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HOMOLOGICAL FINITENESS CONDITIONS FOR MODULES OVER GROUP ALGEBRAS JONATHAN CORNICK AND PETER H. KROPHOLLER A BSTRACT. We develop a theory of modules of type FP∞ over group algebras of hierarchically decomposable groups. This class of groups is denoted HF and contains many different kinds of discrete groups including all countable polylinear groups. Amongst various results, we show that if G is an HF-group and M is a ZG-module of type FP∞ then M has finite projective dimension over ZH for all torsion-free subgroups H of G. We also show that if G is an HF-group of type FP∞ and M is a ZGmodule which is ZF-projective for all finite subgroups F of G, then M has finite projective dimension over ZG. Both of these results have as a special case the striking fact that if G is an HF-group of type FP∞ then the torsion-free subgroups of G have finite cohomological dimension. A further result in this spirit states that every residually finite HF-group of type FP∞ has finite virtual cohomological dimension.

1. I NTRODUCTION Let G be a group and let K be a commutative ring. The primary goal of this paper is to study relationships amongst homological finiteness conditions for KG-modules and we obtain substantial results in the case when G belongs to the class HF of hierarchically decomposable groups which was introduced in [10]. Towards the end of the paper we review the class HF, showing that it contains all countable linear groups. The results are applied to study groups which admit Eilenberg-Mac Lane spaces of finite type. There are two homological finiteness conditions, defined for modules over any ring R, which play a crucial role throughout the paper. First, an R-module M is said to be of type FP∞ if it admits a projective resolution in which every module is finitely generated. Secondly, M is said to have finite projective dimension if it admits a projective resolution of finite length. At first sight these two conditions seem to address different qualitative properties of projective resolutions, and one would not expect either one to imply the other. In this paper we focus attention on circumstances in which the FP∞ property implies finiteness of projective dimension or at least some closely related property. In general, one would naturally expect there to be The first author was supported by an SERC Research Assistantship and a CRM Postdoctoral Fellowship. 1

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many modules of type FP∞ which do not have finite projective dimension, and this phenomenon arises naturally for many Noetherian rings, because then every finitely generated module has type FP∞ . In particular, if G is a non-trivial finite group then every finitely generated ZG-module is of type FP∞ and most have infinite dimension. By contrast, it was shown in [10] that if G is a torsion-free HF-group then every ZG-module of type FP∞ has finite projective dimension. This result is significant partly because the class HF is large. In addition to containing countable linear groups of arbitrary characteristic, (established in §9 below in the positive characteristic case), it was shown in [10] that HF is extension closed, subgroup closed, closed under countable directed unions and closed under amalgamated free products and HNN-extensions. Given these properties one sees that HF contains a large proportion of countable discrete groups presently studied in the literature. However, one shortcoming of the results in [10] is that they give only limited information about ZG-modules of type FP∞ in the case when G is an HF-group which is not torsion-free. When torsion is present, expectations are naturally limited because of the following simple fact: Lemma 1.1. Let G be a group. If every ZG-module of type FP∞ has finite projective dimension then G is torsion-free. Proof. Suppose that G has a non-trivial finite subgroup F. Then the induced module IndG  F Z is of type FP∞ and has infinite projective dimension. Nevertheless, many positive results can be obtained. For example, we shall see that if G is an HF-group and H is a torsion-free subgroup of G then every ZG-module of type FP∞ has finite projective dimension as a ZH-module. The results of this paper are of particular interest when applied to groups of type FP∞ . Recall that a group G is said to be of type FP∞ if the trivial module Z is of type FP∞ as a ZG-module. Groups with this property arise naturally in topology, because a group admits an Eilenberg-Mac Lane space of finite type if and only if it is finitely presented and of type FP∞ . A very special case of results in [10] shows that every torsion-free HF-group of type FP∞ has finite cohomological dimension. The results of this paper allow us to make conclusions for arbitrary HF-groups of type FP∞ . As examples, we can draw the following conclusions about such a group G: • Every torsion-free subgroup of G has finite cohomological dimension. [Corollary B2(1) of §5] This is of interest in relation to the following remark of Bieri, [4, page 186]:

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“I suspect that if G is a group of type FP∞ then . . . the torsion-free subgroups of G are of finite cohomology dimension.” Although the Brown-Geoghegan example [5] is a counterexample to the general assertion, it is nevertheless of interest that the assertion is true for HF-groups. • ZG has finite finitistic dimension, meaning that there is a bound on the projective dimensions of the modules of finite projective dimension. [Corollary C §6] • A ZG-module M has finite projective dimension if and only if its restriction to every finite subgroup has finite projective dimension. [Corollary C §6] It is interesting to compare this with the results of [10] which show that the same conclusion holds when G is an arbitrary HF-group and M is a ZG-module of type FP∞ . • If G is residually finite then it has finite virtual cohomological dimension. [Theorem D §7] These results are special cases of more general theorems for modules over HF-groups which are proved in §§5,6 and 7. In §§2 and 3 we review the Vogel-Mislin complete cohomology theory, already used in [10] in this context, and we establish the crucial vanishing theorem, Theorem A, in §3. A technical device using bounded functions on G-sets is set up in §4: this is based on earlier work of Kropholler and Talelli [12], and proves invaluable as a tool for use with complete cohomology. In §8 we prove that HF contains linear groups in positive characteristic, and in the short conclusion we discuss some simple applications of the theory. 2. C OMPLETE COHOMOLOGY The main tool for proving that certain modules have finite projective dimension is the complete cohomology theory introduced by Vogel [8] and c nR (A, B) for the nth axiomatized by Mislin [13]. In general we write Ext complete Ext group associated with R-modules A and B. The following criterion, proved in [10], is the basic criterion which we need. Lemma 2.1. Let R be a ring and let M be an R-module. Then M has finite c 0R (M, M) = 0. projective dimension if and only if Ext In practice, this criterion is not always easy to apply directly, and so for modules over group algebras we use an embellishment of it which is more convenient. To this end, we first describe some of the background. There is a natural map from the ordinary Ext group to the complete Ext group: c nR (A, B), and in the special case when n = 0 and A = B this ExtnR (A, B)→Ext

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natural map is a ring homomorphism: the ordinary Ext group reduces to the endomorphism ring homR (A, A), and the complete Ext group also admits a natural ring structure. This is particularly useful in the context of of Lemma c 0R (M , M) in 2.1 because it is sufficient to prove that 1 = 0 in the ring Ext

c 0R (M, M) = 0. When R is a group algebra there is order to establish that Ext a useful way of exploiting this observation. Let G be a group and let K be a commutative ring. If M is a KG-module and B is a ZG-module then we can make M ⊗ B into a KG-module using the diagonal action of G. In the first instance we shall apply this when B is a G-operator ring; that is, when B is a ring which admits an action of G by ring automorphisms. In this situation, the ring multiplication µ : B ⊗ B→B is a G-module map which splits the inclusion B→B ⊗ B given by b 7→ b ⊗ 1, and thus B is a ZG-direct summand of B ⊗ B. Based on this, we have the following variation of Lemma 2.1 which will be very useful throughout the paper. Lemma 2.2. Let B be a G-operator ring which is free as an additive group. c 0KG (M, M ⊗ B) = 0 then M ⊗ B has finite If M is a KG-module such that Ext projective dimension. To prove Lemma 2.2 we shall use Vogel’s definition of complete cohomology together with the following well known Lemma. The proof is omitted. Lemma 2.3. If P is a projective KG-module and B is a ZG-module which is free as an additive group then P ⊗ B is projective as a KG-module. c 0KG (M ⊗ Proof of Lemma 2.2. By Lemma 2.1 it suffices to show that Ext B, M ⊗ B) = 0. Consider the following diagram in which the horizontal maps are the natural maps: HomKG (M, M ⊗ B)

/ c0 ExtKG (M, M ⊗ B)



 / c0 ExtKG (M ⊗ B, M ⊗ B ⊗ B)

HomKG (M ⊗ B, M ⊗ B ⊗ B) 

HomKG (M ⊗ B, M ⊗ B)

 / c0 ExtKG (M ⊗ B, M ⊗ B)

To understand the top right hand vertical map we need to briefly review Vogel’s approach to complete cohomology. We refer the reader to [8] and [3] for a detailed discussion of this theory. From this point of view, the comc 0KG (M, M ⊗ B) is defined as follows. First let plete cohomology group Ext

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P∗  M and Q∗  M ⊗ B be projective resolutions of M and M ⊗ B. Then c 0KG (M, M ⊗ B) can be defined to be the group of almost chain maps (of Ext degree zero) from P∗ to Q∗ modulo the group of almost chain homotopies. Now, tensoring with B, and using Lemma 2.3, we obtain projective resolutions P∗ ⊗ B  M ⊗ B and Q∗ ⊗ B  M ⊗ B ⊗ B of M ⊗ B and M ⊗ B ⊗ B. Moreover, any almost chain map P∗ →Q∗ induces an almost chain map P∗ ⊗ B→Q∗ ⊗ B, the same holds for almost chain homotopies, and hence there is an induced map between the complete cohomology groups as in the diagram. In effect, this map amounts to cup product with the identity element of Ext0KG (B, B), hence the notation − ^ 1, and the resulting square at the top of the diagram commutes. The product structure of the ring B induces the vertical maps making the bottom commutative square of the diagram. Now consider the element ι : M→M ⊗ B of HomKG (M, M ⊗ B) defined by m 7→ m ⊗ 1. Following round the left and bottom of this diagram, it is easy to see that this is mapped to the identity element of the endomorphism ring HomKG (M ⊗ B, M ⊗ B) and hence to the identity element of the ring c 0KG (M ⊗B, M ⊗B). On the other hand, following round the top and right, Ext 0

c KG (M, M ⊗ B) which is zero by hypothesis. the element passes through Ext 0 c KG (M ⊗ B, M ⊗ B) and Ext c 0KG (M ⊗ B, M ⊗ B) = 0 as Hence 0 = 1 in Ext required.  Remark 2.4. For purposes in §7 below we shall need a slightly more general formulation of Lemma 2.2. Notice that in the proof we do not need to know c 0KG (M, M ⊗B) itself is zero, but only that the inclusion ι : M→M ⊗B that Ext 0

