{α}} is projective by hypothesis, and hence Q_{α} is projective. So the map from Q_{α} to Q_{β} splits, and its kernel, which is isomorphic to P_{β}/P_{α}, is therefore projective. Furthermore, if β is a limit ordinal, then P_{β} = ⋃_{α<β} P_{α}. So the kernel P_{γ} of P → M has a transfinite filtration

by projective modules. So we have (using transfinite induction)

P_{γ} ≅ ⨁_{α<γ} P_{α+1}/P_{α},

COMPLEXITY AND VARIETIES FOR INFINITE GROUPS, I 13

and therefore P_{γ} is projective. Since M is isomorphic to P/P_{γ}, it has projective dimension at most one.

Theorem 5.7. Suppose that Γ is an LH𝔉-group and k is a commutative ring of coefficients. If M is a kΓ-module such that B ⊗_{k} M has projective dimension at most r, and M ↓_{G} has projective dimension at most r for all finite subgroups G ≤ Γ, then M has projective dimension at most r.

Proof. Replacing M by Ω^{r}(M), we may assume that B ⊗_{k} M and M ↓_{G} are projective.

We first consider the case where Γ is an H𝔉-group. We prove by induction on ordinals α, that if Γ′ is a subgroup of Γ in H_{α}𝔉 then M ↓_{Γ′} is projective. If α = 0, this is true by hypothesis. If α > 0, let X be a contractible Γ′-space as in the definition of H_{α}𝔉-group, and tensor the cellular chains on X with M to obtain an exact sequence of kΓ′-modules

0 → C_{n}(X) ⊗_{ℤ} M → ⋯ → C_{0}(X) ⊗_{ℤ} M → M → 0.

Each C_{j}(X) ⊗_{ℤ} M is a direct sum of modules of the form M ↓_{H}↑^{Γ′} with H in H_{β}𝔉 for some β < α. By the inductive hypothesis, each of these modules is projective, and so it follows that M ↓_{Γ′} has finite projective dimension. Applying Proposition 5.4, we deduce that M ↓_{Γ′} is projective. The case Γ′ = Γ proves the theorem for H𝔉-groups.

Now suppose that Γ is an LH𝔉-group. We argue by induction on the cardinality of Γ. If Γ is countable, then it may be expressed as an ascending union of a chain of finitely generated subgroups Γ_{1} < Γ_{2} < ⋯, with Γ = ⋃_{n=1}^{∞} Γ_{n}. Each Γ_{n} is an H𝔉-group (by definition of LH𝔉-group), and so M ↓_{Γ_{n}} is projective. Applying Lemma 5.6, we see that M has projective dimension at most one. Applying Proposition 5.4, we deduce that M is projective.

If Γ is uncountable, then it may be expressed as an ascending union Γ = ⋃_{α<γ} Γ_{α} (for

some ordinal γ) in such a way that each Γ_{α} has strictly smaller cardinality than Γ. The theorem has already been proved for each Γ_{α}, and so M ↓_{Γ_{α}} is projective. Applying Lemma 5.6, and Proposition 5.4 as before, we again deduce that M is projective.

The above theorem can be used to clarify the role of the module B in the theory.

Corollary 5.8. Let Γ be an LH𝔉-group, k be a commutative ring of coefficients, and M be a kΓ-module. Then B ⊗_{k} M has finite projective dimension as a kΓ-module if and only if the following two conditions hold.

(i) M has finite projective dimension as a k-module, and

(ii) for any kΓ-module N, N ⊗_{k} M has finite projective dimension if and only if the restriction (N ⊗_{k} M)↓_{G} has finite projective dimension for all finite subgroups G ≤ Γ.

14 D. J. BENSON

Proof. First suppose that B ⊗_{k} M has finite projective dimension as a kΓ-module. Since B is a free k-module by Lemma 3.4(i), this implies that M has finite projective dimension as a k-module. If N is another kΓ-module, and N ⊗_{k} M ↓_{G} has finite projective dimension for all finite subgroups G ≤ Γ, then by Theorem 5.7, N ⊗_{k} M has finite projective dimension. So M satisfies conditions (i) and (ii).

Conversely, suppose that M satisfies (i) and (ii). Since M has finite projective dimension as a k-module, by Lemma 3.4(ii), B⊗_{k}M ↓_{G} has finite projective dimension as a kG-module for all finite subgroups G ≤ Γ. So applying condition (ii), we see that B ⊗_{k} M has finite projective dimension as a kΓ-module.

Remark 5.9. These conditions do not mention B. Furthermore, the only properties of B which are used in the proofs in this and the previous section are that B is a kfree commutative kΓ-algebra whose restriction to every finite subgroup is projective. It follows that if B′ is another such, then B ⊗_{k} M has finite projective dimension if and only if B′ ⊗_{k} M does. Using Lemma 4.5, we deduce that B ⊗_{k} M is projective if and only if B′ ⊗_{k} M is projective.

The point of using complete resolutions is that they compute complete cohomology, in the sense of the following proposition.

