THE SUPPORT OF REGULAR ELEMENTS IN
THE COHOMOLOGY RING OF A GROUP
Jon F. Carlson1 and Hans-Werner Henn2
The last few years have witnessed a great deal of research on the algebraic *
*structure of
the cohomology rings of groups. The reason is that many of the ring-theoretic c*
*onsider-
ations such as varieties, associated primes and depth reflect properties of the*
* groups and
also of the topology of their classifying spaces. The evidence for such relati*
*onships has
been building ever since Quillen first proved [Q] that the minimal primes of th*
*e mod-p
cohomology were the radicals of the kernels of restrictions to maximal elementa*
*ry abelian
p-subgroups. More generally, any associated prime of the cohomology ring is in*
*variant
under the Steenrod reduced power operations and hence must be the radical of th*
*e kernel
of restriction to some elementary abelian p-subgroup of the group.
Recently, particularly through Lannes' work [L], it has become evident that *
*the ele-
mentary abelian p-groups play a central role in the theory of unstable modules *
*over the
Steenrod algebra. The new developments in this theory were used in [HLS] to pro*
*ve that
the existence of regular elements (non-divisors of zero) in cohomology implies *
*a detectabil-
ity of mod-p cohomology on the centralizers of certain elementary abelian p-sub*
*groups
determined by the regular element. The main purpose of this note is to strength*
*en that
theorem (Theorem 1, below) by proving that it has a strong converse, that the d*
*etectability
condition requires the existence of certain homogeneous regular elements and he*
*nce that
the set of homogeneous regular elements is characterized by the detectability c*
*ondition.
Let p be a fixed prime and and let G be either a compact Lie group (which ma*
*y be
finite) or a discrete group of finite virtual cohomological dimension such that*
* the mod p
cohomology H*BG = H*(BG; Z=p) of its classifying space BG is noetherian. The se*
*t of
elementary abelian p subgroups of G will be denoted by A(G). For a subgroup H *
*of G
let CG (H) be its centralizer in G. The starting point of this paper is the res*
*ult mentioned
above [HLS, Corollary I.5.7].
Theorem 1. Let G be as above. Assume x 2 H*BG is not a zero divisor. Denote by *
*Cx
the set of elementary abelian p subgroups such that the restriction of x to H*B*
*E is not
nilpotent. Then the map
Y Y
resG;CG(E): H*BG -! H*BCG (E)
E2Cx E2Cx
whose components are the restriction homomorphisms is a monomorphism.
_____________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
We call Cx the support of x and denote it from now on by Supp(x).
Obviously Supp(x) is a subset C of A(G) which has the following two properti*
*es:
i)If E is in C then gEg-1 is in C for all g 2 G and
ii)If E is in C and E E0 then E0 is in C.
Definition. A subset C of A(G) is called closed if and only if C satisfies i) a*
*nd ii) above.
In particular, Supp(x) is closed for each x 2 H*BG. We remark that there is *
*a topology
on the set A(G) whose closed subsets are precisely those which are closed in th*
*e sense of
the definition.
In this note we investigate consequences of the injectivity of the maps
Y Y
resG;CG(E): H*BG -! H*BCG (E)
E2C E2C
where C is any subset of A(G) which we may assume to be closed. Our main result
combined with Theorem 1 above can be stated as follows.
Theorem 2. Let C be a closed subset of A(G). Then the following statements are *
*equiv-
alent:
Q Q
a) resG;CG(E): H*(BG) ! H*(BCG (E)) is injective.
E2C E2C
b) There exists x 2 H*BG such that Supp(x) = C and x is not a divisor of ze*
*ro.
c) If x is any element in H*BG with Supp(x) = C then x is not a divisor of *
*zero.
Of course, the implication (b) ) (a) is just Theorem 1, so we only have to p*
*rove that
(a) ) (c) and (c) ) (b).
As a byproduct of our proof we will also get restrictions for prime ideals of H*
**BG to be
an associated prime ideal.
The implication (c) ) (b) of Theorem 2 follows from
Proposition 3. Let C A(G) be a closed subset. Then there is an element x 2 H*BG
with suppx = C.
Proof:_By [Q] there are only finitely many conjugacy classes of elementary a*
*belian p
subgroups of G. Let E1; : :;:Et be a complete set of representatives of the co*
*njugacy
classes of minimal elements of C.