c KG (M, M ⊗ B). maps to zero under the natural map HomKG (M, M ⊗ B)→Ext Therefore, even under this weaker hypothesis, one can still conclude that M ⊗ B has finite projective dimension. 3. T HE VANISHING THEOREM In order to use the criteria for finiteness of projective dimension described in §2 we need theorems which show that complete cohomology vanishes under certain circumstances. The main goal of his section is to establish such a result. In fact it can be deduced fairly easily from results in [10]. First we recall the following crucial underlying result, ([10, (4.1)(ii)]). Lemma 3.1. Let R be a ring and let M be an R-module of type FP∞ . Then c 0R (M, ) is continuous. the functor Ext As in [10], we say that the functor is continuous to mean that given any direct limit system (Nλ | λ ∈ Λ) of modules over a directed poset Λ, the

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natural map c 0R (M, N )→Ext c 0R (M, lim N ) lim Ext λ λ −→ −→ λ

λ

is an isomorphism. Lemmas 2.1, 2.2 and 3.1 allow us to translate questions about the two basic homological finiteness conditions into questions about complete cohomology, and it turns out that once this translation is made all our results follow from facts about complete cohomology. We use the following vanishing theorem: Theorem A . Suppose that G belongs to HF. Let M be an KG-module of type FP∞ and let N be an KG-module which has finite projective dimension c iKG (M, N) = 0 as an KF-module for all finite subgroups F of G. Then Ext for all i ∈ Z. c iKG (M, ) form a (−∞, ∞)-cohomological functor Proof. The functors Ext from KG-modules to K-modules, and they are continuous by virtue of the fact that M is of type FP∞ over KG — cf. [10, (4.1)]. By way of a conc iKG (M , N) is non-zero for some i. According tradiction, suppose that Ext to [10, (3.2)], there is a finite subgroup F of G and an integer j ≥ i such j c KG that Ext (M, IndG F N) is non-zero. Since N has finite projective dimension as a KF-module, it follows that IndG F N has finite projective dimension as an KG-module. Thus we have a contradiction to the fact that the complete j c KG cohomology Ext (M, ) vanishes on modules of finite projective dimension.  4. B OUNDED FUNCTIONS ON G- SETS In this section we describe a source of G-operator rings which can be used in conjunction with Lemma 2.2. When using ordinary cohomology, coinduced modules play an important role because there is the Eckmann-Shapiro Lemma. But there is no known analogue of this lemma for complete cohomology and although coinduced modules are still useful, they are less effective. Once one accepts that the Eckmann-Shapiro Lemma has to be avoided it becomes natural to use a variant of the coinduced module which has other advantageous properties. The basic ideas behind this were first presented in [12]. Here, we shall make use of G-sets to describe the variant. If ∆ is a G-set (with G acting on either the left or the right) and H is a subgroup of G then we write Hδ for the stabilizer of any δ ∈ ∆ in H. We shall use the following notation: Notation 4.1. We write H{∆} for the set {Hδ | δ ∈ ∆}.

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If ∆ is a right G-set then Z∆, the free abelian group on ∆ is a right ZGmodule which behaves in many respects like an induced module. We make the following crucial definition of our variant coinduced module. Definition 4.2. Let ∆ be a set. We write B(∆, Z) for the ring of bounded Z-valued functions on ∆. If ∆ is a left G-set then a right action of G on B(∆, Z) is defined by φ g (δ ) = φ (gδ ) for all g ∈ G and δ ∈ ∆, and in this way B(∆, Z) becomes a G-operator ring.. If H is a subgroup of G then CoindG H Z can be identified with the ZGmodule comprising all Z-valued functions on the left G-set G/H and IndG HZ can be identified as the submodule comprising functions with finite support. Thus we have natural inclusions G IndG H Z ⊆ B(G/H, Z) ⊆ CoindH Z.

We shall see that B(G/H, Z) has some of the properties of the induced module and some of the properties of the coinduced module. These features make it ideally suited for use with complete cohomology. The main goal of this section is to establish the following result. Proposition 4.3. Let M be an KG-module, let ∆ be a left G-set and let F be a finite subgroup of G. Write B := B(∆, Z). If M has finite projective dimension as an KL-module for all L ∈ F{∆} then M ⊗ B has finite projective dimension as an KF-module. The remainder of this section is devoted to proving this, and then only the Proposition itself is used in subsequent sections. We begin by recalling the following remarkable result which is a special case of N¨obeling’s theorem, [6, Corollary 97.4]. Lemma 4.4. For any set Γ, B(Γ, Z) is free as an additive group. Remark 4.5. This lemma can be deduced very directly from Bergman’s lovely result [6, Lemma 97.2], by observing that B(Γ, Z) is an additively torsion-free commutative ring generated as an additive group by its idempotents. If Γ is an infinite set then the additive group A of all Z-valued functions on Γ is not free abelian — cf. [9, §15, Theorem 21], and this is why Lemma 4.4 is both remarkable and useful. Notice that B(Γ, Z) is precisely the additive subgroup of A generated by idempotents. Using Lemma 4.4 we can establish the properties of B(∆, Z) which are crucial in this paper. Lemma 4.6. Let ∆ be a left G-set and let B denote the G-operator ring B(∆, Z). Then, for each finite subgroup F of G, there is a right F-set ∆0 such that F{∆} = F{∆0 } and B is isomorphic to Z∆0 as a ZF-module.