Proposition 5.10. Suppose that Γ is a group, k is a commutative ring of coefficients, and M is a kΓ-module. Suppose that P_{∗} is a complete resolution of M which splits on tensoring with B. Then for any module N, the cohomology of the complex Hom_{kΓ}(P_{∗}, N) is Êxt_{kΓ}^{∗}(M, N).

Proof. (cf. Cornick–Kropholler [9, 10]). Since B ⊗_{k} P_{∗} splits,

Hom_{kΓ}(P_{∗}, Hom_{k}(B, N)) ≅ Hom_{kΓ}(B ⊗_{k} P_{∗}, N)

is exact. If N is projective, then i^{∗} : Hom_{k}(B, N) → N is a split epimorphism, and so Hom_{kΓ}(P_{∗}, N) is also exact.

Now let N be an arbitrary kΓ-module. Write Ω^{r}M for the image of P_{r} → P_{r−1} (r ∈ ℤ), or equivalently the cokernel of P_{r+1} → P_{r}. Then

Hom_{kΓ}(Ω^{r}M, N) ≅ Ker(Hom_{kΓ}(P_{r}, N) → Hom_{kΓ}(P_{r+1}, N)).

It follows from the above that under this isomorphism, a homomorphism Ω^{r}M → N factors through a projective kΓ-module if and only if it corresponds to an element of

Im(Hom_{kΓ}(P_{r−1}, N) → Hom_{kΓ}(P_{r}, N)).

COMPLEXITY AND VARIETIES FOR INFINITE GROUPS, I 15

Furthermore, if 0 → ΩN → Q → N → 0 is a short exact sequence with Q projective, then examining the long exact sequence obtained by applying Hom_{kΓ}(P_{∗}, −) to this short exact sequence and taking cohomology, we obtain

H^{r}Hom_{kΓ}(P_{∗}, N) ≅ H^{r+1}Hom_{kΓ}(P_{∗}, ΩN).

It follows that the maps

\underline{Hom}_{kΓ}(Ω^{r+i}M, Ω^{i}N) → \underline{Hom}_{kΓ}(Ω^{r+i+1}M, Ω^{i+1}N)

are isomorphisms for i ≥ 0, and so the common value is Êxt_{kΓ}^{r}(M, N).

Corollary 5.11 (Eckmann–Shapiro isomorphisms). Suppose that Γ′ ≤ Γ are groups and k is a commutative ring of coefficients.

(i) If M is a kΓ-module such that B ⊗_{k} M has finite projective dimension, and N is a kΓ′-module, then for all r ∈ ℤ

Êxt_{kΓ}^{r}(M, Hom_{kΓ′}(kΓ, N)) ≅ Êxt_{kΓ′}^{r}(M ↓_{Γ′}, N).

(ii) If M is a kΓ′-module and N is a kΓ-module, then for all r ∈ ℤ

Êxt_{kΓ}^{r}(M ↑^{Γ}, N) ≅ Êxt_{kΓ′}^{r}(M, N ↓_{Γ′}).

Proof. (i) This follows from Propositions 5.2 and 5.10 (cf. Corollary 3.8 of [10]).

(ii) The usual Eckmann–Shapiro isomorphism for \underline{Hom} commutes with the maps used to define Êxt.

Theorem 5.12. Let Γ be an LH𝔉-group, k a field of coefficients of characteristic p and M a kΓ-module such that B ⊗_{k} M has finite projective dimension (resp. is projective). If Êxt_{kΓ}^{0}(k_{E} ↑^{Γ}, M) = 0 for all finite elementary abelian p-subgroups E ≤ Γ then M has finite projective dimension (resp. is projective).

Proof. By Corollary 5.11, the hypotheses imply that Êxt_{kE}^{0}(k, M ↓_{E}) = 0 for all finite elementary abelian p-subgroups E ≤ Γ. Since k is a field of characteristic p and E is a finite p-group, this implies that M ↓_{E} is projective. By Chouinard’s theorem [7], this implies that M ↓_{G} is projective for all finite subgroups G ≤ Γ. By Theorem 5.7, we deduce that M has finite projective dimension. If B ⊗_{k} M is projective, then by Proposition 5.4 we deduce that M is projective.

6. Complete cohomology for finite groups

In this section, we prove a technical theorem which will be used in the proof of the main theorem of Section 7. Roughly speaking, this says that finite projective dimension can

16 D. J. BENSON

be tested by taking complete Ext with modules obtained by inducing finitely generated modules from finite elementary abelian subgroups. We begin with a lemma which enables us to construct module homomorphisms. Recall that if G is a finite group and N is a kG-module which is projective as a k-module, then N has a k-split embedding into a projective kG-module. For example, one can take the map i ⊗ Id_{M} : M → B ⊗_{k} M (here, B ≅ kG) described in Section 5.

Lemma 6.1. Let G be a finite group and k be a commutative ring of coefficients. Let M and N be kG-modules such that N is projective as a k-module, and let γ : N → N′ be a k-split monomorphism of kG-modules. If ϕ : N → M factors through some projective module, then it factors through the map γ.

