DEPTH ANDTHE COHOMOLOGY OF WREATH PRODUCTS
by Jon F. Carlson1 and Hans-Werner Henn2
Department ofMathematics Mathematisches Institut
University of Georgia Universit{t Heidelberg
Athens, GA 30602 Im Neuenheimer Feld 288
USA D-6900 Heidelberg
GERMANY
1. Introduction.
Suppose that G is a finite group and that k is a field of characteristicp > 0*
*. The
cohomology ring, H (G;k), is a finitely generated, graded-commutative k-algebra*
* and
hence is subject to the usual ring-theoretic scrutinies of commutative algebra.*
* Our knowl-
edge of the ideal spectra and associated varieties is reasonably advanced thank*
*s to the
work of Quillen and others (see [1]or [7] for general reference). Questions abo*
*ut depth and
associated primes seem more mysterious.The depth of H (G; k) is defined to be t*
*he length
of the longest regular sequence in the ring. The most general result on depth i*
*s probably
the theorem of Jeanne Duflot ([5],see also [2]) which says that the depth of H *
*(G;k) is
at least equal to the p-rank of the center of a Sylow p-subgroup of G. This is *
*truely a
satisfying result in that it relates a cohomological structure directly to stru*
*cture of the
underlying group. However itdo esnot tell us what the depth is, and there are m*
*any ex-
amples where the depth exceeds Duflot'slower bound. The study of the associated*
* primes
for H (G;k) is in even greater disarray. It is known that they must be invarian*
*t under
the action of the subalgebra generated by the reduced power operations of the S*
*teenrod
algebra [9], which implies that they must be the radicals of the restrictions t*
*o elementary
abelian p-subgroups. But little else is known beyond the only partially verifie*
*d assertions
of the relation to depth in [3].
In this paper we consider the behavior of the depth and of the associated pri*
*mes of
the cohomology ring H (G;k) under the wreath product operation: G ! G o Z=p. In
particular, we show that the depth increases by one and also that the minimal d*
*imension
of!the!associated primes increases by atmost one. This allows us to determine c*
*ompletely
the!depths of the mod-p cohomology rings of the symmetric groups. The example o*
*f an
n-fold!wreath!product ( (Z=p oZ=p) )oZ=p shows that the depth can exceed the*
* p-rank
of!the!center of a p-group by anarbitrarily large amount. The results give some*
* validity
! 1
Partially supported bya grant from NSF.
2Supported by a Heisenberg grant from DFG.
Typeset by AM S-T*
*EX
2 JON F. CARLSON AND HANS-WERNER HENN
to the question in [3] on the relation of depth and the dimensions of associate*
*d primes. It
seems also to be connected to question of detectability of cohomology (see [4],*
* [8]).
The first author would like to thank the Mathematisches Institut of the Unive*
*rsit{t
Heidelberg for its kind hospitality andsupp ort during the period when the resu*
*lts of this
paper were discovered.