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Proof. Let Γ be a set of F-orbit representatives in ∆. Since F has only finitely many subgroups we can partition Γ into a finite family of subsets Γ1 , . . . , Γr such that for each i all elements of Γi have the same stabilizer in F. Let ∆1 , . . . , ∆r be the corresponding partition of ∆ into F-sets. For each i let Fi be the stabilizer of a point of Γi . We first prove that for each i, B(∆i , Z) is isomorphic to IndFFi B(Γi , Z), where B(Γi , Z) is viewed as a trivial ZFi -module. Let Ti be a left transversal to Fi in F, so that F is the disjoint union of cosets tFi . For φ ∈ B(∆i , Z) and f ∈ F, let φ f : Γ→Z be the function defined by φ f (γ) = φ ( f γ). Thus φ f is the restriction of φ f to Γ, and for f , f 0 ∈ F we have (φ f ) f 0 = φ f f 0 . It is easy to check that the map from B(∆i , Z) to IndFFi B(Γi , Z) defined by φ 7→ ∑ φt ⊗ t −1 is a ZF-module isomorphism. t∈T

Now we can define ∆0 as follows. For each i, let Xi be a free basis of B(Γi , Z) and let ∆0 be the disjoint union of the right F-sets Xi × Fi \F. Then ZXi = B(Γi , Z), Z[Xi × Fi \F] can be identified with IndFFi B(Γi , Z), and so we have B(∆, Z) =

r M i=1

B(∆i , Z) ∼ =

r M

IndFFi B(Γi , Z) = Z∆0 ,

i=1

as required.



We need the following result about modules for finite groups which is no doubt well known. Lemma 4.7. Let F be a finite group, and let M be a KF-module. (1) If proj. dimKF M < ∞ then proj. dimKF M = proj. dimK M. (2) If ∆0 is a right F-set such that proj. dimKL M < ∞ for all L ∈ F{∆0 } then proj. dimKF M ⊗ Z∆0 < ∞. Proof. (1) Clearly proj. dimK M ≤ proj. dimKF M. For the reverse inequality, let n := proj. dimK M and consider Ωn M. Then Ωn M is projective as an Kmodule and has finite projective dimension as KF-module. It follows from [2, Corollary 3.6.5] that Ωn M is projective as an KF-module, and hence M has projective dimension at most n. (2) The ZF-module Z∆0 is the direct sum of modules of the form IndFL Z with L ∈ F{∆0 }, one such for each F-orbit of ∆0 . Now M ⊗ IndFL Z with diagonal F-action is isomorphic to the induced KF-module IndFL M and so if proj. dimKL M < ∞ then proj. dimKF M ⊗ IndFL Z < ∞. Part (1) shows that there is a uniform bound on these projective dimensions and hence proj. dimKF M ⊗ Z∆0 < ∞ as required. 

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Proof. Proof of Proposition 4.3 By Lemma 4.6, there is a right F-set ∆0 such that F{∆0 } = F{∆} and B(∆, Z) is ZF-isomorphic to Z∆0 . Thus, if M has finite projective dimension as KL-module for all L ∈ F{∆} then M ⊗ B∼ = M ⊗ Z∆0 has finite projective dimension as an KF-module by Lemma 4.7(2).  5. M ODULES OF TYPE FP∞

OVER GROUP ALGEBRAS OF ARBITRARY HF- GROUPS

Let G be an HF-group and let M be a KG-module of type FP∞ . Then the conclusion of Theorem A states that c 0KG (M, ) vanishes on those modules N which have finite projective (*) Ext dimension over KF for all finite subgroups F of G. It turns out that many of our conclusions about the module M follow solely from this property (*). All the arguments hinge on showing that for certain carefully chosen G-operator rings B, the module M ⊗ B has finite projective dimension. When using the rings of bounded functions described in §4, conclusions can then be drawn from the following general observation. Lemma 5.1. Let ∆ be a G-set and let B = B(∆, Z). If proj. dimKG M ⊗ B = n < ∞ then proj. dimKG M ≤ n for all δ ∈ ∆. δ

Proof. Fix δ ∈ ∆. Let eδ : B→Z be the map given by evaluation at δ . Then 1 ⊗ eδ : M ⊗ B→M is a KGδ -map which splits the natural map M→M ⊗ B, and hence M is a KGδ -module direct summand of M ⊗ B. The Lemma now follows.  It is now easy to establish one of the main conclusions about KG-modules of type FP∞ . c 0KG (M, ) Theorem B . Let ∆ be a G-set and let B = B(∆, Z). Suppose that Ext satisfies (*). Then the following are equivalent: (1) proj. dimKE M < ∞ whenever E is a finite subgroup of a stabiliser Gδ ; (2) proj. dimKF M ⊗ B < ∞ whenever F is a finite subgroup of G; (3) proj. dimKG M ⊗ B < ∞; and (4) there is a natural number n such that proj. dimKH M ≤ n whenever H is a subgroup of a stabliser Gδ . Proof. (1)⇒(2) is immediate from Proposition 4.3. c 0KG (M, M ⊗ B) = 0 and hence M ⊗ B (2)⇒(3). Property (*) shows that Ext has finite projective dimension by Lemma 2.2. (3)⇒(4) follows from Lemma 5.1.