Proof. Let ϕ = ϕ_{1} ∘ ϕ_{2} with ϕ_{1} : P → M, ϕ_{2} : N → P, and with P projective. Since P is projective, we may find a k-module endomorphism θ : P → P such that ∑_{g∈G} g ∘θ ∘g^{−1} = Id_{P}. Let ρ : N′ → N be a k-splitting for γ, and consider the map

α = ∑_{g∈G} g ∘ ϕ_{1} ∘ θ ∘ ϕ_{2} ∘ ρ ∘ g^{−1} : N′ → M.

We have

α ∘ γ = ∑_{g∈G} g ∘ ϕ_{1} ∘ θ ∘ ϕ_{2} ∘ ρ ∘ γ ∘ g^{−1}

= ∑_{g∈G} g ∘ ϕ_{1} ∘ θ ∘ ϕ_{2} ∘ g^{−1}

= ϕ_{1} ∘ (∑_{g∈G} g ∘ θ ∘ g^{−1}) ∘ ϕ_{2}

= ϕ_{1} ∘ ϕ_{2} = ϕ.

Remark 6.2. We only apply this when N′ is projective, but this does not seem to be relevant to the proof of the lemma.

Proposition 6.3. Let G be a finite group and k be a commutative ring of coefficients. If M and N are kG-modules such that N has projective dimension at most r over k, and ϕ : N → M is a kG-module homomorphism which goes to zero in \underline{Hom}_{kG}(Ω^{r}(N), Ω^{r}(M)), then ϕ factors through a kG-module X of projective dimension at most r. If Ω^{r}(N) is finitely generated, then X may also be taken to be finitely generated.

Proof. We choose resolutions and maps

[figure: 0; Ω^{r}(N); d′_{r}; Ω^{r}(ϕ); P′_{r−1}; d′_{r−1}; ϕ_{r−1}; ⋯; d′_{1}; P′_{0}; d′_{0}; ϕ_{0}; N; ϕ; 0; 0; Ω^{r}(M); d_{r}; P_{r−1}; d_{r−1}; ⋯; d_{1}; P_{0}; d_{0}; M; 0]

with P_{0}, …, P_{r−1}, P′_{0}, …, P′_{r−1} projective. Since Ω^{r}(N) is k-projective we can also find a k-projective module N′, and kG-projective modules P′′_{0}, …, P′′_{r−1} (which are finitely generated if Ω^{r}(N) is) so that there is a k-split exact sequence of kG-modules

[figure: 0; Ω^{r}(N); d′′_{r}; P′′_{r−1}; d′′_{r−1}; ⋯; d′′_{1}; P′′_{0}; d′′_{0}; N′; 0.]

By repeated use of the lemma, we obtain a commutative diagram

[figure: 0; Ω^{r}(N); d′′_{r}; P′′_{r−1}; d′′_{r−1}; ψ_{r−1}; ⋯; d′′_{1}; P′′_{0}; d′′_{0}; ψ_{0}; N′; ψ; 0; 0; Ω^{r}(N); d′_{r}; P′_{r−1}; d′_{r−1}; ⋯; d′_{1}; P′_{0}; d′_{0}; N; 0.]

We may enlarge N and the original resolution by adding a split exact sequence of projective modules to make the vertical morphisms injective. If we extend ϕ by the zero map on the new projective summand, this does not affect the hypothesis or the conclusion of the theorem. Then the cokernel of ψ is a kG-module with finite projective dimension.

Since Ω^{r}(ϕ) factors through a projective module, we may use the lemma repeatedly to find maps h_{r−1}, …, h_{0}, h_{−1} as in the following diagram

[figure: 0; Ω^{r}(N); d′′_{r}; P′′_{r−1}; d′′_{r−1}; ψ_{r−1}; h_{r−1}; ⋯; d′′_{1}; P′′_{0}; d′′_{0}; ψ_{0}; h_{0}; N′; ψ; h_{−1}; 0; 0; Ω^{r}(N); d′_{r}; Ω^{r}(ϕ); P′_{r−1}; d′_{r−1}; ϕ_{r−1}; ⋯; d′_{1}; P′_{0}; d′_{0}; ϕ_{0}; N; ϕ; 0; 0; Ω^{r}(M); d_{r}; P_{r−1}; d_{r−1}; ⋯; d_{1}; P_{0}; d_{0}; M; 0]

satisfying h_{i−1} ∘ d′′_{i} + d_{i−1} ∘ h_{i} = ϕ ∘ ψ_{i}, for 0 ≤ i ≤ r − 1. It then follows that

ϕ ∘ ψ ∘ d′′_{0} = d_{0} ∘ ϕ_{0} ∘ ψ_{0}
= d_{0} ∘ (h_{−1} ∘ d′′_{0} + d_{1} ∘ h_{0})
= d_{0} ∘ h_{−1} ∘ d′′_{0}.

18 D. J. BENSON

Since d′′_{0} is surjective, it follows that ϕ ∘ ψ = d_{0} ∘ h_{−1}.

