ON THE STRUCTURE OF MODULES OF TYPE FP_{∞}

PETER H. KROPHOLLER

Abstract. Let G be a group in the class LH𝔉 of locally hierarchically decomposable groups and let R be a strongly G-graded algebra. We provide a characterization of the R-modules of type FP_{∞} under the assumptions that R_{1} is coherent and of finite global dimension.

1. Introduction

Benson [2, 3] has recently made substantial progress in studying modules of type FP_{∞} over certain group algebras by using algebraic homotopy colimits. These were introduced into representation theory by Rickard [16]. Rickard and Benson make use of triangulated categories. Here, we generalize one of Benson’s main results but at the same time we provide a direct argument which avoids much of the category theory as well as some technicalities associated with Benson’s approach.

Let k be a commutative ring, let G be a group. Let R be a strongly G-graded k-algebra: this is a k-algebra which admits a k-module decomposition R = ⨁_{g∈G} R_{g} such that for all g, h ∈ G, R_{g}R_{h} = R_{gh}. For any subgroup H of G, we write R_{H} for the subalgebra supported on H. The simplest example of such an R is the group algebra kG, and this is the case which Benson considered. However it is natural to generalize the theory to the wider setting which includes the possibility of twisted and skew group algebras and indeed, general crossed products. A crossed product is a strongly G-graded k-algebra in which each R_{g} contains a unit. A twisted group algebra is a crossed product R in which k is carried isomorphically onto R_{1}. A skew group algebra is a crossed product R which admits a homomorphism from G to the group of units in which each g is carried to an element of R_{g}.

Here we study R-modules of type FP_{∞}. There are certain natural ways in which such modules arise:

• If H is any subgroup of G and V is an R_{H}-module of type FP_{∞} then V ⊗_{R_{H}} R is an R-module of type FP_{∞}.

• the class of FP_{∞}-modules is extension closed; and

• every direct summand of an FP_{∞} module is again FP_{∞}.

The first remark is a consequence of the fact that the induction functor, − ⊗_{R_{H}} R, is exact and carries projective R_{H}-modules to projective Rmodules. The second and third apply to FP_{∞} modules over any ring or algebra.

For convenience we call a module elementary if it is of the form U ⊗_{R_{E}} R where E is a finite elementary abelian subgroup of G and U is a finitely generated R_{E}-module which is projective over R_{1}. The elementary modules are certainly of type FP_{∞}. Polyelementary modules, the ones which have a series

0 = M_{0} ≤ ⋯ ≤ M_{n} = M

in which the sections M_{i}/M_{i−1} are elementary, also are of type FP_{∞}. Every direct summand of a polyelementary module is of type FP_{∞}.

In the main theorem we adopt the following convention: if P_{∗} ↠ M is a projective resolution of a module M then the nth kernel is the kernel of the map P_{n−1} → P_{n−2} when n ≥ 2, the kernel of the map P_{0} ↠ M when n = 1 and simply M itself if n = 0.

Theorem A. Suppose that G belongs to LH𝔉. Suppose also that R_{1} is coherent and has finite global dimension. Let M be an R-module of type FP_{∞}. Then there is an integer n ≥ 0 such that in any projective resolution P_{∗} ↠ M of finite type, the n^{th} kernel is isomorphic to a direct summand of a polyelementary module.

We use a vanishing theorem for complete cohomology as our main tool in proving Theorem A. We refer the reader to [6, 7, 8, 15] for further information about complete cohomology. First, a definition couched in terms of complete cohomology:

Definition 1.1. An R-module N is said to be stably flat (over R) if and only if Êxt_{R}^{0}(M , N) = 0 for all R-modules M of type FP_{∞}.