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(4)⇒(1) trivially.

c 0KG (M , ) satisfies (*). Then M has the Corollary B . Suppose that Ext following properties: (1) If H is a subgroup of G then proj. dimKH M < ∞ if and only if proj. dimKF M < ∞ for all finite subgroups F of H. (2) If proj. dimK M < ∞ then there is an n such that proj. dimKH M ≤ n for all torsion-free subgroups H of G. (3) Let B := B(G, Z) be the G-operator ring of bounded Z-valued functions on G. Then proj. dimKG M ⊗B < ∞ if and only if proj. dimK M < ∞. (4) If proj. dimK M < ∞ then there exists m such that for all n ≥ m, (a) ExtnKG (M, P) = 0 for all n ≥ m and all projective KG-modules P, and c nKG (M , ) is an isomor(b) the natural map ExtnKG (M , )→Ext phism. Proof. (1) Suppose that M has finite projective dimension over KF for all finite F ≤ H. If F and F 0 are conjugate subgroups of G then proj. dimKF M = proj. dimKF 0 M, so M has finite projective dimension over any finite subgroup which is conjugate to a subgroup of H. Let ∆ be the left G-set G/H of left cosets of H. Finite subgroups of stabilisers are conjugate to finite subgroups of H so (1) of Theorem B is satisfied, and the result follows from Theorem B(1)⇒(4). (2) Choose a left G-set ∆ with the property that G{∆} is the set of torsion-free subgroups of G, and let B := B(∆, Z) as in Theorem B. Then Theorem B(1) is satisfied and hence M ⊗ B has finite projective dimension over KG. The desired conclusion now follows from Lemma 5.1. (3) This follows from Theorem B(1)⇔(3). (4) (a) Let B := B(G, Z). For any KG-module N let Hom(B, N) denote the KG-module comprising all additive homomorphisms from B to N, with diagonal action of G. Then one has a natural isomorphism Hom (M ⊗ B, N) ∼ = Hom (M, Hom(B, N)), KG

KG

and this extends to an isomorphism in (ordinary) cohomology:

Extn M ⊗ B, N ∼ = Extn M, Hom(B, N) . KG

KG

By part (3) of this result, M ⊗
B has finite projective dimension, n and hence ExtKG M ⊗ B, N is zero for all sufficiently large n.

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From the isomorphism, it follows that ExtnKG M, Hom(B, N) is zero for all sufficiently large n. Now suppose that P is a projective KG-module. The natural inclusion Z→B induces a surjective KG-homomorphism Hom(B, P)→P. Thus P, being projective, is a direct summand of Hom(B, P) and hence
ExtnKG (M, P) is a direct summand of ExtnKG M, Hom(B, P) . (b) It follows from Mislin’s satellite functor approach to complete cohomology that if ExtnKG (M, ) vanishes on projective modc nKG (M, ) coincide ules for all n ≥ m then ExtnKG (M, ) and Ext when n ≥ m. 

This completes the proof.

Specialising these results to the case K = Z and M = Z, we can state some results about HF-groups of type FP∞ . Here, we denote the complete b ∗ (G, ), but in the same way as with cohomology of the group G by H ordinary cohomology, these functors can be identified with the functors c ∗ZG (Z, ). Ext Corollary B2 . Let G be an HF-group of type FP∞ . Then (1) the torsion-free subgroups of G have finite cohomological dimension; (2) B := B(G, Z) has finite projective dimension as a ZG-module; and (3) there is an m such that for all n ≥ m (a) H n (G, ZG) = 0, and b n (G, ) is an isomorphism. (b) the natural map H n (G, )→H 6. T HE G EDRICH -G RUENBERG INVARIANTS AND

FINITISTIC

DIMENSION

In this section we study the invariants spli(R) and silp(R) of a ring R introduced by Gedrich and Gruenberg [7], and their relationship with the finitistic dimension fin. dim(R) in the case when R is the group algebra of an HF-group. First, spli(R) is defined to be the supremum of the projective lengths of the injective R-modules, and silp(R) to be the supremum of the injective lengths of the projective R-modules. The finitistic dimension fin. dim(R) of R is the supremum of the projective lengths of the R-modules of finite projective dimension. We also need one further invariant: Definition 6.1. Let K be a commutative ring and let G be a group. Let X denote the class of KG-modules M such that proj. dimKF M < ∞ for all finite subgroups F of G. We write κ(KG) for the supremum of the projective dimensions of the modules in X.