We now enlarge N further by replacing it with N ⊕ P_{0}, and we replace ϕ by (ϕ, d_{0}) and ψ by (ψ, −h_{−1}). The effect of this is that ϕ ∘ ψ is now the zero map, so ϕ factors through the cokernel X of ψ, which still has finite projective dimension.

Lemma 6.4. Let R be a ring and M be a countably presented R-module. If M is flat then it has projective dimension at most one.

Proof. See Lemma 4.4 of Bieri [3].

We shall not need the following, but we include it for the sake of interest.

Proposition 6.5. Let G be a finite group and k be a commutative ring of coefficients. If M is a countably presented kG-module which is projective as a k-module and flat as a kG-module, then M is projective as a kG-module.

Proof. By the lemma, M has projective dimension at most one. Let

0 → P_{1} → P_{0} → M → 0

be a projective resolution of M. Since M is projective as a k-module, this sequence splits as a sequence of k-modules. Since P_{1} is weakly injective (cf. Cartan and Eilenberg [6], X.8.6 and XII.1.1), the sequence splits as a sequence of kG-modules, and so M is projective.

Theorem 6.6. Let G be a finite group and k be a commutative Noetherian ring of coefficients of finite global dimension. If M is a countably generated kG-module then the following are equivalent:

(i) M has finite projective dimension.

(ii) M has finite weak dimension.

(iii) M is a filtered colimit of finitely generated kG-modules of finite projective dimension.

(iv) For every finitely generated kG-module N we have Êxt_{kG}^{0}(N, M) = 0.

Proof. (i) ⇒ (iv): If M has projective dimension at most r, then Ω^{r}(M) is projective, so any homomorphism from Ω^{r}(N) to Ω^{r}(M) factors through a projective module.

(iv) ⇒ (iii): Let I be a small skeleton of the category whose objects are the homomorphisms N → M with N finitely generated, and whose morphisms are the commutative triangles. Let J be the full subcategory of I whose objects are the morphisms N → M with N a finitely generated module of projective dimension at most r.

If N is finitely generated, since k is Noetherian, it follows that Ω^{r}(N) may also be taken to be finitely generated. So by Proposition 6.3, every map from N to M factors through a

COMPLEXITY AND VARIETIES FOR INFINITE GROUPS, I 19

finitely generated module of projective dimension at most r. It follows that J is a cofinal subcategory of I. The module M is the filtered colimit of the functor from I to the module category which sends N → M to N. It follows that M is also the filtered colimit of the same functor on the subcategory J.

(iii) ⇒ (ii): It follows from Corollary 5.5 and the fact that k has finite global dimension, that there is an upper bound on the finite projective dimensions of kG-modules. It follows that for some r ≥ 0, M is a filtered colimit of modules of weak dimension at most r.

(ii) ⇒ (i): Since k is Noetherian, it follows from countable generation that M has a resolution by countably generated projective modules. So each Ω^{r}(M) may be taken to be countably generated. Some Ω^{r}(M) is flat, so by Lemma 6.4, Ω^{r+1}(M) is projective.

Corollary 6.7. Let G be a finite group and k be a commutative Noetherian ring of coefficients of finite global dimension. If M is a countably generated kG-module which does not have finite projective dimension, then there exists a finitely generated kG-module N such that Êxt_{kG}^{0}(N, M) ≠ 0.

Remark 6.8. I do not know whether the hypothesis of countable presentability can be dropped in Proposition 6.5, or countable generation in Theorem 6.6 and Corollary 6.7. But at least if k is a field then every flat kG-module is projective (Bass [1]) and so this hypothesis can be dropped in Proposition 6.5, Theorem 6.6 and Corollary 6.7 in this case.^{1}

7. Induced modules

We begin this section by explaining why, for the purpose of examining finitely presented modules and modules of type F P_{∞}, it suffices to consider countable groups. We wish to apply this to modules for an LH𝔉-group, but the method works in greater generality.

Lemma 7.1. Let Γ be a group and k a commutative ring of coefficients. If M is a finitely presented kΓ-module, then there is a finitely generated subgroup Γ′ ≤ Γ and a finitely presented kΓ′-module M′ such that M is isomorphic to the induced module M′ ↑^{Γ}.

Proof. Let (kΓ)^{r} → (kΓ)^{d} → M → 0 be a finite presentation for M. Then the first map in this sequence is represented by a d × r matrix with entries in kΓ. Only a finite number of elements of Γ can be involved in the entries in this matrix. Let Γ′ be the subgroup of Γ generated by these entries. Then we can use the same matrix over kΓ′ to present a

1Note added in proof: It follows from Corollary 5.5, together with proposition 6 of C. U. Jensen: On the vanishing of lim_{⟵}^{(i)}, J. Algebra 15 (1970), 151–166, that the countability hypothesis can be dropped in Theorem 6.6 and Corollary 6.7, as well as in the definitions of StMod(kΓ) and E^{⊕} in Section 8 and Theorem 12.5.

20 D. J. BENSON

module M′: (kΓ′)^{r} → (kΓ′)^{d} → M′ → 0. Since induction is right exact, M is isomorphic to M′ ↑^{Γ}.

