DEPTH AND THE COHOMOLOGY OF WREATH PRODUCTS
by Jon F. Carlson1 and Hans-Werner Henn2
Department of Mathematics Mathematisches Institut
University of Georgia Universit"at 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 characteristic p >*
* 0. The
cohomology ring, H*(G; k), is a finitely generated, graded-commutative k-algebr*
*a 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 ab*
*out depth and
associated primes seem more mysterious. The depth of H*(G; k) is defined to be *
*the 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 it does not tell us what the depth is, and there are *
*many ex-
amples where the depth exceeds Duflot's lower bound. The study of the associate*
*d primes
for H*(G; k) is in even greater disarray. It is known that they must be invari*
*ant 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 pr*
*imes 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 at most one. This allows us to determine *
*completely
the depths of the mod-p cohomology rings of the symmetric groups. The example o*
*f an
n-fold wreath product (. .(.Z=poZ=p) . .).oZ=p shows that the depth can exceed *
*the p-rank
of the center of a p-group by an arbitrarily large amount. The results give som*
*e validity
_____________1
Partially supported by a grant from NSF.
2Supported by a Heisenberg grant from DFG.
Typeset by AM S-*
*TEX
1
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 Univ*
*ersit"at
Heidelberg for its kind hospitality and support during the period when the resu*
*lts of this
paper were discovered.
2. The cohomology of wreath products.
We begin this section by outlining the structure of the ring H*(G o Z=p; k).*
* We rely
on the fundamental result of Nakaoka, which expresses the ring in terms of the *
*related
spectral sequence. Specifically, there is the extension
1 ! Gp ! G o Z=p ! Z=p ! 1:
The associated Lyndon-Hochschild-Serre spectral sequence has E2 term
E*;*2= H*(Z=p; H*(Gp; k)):
Nakaoka's Theorem [11] says that the spectral sequence collapses at the E2 term*
* and that
E*;*2, which is a ring and a k-algebra, is isomorphic to H*(G o Z=p; k) as a k-*
*algebra.
Now consider first the terms along the fiber
E0;*2= H0(Z=p; H*(Gp; k)) ~=H*(Gp; k)Z=p:
This subring is the ring of invariants of H*(Gp; k) ~=(H*(G; k))p under the pe*
*rmutation
oe of order p which permutes the factors cyclically. The invariants (Z=p-fixed *
*points) come
in two types: norms and traces. The norms are the elements of the form OE(x) = *
*x . . .x
for x 2 H*(G; k). In fact the map OE : H*(G; k) ! (H*p(Gp; k))Z=pis a ring homo*
*morphism
except for a possible sign change in odd degrees. If x1; . .;.xp 2 H*(G; k) the*
*n the trace
of x1 . . .xp is
p-1X
o(x1 . . .xp) = oei(x1 . . .xp);
i=0
which is invariant under the permutation oe. The set of all traces in E0;*2is *
*an ideal of
H*(G o Z=p; k) which we label T r(H*(Gp; k)). It is the annihilator in H*(G o *
*Z=p; k) of
the canonical element in E2;02= H2(Z=p; k) which is the Bockstein of the degre*
*e-one
element j represented by the homomorphism G o Z=p ! Z=p.
Summing up, we have that
E0;*2= T r(H*Gp; k)) OE(H*(G; k)):
Factoring out the ideal of traces we get the exact sequence
(2.1) 0 ! T r(H*(Gp; k)) ! H*(G o Z=p; k) ! H*(Z=p; k) OEH*(G; k) ! 0
Here the base of the spectral sequence E*;02= H*(Z=p; k) is generated as an alg*
*ebra by
j and . With this preparation we are ready to prove our main theorem which read*
*s as
follows.
DEPTH AND THE COHOMOLOGY OF WREATH PRODUCTS 3
Theorem 2.1. Suppose that the depth of H*(G; k) is d. Then the depth of H*(GoZ=*
*p; k)
is d + 1.
Proof. The proof requires a series of steps. First notice that H*(Z=p; k) modul*
*o its radical
is a polynomial ring and hence has depth 1, while OE(H*(G; k)) has depth d, the*
* same
as that of H*(G; k). So the tensor product has depth exactly equal to d + 1. A *
*regular
sequence is given by and OE(x1); . .;.OE(xd) where x1; : :;:xd is any regular *
*sequence for
H*(G; k). So we have that
Lemma 2.2. The depth of H*(Z=p; k) OE(H*(G; k)) is d + 1.
Theorem 16.7 of [10] gives a homological criterion for depth. In the case th*
*at M is a
finitely generated graded A = H*(G; k) module, it says that the depth of M is t*
*he least
integer n such that ExtnA(k; M) 6= 0. From sequence (2.1) and the long exact se*
*quence on
Ext, the theorem is a consequence of the following lemma.
Lemma 2.3. The depth of T r(H*(Gp; k)) as an H*(G o Z=p; k)-module is d + 1.