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Theorem C . Let K be a commutative ring of finite global dimension and let G be an HF-group. Then silp(KG) = spli(KG) = κ(KG) = fin. dim(KG). Proof. We shall establish the chain of inequalities silp(KG) ≤ spli(KG) ≤ κ(KG) ≤ fin. dim(KG) ≤ silp(KG). (1) silp(KG) ≤ spli(KG). Suppose that n := spli(KG) is finite. It follows from [7, Theorem 2.4] that silp(KG) ≤ n + silp(K), and hence silp(KG) is also finite. In this case [7, 1.6] shows that silp(KG) = spli(KG). Thus, in general, we have silp(KG) ≤ spli(KG) as required. (2) spli(KG) ≤ κ(KG). Let I be an injective KG-module. Then I is injective as KF-module for any subgroup F. Now, if F is a finite group, then every injective KF-module has finite projective dimension because K has finite global dimension. Hence I has finite projective dimension over KF for all finite subgroups F of G, and so proj. dimKG ≤ κ(KG) and the required inequality follows. (3) κ(KG) ≤ fin. dim(KG). For this inequality we need to use the assumption that G belongs to HF. Without loss of generality we may assume that n := fin. dim(KG) is finite. Now let M be a module with the property proj. dimKF M < ∞ for all finite F ≤ G. We claim that for all subgroups H of G, proj. dimKG (M ⊗KH KG) ≤ n. We shall prove this by induction on the least ordinal α such that H belongs to the subclass Hα F of HF. We refer the reader to [10] for an explanation of this notation. Note that it suffices to prove that M ⊗KH KG has finite projective dimension because fin. dim(KG) = n. If α = 0 then H is finite so that M has finite projective dimension over KH, by hypothesis, and hence M ⊗KH KG has finite projective dimension over KG. If α > 0 then there is an action of H on a finite dimensional contractible cell complex X in such a way that each cell stabilizer belongs to Hβ F for some β < α. The augmented cellular chain complex of X is an exact sequence of finite length: 0→Cr → · · · →C1 →C0 →Z→0 of ZG-modules. Each Ci is a permutation module, and in fact a direct sum of modules of the form IndH L Z where L is a stabilizer in H of an i-dimensional cell in X. Tensoring with M and applying induction from H to G yields an exact sequence G G G 0→IndG H (M ⊗Cr )→ · · · →IndH (M ⊗C1 )→IndH (M ⊗C0 )→IndH M→0

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of KG-modules. Now, each IndG H (M ⊗ Ci ) is a direct sum of modG ules of the form IndL M where L belongs to Hβ F for some β < α. By induction, each of the modules IndG L M has projective dimension ≤ n, and hence IndG (M ⊗ C ) has projective dimension ≤ n for all H i i. The exact sequence above now guarantees that IndG H M has projective dimension at most n + r, which is finite, and the inequality follows. (4) fin. dim(KG) ≤ silp(KG). We may assume that n := silp(KG) is finite. Let M be a module of finite projective dimension m. Then there is a free module F such that Extm KG (M, F) 6= 0. Since F has injective dimension at most n, it follows that m ≤ n and the inequality follows. This completes the proof.  Corollary C . Let K be a commutative ring of finite global dimension and let G be an HF-group of type FP∞ . Then silp(KG) = spli(KG) = κ(KG) = fin. dim(KG) < ∞. In particular, if M is a KG-module then proj. dimKG M < ∞ if and only if proj. dimKF M < ∞ for all finite subgroups F of G. Proof. In view of Theorem C, it suffices to prove that every injective KGmodule has finite projective dimension. Let I be an injective KG-module and let B := B(G, Z). Since K has finite global dimension and B has finite projective dimension over ZG by Corollary B(2), it follows that I ⊗ B has finite projective dimension over KG. Now the natural inclusion of Z into B induces an inclusion of I into I ⊗B, and since I is injective, it follows that I is a direct summand of I ⊗ B, and hence I has finite projective dimension.  7. I MPLICATIONS OF RESIDUAL FINITENESS AND RELATED HYPOTHESES

The principal goal of this section is to show that every residually finite HF-group of type FP∞ has finite virtual cohomological dimension. To prove this is certainly suffices to show that there is a torsion-free subgroup of finite index, but there seems to be no obvious way of approaching the problem from this direction, so we shall use more sophisticated methods. Proposition 7.1. Let G be a group and let (Bλ | λ ∈ Λ) be a direct limit system of G operator rings. Set B := lim Bλ . Suppose that B and each Bλ −→ λ

are free as additive groups. If M is a KG-module of type FP∞ such that c 0KG (M, M ⊗ B) = 0 Ext then there exists λ ∈ Λ such that M ⊗ Bλ has finite projective dimension.

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Proof. By Remark 2.4 it suffices to find a λ such that the natural inclusion ιλ : M→M ⊗ Bλ maps to zero under the natural map c 0KG (M, M ⊗ B ). HomKG (M, M ⊗ Bλ )→Ext λ Since M is of type FP∞ , the vertical maps in the following diagram are isomorphisms: lim HomKG (M, M ⊗ Bλ ) −→ λ

0

c KG (M, M ⊗ B ) / lim Ext λ −→

λ



HomKG (M, M ⊗ B)

 / c0 ExtKG (M, M ⊗ B).