Lemma 7.2. Let Γ′ ≤ Γ be groups and k be a commutative ring of coefficients. If M′ is a kΓ′-module then M′ ↑^{Γ} has type F P_{∞} over kΓ if and only if M′ has type F P_{∞} over kΓ′.

Proof. The functor

Ext_{kΓ}^{∗}(M′ ↑^{Γ}, −) ≅ Ext_{kΓ′}^{∗}(M′, −)

commutes with filtered colimits of systems of kΓ-modules if and only if M′ ↑^{Γ} has type F P_{∞} as a kΓ-module (see for example Bieri [3]). By the Mackey decomposition theorem, the functor on the right commutes with filtered colimits of systems restricted from systems of kΓ-modules if and only if it commutes with all filtered colimits of systems of kΓ′-modules, namely if and only if M′ has type F P_{∞} as a kΓ′-module.

Proposition 7.3. Let Γ be a group and k be a commutative ring of coefficients. Let M be a kΓ-module of type F P_{∞}. Then there is a finitely generated subgroup Γ′ ≤ Γ and a kΓ′-module M′ of type F P_{∞} such that M′ ↑^{Γ} is isomorphic to M.

Proof. This follows immediately from Lemmas 7.1 and 7.2.

In the same way, we obtain the following.

Proposition 7.4. Let Γ be a group and k a commutative ring of coefficients. Let M be a countably presented kΓ-module. Then there is a countable subgroup Γ′ ≤ Γ and a countably presented kΓ′-module M′ such that M′ ↑^{Γ} is isomorphic to M.

Corollary 7.5. Let Γ be a group and k a commutative Noetherian ring of coefficients. Let M be a countably presented kΓ-module. Then M has a resolution by countably generated projective kΓ-modules. In particular, Ω^{r}M may be taken to be countably presented for all r ≥ 0.

Proof. Suppose that M is induced from a countably presented module M′ for a countable subgroup Γ′. Every countably generated kΓ′-module is also countably generated as a kmodule. Since k is Noetherian, every submodule of a countably generated k-module is again countably generated. It follows that the same is true for kΓ′-modules. So M′ has a resolution by countably generated projective kΓ′-modules. Inducing up to kΓ gives a resolution of M by countably generated projective kΓ-modules.

Theorem 7.6. Let Γ be an LH𝔉-group and k a commutative Noetherian ring of coefficients of finite global dimension. Let M be a countably presented kΓ-module such that for

COMPLEXITY AND VARIETIES FOR INFINITE GROUPS, I 21

every finite elementary abelian subgroup E ≤ Γ and every finitely generated kE-module N we have Êxt_{kΓ}^{0}(N ↑^{Γ}, M) = 0. Then

(i) M ↓_{G} has finite projective dimension for every finite subgroup G ≤ Γ.

(ii) If M has type F P_{∞} then M has finite projective dimension.

(iii) If B ⊗_{k} M has finite projective dimension then M has finite projective dimension.

Proof. By Proposition 7.4, there exists a countable subgroup Γ′ ≤ Γ and a kΓ′-module M′ such that M′ ↑^{Γ}≅ M. Since N has type F P_{∞} as a kE-module, by Corollary 5.11 we have

Êxt_{kΓ}^{0}(N ↑^{Γ}, M) ≅ Êxt_{kΓ′}^{0}(N ↑^{Γ′}, M ↓_{Γ′}).

If the theorem holds for the kΓ′-module M′ then it holds for the kΓ-module M since M′ is a direct summand of M ↓_{Γ′}. So we may assume without loss of generality that Γ is countable. Since M is countably generated it follows that M ↓_{E} is countably generated. So by applying Theorem 6.6 to the statement

Êxt_{kE}^{0}(N, M ↓_{E}) ≅ Êxt_{kΓ}^{0}(N ↑^{Γ}, M) ≅ 0

we deduce that M ↓_{E} has finite projective dimension for every finite elementary abelian subgroup E ≤ Γ. Now applying Chouinard’s theorem [7], we deduce that M ↓_{G} has finite projective dimension for every finite subgroup G ≤ Γ. If M has type F P_{∞} then we may apply Kropholler’s theorem (3.3) to deduce that M has finite projective dimension as a kΓ-module. Finally, if B ⊗_{k} M has finite projective dimension, we may apply Theorem 5.7 to deduce that M has finite projective dimension.

Remark 7.7. Using the remark following Corollary 6.7, if k is a field of coefficients then we may drop the hypothesis of countable generation from Theorem 7.6.

8. Triangulated categories

Let Γ be a group, and k a commutative Noetherian ring of coefficients. We write mod(kΓ) for the category in which the objects are the kΓ-modules of type F P_{∞}, which have finite projective dimension as a k-module. The arrows from M to N in mod(kΓ) are given by Hom_{kΓ}(M, N). We write stmod(kΓ) for the category with the same objects as mod(kΓ), but whose arrows are given instead by Êxt_{kΓ}^{0}(M, N).