Proof. We first show that d + 1 is a lower bound. Consider a regular sequence x*
*1; : :;:xd
of homogeneous elements in H*(G; k) and write H*(G; k) ~=P V as a P -module wh*
*ere
P is the polynomial algebra with polynomial generators x1; : :;:xd. This state*
*ment is
roughly a consequence of the discussion of polynomial extensions and graded mod*
*ules on
page 125 of [10]. Here V is considered as graded vector space which is isomor*
*phic to
H*(G; k)=(x1; : :;:xd) and hence contains a canonical element 1. As (P p )Z=p -*
* modules
we have an isomorphism
T r(H*(Gp; k) ~=(T r(P p ) OEV ) (P p T r(V p )) :
That is, T r(H*(Gp; k))=(T r(P p )OEV ) is isomorphic to P p T r(V p ) via the *
*map which
sends (y1. .y.p)o(v1. .v.p) to the element o((y1v1). .(.ypvp)) for y1; . .y.p
in P and v1; . .;.vp 2 V . To get the lower bound on depth it suffices to show*
* that, as
(P p )Z=p - modules, we have
a) depthT r(P p ) d + 1, and
b) depthP p d + 1.
We claim that the elements OE(x1); : :;:OE(xd), o(x1 1 . . .1) form a regular*
* sequence
for both modules.
We consider case (a) first. We define a lexicographical order on the monomi*
*als in P
induced by declaring x1 > x2 > . .>.xd. We get an induced lexicographical order*
* on P p .
With this order, a basis of T r(P p ) is given by the elements o(m1 m2 . . .mp)*
* where
each mi is a monomial in P , not all the mi are equal and m1 m2 . . .mp is larg*
*er than
any of its cyclic permutations in P p . That the first d elements in the sequen*
*ce above are
regular is easy to see. The action of OE(xi) on o(m1 m2 . . .mp) is given by
OE(xi)o(m1 m2 . . .mp) = o(xim1 xim2 . . .ximp)
and hence after dividing out OE(x1); : :;:OE(xk) the quotient has a vector spac*
*e basis con-
sisting of elements o(m1 m2 . . .mp) as above with the additional condition t*
*hat the
4 JON F. CARLSON AND HANS-WERNER HENN
gcd of m1; : :m:dis not divisible by xi for all i k. In particular OE(xk+1) is*
* still regular
on this quotient (for k + 1 d). Now the action of o(x1 1 . . .1) on a basis *
*element
is given by
o(x1 1 . . .1)o(m1 m2 . . .mp) = io(m1 . . .x1mi . . .mp) :
In particular we get o(x1 1 . . .1)o(m1 m2 . . .mp) = o(x1m1 m2 . . .mp)
modulo terms of smaller lexicograghical order. Note that if x1 does not divide *
*m1 then x1
does not divide any of the mi because m1 mi for all i. In particular o(x1m1 *
*m2
. . .mp) is a nontrivial basis element in o(P p )=(OE(x1); :::; OE(xd)). From t*
*his description
it is obvious that o(x1 1 . . .1) is still regular on o(P p )=(OE(x1); : :;:O*
*E(xd)).
To prove (b) we should note that for each i, the elementary symmetric polyno*
*mials in xi
in the polynomial ring (k[xi])p = k[xi]. .k.[xi] form a set of homogeneous par*
*ameters.
Two of these symmetric polynomials are OE(xi) and o(x1 1 . . .1) and hence th*
*e two
form a regular sequence. Likewise the symmetric polynomials in the individual v*
*ariables
x1; : :;:xd form a set of homogeneous paramenters for the polynomial ring
P p = (k[x1])p . . .(k[xd])p :
It is well known that for a Cohen-Macaulay ring such as this any set of homogen*
*eous param-
eters, taken in any order, is a regular sequence. Consequently the elements OE(*
*x1); : :;:OE(xp),
o(x1. .1.), being members of a homogeneous set of parameters, form a regular se*
*quence.
This proves the lower bound for the depth.
To get the upper bound we note that the element i = o(x1 1 . . .1) is not *
*in the
submodule
(OE(x1); . .;.OE(xd); i)T r(H*(Gp; k))
but any multiple of i by an element of positive degree in (H*(Gp; k))Z=p is in *
*this sub-
module. So there is no regular sequence of length d + 2.
3. Dimensions, Associated Primes and Symmetric Groups.
It is well known that in any reasonable ring the depth is at most equal to t*
*he minimum of
the dimensions of the associated primes in the ring. By the dimension of a prim*
*e p in a co-
homology ring H*(G; k), we mean the Krull dimension of the quotient ring H*(G; *
*k)=p, or
the dimension of its maximal ideal spectrum VG (p). Now the minimal primes in H*
**(G; k)
are always among the associated primes (see [10], (6.5)), and they are in one-t*
*o-one corre-
spondence with the components of the maximal ideal spectrum VG (k). By Quillen *
*[12] ,
the components of the variety are in one-to-one correspondence with the conjuga*
*cy classes
of maximal elementary abelian p-subgroups of G. That is, if E is a maximal elem*
*entary
abelian p-subgroup of G, then the radical of the restriction to E is a minimal *
*prime ideal
in H*(G; k) and its dimension is the rank of E. It follows that the depth of H**
*(G; k) never
exceeds the minimum of the ranks of the maximal elementary abelian p-subgroups *
*of G.