The desired result now follows because the lower right hand group is zero by hypothesis and the natural inclusions ιλ form a compatible family in the direct limit system.  Theorem D . Let G be a residually finite HF-group and let M be a KGmodule of type FP∞ which has finite projective dimension as a K-module. Then there is a subgroup H of finite index in G such that M has finite projective dimension over KH. In particular, every residually finite HF-group of type FP∞ has finite virtual cohomological dimension. Proof. Let (Gλ | λ ∈ Λ) be the family of subgroups of finite index in G. For each λ , let Bλ := B(G/Gλ , Z). Then the Bλ form a direct limit system and the direct limit B is free as a module for every finite subgroup of G. It follows that M ⊗ B is free as KF-module for all finite F ≤ G, and c 0KG (M, M ⊗ B) is zero by Theorem A. Proposition 7.1 shows that hence Ext IndG Gλ M = M ⊗ Bλ has finite projective dimension for some λ , and this is equivalent to the assertion that M has finite projective dimension over Gλ . Thus the result follows with H = Gλ . The last assertion is the special case with K = Z and M = Z.  8. G ROUPS OF AUTOMORPHISMS OF N OETHERIAN

MODULES

The goal of this section is to prove Theorem E . If R is a commutative ring and M is a Noetherian R-module then every countable subgroup of AutR (M) belongs to HF. We first recall from [10], the definiton of the operation H1 on classes of groups.

HOMOLOGICAL FINITENESS CONDITIONS

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Definition 8.1. We write H1 X for the class of all groups G which admit an action on a finite dimensional contractible cell complex in such a way that the stabilizers belong to X. The operation H1 is not a closure operation: the closure operation it generates is H. We now turn to the proof of the theorem. The following result [15, Theorem 6.1] of Wehrfritz allows us to reduce almost at once to the case of groups which are linear in positive characteristic. Theorem 8.2. Let M be a Noetherian module over a commutative ring R. Then there exists a positive integer n, a finite number of fields Fi and an embedding of AutR (M) into ∏i GLn (Fi ). Our goal is to establish the following Proposition which, together with Wehrfritz’s theorem and the fact that all countable H1 LF-groups belong to HF, suffices to yield Theorem E. Proposition 8.3. Let n be a positive integer and let (Fi ) be a finite family of fields all of which have positive characteristic. Then every finitely generated subgroup of ∏i GLn (Fi ) belongs to H1 LF. Here, we use the notation LX to denote the class of locally X-groups. Thus the conclusion of the proposition states that the finitely generated subgroups admit actions on finite dimensional contractible spaces with locally finite stabilizers. Now HF is subgroup closed and extension closed. It is also closed under countable directed unions (see [11, Corollary 3.2.4]), and since it contains all finite groups it therefore contains all countable locally finite groups. From these remarks it is clear that Theorem A follows immediately from the Proposition. In proving the Proposition some easy reductions can be made. Since GLn (Fi ) embeds in SLn+1 (Fi ) we only need to consider the finitely generated subgroups of ∏i SLn (Fi ). Moreover H1 LF is subgroup closed and closed under Cartesian products, and tus it suffices to show that SLn (S) is an H1 LF-group when S is a finitely generated domain of characterisic p. The methods we use are adapted from the work of Alperin and Shalen [1]. Since S is a finitely generated domain, it follows from Noether’s Normalization lemma that S is integral over the polynomial ring F p [x1 , . . . , xm ] for some integer m. We denote by E the field of fractions of S, and by K = F p (x1 , . . . , xm ) the field of fractions of F p [x1 , . . . , xm ]. The following proposition is a characteristic p version of [1, Proposition 1.2]. Proposition 8.4. With the above notation, thereTexists finitely many discrete valuation rings O vi of E, 1 ≤ i ≤ n, so that S ∩ ni=1 O vi ⊂ F, the algebraic closure of F p in E. Furthermore, F is finite.

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JONATHAN CORNICK AND PETER H. KROPHOLLER

Proof. The first assertion is proved exactly the same as in [1]. The second assertion is clear, since F is a finitely generated, algebraic extension of a finite field.  Remark 8.5. The finiteness of F is the key point in the rest of the proof. As in Alperin and Shalen’s paper [1], we have a finite set of buildings {Xvi }ri=1 , one for each valuation vi , each of which admits an action of SLn (E). Consequently, SLn (S) ⊂ SLn (E) acts simplicially on r

X = ∏ Xvr . i=1

It follows from previous remarks that the stabilizer H of a vertex (x0 , . . . , xr ) ∈ X is such that H ⊂ SLn (S) ∩

r \

a−1 i SLn (O vi )ai , ai ∈ SLn (E), 1 ≤ i ≤ r.