This category is not quite big enough to perform the constructions we need to use in order to apply Rickard’s idempotent functor approach, because we shall need to take countable direct sums. So we define Mod(kΓ) to be the category in which the objects are the countably presented kΓ-modules M such that B ⊗_{k} M has finite projective dimension. By Corollary 7.5, if M is in Mod(kΓ) then so is Ω^{r}M for r ≥ 0. The arrows are given as

22 D. J. BENSON

before by Hom_{kΓ}(M, N). We write StMod(kΓ) for the category with the same objects as Mod(kΓ), but whose arrows are given instead by Êxt_{kΓ}^{0}(M, N). Note that since B is free as a k-module, every module in Mod(kΓ) has finite projective dimension as a k-module. By Corollary 3.6, if Γ is an LH𝔉-group, then mod(kΓ) is a full subcategory of Mod(kΓ), and stmod(kΓ) is a full subcategory of StMod(kΓ).

Remark 8.1. As in Sections 6 and 7, if k is a field of coefficients then we can lift the restriction of countable presentation in the definition of StMod(kΓ).

Proposition 8.2. Let Γ be a group and k a commutative ring of coefficients. For modules M and N in StMod(kΓ) the following are equivalent:

(i) M is isomorphic to N in StMod(kΓ).

(ii) There exist an integer r ≥ 0 and projective kΓ-modules P and Q such that

Ω^{r}(M) ⊕ P ≅ Ω^{r}(N) ⊕ Q.

More generally, given maps M → N and N → M in StMod(kΓ) whose composite is the identity map on N, they are represented, for some r ≥ 0, by an isomorphism

Ω^{r}(M) ⊕ P ≅ Ω^{r}(N) ⊕ Q

where P is projective, but Q need not be.

Proof. It is clear from the definition of Êxt_{kΓ}^{0}(M, N) that (i) implies (ii). Conversely, if M and N are isomorphic in StMod(kΓ), then there exists an integer r and maps ϕ : Ω^{r}(M) → Ω^{r}(N) and ψ : Ω^{r}(N) → Ω^{r}(M) such that ψ ∘ ϕ = Id_{Ω^{r}(M)} + π_{1} ∘ θ_{1}, and ϕ ∘ ψ = Id_{Ω^{r}(N)} + π_{2} ∘ θ_{2}. Here, π_{1} : P_{1} → Ω^{r}(M), π_{2} : P_{2} → Ω^{r}(N), θ_{1} : Ω^{r}(M) → P_{1}, θ_{2} : Ω^{r}(N) → P_{2}, and P_{1} and P_{2} are projective. In order to prove the last part of the proposition as well, let us for the moment only assume the condition on ϕ ∘ ψ, and impose the condition on ψ ∘ ϕ later. By enlarging Ω^{r}(M) if necessary by adding a projective module P, we may assume that ϕ is surjective. So we may factor π_{2} as ϕ ∘ π′_{2} with π′_{2} : P_{2} → Ω^{r}(M). Hence

Id_{Ω^{r}(N)} = ϕ ∘ (ψ − π′_{2} ∘ θ_{2})

and so ϕ is a split surjection. It follows that Ω^{r}(M) is isomorphic to Ω^{r}(N) ⊕ Q, where Q = Ker(ψ − π′_{2} ∘ θ_{2}).

Finally, if we also impose the condition on ψ ∘ ϕ, then on the submodule Q of Ω^{r}(M), the identity map can be written as

ψ ∘ ϕ − π_{1} ∘ θ_{1} = π′_{2} ∘ θ_{2} ∘ ϕ − π_{1} ∘ θ_{1}.

COMPLEXITY AND VARIETIES FOR INFINITE GROUPS, I 23

It follows that the identity map on Q is a difference of maps which factor through a projective module, so Q is projective.

Next, let Mod_{c}(kΓ) be the full subcategory of Mod(kΓ) consisting of the cofibrant (countably presented) kΓ-modules (see Definition 4.1), and let StMod_{c}(kΓ) be the corresponding full subcategory of StMod(kΓ). We make similar definitions for mod_{c}(kΓ) and stmod_{c}(kΓ).

Lemma 8.3. Let Γ be a group and k a commutative ring of coefficients. The inclusion of the subcategory StMod_{c}(kΓ) into StMod(kΓ) is an equivalence of categories. If Γ is an LH𝔉-group then the inclusion of the subcategory stmod_{c}(kΓ) into stmod(kΓ) is an equivalence of categories.

Proof. Let M be a module in StMod(kΓ) (resp. stmod(kΓ)). First, if B ⊗_{k} M is projective, then it has a countably generated (resp. finitely generated) projective summand which contains the image of i ⊗ Id_{M} : M → B ⊗_{k} M. So B̄ ⊗_{k} M plus a projective module is isomorphic to a direct sum of a countably generated (resp. finitely generated) module and a projective module. It therefore gives rise to a well defined object in StMod_{c}(kΓ), which we may as well continue to write as B̄ ⊗_{k} M.

If B⊗_{k}M has finite projective dimension, then for some r ≥ 0, B⊗_{k}Ω^{r}(M) is projective. Then M is isomorphic in StMod(kΓ) to B̄ ⊗_{k} ⋯ ⊗_{k} B̄ ⊗_{k} Ω^{r}(M) (r copies of B̄), which lies in StMod_{c}(kΓ).

