ON COMPLETE RESOLUTIONS

JONATHAN CORNICK AND PETER H. KROPHOLLER In memoriam Thomas Václav Gedrich 12th August 1957 - 17th July 1995

Abstract. We develop simple criteria for constructing and using complete resolutions for modules over group algebras and strongly group graded rings.

1. Introduction

The development of this paper has been strongly influenced by the work [9] of Gedrich and Gruenberg in which the concepts of complete resolutions and the invariants spli and silp, amongst others, were introduced into the study of infinite group rings. Here we combine their methods with the complete cohomology theory axiomatised by Mislin [14] to develop simple criteria for constructing complete resolutions for modules over group algebras and strongly graded rings, and show that complete resolutions can be used to compute complete cohomology.

Tom Gedrich followed our research project with great enthusiasm characteristic of his lively personality, and so we have chosen to dedicate this paper to his memory.

Throughout this paper S will denote a ring, A and B will be right Smodules, G will be a group and k will be a commutative ring. Following Mislin [14] the nth complete Ext group associated to A and B is defined as

Êxt_{S}^{n}(A, B) = lim_{j≥0^{⟶}} S^{−j}Ext_{S}^{n+j}(A, B),

1991 Mathematics Subject Classification. 20J.

Key words and phrases. complete resolutions, group rings, cohomology.

The first author is supported by a postdoctoral fellowship from the ETH.

2 JONATHAN CORNICK AND PETER H. KROPHOLLER

where S^{−j}Ext_{S}^{n+j}(A , ) denotes the j-th left satellite of the functor Ext_{S}^{n+j}(A, ,). Alternative (but equivalent) definitions were introduced by Benson and Carlson [3] and Vogel (see [10]).

It is known that proj. dim_{S}(A) < ∞ if and only if Êxt_{S}^{0}(A, A) = 0 [11] and generalizations of this result have been used to study cohomological properties of modules over group algebras and strongly group graded rings ( [6], [7], [12], [13],). Important information about A is therefore contained in the complete Ext groups but unfortunately none of the aforementioned definitions lend themselves very easily to actual computation.

When G is a group and M is a kG-module we call Ĥ^{∗}(G , M) = Êxt_{kG}^{∗}(k, M) the complete cohomology groups of G with coefficients in M. As the notation suggests these cohomology groups are canonically isomorphic to Tate-Farrell cohomology when G is a group of finite virtual cohomological dimension [14, Lemma 3.1] and this was in fact original motivation for inventing complete cohomology. Recall that Tate-Farrell cohomology is defined using complete resolutions. If G is a group of finite virtual cohomological dimension over k then a complete resolution of k is an acyclic complex of projective kG-modules P = (P_{∗}, δ) which agrees with a projective resolution of k above a certain dimension, and Ĥ^{∗}(G, M) is defined to be H^{∗}(Hom_{kG}(P, M)).

The purpose of this paper is to show that complete cohomology can be calculated using complete resolutions under more general circumstances, but not always. The advantage with this approach is that we then have computational devices such as the Eckmann-Shapiro lemma and certain spectral sequences.

Complete cohomology vanishes on projective modules and so we superficially modify the definition of complete resolution.

ON COMPLETE RESOLUTIONS 3

Definition 1.1. Let S be a ring and let A be an S-module. Then a complete resolution of A is an acyclic complex of projective S-modules P = (P_{∗}, δ), indexed by the integers, such that

1. P coincides with a projective resolution of M in sufficiently high dimensions.

2. Hom_{S}(P, Q) is acyclic for every projective S-module Q.

As previously mentioned only the first part of this definition is used in defining complete resolutions in Tate-Farrell cohomology. However, if H is a subgroup of finite index in G with cd_{k} H < ∞, then P is split as a sequence of kH-modules and the Eckmann-Shapiro lemma implies that Ĥ^{n}(G, M) = 0 for every n, when M is a (direct summand of a) module (co)induced from H and in particular if M is projective.

In Section 2 we investigate some basic properties of complete resolutions. Mislin has shown that complete cohomology satisfies a certain universal property with respect to cohomological functors which vanish on projective modules. Using this fact together with a dimension shifting lemma also due Mislin we prove

Theorem 1.2. Let P = (P_{∗}, δ) be a complete resolution of the Smodule A and let B be an arbitrary S-module. Then

Êxt_{S}^{∗}(A, B) ≅ H^{∗}(Hom_{S}(P_{∗}, B)).

We shall mainly be concerned with modules over strongly group graded rings, and in particular when the grading group belongs to the class H𝔉.

Definition 1.3. Let 𝔉 denote the class of finite groups. Then

1. H_{1}𝔉 is the class of groups which admit a cellular action on a finite dimensional contractible cell complex with isotropy groups in 𝔉.

4 JONATHAN CORNICK AND PETER H. KROPHOLLER

2. H𝔉 is the smallest class of groups containing 𝔉 such that G belongs to H𝔉 whenever it acts cellularly on a finite dimensional contractible complex with isotropy groups already in H𝔉.