First we show that there exist elements i 2 H*G such that Supp i consists pr*
*ecisely
of those E in A(G) such that Ei is conjugate to a subgroup of E. For this we *
*consider
THE SUPPORT OF REGULAR ELEMENTS IN THE COHOMOLOGY RING OF A GROUP 3
D = {E 2 A(G) | E =2C} and let D1; : :;:Ds be representatives of the conjugacy *
*classes of
maximal elements of D. Notice that Ei 6 gEjg-1 for any j 6= i and g 2 G. By Qui*
*llen's
results in [Q] this means that the radical of the kernel of restriction to gEjg*
*-1 is not
contained in the radical of kernel of restriction to Ei and therefore there exi*
*sts an element
flij2 H*(G; k) which is homogeneous and which has the property that resG;Ei(fli*
*j) is not
a divisor of zero while resG;Ej(flij) = 0. Likewise Ei 6 gDjg-1 for any j and a*
*ny g 2 G.
So for j = 1; : :;:s there is a homogeneous element fl0ij2 H*(G; k) such that r*
*esG;Ei(fl0ij)
Qs Q
is a regular element but resG;Dj(fl0ij) = 0. Let i = fl0ij. flij. Then th*
*e support of i
i=1 i6=j
consists precisely of those E in A(G) such that Ei is conjugate to a subgroup o*
*f E.
Now we may assume that all i arePof the same degree, by replacing each i by *
*some
power of itself. Then consider = ti=1i. Because the Ei's run through a compl*
*ete set
of representatives of conjugacy classes of minimal elements of C we find
[
Supp = Supp i= C :
The implication (a) ) (c) is a straightforward consequence of the fact that *
*if x is an
element with Supp(x) = C, then by the following proposition, resG;CG(E)(x) is n*
*ot a divisor
of zero.
Proposition 4 (cf. [BH]). Assume y 2 H*BG is an element whose restriction to s*
*ome
non-trivial central subgroup C is not nilpotent. Then y is not a zero divisor.
Proof:_Take any non-trivial z 2 H*BG and assume yz = 0. Consider the homomorphi*
*sm
: C x G ! G, induced by multiplication in G. Then we get 0 = *(yz) = *(y) . *(*
*z).
Now we write *(y) = resG;C(y) 1 + "ywith "y2 H*BC eH*BG, and *(z) = zi with
zi 2 HiBC H*BG. Then z0 = 1 z is non-trivial and so thereLis a maximal i0 su*
*ch
that zi06= 0. Then *(yz) = (resG;C(y) 1) . zi0+ w with w 2 i<|y|+i0HiBC H*B*
*G.
Consequently we get (resG;C(y) 1) . zi0= 0 and hence resG;C(y) is a divisor of*
* zero in
H*BC. Because C is finite abelian this is equivalent to say that resG;C(y) is n*
*ilpotent in
contradiction to our assumption.
Now we turn to associated primes in H*BG. First we recall from [D2] or [LS]*
* that
associated primes of H*BG are invariant with respect to the actions of the Stee*
*nrod
reduced power operations P i. By [Q], such invariant prime ideals " are of the *
*followingp_
form: There exists an elementary abelian p-subgroup E of G such that " = res-1G*
*;E( 0).
The following result is here presented as a consequence of Proposition 3 and*
* Proposition
4. It could have also been deduced from the results of Duflot in [D1, D2].
p __
Proposition 5. Assume " = res-1G;E( 0) is an associated prime. Then E contain*
*s all
central elements of G of order p.
Proof:_For x 2 G central of order p consider , the subgroup generated by x. *
* Then
{D 2 A(G) | D} is closed. Assume 6 E. By Proposition 3 we find 2 H*BG
4 JON F. CARLSON AND HANS-WERNER HENN
such that resG; is not nilpotent and resG;E = 0, i.e. 2 ". By Proposition 4*
* we find
that is not a zero divisor, i.e 62 " and we have arrived at a contradiction.