If Γ is an LH𝔉-group, the same argument applies to stmod_{c}(kΓ) and stmod(kΓ) by Corollary 3.6.

Remark 8.4. From now on, we use the above convention that for a module M in the subcategory StMod_{c}(kΓ) (resp. stmod_{c}(kΓ)), B̄ ⊗_{k} M denotes the object in StMod_{c}(kΓ) (resp. stmod_{c}(kΓ) which it determines as in the proof of the above lemma.

Lemma 8.5. Let Γ be a group and k a commutative ring of coefficients. For modules M and N in StMod_{c}(kΓ), the natural map \underline{Hom}_{kΓ}(M, N) → Êxt_{kΓ}^{0}(M, N) is an isomorphism.

Proof. It follows from Lemma 4.2 that the map

\underline{Hom}_{kΓ}(M, N) → \underline{Hom}_{kΓ}(ΩM, ΩN)

is an isomorphism.

Since it does not make sense to ask whether a morphism in StMod(kΓ) is injective or surjective, we do not expect this to be an abelian category. The correct formalism is that of triangulated categories. For background on triangulated categories, we refer the reader

24 D. J. BENSON

to the Appendix of [19] or Chapter 10 of [12]. The substitute for a short exact sequence in an abelian category is a distinguished triangle in a triangulated category. The translation functor on StMod(kΓ) plays the role of an inverse to the functor Ω, and is defined as follows. Replace M by a module M_{0} in StMod_{c}(kΓ) isomorphic to it, and define τ(M) = B̄ ⊗_{k} M_{0}. This is regarded as an element of StMod_{c}(kΓ) as in the proof of Lemma 8.3 The distinguished triangles in StMod(kΓ) are the triangles which are isomorphic to triangles of the form

M → M′ → M′′ → τ(M)

where

0 → M ⟶^{ϕ} M′ ⟶^{ψ} M′′ → 0

is a short exact sequence whose modules lie in StMod_{c}(kΓ). Given such a short exact sequence, we can turn M′ → M′′ into an injective map by replacing M′′ by the module M′′ ⊕ (B ⊗_{k} M′) (which is isomorphic to it in StMod_{c}(kΓ)). Writing M′′′ for the cokernel, the diagram

[figure: 0; 0; 0; M; B ⊗_{k} M′; 0; M′; M′′ ⊕ (B ⊗_{k} M′); M′′′ → 0; M′′; M′′; 0; 0]

shows that there is a short exact sequence

0 → M → B ⊗_{k} M′ → M′′′ → 0

and so M′′′ is isomorphic in StMod_{c}(kΓ) to τ(M) = B̄ ⊗_{k} M. This explains what to take for the third morphism in the triangle

M → M′ → M′′ → τ(M).

When it is more convenient to do so, we shall omit the last term of a triangle and just write

M → M′ → M′′ → .

COMPLEXITY AND VARIETIES FOR INFINITE GROUPS, I 25

The remaining axioms for a triangulated category are easy to verify for StMod(kΓ). For example, the octahedral axiom corresponds to the third isomorphism theorem in the module category.

If Γ is an LH𝔉-group, then the subcategory stmod(kΓ) is a thick subcategory of StMod(kΓ). In other words, it is triangulated with the same definition for the distinguished triangles, and it is closed under taking direct summands. As in the proof of Lemma 8.3, if M has type F P_{∞} then although B̄ ⊗_{k} M may not have type F P_{∞}, it is isomorphic to a direct sum of a module of type F P_{∞} and a projective module. It follows that the translation τ may be defined in the same way on stmod(kΓ) as on StMod(kΓ).

Countable direct sums in StMod(kΓ) need some discussion. It is possible that a countable direct sum of modules M_{i} in StMod(kΓ) is no longer in StMod(kΓ), because there need be no upper bound on the projective dimensions of the modules B ⊗_{k} M_{i}. However, StMod_{c}(kΓ) is closed under countable direct sums, and so StMod(kΓ) has countable direct sums in the categorical sense.

9. Derived categories

This section and the next are not necessary to the logical structure of this paper, but is included for the sake of providing another point of view on the categories stmod(kΓ) and StMod(kΓ). Namely, we display these categories in a natural way as quotients of appropriate derived categories D^{b}(mod(kΓ)) and D^{b}(Mod(kΓ)).

Let Γ be a group and k a commutative ring of coefficients. Following Verdier’s Appendix to SGA4\frac{1}{2} [12] (but reversing the arrows), we write K^{+,b}(Mod(kΓ)) for the category whose objects are chain complexes of objects in Mod(kΓ), which are bounded below in the sense that they are zero in large enough negative degrees (this corresponds to the first index +), and whose homology is bounded both above and below (second index b). The arrows are chain homotopy classes of maps. We write K^{b,b}(Mod(kΓ)) for the full subcategory whose objects are chain complexes bounded both above and below, and K^{b,ϕ}(Mod(kΓ)) for the full subcategory with the further restriction that the homology is zero (i.e., exact complexes). The derived category D^{b}(Mod(kΓ)) is the quotient category K^{b,b}(Mod(kΓ))/K^{b,ϕ}(Mod(kΓ)), formed by formally inverting quasi-isomorphisms (maps inducing isomorphisms in homology).

