TATE COHOMOLOGY FOR ARBITRARY GROUPS VIA SATELLITES
Guido Mislin
ETH Mathematik, Z"urich, Switzerland, and Department
of Mathematics, Ohio State University, Columbus, Ohio
Abstract. We define cohomology groups ^Hn(G; M), n 2 Z, for an arbitrary*
* group G
and G-module M, using the concept of satellites. These cohomology groups g*
*eneralize the
Farrell-Tate groups for groups of finite virtual cohomological dimension a*
*nd 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 ar*
*e 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
H^n(G; M) = limS-j Hn+j (G; M)
-!j0
with S-j Hn+j (G; ?) denoting the j-th left satellite of the functor Hn+j (G; ?*
*). These
general Tate groups are shown to agree, if applicable, with the ones obtained b*
*y the various
generalizations mentioned above. The family H^o= {H^n(G; ?); n 2 Z} forms a con*
*nected
sequence of functors and is as such characterized by a natural universal proper*
*ty, which
identifies it with what we call the completion with respect to projective modul*
*es, or short,
the P-completion of ordinary cohomology (cf. section 3).
In the first section we recall, to fix our terminology and notation, some ba*
*sic facts on
satellites. Section two is devoted to an axiomatic description of the P -compl*
*etion of a
cohomological functor, leading to our definition of the general Tate groups. I*
*n section
three we compare the definition to various other ones and discuss a few example*
*s. 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 pro-
jective completion for connected sequences of functors.
_____________
1991 Mathematics Subject Classification. Primary 18Gxx, 55Uxx; Secondary 20J*
*xx.
Key words and phrases. cohomology functors, Tate cohomology.
Typeset by AM S-*
*TEX
1
2 GUIDO MISLIN
Section 1 : Satellites
The basic references here are [CE] and [HS]. We will quickly review the fact*
*s of which
we make use. Let be an associative ring with 1 and M a (left) -module. We writ*
*e F M
for the free -module on the underlying set of M, and M for the kernel of the ob*
*vious
map F M ! M. If T denotes an additive functor from -modules to abelian groups, *
*then
S-1 T (M) = ker(T (M) ! T (F M))
defines a new additive functor S-1 T , the left satellite of T . For n 1 one d*
*efines induc-
tively S-n T = S-1 (S-n+1 T ) and n M = (n-1M) with the convention that S0T = T
and 0M = M, respectively. Each short exact sequence of -modules A0! A ! A00gives
rise to a connecting homomorphism S-n T (A00) ! S-n+1 T (A0), n > 0, in such a *
*way that
in the long sequence
. .!.S-n T A0! S-n T A ! S-n T A00! S-n+1 T A0! . .!.T A00;
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 P ! F P is a split monomorphism. As a result, on*
*e has
for general M and n > k 0 natural isomorphisms
(1.1) S-n T M ~=S-n+k T kM:
A connected sequence of functors V 0 = {V -n; n 0} is called of cohomological*
* type, if
the long sequence
. .!.V -nA0! V -nA ! V -nA00! V -n+1A0! . .!.V 0A00;
associated with any short exact sequence A0 ! A ! A00, is exact; in the termino*
*logy of
[GG] such a V 0 is called a (-1; 0)-cohomological functor. For instance, if T *
*is additive
and half exact, then S0 T is of cohomological type (cf.[CE]). Any natural tran*
*sformation
OE : U ! V of additive functors extends uniquely to a morphism OE0 : S0 U ! *
*S0 V
of connected sequences of functors. More generally, if V 0 is any connected s*
*equence
of (additive) functors, then any natural transformation : V 0 ! V 0extends un*
*iquely
to V 0 ! S0 V ; in particular, by taking for the the identity of V 0, one o*
*btains a
morphism V 0 ! S0 V which we call the canonical one. The connected sequence o*
*f left
satellites of a half exact functor can be characterized as follows (cf.[CE], II*
*I 5:2).
Theorem 1.2. Let U0 and V 0 denote connected sequences of (additive) functors*
* and
OE0: U0 ! 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) OE0 extends uniquely to OE0 : U0 ! V 0 and OE0 factors uniquely thro*
*ugh the
canonical morphism U0 ! S0 U0
(2) if U0 is half exact and OE0 is an equivalence then the induced morphism *
*S0 U0 !
V 0 is an equivalence.