For motivation, this definition may be compared with the criterion of Lazard [13] which states that an R-module N is flat if and only if every map from a finitely presented module M to N factors through a projective module. Notice that by writing \underline{Hom}_{R}(M, N) for the quotient of Hom_{R}(M, N) by the subgroup comprising maps which factor through projective modules, Lazard’s theorem may be expressed as follows: N is flat if and only if \underline{Hom}_{R}(M, N) = 0 for all finitely presented M. The analogy with our definition becomes closer if we consider the stable category of R-modules: the objects are ordinary R-modules and the morphisms from M to N are the elements of Êxt_{R}^{0}(M, N). Because the complete cohomology involves a limiting process, our definition of stably flat would have been badly behaved had we chosen the test modules M to be arbitrary finitely presented modules, and it seems natural in the stable context that the test modules should be the FP_{∞} modules. Although most of the algebras considered in this paper are unlikely to be coherent, it is useful to notice the analogy between the definitions of flat and stably flat becomes closer for coherent rings, these being the rings for which the classes of finitely presented modules and FP_{∞} modules coincide.

Theorem B. Let G be an LH𝔉-group and let R be a strongly G-graded k-algebra such that R_{1} is coherent. Let N be an R-module. Then the following are equivalent:

1. N is stably flat as an R-module;

2. N is stably flat as an R_{E}-module for all finite elementary abelian subgroups E of G.

When G is finite, the same result holds without the assumption that R_{1} is coherent.

To give an idea of how Theorem B plays a rôle in proving Theorem A, one can draw a very naïve analogy with a situation for finite dimensional vector spaces. Let ℝ^{n} be a Euclidean space, let U be a subspace and let S be a subset of U. Then the orthogonal complement of U is contained in the subspace S^{⊥} of vectors orthogonal to the elements of S. If we know that U^{⊥} = S^{⊥} then it follows that S spans U.

Here the situation has a striking similarity. In place of Euclidean space we have the category of all R-modules, and within it, in place of U, the category of R-modules of type FP_{∞}. Within the category of R-modules of type FP_{∞} there is the collection of modules induced from subalgebras supported on finite elementary abelian subgroups. Now we can define two classes of modules which are in a very loose sense the orthogonal complements namely:

1. those modules N such that Êxt_{R}^{0}(M , N) = 0 for all modules M of type FP_{∞}; and

2. those modules N such that Êxt_{R}^{0}(C , N) = 0 for all modules C of the form V ⊗_{R_{E}} R where E is a finite elementary abelian subgroup and V is a finitely presented R_{E}-module.

Clearly class (1) is contained in class (2).

There is a version of the Eckmann-Shapiro lemma for complete cohomology which can be proved in just the same way as for ordinary cohomology:

Lemma 1.2. Let H be a subgroup of G, let V be an R_{H}-module and let N be an R-module. Then for all n, there is a natural isomorphism between Êxt_{R}^{n}(V ⊗_{R_{H}} R, N) and Êxt_{R_{H}}^{n}(V , N).

In view of this, (2) is the class of those N such that Êxt_{R_{E}}^{0}(V , N) = 0 for all finite elementary abelian subgroups E and finitely presented R_{E}-modules V. Class (1) is, by definition, the class of stably flat Rmodules. The conclusion of Theorem B asserts that classes (1) and (2) coincide. We are able to deduce that, in some sense, the special collection of modules used to define class (2) spans the class of modules of type FP_{∞}. The analogy can be made more precise by using the theory of triangulated categories: according to Rickard and Benson, Theorem B allows us to conclude that the category of R-modules of type FP_{∞} is generated as a thick subcategory by the class of special modules. In order to make this rigorous, one has to work with triangulated categories, and for this it is important that R_{1} has finite global dimension. Here we give direct proofs without using triangulated categories, but we have nevertheless been unable to avoid the additional hypothesis on R_{1} in Theorem A.