Remarks 6. We end the paper by pointing out a few of the easier applications of*
* Theorem
1 and 2.
i) It is not difficult to see that if the cohomology ring H*BG is Cohen-Maca*
*ulay then all
of the cohomology must be detected on the centralizers of the maximal elementar*
*y abelian
p-subgroups, and moreover that the maximal elementary abelian p-subgroups must *
*all
have the same rank r. For suppose that VG is the maximal ideal spectrum of H**
*BG,
and let VG (i) denote the set of all maximal ideals containing an element i 2 H*
**BG.
By Proposition 3, it is possible to find a homogeneous element i with the prope*
*rty that
resG;E(i) 6= 0, for an elementary abelian p-group E, if and only if E has maxim*
*al rank r.
So dim VG (i) = r-1 and we can find i2; : :;:ir such that VG (i)\VG (i2)\. .\.V*
*G (ir) = 0.
That is, each iiis chosen so that dim(VG (i)\. .\.VG (ii)) = r-i, or the dimens*
*ion decreases
by one with each additional intersection. Then i; i2; : :;:ir is a system of pa*
*rameters for
H*BG and hence, by the Cohen-Macaulay assumption, is a regular sequence (see Th*
*eorem
17.4 of [M]). In particular, i is regular and Supp (i) is precisely the element*
*ary abelian
p-subgroups of rank r.
ii) The result of (i) above can also be shown using the methods involving tr*
*ansfer
maps as in [C]. For assuming that H*BG is Cohen-Macaulay, we may choose a set of
parameters in which one of them is a sum of transfers from the centralizers of *
*the elementary
abelian p-subgroups of maximal rank r ([C], Proposition 2.4). By Frobenius reci*
*procity,
this element would annihilate any element whose restriction to the centralizer *
*of every
elementary abelian p-subgroup of rank r was zero.
As in [C] it is tempting to ask if the converse of the statement in (i) is c*
*orrect. Or
more generally, if the cohomology H*BG is detected on the centralizers of the e*
*lementary
abelian p-subgroups of rank t, is it necessary that the depth of H*BG be at lea*
*st t? It is
not difficult to show that this question is equivalent to Question 3.1 of [C].
iii) Still another proof of the statement of (i) can be given by using Corol*
*lary I.5.5 of
[HLS]. We do not reproduce the statement of Corollary I.5.5 here, but would lik*
*e to point
out that it can be considered as a technical sharpening of Corollary I.5.7 of [*
*HLS], i.e. of
Theorem 1 of this paper.
iv) Theorems 1 and 2 can be used to prove the detectability of cohomology in*
* certain
circumstances. An example is given in (II, 5.4.2) of [HLS]. There the mod-2 coh*
*omology
of the group G = GL(2; q), q 3 (mod 4), is considered. In this case each max*
*imal
elementary abelian 2 subgroup of G is conjugate to the subgroup E of diagonal m*
*atrices
of order 2. By exhibiting an element of H*BG which is not a divisor of zero and*
* whose
support consists precisely of the conjugates of E, one sees via Theorem 1 that *
*the restriction
map from H*BG to H*BCG (E) is injective.
The Sylow 2-subgroup of G is a semi-dihedral group whose mod-2 cohomology ha*
*s depth
one while the cohomology of G has depth 2. Other examples of this phenomenon ar*
*e the
mod-2 cohomology of the Mathieu group M11 [BC], and SL(3; q), q congruent to 3 *
*modulo
4 (see [MP]).
v) In Section I.5 of [HLS] there is another application of these ideas to in*
*finite groups.
Dwyer has shown that for some n, the mod-2 cohomology of BGL n; Z 1_2is not d*
*e-
THE SUPPORT OF REGULAR ELEMENTS IN THE COHOMOLOGY RING OF A GROUP 5
tected on the centralizer of the subgroup E of all diagonal matrices of order 2*
*. It is not
difficult to exhibit an element in H*BGL(n; Z 1_2) which is the restriction of*
* an element
in H*BGL(n; C) and whose support is again precisely the conjugates of E. By The*
*orem
1 this element must be a zero divisor and thus it gives an example which is con*
*trary to a
conjecture of Quillen ([Q], page 591) in that H*BGL n; Z 1_2can not be a free*
* module
over H*BGL(n; C).
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Jon F. Carlson Hans-Werner Henn
Department of Mathematics Mathematisches Institut
University of Georgia der Universit"at
Athens, GA 30602 Im Neuenheimer Feld 288
USA D-69120 Heidelberg
Fed. Rep. of Germany