With regards to associated primes, it was speculated in [3] that the depth o*
*f any mod-
p cohomology ring H*(G; k) might coincide with the minimum of the dimensions of*
* the
associated primes. The following analysis and examples support the speculation*
*, even
though the evidence is not strong.
DEPTH AND THE COHOMOLOGY OF WREATH PRODUCTS 5
Proposition 3.1. (a) If a is the minimum of the dimensions of the associated pr*
*imes of
H*(G; k), then the minimum of the dimensions of the associated primes of H*(G o*
* Z=p; k)
is at most a + 1.
(b) Suppose that m is the minimum of the dimensions of the components of the*
* variety
of H*(G; k). Then m + 1 is the minimum of the dimension of the components of t*
*he
maximal ideal spectrum of H*(G o Z=p; k).
Proof. Suppose that i 2 H*(G; k) is an element whose annihilator is a prime p w*
*ith
H*(G; k)=p of minimal dimension a. Then we claim that (in the notation of the *
*last
section) the annihilator p0 of OE(i) has dimension a + 1. This is because T r(H*
**(Gp; k))
annihilates and hence
H*(G o Z=p; k)=p0~= OE(H*(G; k)=p) H*(Z=p; k):
This proves (a).
For (b) we use Quillen's correspondence of the components of VG (k) with the*
* maximal
elementary abelian p-subgroups of G. Let y 2 G o Z=p be an element of order p i*
*n G o Z=p
that cyclically permutes the factors of Gp = G x . .x.G. Then the centralizer o*
*f y is the
direct product Gx < y > where G = {(g; . .;.g)|g 2 G} ~=G is the diagonal subgr*
*oup.
Let E be a maximal elementary abelian p-subgroup of rank m in G. Then Ex < y > *
*is
contained in no larger elementary abelian p-subgroup of G o Z=p.
Finally we consider the case of the symmetric groups Sn. For notation let W1*
* = Z=p
and inductively define the n-fold wreath product Wn = Wn-1 o Z=p.
Corollary 3.2. Suppose that n has p-adic expansion n = b0 + b1p + . .+.btpt. Th*
*en
depth (H*(Sn; k)) = b1 + 2b2 + . .+.tbt:
This number is equal to the minimum of the dimensions of the components of H*(S*
*n; k)
and also the minimum of the dimensions of the associated primes.
Proof. The point is that a Sylow p-subgroup P of Sn is isomorphic to
W1b1x W2b2x . .x.Wtbt
which has the prescribed depth by Theorem 2.1. But now the depth of H*(G; k) is*
* at least
equal to the depth of H*(P; k) by known arguments (see [7], proof of 10.3.1). T*
*he reverse
inequality comes from Proposition 3.1 and the fact that G has a maximal element*
*ary
abelian p-subgroup of rank b1 + . .+.tbt. The statement about the dimensions o*
*f the
components is a consequence of Quillen's Theorem and the fact that Sn has a max*
*imal
elementary abelian p-subgroup of the prescribed rank.
6 JON F. CARLSON AND HANS-WERNER HENN
References
1.D. J. Benson, Representations and Cohomology II: Cohomology of Groups and Mo*
*dules, Cambridge
Studies in Advanced Mathematics 31 (1991), Cambridge University Press, Cambr*
*idge.
2.C. Broto and H.-W. Henn, Some remarks on central elementary abelian p-subgro*
*ups and cohomology
of classifying spaces, Quarterly Journal of Mathematics 44 (1993), 155-163.
3.J. F. Carlson, Depth and transfrer maps in the cohomology of groups, Math. Z*
*eit (to appear).
4.J. F. Carlson and H.-W. Henn, The support of regular elements of cohomology *
*rings (to appear).
5.J. Duflot, Depth and equivariant cohomology, Comm. Math. Helvetici 56 (1981)*
*, 627-637.
6.J. Duflot, The associated primes of H*G(x), J. Pure and Appl. Algebra 30 (19*
*83), 131-141.
7.L. Evens, The Cohomology of Groups (1991), Oxford University Press, New York.
8.H.-W. Henn, J. Lannes and L. Schwartz, Localization of unstable A-modules an*
*d equivariant mod-p
cohomology (to appear).
9.P. S. Landweber and R. E. Stong, The depth of rings of invariants over finit*
*e fields, Number Theory,
New York, 1984-1985; Lecture Notes in Math, vol. 1240, Springer-Verlag, Berl*
*in, 1987.
10.H. Matsumura, Commutative Ring Theory (1986), Cambridge University Press, Ca*
*mbridge.
11.M. Nakaoka, Decomposition theorems for homology groups of symmetric groups, *
*Ann. of Math. 73
(1961), 229-257.
12.D. Quillen, The spectrum of an equivariant cohomology ring I, II, Ann. Math.*
* 94 (1971), 549-602.