TATE COHOMOLOGY FOR ARBITRARY GROUPS VIA SATELLITES

Guido Mislin

ETH Mathematik, Zürich, Switzerland, and Department of Mathematics, Ohio State University, Columbus, Ohio

Abstract. We define cohomology groups Ĥ^{n}(G; M), n ∈ ℤ, for an arbitrary group G and G-module M, using the concept of satellites. These cohomology groups generalize the Farrell-Tate groups for groups of finite virtual cohomological dimension and form a connected sequence of functors, characterized by a natural universal property.

The classical Tate cohomology groups of finite groups have been generalized to larger classes of groups by several authors ([BC], [F], [GG], [I]). The definition of Benson and Carlson in [BC] makes sense for an arbitrary group, but no formal properties are discussed there. We propose here a different definition for Tate cohomology groups for an arbitrary group G and G-module M, which takes the form

Ĥ^{n}(G; M) = \underrightarrow{lim}_{j≥0} S^{−j}H^{n+j}(G; M)

with S^{−j}H^{n+j}(G; ?) denoting the j-th left satellite of the functor H^{n+j}(G; ?). These general Tate groups are shown to agree, if applicable, with the ones obtained by the various generalizations mentioned above. The family Ĥ^{•} = {Ĥ^{n}(G; ?); n ∈ ℤ} forms a connected sequence of functors and is as such characterized by a natural universal property, which identifies it with what we call the completion with respect to projective modules, or short, the P-completion of ordinary cohomology (cf. section 3).

In the first section we recall, to fix our terminology and notation, some basic facts on satellites. Section two is devoted to an axiomatic description of the P -completion of a cohomological functor, leading to our definition of the general Tate groups. In section three we compare the definition to various other ones and discuss a few examples. Section four contains a detailed comparison with the definition proposed by Benson and Carlson.

We thank Karl Gruenberg for enlightening discussions concerning the concept of projective completion for connected sequences of functors.

Typeset by AMS-TEX

1991 Mathematics Subject Classification. Primary 18Gxx, 55Uxx; Secondary 20Jxx.

Key words and phrases. cohomology functors, Tate cohomology.

Section 1 : Satellites

The basic references here are [CE] and [HS]. We will quickly review the facts of which we make use. Let Λ be an associative ring with 1 and M a (left) Λ-module. We write F M for the free Λ-module on the underlying set of M, and ΩM for the kernel of the obvious map F M → M. If T denotes an additive functor from Λ-modules to abelian groups, then

S^{−1}T (M) = ker (T (ΩM) → T (F M))

defines a new additive functor S^{−1}T, the left satellite of T. For n ≥ 1 one defines inductively S^{−n}T = S^{−1}(S^{−n+1}T) and Ω^{n}M = Ω(Ω^{n−1}M) with the convention that S^{0}T = T and Ω^{0}M = M, respectively. Each short exact sequence of Λ-modules A′ → A → A′′ gives rise to a connecting homomorphism S^{−n}T (A′′) → S^{−n+1}T (A′), n > 0, in such a way that in the long sequence

⋯ → S^{−n}T A′ → S^{−n}T A → S^{−n}T A′′ → S^{−n+1}T A′ → ⋯ → T A′′,

the composition of any two consecutive homomorphisms is zero. Thus the family S^{≤0}T = {S^{−n}T; n ≥ 0} forms a connected sequence of functors. Obviously, S^{−1}T (P) = 0 for all projective modules P, because ΩP → F P is a split monomorphism. As a result, one has for general M and n > k ≥ 0 natural isomorphisms

(1.1)  S^{−n}T M ≅ S^{−n+k}T Ω^{k}M.

A connected sequence of functors V^{≤0} = {V^{−n}; n ≥ 0} is called of cohomological type, if the long sequence

⋯ → V^{−n}A′ → V^{−n}A → V^{−n}A′′ → V^{−n+1}A′ → ⋯ → V^{0}A′′,

associated with any short exact sequence A′ → A → A′′, is exact; in the terminology of [GG] such a V^{≤0} is called a (−∞, 0)-cohomological functor. For instance, if T is additive and half exact, then S^{≤0}T is of cohomological type (cf.[CE]). Any natural transformation ϕ : U → V of additive functors extends uniquely to a morphism ϕ^{≤0} : S^{≤0}U → S^{≤0}V of connected sequences of functors. More generally, if V^{≤0} is any connected sequence of (additive) functors, then any natural transformation ψ : V^{0} → V^{0} extends uniquely to V^{≤0} → S^{≤0}V; in particular, by taking for ψ the the identity of V^{0}, one obtains a morphism V^{≤0} → S^{≤0}V which we call the canonical one. The connected sequence of left satellites of a half exact functor can be characterized as follows (cf.[CE], III 5.2).