We refer the reader to [11] and [12] for a detailed discussion of these classes. However we point out that H𝔉 is a very large class which is extension closed and contains all countable linear and countable soluble groups. The class H_{1}𝔉 is smaller but includes all groups of finite virtual cohomological dimension and seems to be a natural setting for studying such groups.

In Section 3 we prove that complete resolutions for modules over strongly graded rings can be constructed under certain conditions, and in particular prove the following theorem.

Theorem 1.4. Let G be a group in the class H𝔉, let R be a strongly G-graded ring and let M be an R-module such that proj. dim_{R_{1}} M < ∞. Suppose that either

1. G ∈ H_{1}𝔉, or

2. M is a module of type FP_{∞}. Then M has a complete resolution.

There are various corollaries to this theorem, including a version of Shapiro’s Lemma for modules over strongly graded rings and the existence of certain spectral sequences.

If A is a module of type FP_{∞} then by definition it has a projective resolution by finitely generated projective modules, and this is well known to be equivalent to the property that Ext_{S}^{∗}(A, ) commutes with direct limits.

Since Êxt_{S}^{∗}(A, ) also commutes with direct limits if A is of type FP_{∞} [11] it seems reasonable to expect that such a module will have a complete resolution by finitely generated projective modules if a complete

ON COMPLETE RESOLUTIONS 5

resolution exists. We prove in Section 4 that this is indeed the case if M is an R-module of type FP_{∞} satisfying the hypotheses of Theorem 1.4. One consequence of this result is that if G is a group in H𝔉 of type FP_{∞} then Ĥ^{0}(G, ℤ) is finite.

In Section 5 we construct examples to show that complete resolutions do not always exist. In particular complete resolutions may not exist for modules over group algebras if the hypotheses of Theorem 1.4 are weakened.

The final section is a discussion of certain (unanswered) questions which arise from the results in this paper.

2. Properties

To show that complete resolutions can be used to compute complete cohomology we first recall some general properties of complete cohomology functors which were established in [14, Section 2]. A sequence of additive functors T^{∗} = {T^{n}|n ∈ ℤ} is a (−∞, ∞)-cohomological functor from S-modules to abelian groups if there is a natural long exact sequence

⋯ ⟶T^{n}B′⟶T^{n}B⟶T^{n}B′′⟶T^{n+1}B′⟶ ⋯

associated to any short exact sequence B′ ↣ B ↠ B′′ of S-modules.

Definition 2.1. A (−∞, ∞)-cohomological functor T^{∗} = {T^{n}|n ∈ ℤ} is called P -complete if T^{n}(P) = 0 for every projective module P and for every n.

A morphism U^{∗}→V^{∗} of (−∞, ∞)-cohomological functors is called a P -completion if

1. V^{∗} is P -complete, and

2. every morphism U^{∗}→W^{∗} into a P -complete cohomological functor W^{∗} factors uniquely through U^{∗}→V^{∗}.

6 JONATHAN CORNICK AND PETER H. KROPHOLLER

We regard Ext_{S}^{∗}(A, ) as (−∞, ∞)-cohomological functor with the convention that Ext_{S}^{n}(A, ) = 0 for n < 0. The following theorem is then a restatement of [14, Theorem 2.2].

Theorem 2.2. Êxt_{S}^{∗}(A, ) is the (unique) P -completion of Ext_{S}^{∗}(A, ).

We shall also need the following lemma which follows directly from [14, Lemma 2.5].

Lemma 2.3. If θ^{∗} : Ext_{S}^{∗}(A, )→V^{∗} is a morphism of (−∞, ∞)-cohomological functors such that V^{∗} is P -complete and θ^{i} is an equivalence for i ≥ n (a fixed integer), then the induced morphism Êxt_{S}^{∗}(A , )→V^{∗} is an equivalence.

Now suppose that P = (P_{∗}, δ) is a complete resolution of the Smodule A. It is obvious that H^{∗}(Hom_{S}(P, )) is a P -complete (−∞, ∞)- cohomological functor from S-modules to abelian groups, and so we construct a morphism

Ext_{S}^{∗}(A, )→H^{∗}(Hom_{S}(P, )).

This will follow from the following general result.

Lemma 2.4. Let P = (P_{∗}, δ) be an exact sequence of projective Smodules such that Hom_{S}(P, Q) is acyclic for every projective S-module Q. Suppose that Q = (Q_{∗}, d) is a complex of projective S-modules and that there is a partial chain map ϕ = (ϕ_{i≥n}) : P→Q.

⋯ P_{n+1} ⟶^{δ_{n+1}} P_{n} ⟶^{δ_{n}} P_{n−1} ⟶^{δ_{n−1}} ⋯
↓^{ϕ_{n+1}} ↓^{ϕ_{n}}
⋯ Q_{n+1} ⟶^{d_{n+1}} Q_{n} ⟶^{d_{n}} Q_{n−1} ⟶^{d_{n−1}} ⋯

Then ϕ may be extended to a chain map in all dimensions which is unique up to chain homotopy.

In particular

ON COMPLETE RESOLUTIONS 7

1. If Q ↠ A is a projective resolution of A and P is a complete resolution of A which coincides with Q in sufficiently high dimension then there is a chain map ϕ : P→Q (where we assume that Q_{i} = 0 if i < 0.