i=1

Lemma 8.6. H is an extension of a locally finite group, by a product of groups each of which act irreducibly on a a finite dimensional vector space over E. Proof. Since H < SLn (E) it acts on V = E n . Let 0 ≤ V j ≤ V j−1 ≤ . . . ≤ V1 = V, j ≤ n be a composition series for the action of H on V . If we denote by Hi the image of H in GL(Vi /Vi+1 ) then Hi acts irreducibly on Vi /Vi+1 . The kernel of the natural homomorphism ρ : H−→H0 × H1 × . . . × H j is a unipotent subgroup of SLn (E) and hence locally finite since char(E) = p > 0.  Lemma 8.7. The groups Hi in the above lemma are finite. Proof. Let ph be the characteristic polynomial of h ∈ H. Since H ⊂ SLn (S) ∩

r \

A−1 i SLn (O vi )Ai , A ∈ SLn (E), 1 ≤ i ≤ r,

i=1

it follows from 8.4 that the coefficients of ph are elements of F, the algebraic closure of F p in E. Now suppose that hi ∈ Hi , and ρ(h) = hi . If hi has minimal polynomial ph , then ph is a factor of ph . In particular the i i coefficients of ph are elements of F. Since F is finite we have i

Trace Hi = {Trace h|h ∈ Hi }

HOMOLOGICAL FINITENESS CONDITIONS

17

is finite. Since Hi is acting irreducibly it follows from a result of Burnside that Hi is finite by corollary 1.23 of [14].  Putting all of this together we can now prove Proposition 8.3 Proof. We have reduced to showing that SLn (S) is an H1 LF-group, when S is finitely generated domain of positive characteristic. In this case SLn (S) acts on a contractible simplicial complex X, and it follows from Lemmas 8.6 and 8.7 that the stabilisers of this action are locally finite.  9. C ONCLUDING R EMARK As an application of the results of this paper, we have Theorem F . Every linear group of type FP∞ has finite virtual cohomological dimension. This was proved in [10], but only for groups linear in characteristic zero. The methods of [10] were insufficient for groups linear in positive characteristic not only because of the need to establish that they belong to HF, but also because in general, finitely generated linear groups are not virtually torsion-free. Here, we can deduce Theorem F easily. Let G be a linear group of type FP∞ . Then G belongs to HF by Theorem E, and since all finitely generated linear groups are residually finite, G is also residually finite. Thus Theorem F follows from Theorem D. R EFERENCES [1] R. C. Alperin and P. B. Shalen, ‘Linear Groups of Finite Cohomological Dimension’, Invent. Math., 66 (1982), 89–98. [2] D. J. Benson, Representations and Cohomology I: Basic representation theory of finite groups and associative algebras, Cambridge Studies in Advanced Mathematics 30, (Cambridge U.P. 1991). [3] D. J. Benson and J. Carlson, ‘Products in negative cohomology’, J. Pure Appl. Algebra 82 (1992), 107–130. [4] R. Bieri, ‘Homological dimension of discrete groups’, Queen Mary College Mathematics Notes, Second Edition, 1982. [5] K. S. Brown and R. Geoghegan, ‘An infinite dimensional FP∞ -group’, Invent. Math. 77 (1984), 367–381. [6] L. Fuchs, Infinite Abelian Groups, Volume II, (Academic Press, New York, 1973). [7] T. Gedrich and K. Gruenberg, ‘Complete cohomological functors on groups’, Topology and its Applications 25 (1987), 203–223. [8] F. Goichot, ‘Homologie de Tate-Vogel e´ quivariante’, J. Pure Appl. Algebra 82 (1992), 39–64. [9] I. Kaplansky, Infinite Abelian Groups, (University of Michigan Press, Second Edition, 1969). [10] P. H. Kropholler, ‘On groups of type FP∞ ’, J. Pure Appl. Algebra 90 (1993), 55–67. .

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[11] P. H. Kropholler, ‘Hierarchical decompositions, generalized Tate cohomology, and groups of type FP∞ ’, in: (eds. A. Duncan, N. Gilbert, and J. Howie) Proceedings of the Edinburgh Conference on Geometric Group Theory, 1993, (Cambridge U. P. 199?). [12] P. H. Kropholler and O. Talelli, ‘On a property of fundamental groups of graphs of finite groups’, J. Pure Appl. Algebra 74 (1991), 57-59. [13] G. Mislin, ‘Tate cohomology for arbitrary groups via satellites’, Topology and its Applications, [14] B. A. F. Wehrfritz, Infinite linear groups, Ergebnisse der Mathematic und ihrer Grenzgebiete 76 Springer, Berlin, 1973. [15] B. A. F. Wehrfritz, Lectures around complete local rings, Queen Mary College Mathematics Notes, 1979. C ENTRE DE R ECERCA M ATEMATICA , I NSTITUT D ’E STUDIS C ATALANS , A PARTAT 50, E 08193 B ELLATERRA , S PAIN E-mail address: ICRM6@@cc.uab.es S CHOOL OF M ATHEMATICAL S CIENCES , Q UEEN M ARY M ILE E ND ROAD , L ONDON E1 4NS E-mail address: P.H.Kropholler@@qmw.ac.uk

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