Theorem 1.2. Let U^{≤0} and V^{≤0} denote connected sequences of (additive) functors and ϕ^{0}: U^{0} → V^{0} a natural transformation. If V^{≤0} is of cohomological type and satisfies V^{−n}(P) = 0 for all n > 0 and all projective P, then the following holds:

(1) ϕ^{0} extends uniquely to ϕ^{≤0} : U^{≤0} → V^{≤0} and ϕ^{≤0} factors uniquely through the canonical morphism U^{≤0} → S^{≤0}U^{0}

(2) if U^{0} is half exact and ϕ^{0} is an equivalence then the induced morphism S^{≤0}U^{0} → V^{≤0} is an equivalence.

Section 2 : P-complete functors

We will follow the terminology of [GG] and call a connected sequence of additive functors T^{•} = {T^{n}; n ∈ ℤ} a (−∞, +∞)-cohomological functor, if the long sequence

⋯ → T^{n}A′ → T^{n}A → T^{n}A′′ → T^{n+1}A′ → ⋯

associated with any short exact sequence A′ → A → A′′ of Λ-modules is exact. A typical example is given by ordinary cohomology H^{•}={H^{n}(G; ?); n ∈ ℤ}, with the convention that H^{n}(G; ?) = 0 for n < 0.

Definition 2.1. A (−∞, +∞)-cohomological functor T^{•} = {T^{n}; n ∈ ℤ} is called P - complete, if T^{n}(P) = 0 for every n and every projective module P. A morphism U^{•} → V^{•} of (−∞, +∞)-cohomological functors is called a P -completion, if V^{•} is P -complete and if every morphism U^{•} → W^{•} into a P -complete cohomological functor W^{•} factors uniquely through U^{•} → V^{•}.

If G is a finite group, then the classical Tate groups Ĥ^{•} = {Ĥ^{n}(G; ?); n ∈ ℤ} form a P - complete cohomological functor and the natural morphism H^{•} → Ĥ^{•} is a P -completion (see also 3.1). More generally, if the (−∞, +∞)-cohomological functor U^{•} admits a “terminal completion” U^{•} → V^{•} in the sense of [GG] then it follows that U^{•} → V^{•} is a P -completion (we will discuss this in section 3). Since not every U^{•} admits a “terminal completion”, one can, in view of the following theorem, think of the P -completion as a natural generalization of the “terminal completion” of [GG].

Theorem 2.2. Every (−∞, +∞)-cohomological functor T^{•} = {T^{n}; n ∈ ℤ} admits a unique P -completion τ^{•} : T^{•} → T̂^{•}.

Proof. For every n ∈ ℤ we can form the (−∞, n)-cohomological functor S^{≤0}T^{n} which extends to a (−∞, +∞)-cohomological functor T^{•}⟨n⟩ by putting

(2.3)  T^{j}⟨n⟩ = \{ \begin{matrix}S^{j−n}T^{n}, & if j < n \\ T^{j}, & if j ≥ n. \end{matrix}

The identity transformation T^{n} → T^{n} extends uniquely to T^{≤n} → S^{≤0}T^{n}, and we extend it further to τ_{n}^{•} : T^{•} → T^{•}⟨n⟩ by putting τ_{n}^{j} = Id_{T^{j}} for j > n. Similarly, for any m ≥ n the identity T^{m} → T^{m} extends uniquely to a morphism τ_{n,m}^{•} : T^{•}⟨n⟩ → T^{•}⟨m⟩ satisfying τ_{n,m}^{j} = Id_{T^{j}} for each j ≥ m. We define now

T̂^{•} = \underrightarrow{lim} \{T^{•}⟨n⟩; τ_{n,m}^{•}\}.

Because τ_{n,m}^{•} ∘ τ_{n}^{•} = τ_{m}^{•} for m ≥ n, we obtain a natural morphism

τ^{•} = \underrightarrow{lim} τ_{n}^{•} : T^{•} → T̂^{•} .