2. Any two complete resolutions of the S-module A are chain homotopy equivalent.

Proof. We proceed by induction. Assume that ϕ_{k} has been constructed for k ≥ i (of course this is true if k ≥ n which starts the induction), then we need to construct a map ϕ_{i−1} : P_{i−1}→Q_{i−1} such that ϕ_{i−1}δ_{i} = d_{i}ϕ_{i}. The sequence

Hom_{S}(P_{i−1}, Q_{i−1})⟶ Hom_{S}(P_{i}, Q_{i−1})⟶ Hom_{S}(P_{i+1}, Q_{i−1})

is exact at Hom_{S}(P_{i}, Q_{i−1}) by hypothesis. The element d_{i}ϕ_{i} ∈ Hom_{S}(P_{i}, Q_{i−1}) maps to d_{i}ϕ_{i}δ_{i+1} = d_{i}d_{i+1}ϕ_{i+1} = 0 ∈ Hom_{S}(P_{i+1}, Q_{i−1}) and so there exists

ϕ_{n−1} ∈ Hom_{S}(P_{n−1}, Q_{n−1})

such that ϕ_{i−1}δ_{i} = d_{i}ϕ_{i} as required.

Now suppose that ϕ′ is another extension of ϕ. We must show that for all i there exists s_{i} ∈ Hom_{S}(P_{i}, Q_{i+1} such that ϕ_{i} − ϕ′_{i} = d_{i+1}s_{i} + s_{i−1}δ_{i}. Clearly we may choose s_{i} = 0 if i ≥ n − 1 and proceeding by induction assume that s_{k} has been constructed for k ≥ i. The sequence

Hom_{S}(P_{i−1}, Q_{i})⟶ Hom_{S}(P_{i}, Q_{i})⟶ Hom_{S}(P_{i+1}, Q_{i})

is exact at Hom_{S}(P_{i}, Q_{i}) by hypothesis. The element (ϕ_{i}−ϕ′_{i})−d_{i+1}s_{i} ∈ Hom_{S}(P_{i}, Q_{i}) maps to

((ϕ_{i} − ϕ′_{i}) − d_{i+1}s_{i})δ_{i+1} = d_{i+1}((ϕ_{i+1} − ϕ′_{i+1}) − s_{i}δ_{i+1})
= d_{i+1}d_{i+2}s_{i+1}
= 0 ∈ Hom_{S}(P_{i+1}, Q_{i}),

and so there exists s_{i−1} ∈ Hom_{S}(P_{i−1}, Q_{i}) such that s_{i−1}δ_{i} = (ϕ_{i} −ϕ′_{i})− d_{i+1}s_{i} as required.

8 JONATHAN CORNICK AND PETER H. KROPHOLLER

The two specific cases follow easily. For the second note that two projective resolutions Q ↠ A and Q′ ↠ A are chain homotopy equivalent and so without loss of generality we may assume they are equal. Furthermore we may assume that two complete resolutions P and P′ coincide with Q above dimension n. Then we have partial chain maps ϕ = (ϕ_{i≥n}) : P→P′ and θ = (θ_{i≥n}) : P′→P which extend to chain maps which are chain homotopy inverses.

Proof of Theorem 1.2. Suppose that Q ↠ A is a projective resolution of A and P is a complete resolution of A which coincides with Q in dimensions ≥ n. Then by the previous lemma there is a chain map ϕ : P→Q which induces a morphism

Ext_{S}^{∗}(A, )→H^{∗}(Hom_{S}(P, ))

of (−∞, ∞)-cohomological functors. Since ϕ is the identity map in dimensions n and above it is clear that

Ext_{S}^{i}(A, ) ≅ H^{i}(Hom_{S}(P, )) for all i > n,

and so the result follows from Theorem 2.2 and Lemma 2.3.

Corollary 2.5. If A has a complete resolution P then

1. If Q is a projective module then Ext_{S}^{i}(A, Q) = 0 in sufficiently high dimensions.

2. If I is an injective module then Êxt_{S}^{∗}(A, I) = 0.

Proof. 1. If P coincides with a projective resolution in dimensions ≥ n then we have seen in the proof of the theorem that Ext_{S}^{i}(A, ) = Êxt_{S}^{i}(A, ) for i > n, and so statement follows because complete cohomology vanishes on projective modules.

2. Let K_{i} = Ker(P_{i}→P_{i−1}) denote the ith kernel in P. Then it is easy to see that Êxt_{S}^{n}(A, B) = Ext_{S}^{1}(K_{n−2}, B) for any module B and so Êxt_{S}^{n}(A, I) = 0.

3. Constructing complete resolutions

Throughout this section R = ⊕_{g∈G}R_{g} will be a strongly G-graded k-algebra, and M will be an R-module. If H is a subgroup of G then R_{H} is the subring of R supported on H, and R is projective as both a right and left R_{H}-module [7, Lemma 6.2]

If V is a kG-module then M⊗V is an R-module via the semi-diagonal action defined by then (m ⊗ v)r := mr ⊗ vg for m ∈ M, v ∈ V and r ∈ R_{g}. Similarly there is a semi-diagonal action of R on hom_{k}(V, M) defined by v[ϕr] := ((vg^{−1})ϕ)r for ϕ ∈ hom_{k}(V, M), v ∈ V and r ∈ R_{g}.