TATE COHOMOLOGY FOR ARBITRARY GROUPS VIA SATELLITES 3
Section 2 : P-complete functors
We will follow the terminology of [GG] and call a connected sequence of addi*
*tive functors
T o= {T n; n 2 Z} a (-1; +1)-cohomological functor, if the long sequence
. .!.T nA0! T nA ! T nA00! T n+1A0! . . .
associated with any short exact sequence A0! A ! A00of -modules is exact. A typ*
*ical
example is given by ordinary cohomology Ho={Hn (G; ?); n 2 Z}, with the convent*
*ion
that Hn (G; ?) = 0 for n < 0.
Definition 2.1. A (-1; +1)-cohomological functor T o = {T n; n 2 Z} is called P*
* -
complete, if T n(P ) = 0 for every n and every projective module P . A morphism*
* Uo ! V o
of (-1; +1)-cohomological functors is called a P -completion, if V ois P -compl*
*ete and if
every morphism Uo ! W ointo a P -complete cohomological functor W ofactors uniq*
*uely
through Uo ! V o.
If G is a finite group, then the classical Tate groups ^Ho= {H^n(G; ?); n 2 *
*Z} form a P -
complete cohomological functor and the natural morphism Ho ! ^Hois a P -complet*
*ion (see
also 3:1). More generally, if the (-1; +1)-cohomological functor Uo admits a "t*
*erminal
completion" Uo ! V oin the sense of [GG] then it follows that Uo ! V ois a P -c*
*ompletion
(we will discuss this in section 3). Since not every Uo admits a "terminal comp*
*letion", 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 (-1; +1)-cohomological functor T o = {T n; n 2 Z} admits a
unique P -completion oo : T o! ^T.o
Proof. For every n 2 Z we can form the (-1; n)-cohomological functor S0 T n wh*
*ich
extends to a (-1; +1)-cohomological functor T o by putting
aeSj-n T n; if j < n
(2.3) T j =
T j; if j n:
The identity transformation T n! T nextends uniquely to T n ! S0 T n, and we e*
*xtend
it further to oon: T o! T o by putting ojn= IdTj for j > n. Similarly, for a*
*ny m n
the identity T m ! T m extends uniquely to a morphism oon;m: T o ! T o sa*
*tisfying
ojn;m= IdTj for each j m. We define now
^T o= lim{T o; oo :}
-! n;m
Because oon;mO oon= oomfor m n, we obtain a natural morphism
oo = lim-!oon: T o! ^T:o
The exactness of lim-!implies that ^T ois a (-1; +1)-cohomological functor. By *
*our defi-
nition, we have for any M
T^j(M) = limS-k T j+k(M)
-!k0
4 GUIDO MISLIN
so that for P projective ^T(jP ) = 0 for any j, because S-k T j+k(P ) = 0 for k*
* > 0. Thus
T^o is P -complete. For the universal property of oo consider T o! V o with V *
*o a P -
complete (-1; +1)-cohomological functor. Then each T n ! V n extends uniquely *
*to
S0 T n! S0 V n, and S0 V n~= V 0 by (1:2). In this way we obtain for each n*
* a unique
morphism T o ! V ofactoring T o! V oas T o! T o ! V o. As a result, T o! *
*V o
factors uniquely through oo. The uniqueness of the P -completion is a consequen*
*ce of its
definition.
The following two lemmas are useful for computations.
Lemma 2.4. If T ois a (-1; +1)-cohomological functor and n0 2 Z satisfies T n(P*
* ) = 0
for all n no and all P projective, then on (M) : T n(M) ! ^T(nM) is an isomorp*
*hism for
all n n0 and ^T ois naturally equivalent to T o.
Proof. Because T m(P ) = 0 for P projective and m n0 we have, similarly as in *
*(1:1),
for all n n0 and all k 0 natural isomorphisms S-k T n+k(M) ~=T n+k(kM), and a*
*lso
T n+k(kM) ~=T n(M). As a result,
^T n(M) = limS-k T n+k(M) ~=T n(M):
-!k0
For n no, T o is P -complete and thus T o! T o induces ^T o! T o, whi*
*ch is
inverse to the natural map T o ! ^T.o
Lemma 2.5. If OEo : T o! V ois a morphism of (-1; +1)-cohomological functors wi*
*th
V o P -complete and if OEn : T n ! V n is an equivalence for n no, then the in*
*duced
morphism ^T o! V ois an equivalence.