The exactness of \underrightarrow{lim} implies that T̂^{•} is a (−∞, +∞)-cohomological functor. By our definition, we have for any M

T̂^{j}(M) = \underrightarrow{lim}_{k≥0} S^{−k}T^{j+k}(M)

so that for P projective T̂^{j}(P) = 0 for any j, because S^{−k}T^{j+k}(P) = 0 for k > 0. Thus T̂^{•} is P -complete. For the universal property of τ^{•} consider T^{•} → V^{•} with V^{•} a P - complete (−∞, +∞)-cohomological functor. Then each T^{n} → V^{n} extends uniquely to S^{≤0}T^{n} → S^{≤0}V^{n}, and S^{≤0}V^{n} ≅ V^{≤0} by (1.2). In this way we obtain for each n a unique morphism T^{•}⟨n⟩ → V^{•} factoring T^{•} → V^{•} as T^{•} → T^{•}⟨n⟩ → V^{•}. As a result, T^{•} → V^{•} factors uniquely through τ^{•}. The uniqueness of the P -completion is a consequence of its definition.

The following two lemmas are useful for computations.

Lemma 2.4. If T^{•} is a (−∞, +∞)-cohomological functor and n_{0} ∈ ℤ satisfies T^{n}(P) = 0 for all n ≥ n_{o} and all P projective, then τ^{n}(M) : T^{n}(M) → T̂^{n}(M) is an isomorphism for all n ≥ n_{0} and T̂^{•} is naturally equivalent to T^{•}⟨n⟩.

Proof. Because T^{m}(P) = 0 for P projective and m ≥ n_{0} we have, similarly as in (1.1), for all n ≥ n_{0} and all k ≥ 0 natural isomorphisms S^{−k}T^{n+k}(M) ≅ T^{n+k}(Ω^{k}M), and also T^{n+k}(Ω^{k}M) ≅ T^{n}(M). As a result,

T̂^{n}(M) = \underrightarrow{lim}_{k≥0} S^{−k}T^{n+k}(M) ≅ T^{n}(M).

For n ≥ n_{o}, T^{•}⟨n⟩ is P -complete and thus T^{•} → T^{•}⟨n⟩ induces T̂^{•} → T^{•}⟨n⟩, which is inverse to the natural map T^{•}⟨n⟩ → T̂^{•}.

Lemma 2.5. If ϕ^{•} : T^{•} → V^{•} is a morphism of (−∞, +∞)-cohomological functors with V^{•} P -complete and if ϕ^{n} : T^{n} → V^{n} is an equivalence for n ≥ n_{o}, then the induced morphism T̂^{•} → V^{•} is an equivalence.

Proof. We apply a “dimension shifting” argument as follows. Since T̂^{•} and V^{•} are P - complete, they satisfy for any k ∈ ℤ and any M

T̂^{k}(M) ≅ T̂^{k+1}(ΩM), V^{k}(M) ≅ V^{k+1}(ΩM).

Thus it suffices to show that T̂^{k} → V^{k} is an equivalence for k ≥ n_{0}. As T^{k}(P) = 0 for k ≥ n_{0}, we know from (2.4) that T^{k} ≅ T̂^{k} for k ≥ n_{0}, and the conclusion follows, since ϕ^{k} is an equivalence for k ≥ n_{0}.

Section 3 : Examples

Let G be an arbitrary group and consider the (−∞, +∞)-cohomological functor H^{•} = {H^{n}(G; ?); n ∈ ℤ} given by ordinary cohomology with H^{n}(G; ?) = 0 for n < 0. It has a P -completion H^{•} → Ĥ^{•} and we call the associated groups Ĥ^{n}(G; M) the n-th Tate cohomology groups of G with coefficients in the G-module M. Thus for any n ∈ ℤ,

Ĥ^{n}(G; M) = \underrightarrow{lim}_{k≥0} S^{−k}H^{k+n}(G; M),

and the morphism H^{•} → Ĥ^{•} into Tate cohomology is universal with respect to morphisms H^{•} → V^{•} into P -complete (−∞, +∞)-cohomological functors V^{•}. These Tate groups generalize the classical Tate groups for finite groups. More generally, the following holds.

Lemma 3.1. Let G denote a group of finite virtual cohomological dimension. Then the P -completion of H^{•} = {H^{n}(G; ?); n ∈ ℤ} is naturally equivalent to Farrell cohomology.

Proof. Consider the natural morphism H^{•} → F^{•} from ordinary cohomology to Farrell cohomology F^{•}. Since F^{•} is a P -complete (−∞, +∞)-cohomological functor and since H^{n} → F^{n} is an equivalence for n > vcd(G), we infer from (2.5) that Ĥ^{•} ≅ F^{•}.