Definition 3.1. Let G be a group. Then 1. B(G, ℤ) is the ring of bounded functions from G to ℤ.

2. B(G, k) := B(G, ℤ) ⊗ ℸ is the k-algebra of functions from G to k which take only finitely many values.

There is a natural action of G on B(G, k) defined by ϕ^{g}(g′) = ϕ(gg′) for g, g′ ∈ G and ϕ ∈ B(G, k) and so M ⊗ B(G, k) is an R-module via the semi-diagonal action. The reader is referred to [7, Sections 3 and 9] for a more detailed discussion of semi-diagonal actions and the k-algebra B(G, k).

It is perhaps not surprising that B(G, k) should play some part in the construction of complete resolutions. We identify Coind_{1}^{G}k with the ring of all functions from G to k, and Ind_{1}^{G}k with the ideal of functions with finite support and so there are inclusions of kG-modules

Ind_{1}^{G}k ⊆ B(G, k) ⊆ Coind_{1}^{G}k

which are equalities if G is finite but not in general. If Γ is a group of finite virtual cohomological dimension over k and N is a normal

10 JONATHAN CORNICK AND PETER H. KROPHOLLER

subgroup of finite index with cd_{k} N < ∞ then there are equalities

Ind_{1}^{Γ/N}k = Coind_{1}^{Γ/N}k

and one uses this fact to construct complete resolutions over Γ [4, Chapter 10].

The following results illustrate that we use B(G, k) because it shares some properties with Ind_{1}^{G}k and some with Coind_{1}^{G}k .

Lemma 3.2. B(G, k) is free as a k-module.

Proof. If k = ℤ then this can be deduced from [8, Lemma 97.2]. The general case follows directly from the definition.

Lemma 3.3. There is a k-split inclusion k ↣ B(G, k) of kG-modules, and so the cokernel \overline{B}(G, k) of this map is a kG-module which is free over k.

Proof. The inclusion of k into B(G, k) as the constant functions is clearly a kG-homomorphism. This map is split by the k-module map B(G, k)→k defined by ϕ ↦ ϕ(1). Thus, in particular, \overline{B}(G, ℤ) is free and the result follows because \overline{B}(G, k) = \overline{B}(G, ℤ) ⊗ ℸ.

If X is a projective R-module then there is an R-module map

hom_{k}(B(G, k), X)→X

defined by θ ↦ θ(1), where θ ∈ hom_{k}(B(G, k), X) and 1 is the multiplicative identity element of B(G, k) (that is the function which maps every element in G to the multiplicative identity of k.) This map splits because X is projective and so we have

Lemma 3.4. X is a direct summand of hom_{k}(B(G, k), X) as R-modules.

We are now in a position to prove the following theorem.

ON COMPLETE RESOLUTIONS 11

Theorem 3.5. If R is a strongly graded ring and M is an R-module such that proj. dim_{R} M ⊗ B(G, k) < ∞ then M has a complete resolution.

Proof. To simplify the notation we put B := B(G, k) and \overline{B} := \overline{B}(G, k). The functor ⊗B is exact because B is k-free, and so we may replace M by a suitable kernel in a projective resolution of M over R and assume that M ⊗ B is a projective R-module.

Let V_{i} denote the kG-module \overline{B}^{⊗i} ⊗ B ( with the convention that \overline{B}^{⊗0} = k). By tensoring the k-split sequence

k ↣ B ↠ \overline{B}

with V_{i} we have short exact sequences

\overline{B}^{⊗i} ↣ V_{i} ↠ \overline{B}^{⊗i+1}.

for every i ≥ 0. We splice these sequences together to obtain a long exact sequence of kG-modules

0⟶k⟶V_{0}⟶V_{1}⟶ · · · .

Since M ⊗ B is projective it follows from [7, Corollary 3.3] that M ⊗ B ⊗ V is also projective over R for any kG-module V which is k-projective (with R acting semi-diagonally). Consequently M ⊗ V_{i} is projective for every i ≥ 0 because \overline{B} is k-projective (Lemma 3.3), and so there is a long exact sequence of R-modules

0⟶M⟶M ⊗ V_{0}⟶M ⊗ V_{1}⟶ ⋯,

in which all the modules (other than M) are projective.

Now let Q ↠ M be a projective resolution of M as an R module. Setting P_{i} = Q_{i} for i ≥ 0, and P_{−i} = M ⊗ V_{i−1} for i ≥ 1 we have an exact sequence P of projective R-modules which coincides with a projective resolution of M above dimension 0.

12 JONATHAN CORNICK AND PETER H. KROPHOLLER

Let X be a projective R-module. To prove that P is a complete resolution of M we must show that Hom_{R}(P, X) is acyclic and by Lemma 3.4 it suffices to show that Hom_{R}(P, hom_{k}(B, X)) is acyclic. But

Hom_{R}(P, hom_{k}(B, X)) ≅ Hom_{R}(P ⊗ B, X)

and we claim that P ⊗ B is split.