2. The kG-module B, cofibrant modules, and the reduction of Theorem A to the cofibrant case

In this section we fix a group G, and a strongly G-graded k-algebra R. We write B := B(G, k) for the kG-module of functions from G to k which take only finitely many values. This module is a commutative k-algebra with the obvious definition of ring operations, and G acts on it as a group of algebra automorphisms. It has the remarkable property that it is free as a kH-module whenever H is a finite subgroup of G. This property was first exploited in [12] and subsequently in [5, 6, 7]. Proposition 9.2 of [6] asserts that if G belongs to H𝔉 and M is an Rmodule of type FP_{∞} then M ⊗ B has finite projective dimension over R. As usual the unembellished ⊗ stands for ⊗_{k} and M ⊗ B is endowed with the diagonal action of R, (m ⊗ b)r = mr ⊗ bg for r ∈ R_{g}, (see Section 3 of [6]). Following Benson, (Section 4 of [2], Section 10 of [3]) we say that an R-module M is cofibrant if M ⊗B is projective. If M ⊗B has projective dimension n then for any projective resolution P_{∗} ↠ M of finite type, the n^{th} kernel M′ is cofibrant, because P_{∗} ⊗ B ↠ M ⊗ B is a projective resolution of M ⊗ B in which the n^{th} kernel is M′ ⊗ B. The first step in proving Theorem A is the elementary observation that any R-module M of type FP_{∞} is isomorphic to a module of the form L ⊗_{R_{H}} R for some finitely generated subgroup H of G and some R_{H}- module L of type FP_{∞}. Given this, it suffices to prove Theorem A for H𝔉-groups when Proposition 9.2 of [6] comes into play. Hence it is enough to prove the following result:

Theorem A′ . Assume that G belongs to LH𝔉 and that R_{1} is coherent and has finite global dimension. Let M be an R-module. Then the following are equivalent:

1. M is cofibrant and of type FP_{∞};

2. M is a direct summand of a polyelementary module.

Cofibrant modules play an important rôle in handling complete cohomology. Theorem 3.5 of [7] shows that every cofibrant module M admits a complete resolution P_{∗} in which M is the zeroeth kernel: M = ker(P_{−1} → P_{−2}). When complete resolutions exist in the sense of [7], they can be used to compute complete cohomology, and there is the following useful consequence:

Lemma 2.1. Let M and N be R-modules. If M is cofibrant then the natural map \underline{Hom}_{R}(M, N) → Êxt_{R}^{0}(M , N) is an isomorphism.

We refer the reader to [7] for a detailed discussion of complete resolutions. It is a useful point of view to consider three different categories of R-modules. First there is the ordinary category Mod-R of right Rmodules, and then there are two further categories \underline{Mod}-R and M̂od-R both of which have the same objects as Mod-R. In \underline{Mod}-R, the morphisms from M to N are the elements of \underline{Hom}_{R}(M, N) and in the stable category, the morphisms are the elements of the complete cohomology group Êxt_{R}^{0}(M, N). Throughout most of this paper we work in the ordinary category, but occasionally it is useful to view matters in the stable category: this is particularly important when proving Theorem B.

3. Generalities

In this section we work with modules over an arbitrary ring or algebra. The following result can be conveniently derived from a variation on Schanuel’s Lemma:

Lemma 3.1. Let M′′ ↣^{ι} M ↠^{π} M′ be any short exact sequence of modules in which π factors through a projective module Q. Then M is isomorphic to a direct summand of Q ⊕ M′′.

Here is the general form of Schanuel’s Lemma which was pointed out to the author by Jon Alperin:

Lemma 3.2. Suppose given two short exact sequences K ↣ L ↠^{π} M

and K′ ↣ L′ ↠^{π′} M of modules, and suppose there are maps ϕ : L → L′ and ϕ′ : L′ → L such that π′ϕ = π and πϕ′ = π′. Then L′ ⊕ K is isomorphic to L ⊕ K′.

Proof. This can be proved by the same pull-back argument as for the classical version of Schanuel’s lemma, as described in Part III of Kaplansky’s notes [9].

Proof of Lemma 3.1. Suppose that π is the composite M →^{ϕ} Q →^{ψ} M′.

Then we have a short exact sequence ker ψ ↣ Q ↠^{ψ} M′. Let ϕ′ : Q → M be a factorization of ψ through π. Then Lemma 3.2 applies to show that M ⊕ ker ψ is isomorphic to Q ⊕ M′′.