Writing Proj(kΓ) for the full subcategory of Mod(kΓ) whose objects are the (countably generated) projective modules, we can form the categories K^{+,b}(Proj(kΓ)), K^{b,b}(Proj(kΓ)), K^{b,ϕ}(Proj(kΓ)) and D^{b}(Proj(kΓ)) in the same way. It is easy to see that K^{b,ϕ}(Proj(kΓ)) is the trivial category with one object and one arrow, so that K^{b,b}(Proj(kΓ)) and D^{b}(Proj(kΓ))

26 D. J. BENSON

are equivalent. Since Mod(kΓ) has enough projectives, we may replace objects by projective resolutions. Given any object X_{∗} in K^{b,b}(Mod(kΓ)), there is an object P_{∗} in K^{+,b}(Proj(kΓ)) and a map of chain complexes P_{∗} → X_{∗} inducing an isomorphism on homology. Such a map is called a projective resolution of X_{∗}. One can show (cf. Proposition II.1.4 of [12], Appendix) that taking projective resolutions induces an equivalence of categories

D^{b}(Mod(kΓ)) → K^{+,b}(Proj(kΓ)).

Now, D^{b}(Proj(kΓ)) is a thick subcategory of D^{b}(Mod(kΓ)), and so we may form the quotient category

D^{b}(Mod(kΓ))/D^{b}(Proj(kΓ)).

Under the above equivalence of categories, this quotient is equivalent to

K^{+,b}(Proj(kΓ))/K^{b,b}(Proj(kΓ)).

In this quotient, truncating a chain complex of projectives by replacing the (finitely many) modules below a given degree by zero results in an isomorphic complex. In this way, we can truncate above the homology, so that every complex is isomorphic to a complex which is exact except in a single degree, and zero below that degree. Such a complex is a projective resolution of a module in Mod(kΓ), possibly shifted in degree.

There is a functor

Mod(kΓ) → K^{+,b}(Proj(kΓ))

which sends a module to a resolution by finitely generated projective modules. Such a resolution is well defined up to chain homotopy equivalence, namely up to isomorphism in K^{+,b}(Proj(kΓ)).

Any homomorphism Ω^{r}M → Ω^{r}N in Mod(kΓ) gives rise to a homomorphism in the quotient category K^{+,b}(Proj(kΓ))/K^{b,b}(Proj(kΓ)), by truncating resolutions as described above; if the map factors through a projective module then it goes to zero. This process is compatible with the natural maps

\underline{Hom}_{kΓ}(Ω^{r}M, Ω^{r}N) → \underline{Hom}_{kΓ}(Ω^{r+1}M, Ω^{r+1}N),

and so we obtain a well defined functor

Φ : StMod(kΓ) → K^{+,b}(Proj(kΓ))/K^{b,b}(Proj(kΓ)).

If a homomorphism Ω^{r}M → Ω^{r}N goes to the zero homomorphism under Φ, then for some s ≥ 0 the image in \underline{Hom}_{kΓ}(Ω^{r+s}M, Ω^{r+s}N) is zero. Furthermore, every homomorphism between resolutions in the quotient category is induced by a map from Ω^{r}M to Ω^{r}N for some r ≥ 0. It follows that Φ induces an isomorphism on Hom sets. Every object in the quotient category can be truncated as described above to give a projective resolution of some module M sitting in some degree d ≥ 0, and is hence isomorphic to the image under Φ of the translate τ^{d}(M) in StMod(kΓ). It follows that Φ is an equivalence of categories.

If Γ is an LH𝔉-group, exactly the same discussion applies to the categories mod(kΓ) of kΓ-modules of type F P_{∞}, and its full subcategory proj(kΓ) of finitely generated projective kΓ-modules. We obtain equivalences

D^{b}(mod(kΓ)) → K^{+,b}(proj(kΓ)),
D^{b}(mod(kΓ))/D^{b}(proj(kΓ)) → K^{+,b}(proj(kΓ))/K^{b,b}(proj(kΓ))

and

stmod(kΓ) → K^{+,b}(proj(kΓ))/K^{b,b}(proj(kΓ)).

These equivalences are compatible with the inclusions in the larger versions. Summarizing, we have the following theorem.

Theorem 9.1. Let Γ be a group and k be a commutative Noetherian ring of coefficients. Then there is an equivalence of categories

StMod(kΓ) → D^{b}(Mod(kΓ))/D^{b}(Proj(kΓ)).

If Γ is an LH𝔉-group, there is an equivalence of categories

stmod(kΓ) → D^{b}(mod(kΓ))/D^{b}(proj(kΓ)).

These equivalences are compatible with the inclusions induced by the inclusion of mod(kΓ) as a full subcategory of Mod(kΓ).

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

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