If i ≥ 0 then the kernel of P_{i} ⊗ B→P_{i−1} ⊗ B is projective because M ⊗ B is projective over R.

If i < 0 then the kernel of P_{i} ⊗ B→P_{i−1} ⊗ B has the form

M ⊗ \overline{B}^{⊗(−1−i)} ⊗ B ≅ M ⊗ B ⊗ \overline{B}^{⊗(−1−i)}

which is also projective over R.

Theorem 1.4 now follows immediately from [7, Proposition 9.2 ] which we state below:

Proposition 3.6. Let G be a group in the class H𝔉, let R be a strongly G-graded ring and let M be an R-module such that proj. dim_{R_{1}} M < ∞. Suppose that either

1. G ∈ H_{1}𝔉, or

2. M is a module of type FP_{∞}. Then

proj. dim_{R} M ⊗ B < ∞

where R acts semi-diagonally.

Corollary 3.7. Let G, R and M be as in the theorem and let H be a subgroup of G. Then the complete resolution of M constructed in the theorem is also a complete resolution of M as an R_{H}-module.

Proof. The first condition for a complete resolution is clearly satisfied since R is projective as an R_{H}-module. If Y is a projective R_{H}-module then repeating the argument in the proof of the theorem shows that

ON COMPLETE RESOLUTIONS 13

Y is a direct summand of Hom_{k}(B(G, k), Y) as R_{H}-modules and the result follows as before.

Corollary 3.8 (Eckmann-Shapiro Lemma). Let G, H, R and M be as in the previous corollary, and suppose that N is an R_{H}-module, then there are natural isomorphisms

Êxt_{R_{H}}^{n}(M , N) ≅ Êxt_{R}^{n}(M , Hom_{R_{H}}(R, N))

for every integer n.

Proof. We have shown above that the complete resolution P constructed in the theorem is a complete resolution for M as an R_{H}-module. The result then follows from the natural isomorphisms Hom_{R_{H}}(P_{i}, N) ≅ Hom_{R}(P_{i}, Hom_{R_{H}}(R, N)).

Another application of the theorem is to construct spectral sequences analagous to the Lyndon-Hochschild-Serre spectral sequence in ordinary cohomology (eg [2, Chapter 3.5]).

Corollary 3.9. Let G be a group in H𝔉, let k be a commutative ring of finite global dimension, and let H be a normal subgroup of G.

Suppose that M_{1} is a k[G/H]-module of finite projective dimension, M_{2} is a kG-module which is of type FP_{∞} (or arbitrary if G ∈ H_{1}𝔉), and M_{3} is any kG-module.

Then there is a first and fourth quadrant spectral sequence E_{2}^{p,q} = Ext_{k[G/H]}^{p}(M_{1}, Êxt_{kH}^{q}(M_{2}, M_{3}))⟹Êxt_{kG}^{p+q}(M_{1} ⊗ M_{2}, M_{3})

which converges because proj. dim_{k[G/H]} M_{1} < ∞.

Proof. Let Q ↠ M_{1} be a (finite length) projective resolution of M_{1} as a k[G/N]-module, and let P be a complete resolution of M_{2} as a kG-module. Then by the first corollary P is also a complete resolution of M_{2} as a kN-module. We claim that Q ⊗ P is a complete resolution of M_{1} ⊗ M_{2} as a kG-module.

14 JONATHAN CORNICK AND PETER H. KROPHOLLER

Q ⊗ P is clearly an acyclic complex of projective kG-modules and it coincides with a projective resolution of M_{1} ⊗ M_{2} in sufficiently high dimensions because P coincides with a projective resolution of M_{2} in sufficiently high dimensions and Q has finite length.

Let X be a projective kG-module. There are isomorphisms

Hom_{k[G/N]}(Q_{p}, Hom_{kN}(P_{q}, )) ≅ Hom_{kG}(Q_{p} ⊗ P_{q}, )

and so Hom_{kG}(Q ⊗ P, X) is acyclic because Hom_{kN}(P, X) is acyclic. Therefore we obtain the stated spectral sequence because

H^{∗}(Hom_{kN}(P, M_{3})) ≅ Êxt_{kN}^{∗}(M_{2}, M_{3})

and

H^{∗}(Hom_{kG}(Q ⊗ P, M_{3})) ≅ Êxt_{kG}^{∗}(M_{1} ⊗ M_{2}, M_{3}).

Finally we point out that there is another way to construct complete resolutions which was discovered by Gedrich and Gruenberg [9]. Let S be a ring, then spli(S) is defined to be the supremum of the projective lengths of the injective S-modules, and silp(S) is the supremum of the injective lengths of the projective S-modules.

Unfortunately their terminology is slightly different from ours, they define a complete resolution using only the first part of our definition. However using their results we have:

Theorem 3.10. Let S be a ring. Then the following are equivalent. 1. spli(S) = silp(S) < ∞.