We shall also need the following elementary fact. Here, for a class of modules 𝔛, we use the term hyper-𝔛 module to mean a module E which possesses an ascending chain of submodules E_{α} indexed by ordinals α in such a way that E_{0} = 0, for each successor α, E_{α}/E_{α−1} belongs to 𝔛, for each limit ordinal α, E_{α} = ∪_{β<α}E_{β}, and E = ∪_{α}E_{α}. Such an ascending chain is referred to as a continuous chain with sections in 𝔛.

Lemma 3.3. Let 𝔛 be a class of finitely presented modules. Then every hyper-𝔛 module is locally poly-𝔛.

Proof. Let E be a hyper-𝔛 module and let (E_{α}) be a continuous chain of submodules with sections in 𝔛. For a finitely generated submodule M of E, we work by induction on the the least ordinal α such that M is contained in E_{α}. If α = 0 then there is nothing to prove, so we may assume that α is a successor ordinal. Let N be a finitely generated submodule of E_{α} which contains M and generates E_{α} modulo E_{α−1}. Then N/N ∩ E_{α−1}≅E_{α}/E_{α−1} is finitely presented and so N ∩ E_{α−1} is finitely generated. By induction, N ∩ E_{α−1} is contained in a poly-𝔛 submodule L of E_{α−1}, and M is contained in the poly-𝔛 submodule N + L.

4. Rickard’s Method and the proof of Theorem A′

Here, we assume the hypotheses of Theorem A′, that G is an LH𝔉group and R is a strongly G-graded k-algebra such that the subalgebra R_{1} is coherent and has finite global dimension. The fact that every direct summand of a polyelementary module is both cofibrant and of type FP_{∞} holds without the assumptions on G and R_{1}. For the converse, let M be a cofibrant R-module of type FP_{∞}. We construct a chain M = M_{0} ⊆ M_{1} ⊆ M_{2} ⊆ … inductively so that for each n ≥ 0 there is a short exact sequence C_{n} ↣ M_{n} ⊕ P_{n} ↠ M_{n+1} in which

1. C_{n} is a direct sum of elementary modules;

2. P_{n} is projective; and

3. every map from an elementary module to M_{n} factors through C_{n}. Set M_{0} = M. Suppose that n ≥ 0 and that M_{n} has been constructed. Consider the pointed category whose objects are ordered pairs (C, ϕ) where C is an elementary module and ϕ is a homomorphism from C to M, and whose morphisms are the obvious commutative triangles. Choose a set 𝔛 containing at least one object of this category from each isomorphism class. Set C_{n} := ⨁_{(C,ϕ)∈𝔛} C and use the maps ϕ associated to each object to define a map C_{n} → M_{n}. Properties (1) and (3) are now guaranteed. Since C_{n} is cofibrant, C_{n} ⊗B is projective and we can set P_{n} := C_{n} ⊗ B. Finally, M_{n+1} can be defined as the cokernel of the inclusion C_{n} → M_{n} ⊕P_{n}, or in other words the pushout, and since the map C_{n} → P_{n} is an inclusion, it follows that the induced map M_{n} → M_{n+1} is also injective and we regard M_{n} as a submodule of M_{n+1}.

Let M_{∞} be the union of the chain of modules M_{n}. This is a concrete construction of Rickard’s algebraic homotopy colimit. There is a natural inclusion of M into M_{∞} and the next step towards understanding M depends on showing that this inclusion represents zero in the complete cohomology group Êxt_{R}^{0}(M, M_{∞}). For this it suffices to prove the following result:

Lemma 4.1. M_{∞} is stably flat.

Proof. By Theorem B, it suffices to prove that M_{∞} is stably flat over R_{E} for each finite elementary abelian subgroup E of G. In other words we must show that Êxt_{R}^{0}(U ⊗_{R_{E}} R, M_{∞}) = 0 whenever U is a module of type FP_{∞} over R_{E} for such an E. Since R_{1} is assumed to have finite global dimension, every such module U is stably isomorphic to a cofibrant module V, and it suffices to prove that Êxt_{R}^{0}(V ⊗_{R_{E}} R, M_{∞}) = 0. One can do this because U has a complete resolution in the sense of [7] and one can take V to be the zeroeth kernel in one such resolution. Therefore we must show that Êxt_{R}^{0}(C , M_{∞}) = 0 whenever C is an elementary module. By Lemma 2.1, we only need to prove that \underline{Hom}_{R}(C, M_{∞}) = 0 for all elementary C. But given any homomorphism from C to M_{∞}, there is an n such that its image lies in M_{n}, and we see from the construction of our sequences of modules the composite C → M_{n} ↪ M_{n+1} already factors through the projective module P_{n}.