Proof. We apply a "dimension shifting" argument as follows. Since T^oand V o a*
*re P -
complete, they satisfy for any k 2 Z and any M
^T(kM) ~=^T k+1(M); V k(M) ~=V k+1(M):
Thus it suffices to show that ^T k! V k is an equivalence for k n0. As T k(P *
*) = 0 for
k n0, we know from (2:4) that T k~=T^kfor k n0, and the conclusion follows, s*
*ince OEk
is an equivalence for k n0.
Section 3 : Examples
Let G be an arbitrary group and consider the (-1; +1)-cohomological functor *
*Ho =
{Hn (G; ?); n 2 Z} given by ordinary cohomology with Hn (G; ?) = 0 for n < 0. *
*It has
a P -completion Ho ! H^o and we call the associated groups H^n(G; M) the n-th T*
*ate
cohomology groups of G with coefficients in the G-module M. Thus for any n 2 Z,
^Hn(G; M) = limS-k Hk+n (G; M);
-!k0
and the morphism Ho ! ^Hointo Tate cohomology is universal with respect to morp*
*hisms
Ho ! V o into P -complete (-1; +1)-cohomological functors V o. These Tate grou*
*ps
generalize the classical Tate groups for finite groups. More generally, the fol*
*lowing holds.
TATE COHOMOLOGY FOR ARBITRARY GROUPS VIA SATELLITES 5
Lemma 3.1. Let G denote a group of finite virtual cohomological dimension. Then*
* the
P -completion of Ho = {Hn (G; ?); n 2 Z} is naturally equivalent to Farrell coh*
*omology.
Proof. Consider the natural morphism Ho ! F ofrom ordinary cohomology to Farrell
cohomology F o. Since F ois a P -complete (-1; +1)-cohomological functor and s*
*ince
Hn ! F nis an equivalence for n > vcd(G), we infer from (2:5) that ^Ho~=F o.
If G is a group such that for some integer n0 one has Hn (G; P ) = 0 for for*
* all n n0
and all P projective, then Ho = {Hn (G; ?); n 2 Z} admits a "terminal completio*
*n" T oin
the sense of [GG] which is given by a morphism Ho ! T osuch that Hn (G; ?) ! T *
*nis
an equivalence for n n0, and T ois actually given by Ho (loc. cit.); sinc*
*e Ho
is P -complete, the natural morphism Ho ! H^o is an equivalence by (2:5), a*
*nd the
"terminal completion" T ois therefore naturally equivalent to the P -completion*
* ^Ho.
It is clear from the definition of the P -completion that for an arbitrary g*
*roup G the
following two conditions are equivalent:
(i)Ho(G; ?) ! ^Ho(G; ?) is an equivalence
(ii)Hn (G; P ) = 0 for all n and all projective P.
There are indeed examples of groups satisfying these conditions. In [BG] a fin*
*itely pre-
sented group of type F P1 satisfying (ii) is described. It is easy to check th*
*at an infinite
free abelian group of countable rank satisfies the condition (ii) too. The foll*
*owing_theorem
provides further examples, which might help to understand the cohomology of Gln*
*(Q )
with Fp-coefficients, a problem which is closely related to a conjecture of Fri*
*edlander and
Milnor [M].
__
Theorem 3.2. Let K Q be a subfield of the algebraic closure of the rational nu*
*mbers
and let j 1. Then the P -completion
Ho(Glj(K); ?) ! ^Ho(Glj(K); ?)
is an equivalence.
S
Proof. We can write K as a countable union OSi(Ki) with each Ki K a number fi*
*eld
and OSi(Ki) Ki the ring of Si-integers, Si a finite set of primes of Ki, and i*
* 2 N.
Without loss of generality we may assume that OSi(Ki) OSi+1(Ki+1) for all i. *
*Note
that vcd(Glj(OSi(Ki)) = ni< 1 and limi!1 ni= 1. Let P be a projective G = Glj(K*
*)-
module. We obtain then a short exact sequence
(3.3) lim-1Hn-1 (Glj(OSi(Ki)); P ) ! Hn (Glj(K); P ) ! lim-Hn (Glj(OSi(Ki));*
* P )
i2N i2N
By a result of Borel and Serre ([BS]) the groups Glj(OSi(Ki)) are virtual duali*
*ty groups
of dimension ni= vcd(Glj(OSi(Ki)), and therefore
Hm (Glj(OSi(Ki)); P ) = 0 for m 6= ni:
Since limi!1 ni = 1 we infer from (3:3) that Hn (Glj(K); P )=0 for all n, provi*
*ng our
assertion.