2. Every S-module has a complete resolution.

Proof. First note that if spli(S) and silp(S) are both finite then they are equal [9, Section 1.6] and in this case every module has a complete resolution (in our sense) [9, Section 4].

ON COMPLETE RESOLUTIONS 15

By standard arguments, to show that silp(S) is finite it suffices to show that every projective S-module has finite injective dimension, and similarly for spli(S). If every S-module has a complete resolution then in particular every injective module I has a complete resolution. Thus by Corollary 2.5, Êxt_{S}^{0}(I, I) = 0 and so I has finite projective dimension [11].

Now let M be an arbitrary S-module and let Q be a projective Smodule. Then by Corollary 2.5 there exists an n such that Ext_{S}^{i}(M, Q) = 0 for all i ≥ n. We claim that n can be chosen to be independent of the module M from which it will follow that Q has finite injective dimension ≤ n.

Suppose for a contradiction that for every non-negative integer j there exists a module M_{j} and an integer j_{M} ≥ j such that Ext_{S}^{j_{M}} (M_{j}, Q) ≠ 0. Then the module M = ⊕_{j≥0}M_{j} has a complete resolution by hypothesis, but there does not exist an integer n such that Ext_{S}^{i}(M , Q) = 0 for all i ≥ n contradicting the result in the Corollary.

Remark 3.11. We do not know of a ring S such that one of these dimensions is finite and the other infinite.

If S = kG where G is an H𝔉-group of type FP_{∞} and k is a commutative ring of finite global dimension then spli(S) = silp(S) < ∞ [6, Corollary C], and so every module over S has a complete resolution.

4. Complete Resolutions of Finite Type

The main goal of this section is to show that if M is a module of type FP_{∞} over a strongly graded ring R, which satisfies the hypotheses of Theorem 3.5, then M has a complete resolution P = (P_{∗}, δ) of finite type, i.e. such that P_{i} is a finitely generated free R-module for every i.

Lemma 4.1. Let K be a finitely generated R-module such that K ⊗ B is a projective R-module. Then there is a short exact sequence K ↣

16 JONATHAN CORNICK AND PETER H. KROPHOLLER

P ↠ L, where P and L are finitely generated R-modules and P is free, such that

K ⊗ B ↣ P ⊗ B ↠ L ⊗ B

is split.

Proof. Recall from Section 3 that there is a k-split short exact sequence k ↣ B ↠ \overline{B} and so by tensoring this sequence with K we obtain a short exact sequence of R-modules

K ↣ K ⊗ B ↠ K ⊗ \overline{B},

which splits upon tensoring with B.

Since K ⊗ B is projective over R we can choose a projective complement Q and so there is a short exact sequence

K ↣ K ⊗ B ⊕ Q ↠ K ⊗ \overline{B} ⊕ Q,

which also splits upon tensoring with B.

Finally since K is finitely generated and K ⊗ B ⊕ Q is free we may choose a finitely generated free direct summand P of K ⊗ B ⊕ Q such that K ↣ P and K ⊗ B ↣ P ⊗ B is split. The result follows if we set L = P/K.

Theorem 4.2. If G is a group in H𝔉, R is a strongly G-graded ring and M is an R-module of type FP_{∞} such that proj. dim_{R_{1}} M < ∞, then M has a complete resolution of finite type.

Proof. We have seen in the previous section that these conditions on M imply that proj. dim_{R} M ⊗ B < ∞. Once again we may replace M by a kernel in a projective resolution (which is still of type FP_{∞}) and assume that M ⊗ B is projective over R.

We construct an acyclic complex P of projective R-modules, which coincides with a projective resolution M above dimension zero, in such

ON COMPLETE RESOLUTIONS 17

a way that P ⊗ B is split. As in the proof of Theorem 3.5 this will be sufficient to show that P is a complete resolution.

For reasons which will become apparent we denote M by K_{−1}. The module M is finitely generated, and so by repeated applications of Lemma 4.1 we have for very i ≥ 1 a short exact sequence of R-modules

K_{−i} ↣ P_{−i} ↠ K_{−i−1}

such that P_{−i} is a finitely generated free R-module, and such that K_{−i} ⊗ B and K_{−i−1} ⊗ B are projective.

By splicing these short exact sequences together we obtain a long exact sequence

M ↣ P_{−1}⟶P_{−2}⟶ ⋯

in which every P_{−i} is finitely generated and free.

Since M is of type FP_{∞} we can choose a projective resolution

⋯ ⟶P_{2}⟶P_{1}⟶P_{0} ↠ M

such that every P_{i} is finitely generated and free, and which splits upon tensoring with B because M ⊗ B is projective. We splice these sequences together to obtain a long exact sequence

P := ⋯ ⟶P_{2}⟶P_{1}⟶P_{0}⟶P_{−1}⟶P_{−2}⟶ ⋯

of finitely generated free R-modules which coincides with a projective resolution of M above dimension zero. This resolution has been constructed so that the kernels K_{i} in this sequence are such that K_{i} ⊗ B are projective, and so P ⊗ B is split.

Corollary 4.3. Let G be a group in H𝔉, let k be a commutative noetherian ring, and let R be a strongly G-graded k-algebra.