We continue the proof of Theroem A′, using this lemma. Note that by Lemma 2.1, the natural map \underline{Hom}_{R}(M, M_{∞}) → Êxt_{R}^{0}(M , M_{∞}) is an isomorphism and this together with the above Lemma implies that the inclusion M → M_{∞} factors through a projective module Q:

M →^{α} Q →^{β} M_{∞}.

Obviously we may assume that Q is free, and since M is finitely generated there will be a summand of Q of finite rank through which the inclusion factors. Thus we may assume that Q is finitely generated. In that case the image of Q in M_{∞} lies in some M_{n}. For the remainder of the proof, fix an n for which this happens: now we have a factorization

M →^{α} Q →^{β} M_{n}.

The composite of the obvious sequence of surjections,

M ⊕ P_{0} ⊕ P_{1} ⊕ … P_{n−1} ↠ M_{1} ⊕ P_{1} ⊕ … P_{n−1} ↠ … ↠ M_{n}

has a kernel K which admits a filtration 0 = K_{−1} ≤ K_{0} ≤ ⋯ ≤ K_{n−1} = K where K_{i}/K_{i−1} is isomorphic to C_{i}. Writing P = P_{0} ⊕ P_{1} ⊕ … P_{n−1}, we now have a short exact sequence K ↣ M ⊕ P ↠ M_{n} in which the second map factors through the projective module P ⊕ Q. Hence, by Lemma 3.1, M is isomorphic to a direct summand of K ⊕ P ⊕ Q. The conclusion, that M is a direct summand of a polyelementary module now follows from the fact that K is hyperelementary, using Lemma 3.3. This completes the proof of Theorem A′.

5. Chouinard’s theorem

In this section we sketch a proof of a generalization of Chouinard’s theorem [4] first proved by Aljadeff and Ginosar [1]. This will provide motivation for the study of stably flat modules in Section 6. The proof is based on an unpublished argument of David Benson which he devised for proving the classical theorem, and it has the advantage that it handles finitely and infinitely generated modules on an equal footing. Underlying the proof is the fundamental result of Serre [17] which states that if G is a finite p-group (p a prime), which is not elementary abelian, then there is a sequence of non-zero elements x_{1}, …, x_{n} of H^{1}(G, ℤ/pℤ) such that the product βx_{1} … βx_{n} of their Bocksteins is zero in H^{2n}(G, ℤ). Serre [18] shows that the number n of Bocksteins can be bounded by \frac{p^{d}−1}{p−1} where d is the minimum number of generators

of G. Recently, Minh [14] has discovered the bound (p + 1)p^{[^{\frac{d}{2}}]}.

Theorem 5.1 (Aljadeff and Ginosar). Let G be a finite group, let R be a strongly G-graded k-algebra, and let N be an R-module. Then the following are equivalent:

1. N is projective over R.

2. N is projective over R_{E} for each elementary abelian subgroup E of G.

Proof. We first prove the theorem in the special case when G is a pgroup for some prime p. By induction we may assume that N is projective over R_{H} for all proper subgroups H of G, and clearly we may also assume that G is not elementary abelian. In this case Serre’s theorem supplies a sequence of non-zero elements x_{1}, …, x_{n} of H^{1}(G, ℤ/pℤ) such that the product βx_{1} … βx_{n} is zero in H^{2n}(G, ℤ). Now H^{1}(G, ℤ/pℤ) is naturally isomorphic with Hom(G, ℤ/pℤ) and so each x_{i} can be regarded as a surjective homomorphism having a kernel H_{i} of index p in G. Also, elements of H^{j}(G, ℤ) = Ext_{ℤG}^{j}(ℤ, ℤ) can be represented by exact sequences of length j + 2 which begin and end with ℤ. For each i, the Bockstein βx_{i} is represented by the four term sequence

0 → ℤ → ℤ[H_{i}\G] → ℤ[H_{i}\G] → ℤ → 0.