If G is a group such that for some integer n_{0} one has H^{n}(G; P) = 0 for for all n ≥ n_{0} and all P projective, then H^{•} = {H^{n}(G; ?); n ∈ ℤ} admits a “terminal completion” T^{•} in the sense of [GG] which is given by a morphism H^{•} → T^{•} such that H^{n}(G; ?) → T^{n} is an equivalence for n ≥ n_{0}, and T^{•} is actually given by H^{•}⟨n_{0}⟩ (loc. cit.); since H^{•}⟨n_{0}⟩ is P -complete, the natural morphism H^{•}⟨n_{0}⟩ → Ĥ^{•} is an equivalence by (2.5), and the “terminal completion” T^{•} is therefore naturally equivalent to the P -completion Ĥ^{•}.

It is clear from the definition of the P -completion that for an arbitrary group G the following two conditions are equivalent:

(i) H^{•}(G; ?) → Ĥ^{•}(G; ?) is an equivalence (ii) H^{n}(G; P) = 0 for all n and all projective P.

There are indeed examples of groups satisfying these conditions. In [BG] a finitely presented group of type F P_{∞} satisfying (ii) is described. It is easy to check that an infinite free abelian group of countable rank satisfies the condition (ii) too. The following theorem provides further examples, which might help to understand the cohomology of Gl_{n}(\overline{ℚ}) with 𝔽_{p}-coefficients, a problem which is closely related to a conjecture of Friedlander and Milnor [M].

Theorem 3.2. Let K ⊂ \overline{ℚ} be a subfield of the algebraic closure of the rational numbers and let j ≥ 1. Then the P -completion

H^{•}(Gl_{j}(K); ?) → Ĥ^{•}(Gl_{j}(K); ?)

is an equivalence.

Proof. We can write K as a countable union ⋃ O_{S_{i}}(K_{i}) with each K_{i} ⊂ K a number field and O_{S_{i}}(K_{i}) ⊂ K_{i} the ring of S_{i}-integers, S_{i} a finite set of primes of K_{i}, and i ∈ ℕ. Without loss of generality we may assume that O_{S_{i}}(K_{i}) ⊂ O_{S_{i+1}}(K_{i+1}) for all i. Note that vcd(Gl_{j}(O_{S_{i}}(K_{i})) = n_{i} < ∞ and lim_{i→∞} n_{i} = ∞. Let P be a projective G = Gl_{j}(K)- module. We obtain then a short exact sequence

(3.3)  \underleftarrow{lim}_{i∈ℕ}^{1}H^{n−1}(Gl_{j}(O_{S_{i}}(K_{i})); P) → H^{n}(Gl_{j}(K); P) → \underleftarrow{lim}_{i∈ℕ} H^{n}(Gl_{j}(O_{S_{i}}(K_{i})); P)

By a result of Borel and Serre ([BS]) the groups Gl_{j}(O_{S_{i}}(K_{i})) are virtual duality groups of dimension n_{i} = vcd(Gl_{j}(O_{S_{i}}(K_{i})), and therefore

H^{m}(Gl_{j}(O_{S_{i}}(K_{i})); P) = 0 for m ≠ n_{i}.

Since lim_{i→∞} n_{i} = ∞ we infer from (3.3) that H^{n}(Gl_{j}(K); P)=0 for all n, proving our assertion.

Section 4 : A comparison with the Benson-Carlson groups.

Let G denote an arbitrary group and M, N be two G-modules. The group of projective homotopy classes [M, N] is, by definition, the factor group of Hom_{G}(M, N) modulo the subgroup consisting of those G-homomorphisms M → N, which may be factored through a projective module. The functor Ω induces a homomorphism [M, N] → [ΩM, ΩN] and one can define functors BC^{n}(G; ?), n ∈ ℤ, by putting

BC^{n}(G; M) = \underrightarrow{lim}_{k,k+n≥0}[Ω^{n+k}ℤ, Ω^{k}M].

It was observed in [BC] that for groups G of finite virtual cohomological dimension one has BC^{n}(G, M) ≅ Ĥ^{n}(G; M), which are just the Farrell cohomology groups. To deal with the case of an arbitrary group G, we first define a natural transformation Ĥ^{n}(G; ?) → BC^{n}(G; ?). If one uses for the definition of H^{n}(G; ?) the projective resolution

(4.1)  ⋯ → P^{n} → P^{n−1} → ⋯ → P^{0} → ℤ

with im(P^{n} → P^{n−1}) = Ω^{n}ℤ, n ≥ 1, then we see that there is a natural surjective homomorphism

H^{n}(G; M) → [Ω^{n}ℤ, M], n ≥ 0.