Suppose that M_{1} is an R-module of type FP_{∞} such that proj. dim_{R_{1}} M_{1} < ∞, and M_{2} is an R-module which is finitely generated over k. Then Êxt_{R}^{n}(M_{1}, M_{2}) is a finitely generated k-module

18 JONATHAN CORNICK AND PETER H. KROPHOLLER

Proof. It follows from the theorem that M_{1} has a complete resolution P of finite type. Êxt_{R}^{n}(M_{1}, M_{2}) is a subquotient of the finitely generated k-module Hom_{R}(P_{n}, M_{2}) and the result follows because k is noetherian.

Corollary 4.4. If G is a group in H𝔉 of type FP_{∞} then Ĥ^{0}(G, ℤ) is finite.

Proof. The ℤ𝔾-module ℤ satisfies the hypotheses of the previous corollary and thus Ĥ^{0}(G, ℤ) is a finitely generated abelian group. However it is known that Ĥ^{0}(G, ℤ) is a torsion group under these conditions [11] and the result follows.

5. Counterexamples

If A is an S-module of finite projective dimension then it is clear from the definition that the sequence of zero modules is a complete resolution of A and this is reflected in the fact that Êxt_{S}^{∗}(A, ) ≡ 0. However the next example illustrates that even if A has finite projective dimension one can not assume that any acyclic complex of projective modules which coincides with a projective resolution of A in high enough dimension is automatically a complete resolution of A (as we have seen is the case for Tate-Farrell cohomology).

Example 5.1. Let k be a field, let R = k⟨x_{1}, x_{2}, x_{3}, … ⟩ be the free k-algebra on countably many variables and let I be the ideal of R generated by the set {x_{i+1}x_{i}|i ≥ 0} . Then S = R/I has as a k-basis the set

{x_{i_{1}}x_{i_{2}} … x_{i_{n}}|1 ≤ i_{j} < ∞, 1 ≤ n < ∞, i_{j} ≠ i_{j+1} − 1}

and in particular

r. ann_{S}(x_{1}) = 0, r. ann_{S}(x_{i}) = x_{i−1}S if i > 1.

ON COMPLETE RESOLUTIONS 19

For every i we have right S-module homomorphisms λ_{x_{i}} : S ↣ S defined by s ↦ x_{i}s for s ∈ S, which is a monomorphism if i = 1 and has kernel x_{i−1}S if i > 1. The cokernel of the inclusion x_{i}S ↣ S is S/x_{i}S ≅ x_{i+1}S and so there are short exact sequences

x_{i}S ↣ S ↠ x_{i+1}S

for every i.

Splicing these sequences together we have a long exact sequence

S := 0 ⟶ S ⟶^{λ_{x_{1}}} S ⟶^{λ_{x_{2}}} S ⟶ ⋯

which we may truncate to a projective resolution of x_{i}S for every i However the sequence Hom_{S}(S, S) :=

⋯ ⟶ Hom_{S}(S, S) ⟶^{λ_{x_{2}}^{∗}} Hom_{S}(S, S) ⟶^{λ_{x_{1}}^{∗}} Hom_{S}(S, S) ⟶ 0

may be identified with the sequence

⋯ ⟶ S ⟶^{ρ_{x_{2}}} S ⟶^{ρ_{x_{1}}} S ⟶ 0

where ρ_{x_{i}} : S→S is defined by s ↦ sx_{i}. Therefore Hom_{S}(S, S) is not exact because ρ_{x_{1}} is not onto (x_{1} does not have a left inverse) and so S is not a complete resolution of x_{i}S.

However to find examples of modules which never have complete resolutions one must concentrate on modules of infinite projective dimension. To this end we have the following proposition which essentially says that modules for which the Eckmann-Shapiro lemma fails can not have complete resolutions.

Proposition 5.2. Let k be a commutative ring of finite global dimension and let S be a k-projective k-algebra. If A is a non zero S-module such that Ext_{S}^{n}(A, Q) = 0 for every projective S-module Q and for every n ≥ 0, (so that necessarily proj. dim_{S} A = ∞,) then A does not have a complete resolution.

20 JONATHAN CORNICK AND PETER H. KROPHOLLER

Proof. If Ext_{S}^{∗}(A, ) vanishes on projective modules then it follows from Lemma 2.3 that

Ext_{S}^{∗}(A, ) ≅ Êxt_{S}^{∗}(A, ).

Suppose for a contradiction that A has a complete resolution P as an S-module. Since S is projective over k, it follows that P is an acyclic complex of projective k-modules which coincides with a projective resolution of A as a k-module in sufficiently high dimensions. Furthermore P is split as a sequence of k-modules because k has finite global dimension, and so P is a complete resolution of A over k.

Therefore we have

Êxt_{k}^{0}(A, A) ≅ H^{0}(hom_{k}(P, A))
≅ H^{0}(Hom_{S}(P, hom_{k}(S, A)))
≅ Êxt_{S}^{0}(A, hom_{k}(S, A))
≅ Ext_{S}^{0}(A, hom_{k}(S, A))
≅ hom_{k}(A, A).