Let β_{i} denote the image of βx_{i} in H^{2}(G, k): this element is represented by the exact sequence

0 → k → k[H_{i}\G] → k[H_{i}\G] → k → 0.

The Yoneda product β_{1} … β_{n} is represented by the sequence of length 2n + 2 which is obtained by concatenating these sequences:

0 → k → k[H_{n}\G] → ⋯ → k[H_{1}\G] → k → 0.

So this sequence represents the zero element of Ext_{kG}^{2n}(k, k). Tensoring with N yields the exact sequence the exact sequence

0 → N → N ⊗ k[H_{n}\G] → ⋯ → N ⊗ k[H_{1}\G] → N → 0.

This sequence has two properties, for not only does it represent the zero element of Ext_{kG}^{2n}(N , N) but in addition, each of the intermediate modules, N ⊗ k[H_{i}\G] is isomorphic to the induced module N ⊗_{R_{H_{i}}} R and is therefore projective over R. Given these two properties, it can be shown that the left hand monomorphism is split, and therefore N is projective as required.

Now let G be an arbitrary finite group. We may assume that N is projective over R_{P} for every Sylow subgroup P of G. Let p_{1}, …, p_{m} be the distinct prime divisors of |G|. For each i let P_{i} be a Sylow p_{i}- subgroup. Since the indices |G:P_{1}|, …, |G:P_{m}| have greatest common

divisor 1, we can choose integers ℓ_{i} such that ∑_{i=1}^{m} ℓ_{i}|G:P_{i}| = 1. Let

σ_{i} ∈ k[P_{i}\G] denote the sum of the cosets of P_{i}: this is invariant and is carried to |G : P_{i}| by the augmentation map k[P_{i}\G] → k. Hence the map

k → k[P_{1}\G] ⊕ ⋯ ⊕ k[P_{m}\G]

given by 1 ↦ (ℓ_{1}σ_{1}, …, ℓ_{m}σ_{m}) splits the augmentation map

k[P_{1}\G] ⊕ ⋯ ⊕ k[P_{m}\G] → k

and the trivial module k is a direct summand of k[P_{1}\G]⊕⋯⊕k[P_{m}\G]. Tensoring with N we conclude that N is a direct summand of the projective module N ⊗ k[P_{1}\G] ⊕ ⋯ ⊕ N ⊗ k[P_{m}\G].

6. Stably flat modules and the proof of Theorem B

First we record a result which is to be applied to the concatenated sequences that arise from Serre’s product of Bocksteins theorem.

Lemma 6.1. Let R be a ring and let L, M, N be R-modules. Suppose given an exact sequence

0 → L → K_{1} → ⋯ → K_{m} → M → 0

of R-modules beginning with L and ending with M in which the intermediate modules K_{i} have the property that Êxt_{R}^{j}(K_{i} , N) = 0 for all j ≥ 0. Then Yoneda product with the element ξ ∈ Êxt_{R}^{m}(M , L) represented by the exact sequence induces an isomorphism Êxt_{R}^{0}(L, N) → Êxt_{R}^{m}(M , N).

Proof. This is a standard dimension shifting argument and we omit the details.

Lemma 6.2. Let G be a group and let H be a subgroup of finite index. Let R be a strongly G-graded k-algebra. Then there is a natural isomorphism Êxt_{R}^{n}(M , N ⊗_{R_{H}} R)≅Êxt_{R_{H}}^{n}(M , N) for all n, and for all R-modules M and R_{H}-modules N. Moreover, if N is a stably flat R_{H}- module then N ⊗_{R_{H}} R is stably flat over R.