6 GUIDO MISLIN
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 proj*
*ective
homotopy classes [M; N] is, by definition, the factor group of HomG (M; N) modu*
*lo the
subgroup consisting of those G-homomorphisms M ! N, which may be factored throu*
*gh
a projective module. The functor induces a homomorphism [M; N] ! [M; N] and
one can define functors BCn (G; ?), n 2 Z, by putting
BCn (G; M) = lim-![n+k Z; kM]:
k;k+n0
It was observed in [BC] that for groups G of finite virtual cohomological dimen*
*sion one
has BCn (G; M) ~=H^n(G; M), which are just the Farrell cohomology groups. To de*
*al with
the case of an arbitrary group G, we first define a natural transformation H^n(*
*G; ?) !
BCn (G; ?). If one uses for the definition of Hn (G; ?) the projective resoluti*
*on
(4.1) . .!.P n! P n-1! . .!.P 0! Z
with im(P n ! P n-1) = nZ, n 1, then we see that there is a natural surjective
homomorphism
Hn (G; M) ! [nZ; M], n 0:
Passing to limits, one obtains a surjective map
(4.2) lim-!Hn+k (G; kM) ! lim-![n+k Z; kM]
k|n| k|n|
which is well-defined for any n 2 Z. Note that the image of the connecting hom*
*omor-
phism Hn+k (G; kM) ! Hn+k+1 (G; k+1M) associated with the short exact sequence
k+1M ! F kM ! kM is, by definition, equal to S-1 Hn+k+1 (G; kM) and, by shift-
ing dimensions, S-1 Hn+k+1 (G; kM) ~=S-k Hn+k+1 (G; M) so that
(4.3) lim-!Hn+k (G; kM) ~=H^n+1(G; M):
k|n|
Using the natural isomorphisms H^n(G; M) ~=H^n+1(G; M), we see that (4:2) toget*
*her
with (4:3) gives rise to a surjection, natural in M,
n(G; M) : ^Hn(G; M) ! BCn (G; M);
which is defined for every n 2 Z.
Theorem 4.4. The natural transformations n(G; ?): ^Hn(G; ?) ! BCn (G; ?) are eq*
*uiv-
alences for all n 2 Z.
___
Proof. Only the injectivity of n(G; M) needs still to be checked. Let x 2 ^Hn(G*
*; M) be
in the kernel of n so that it may be represented by an __x2 Hn+k (G; kM) for so*
*me
TATE COHOMOLOGY FOR ARBITRARY GROUPS VIA SATELLITES 7
__
n + k > 0 such that the image of x in [n+k Z; kM] is zero. Using the resolution*
* (4:1) we
can represent __xby a cocycle x : P n+k! kM, which factors through n+k Z P n+k*
*-1,
yielding a representative y : n+k Z ! kM of the image of __xin [n+k Z; kM]. Con*
*sider
the commutative diagram
P n+k+1 ----! n+k+1 Z ----! P n+k ----! n+k Z
? ? ? ?
z?y ?yy ?y ?yy
k+1M ________k+1M ----! F kM ----! kM:
Since by our assumption y factors through a projective module, it may be factor*
*ed through
F kM ! kM. But this implies that y may be extended over n+k+1 Z ! P n+k
and thus the_cocycle z representing ffi__x2 Hn+k+1 (G; k+1M) is actually a cobo*
*undary.
Therefore __x= 0, proving the theorem.
Remark. By transport of structure, one can use the equivalences of (4:4) to def*
*ine a
(-1; +1)-cohomological functor BCo = {BCn (G; ?); n 2 Z}, equivalent to the P -*
*com-
pletion of ordinary cohomology. The resulting connecting homomorphisms, associ*
*ated
with short exact sequences A0 ! A ! A00, correspond then to maps BCn (G; A00) !
BCn+1 (G; A0) induced from the obvious maps [n+k Z; kA00] ! [n+k+1 Z; kA0], whi*
*ch
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-arithmetic,*
* Topology 15 (1976),
211-232.
[BG] K.S.Brown and R.Geoghegan, An infinite-dimensional torsion-free FP1 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 Text*
*s in Mathematics,
Springer Verlag, 1970.
[I] B.M.Ikenaga, Homological dimension and Farrell cohomology, J. Pure Appl. A*
*lgebra 87 (1984),
422-457.
[M] J.Milnor, On the homology of Lie groups made discrete, Comm. Math. Helv. 5*
*8 (1983), 73-85.
November 1992
E-mail address: mislin@math.ethz.ch and mislin@function.mps.ohio-state.edu