But Êxt_{k}^{0}(A, A) = 0 because k has finite global dimension and Hom_{k}(A, A) ≠ 0 because it contains the identity map and we have our contradiction.

We illustrate this proposition with the following examples of group algebras for which the trivial module does not satisfy the hypotheses of Theorem 1.4.

Example 5.3. Suppose G is a group which is 1. free abelian of infinite rank, or

2. Gl_{n}(K) where K is a subfield of the algebraic closure of ℚ, or

3. the Brown-Geoghegan group ⟨x_{1}, x_{2}, … |x_{i}^{−1}x_{n}x_{i} = x_{n+1}, i < n⟩.

Then H^{n}(G, Q) = 0 for every projective ℤ𝔾-module Q and for all n ≥ 0. This is obvious for a free abelian group of infinite rank. For the other cases the reader is referred to [14, Theorem 3.2] and [5] respectively.

ON COMPLETE RESOLUTIONS 21

We deduce from the proposition that ℤ cannot have a complete resolution as a ℤ𝔾-module. In the first two cases G is in H𝔉 but not of type FP_{∞} (or even finitely generated) and in the third case G is of type FP_{∞} but not in H𝔉 [11].

6. Questions

The proof of Lemma 2.4 does not require P to be exact and so the results in Section 2 still hold if one weakens Definition 1.1 by allowing P to have homology. (Of course it must still be exact in high dimensions to coincide with a projective resolution).

Thus if G is a group as in Example 5.3 and Q ↠ ℤ is a ℤ𝔾-projective resolution then Q would be a complete resolution of ℤ with the weakened definition.

It is unclear how useful this observation might be. In the example Q does not restrict to a complete resolution over ℤ, and as we have seen this means that the Eckmann-Shapiro lemma fails even for modules coinduced from the trivial group.

One of the motivations for writing this paper was to find methods for computing Êxt_{S}^{0}(A, A) and in particular Ĥ^{0}(G, ℤ) when G is a group of type FP_{∞} in the class H𝔉. When G is torsionfree Ĥ^{0}(G, ℤ) = ⊬ and cd G < ∞, so in general how is the torsion in G reflected in Ĥ^{0}(G, ℤ) which we now know to be finite?

Let m denote the least common multiple of the orders of the finite subgroups of G and let n be the exponent of Ĥ^{0}(G, ℤ). Then one has ring homomorphisms

Ĥ^{0}(G, ℤ)⟶ℍ̂^{0}(𝔽, ℤ) = ℤ/|𝔽|ℤ

for every finite subgroup F of G, and since this is a unital homomorphism it follows that m divides n [11]. One might guess that that m = n, but this is not always true as demonstrated by examples of

Adem and Carlson [1].

References

[1] A. Adem and J. F .Carlson, Discrete groups with large exponents in cohomology, J. Pure Appl. Algebra 66 (1990), 111-120.

[2] D. J. Benson, Representations and Cohomology 2: Cohomology of groups and modules, Cambridge Studies in Advanced Mathematics 31, (Cambridge U.P. 1991).

[3] D. J. Benson and J. Carlson, ‘Products in negative cohomology’, J. Pure Appl. Algebra 82 (1992), 107–130.

[4] K. S. Brown, ‘Cohomology of Groups’, Graduate Texts in Math. 87 (Springer, Berlin, 1982).

[5] K. S. Brown and R. Geoghegan, ‘An infinite dimensional FP_{∞}-group’, Invent. Math. 77 (1984), 367–381.

[6] J. Cornick and P. H. Kropholler, Homological finiteness conditions for modules over group algebras, (preprint 261, CRM Barcelona, 1994).

[7] J. Cornick and P. H. Kropholler, Homological finiteness conditions for modules over strongly group-graded rings, to appear Math. Proc. Camb. Phil. Soc. .

[8] L. Fuchs, Infinite Abelian Groups, Volume II, (Academic Press, New York, 1973).

[9] T. V. Gedrich and K. W. Gruenberg, Complete cohomological functors on groups, Topology and its Applications 25 (1987), 203–223.

[10] F. Goichot, Homologie de Tate-Vogel équivariante, J. Pure Appl. Algebra 82 (1992), 39–64.

[11] P. H. Kropholler, On groups of type FP_{∞}, J. Pure Appl. Algebra 90 (1993), 55–67.

[12] P. H. Kropholler, Hierarchical decompositions, generalized Tate cohomology, and groups of type FP_{∞}, (eds. A. Duncan, N. Gilbert, and J. Howie) Proceedings of the Edinburgh Conference on Geometric Group Theory, 1993, (Cambridge U. P. 1994).

[13] P. H. Kropholler, Cohomological finiteness conditions, (eds. C. M. Campbell et al.) Groups ’93 Galway/St. Andrews vol.1, LMS Lecture Notes 211 (1995), 274-304.

[14] G. Mislin, Tate cohomology for arbitrary groups via satellites, Topology and its Applications 56 (1994), 293-300.

Departement Mathematik, ETH Zuerich, 8092 Zuerich

E-mail address: cornick@math.ethz.ch

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

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