Proof. Lemma 6.4 of [6] shows that the induction functor − ⊗_{R_{H}} R is right adjoint to the restriction functor in the case when H has finite index, and since it also carries projective R_{H}-modules to projective R-modules, one obtains the required form of the Eckmann-Shapiro Lemma. The second assertion now follows directly.

The next result establishes the special case of Theorem B when G is finite, and for this part, the assumption that R_{1} is coherent is not used.

Lemma 6.3. Let G be a finite group, let R be a strongly G-graded kalgebra, and let N be an R-module. If N is stably flat over R_{E} for all elementary abelian subgroups E of G then N is stably flat over R.

Proof. As in the proof of Chouinard’s theorem, we first deal with the case when G is a p-group, and we may assume inductively that N is stably flat over R_{H} for all proper subgroups H of G, and that G is not elementary abelian. Again, we choose a sequence of non-zero elements x_{1}, …, x_{n} of H^{1}(G , ℤ/pℤ) = Hom(G, ℤ/pℤ) such that the product βx_{1} … βx_{n} of their Bocksteins is zero in H^{2n}(G, ℤ). Regarding each x_{i} as a surjective homomorphism, let H_{i} denote the kernel. As before, we have an exact sequence

0 → N → N ⊗ k[H_{n}\G] → ⋯ → N ⊗ k[H_{1}\G] → N → 0.

which represents the zero element of Êxt_{R}^{2n}(N, N). Moreover the inductive hypothesis together with Lemma 6.2 shows that the intermediate modules here are all stably flat, and hence the result follows by applying Lemma 6.1. Now in the general case, we reduce to the Sylow subgroups in exactly the same way as in the proof of Chouinard’s theorem.

The next result is a key ingredient in proving Theorem B, and it is here that the coherence of R_{1} enters into the argument.

Lemma 6.4. Suppose that R_{1} is coherent. Let N be an R-module. If H is a finite subgroup of G such that N is stably flat over R_{H}, then N ⊗_{R_{H}} R is stably flat over R.

Proof. Since R_{1} is coherent and H is finite, then R_{H} is also coherent, because R_{H} is finitely generated and projective as a left R_{1}-module. Viewed as an R_{H}-module, we may write N as a filtered colimit of finitely presented modules, N = lim_{→}_{λ} V_{λ}. The coherence of R_{H}implies

that each V_{λ} is of type FP_{∞} and since N is stably flat over R_{H}, each of the natural maps V_{λ} → N represents zero in Êxt_{R_{H}}^{0}(V_{λ}, N). In turn, this implies that each of the induced natural maps V_{λ}⊗_{R_{H}} R → N⊗_{R_{H}} R represents the zero morphism in Êxt_{R}^{0}(V_{λ}⊗_{R_{H}} R, N ⊗_{R_{H}} R) Now suppose given a stable morphism from M to N ⊗_{R_{H}} R, where M is an R-module of type FP_{∞}. Then, for some λ this morphism factors through the natural map V_{λ}⊗_{R_{H}} R → N ⊗_{R_{H}} R, and hence it is zero as required.

In [10], it is shown that for any module M of type FP_{∞}, the functors Êxt_{R}^{n}(M , ) commute with ‘direct limits’ (i.e. colimits over directed posets). Essentially the same argument proves the next lemma:

Lemma 6.5. The class of stably flat modules is closed under arbitrary filtered colimits, and under direct sums.

The following is again elementary, although it does not address the question what happens for negative n.

Lemma 6.6. Let N be a stably flat module and let M be a module of type FP_{∞}. Then Êxt_{R}^{n}(M , N) = 0 for all n ≥ 0.

Proof. Fix n ≥ 1. Choose a projective resolution P_{∗} ↠ M of finite type and let M′ denote the n^{th} kernel, so that there is an exact sequence

0 → M′ → P_{n−1} → ⋯ → P_{0} → M → 0.

Now M′ is also of type FP_{∞} and a simple dimension shifting arguement shows that Êxt_{R}^{n}(M , N) is isomorphic to Êxt_{R}^{0}(M′, N). But the latter group is zero because N is stably flat.

