CENTRALIZERS OF ELEMENTARY ABELIAN p - SUBGROUPS
AND MOD - p - COHOMOLOGY OF PROFINITE GROUPS
by
Hans-Werner Henn
1. Introduction
1.1!Let G be a profinite group and p be a fixed prime. In this paper we will be*
* concerned
with Hc(G;Fp), the continuous cohomology of G with coefficients in the trivial *
*module Fp.
We will abbreviate Hc(G;Fp) by H (G;Fp),or simply by H G if p is understood fro*
*m the
context. Werecall that if G is the (inverse) limit of finite groups Githen H G *
*= colimH Gi.
Throughout this paper we will assume that H Gis finitely generated as Fp - alg*
*ebra. By
work of Lazard [La] it is knownthat this holds for many interesting groups, for*
* example
for profinite p - analytic groups like GL(n; Zp),the general linear groups over*
* the p - adic
integers. In case H G is finitely generated as Fp - algebra Quillen has shown[Q*
*1] that
there are only finitely many conjugacy classes of elementary abelian p - subgro*
*ups of G (i.e.
groups isomorphic to (Z=p)n for some natural number n). In other words, the fo*
*llowing
category A(G) is equivalentto a finite category: objects of A(G) are all elemen*
*tary abelian p
- subgroups of G,and if E1 and E2 are elementary abelian p - subgroups of G,the*
*n the set of
morphisms from E1 to E2 in A(G) consistsprecisely of those homomorphisms ff : E*
*1 ! E2
of abelian groups for which there existsan element g 2 G with ff(e) = geg1 8e *
*2 E1. The
category A(G) plays an important role both in Quillen's results and in the work*
* presented
here.
This category entered into Quillen's work as follows. Theassignment E 7! H E e*
*xtends
to a functor from the opposite category A(G)opto graded Fp- algebras and the re*
*striction
homomorphisms H G !H E (for E running through the elementaryab elian p - subgr*
*oups
of G) induce a canonical map of algebrasq : H G ! limA(G)opH E.
THEOREM 1.2[Q1]. Let G be a profinite group and assume H G is a finitely genera*
*ted Fp
- algebra. Then the canonical mapq : H G ! limA(G)opH E is an F - isomorphism,*
* in
other words q has the following properties.
ffl If x 2 Kerq, then x is nilpotent.
ffl If y 2limA(G)opH E then there exists an integer n with ypn2 Imq.
1.3!In our main result we use the full subcategory A (G) of A(G) whose ob jects*
* are all
elementary abelian p - subgroups exceptthe trivial subgroup. The centralizer C*
*G(E) of
an elementary abelian p - subgroup E is aclosed subgroup and hence inherits a n*
*atural
profinite structure from G. The assignment E 7!H CG (E) extends to a functor fr*
*om A(G)
to graded Fp - algebras and therestriction homomorphisms H G ! H CG (E) (for E
!
running!through!the non-trivial elementary abelian p - subgroups of G) induce a*
* canonical
map!ae!: H G ! limA(G)H CG (E). Our main result reads as follows.
!
THEOREM!1.4.!Let G be a profinite group and assume H G is a finitely generated *
*Fp
-!algebra.!Then the canonical map ae : H G ! limA(G)H CG (E) has finite kenel *
*and
cokernel.!!
!
1.5.!Remarks/Questions:!
a) The map ae of Theorem 1.4 is an actual isomorphism if G is a finite group or*
* a compact Lie
group (in the latter case H G has to be interpreted as the mod p cohomology of *
*the classifying
space BG) provided G contains any elements of order p. This was first proved by*
* Jackowski
and McClure [JM] and thenreproved and extended to certain classes of unstable a*
*lgebras
over the Steenrod algebra by Dwyer and Wilkerson [DW]. One's first reaction mig*
*ht be that
because the continuous cohomology ofa profinite group is the colimit of the coh*
*omology of
finite groups,the profinite case should be a direct consequence of the finite c*
*ase by passing to
appropriate (co-)limits. However, one gets confronted witha subtle problem of i*
*nterchanging
limits and colimits,and this has the effect that ae need not be an isomorphism *
*for a profinite
group.
In fact, our proof requires a different approach: in [He] we investigated an a*
*ppropriately
defined map ae for any unstable algebra K over the Steenrod algebra which isfin*
*itely generated
as Fp - algebra and showed that this map has always finite kernel andcokernel (*
*s.Theorem
2.5 below). In section 2 we explain thisalgebraic result and show how Theorem 1*
*.4 can be
deduced from it. We emphasize that we do not know any proof of Theorem 1.4 whic*
*h does
not use the Steenrod algebra, in particular Lannes' T - functor in a crucial wa*
*y.
b) Obviously Theorem 1.4 gives more precise information than Theorem 1.2, but o*
*n the other
hand its applicability is more limited. For example, thema jor reason for worki*
*ng with A (G)
instead of A(G) was to avoid the appearance of H G in the limit. However, if G *
*contains
central elements of order p,then H G does appear in the limit anyway,and Theore*
*m 1.4 is
not very useful. In other cases the functor E 7!H CG (E) may be too complicated*
* tob e
evaluated. However, in these cases Theorem 1.4 may still be of some theoretical*
* interest (see
2.10 and 2.11 below for examples).
Hans-Werner Henn
Mathematisches Institut der Universit{t
Im Neuenheimer Feld 288
D-69120 Heidelb erg
Fed. Rep. ofGermany