Passing to limits, one obtains a surjective map

(4.2)  \underrightarrow{lim}_{k≥|n|} H^{n+k}(G; Ω^{k}M) → \underrightarrow{lim}_{k≥|n|} [Ω^{n+k}ℤ, Ω^{k}M]

which is well-defined for any n ∈ ℤ. Note that the image of the connecting homomorphism H^{n+k}(G; Ω^{k}M) → H^{n+k+1}(G; Ω^{k+1}M) associated with the short exact sequence Ω^{k+1}M → F Ω^{k}M → Ω^{k}M is, by definition, equal to S^{−1}H^{n+k+1}(G; Ω^{k}M) and, by shifting dimensions, S^{−1}H^{n+k+1}(G; Ω^{k}M) ≅ S^{−k}H^{n+k+1}(G; ΩM) so that

(4.3)  \underrightarrow{lim}_{k≥|n|} H^{n+k}(G; Ω^{k}M) ≅ Ĥ^{n+1}(G; ΩM).

Using the natural isomorphisms Ĥ^{n}(G; M) ≅ Ĥ^{n+1}(G; ΩM), we see that (4.2) together with (4.3) gives rise to a surjection, natural in M,

θ^{n}(G; M) : Ĥ^{n}(G; M) → BC^{n}(G; M),

which is defined for every n ∈ ℤ.

Theorem 4.4. The natural transformations θ^{n}(G; ?) : Ĥ^{n}(G; ?) → BC^{n}(G; ?) are equivalences for all n ∈ ℤ.

Proof. Only the injectivity of θ^{n}(G; M) needs still to be checked. Let \overline{\overline{x}} ∈ Ĥ^{n}(G; M) be in the kernel of θ^{n} so that it may be represented by an \overline{x} ∈ H^{n+k}(G; Ω^{k}M) for some n + k > 0 such that the image of \overline{x} in [Ω^{n+k}ℤ, Ω^{k}M] is zero. Using the resolution (4.1) we can represent \overline{x} by a cocycle x : P^{n+k} → Ω^{k}M, which factors through Ω^{n+k}ℤ ⊂ P^{n+k−1}, yielding a representative y : Ω^{n+k}ℤ → Ω^{k}M of the image of \overline{x} in [Ω^{n+k}ℤ, Ω^{k}M]. Consider the commutative diagram

P^{n+k+1} ⟶ Ω^{n+k+1}ℤ ⟶ P^{n+k} ⟶ Ω^{n+k}ℤ
z↓  ↓^{Ωy}  ↓  ↓^{y}
Ω^{k+1}M  Ω^{k+1}M ⟶ F Ω^{k}M ⟶ Ω^{k}M.

Since by our assumption y factors through a projective module, it may be factored through F Ω^{k}M → Ω^{k}M. But this implies that Ωy may be extended over Ω^{n+k+1}ℤ → P^{n+k} and thus the cocycle z representing δ\overline{x} ∈ H^{n+k+1}(G; Ω^{k+1}M) is actually a coboundary. Therefore \overline{\overline{x}} = 0, proving the theorem.

Remark. By transport of structure, one can use the equivalences of (4.4) to define a (−∞, +∞)-cohomological functor BC^{•} = {BC^{n}(G; ?); n ∈ ℤ}, equivalent to the P -com-pletion of ordinary cohomology. The resulting connecting homomorphisms, associated with short exact sequences A′ → A → A′′, correspond then to maps BC^{n}(G; A′′) → BC^{n+1}(G; A′) induced from the obvious maps [Ω^{n+k}ℤ, Ω^{k}A′′] → [Ω^{n+k+1}ℤ, Ω^{k}A′], which are defined as soon as n + k and k are both ≥ 0.

References

[BC] D.J.Benson and J.F.Carlson, Products in negative cohomology, J. Pure Appl. Algebra 82 (1992), 107–129.

[BS] A.Borel and J-P.Serre, Cohomologie d’immeubles et de groupes S-arithmétic, Topology 15 (1976), 211–232.

[BG] K.S.Brown and R.Geoghegan, An infinite-dimensional torsion-free F P_{∞} group, Invent. math. 77 (1984), 367–381.

[CE] H.Cartan and S.Eilenberg, Homological algebra, Princeton University Press, Princeton, 1956.

[F] F.T.Farrell, An extension of Tate cohomology to a class of infinite groups, J. Pure Appl. Algebra 10 (1977), 153–16.

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

[HS] P.J.Hilton and U.Stammbach, A course in homological algebra, Graduate Texts in Mathematics, Springer Verlag, 1970.

[I] B.M.Ikenaga, Homological dimension and Farrell cohomology, J. Pure Appl. Algebra 87 (1984), 422–457.

[M] J.Milnor, On the homology of Lie groups made discrete, Comm. Math. Helv. 58 (1983), 73–85.

November 1992 E-mail address: mislin@math.ethz.ch and mislin@function.mps.ohio-state.edu