A further dimension shifting argument then yields:

Corollary 6.7. Suppose given an exact sequence

0 → N_{m} → ⋯ → N_{0} → N → 0

in which each N_{i} is stably flat. Then N is stably flat.

Proposition 6.8. Suppose that R_{1} is coherent. Let N be an R-module which is stably flat over R_{H} for all finite subgroups H of G. Then N ⊗_{R_{K}} R is stably flat over R for all LH𝔉-subgroups K of G.

Proof. Let 𝔛 be the set of subgroups K such that N ⊗_{R_{K}} R is stably flat over R. First note that all finite subgroups belong to 𝔛 by Lemma 6.4. We next show that all H𝔉-subgroups of G belong to 𝔛. Let K be a subgroup of G and suppose given a finite dimensional contractible K-CW-complex X with cell stabilizers in 𝔛. The augmented cellular chain complex over k is an exact sequence

0 → C_{m} → ⋯ → C_{0} → k → 0

of kK-modules. Tensoring with N yields a sequence of R_{K}-modules:

0 → N ⊗ C_{m} → ⋯ → N ⊗ C_{0} → N → 0.

Induction to R yields the exact sequence

0 → (N ⊗ C_{m}) ⊗_{R_{K}} R → ⋯ → (N ⊗ C_{0}) ⊗_{R_{K}} R → N ⊗_{R_{K}} R → 0.

For 0 ≤ i ≤ m, C_{i} is a permutation module with stabilizers in 𝔛. Hence it follows that each (N ⊗ C_{i}) ⊗_{R_{K}} R is a direct sum of modules of the form N ⊗_{R_{H}} R where H belongs to 𝔛, and these are assumed to be stably flat. (Compare with the proof of lemma 7.1 of [6] and the treatment of the exact sequence (8.1) of [6].) Since the class of stably flat modules is closed under arbitrary direct sums (Lemma 6.5), each (N ⊗ C_{i}) ⊗_{R_{K}} R is stably flat. Corollary 6.7 shows that N ⊗_{R_{K}} R is stably flat. This argument suffices to show that all H𝔉-subgroups belong to 𝔛.

Finally, if K is a subgroup of G and every finitely generated subgroup of K belongs to 𝔛 then K also belongs to 𝔛, because N ⊗_{R_{K}} R can be regarded as the colimit of the modules N ⊗_{R_{H}} R as H runs through the finitely generated subgroups of K and we may again appeal to Lemma 6.5. This allows us to conclude that N ⊗_{R_{K}} R is stably flat whenever K is an LH𝔉-subgroup as required.

Proof of Theorem B.

(1) ⇒ (2) This follows from the Eckmann-Shapiro Lemma, (1.2).

(2) ⇒ (1) Lemma 6.3 combined with Proposition 6.8 yield this implication.

7. The rôle of k

The commutative coefficient ring k underlies every result of this paper and the reader may wonder what rôle it plays. So here is an explanation. The point is that once k is chosen it becomes conceivable that the results hold for a class of groups which is larger than LH𝔉 and whose definition depends on k.

Let (H_{1}, k) denote the operation on classes of groups which is defined as follows. If 𝔛 is a class of groups then (H_{1}, k)𝔛 denotes the class of groups G for which there is a finite dimensional k-acyclic G-CW-complex X with stabilizers in 𝔛. Let (H, k)𝔛 be the smallest (H_{1}, k)- closed class containing 𝔛. When k = ℤ, these two operations coincide with the original operations H_{1} and H. But for other commutative rings it is conceivable that the class (H, k)𝔛 could be larger than H𝔛.

Remark 7.1. Every result of this paper in which it is assumed that the group G belongs to LH𝔉 holds for L(H, k)𝔉-groups.

At present there are few groups known to be outside the class LH𝔉 and so we do not know if or how the classes L(H, k)𝔉 can be distinguished from one another.

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School of Mathematical Sciences, Queen Mary and Westfield College, Mile End Road, London E1 4NS

E-mail address: P.H.Kropholler@qmw.ac.uk