The image of the transfer map
D. J. Benson*
Mathematical Institute
24-29 St. Giles
Oxford OX1 3LB
Great Britain
1 Introduction
In this paper, we prove a generalization of a theorem of Carlson [1]:
Theorem 1.1 Suppose that G is a finite group, and k is a field of characteris*
*tic p. Let
H be a collection of subgroups of G. Denote by K the collection of subgroups K *
*of G with
the property that the Sylow p-subgroups of the centralizer CG (K) are not conju*
*gate to a
subgroup of any of the groups in H.
Let J be the ideal in H*(G; k) given by the sum of the images of transfer fr*
*om subgroups
in H, and J0 be the ideal in H*(G; k) given by the intersection of thepkernels_*
*ofprestriction_
to subgroups in K. Then the ideals J and J0 have the same radical, J = J0.
The theorem of Carlson is the special case of the above theorem in which H i*
*s taken
to be the collection of subgroups whose index is divisible by p. In this case,*
* K is the
collection of subgroups whose centralizer contains some Sylow p-subgroup of G, *
*and J0 is
the kernel of restriction to the center of a Sylow p-subgroup.
Theorem 1.2 (Carlson) Suppose that G is a finite group, and k is a field of *
*character-
istic p. Let J be the ideal in H*(G; k) spanned by the transfers from subgroups*
* of index
divisiblepby_p,pand_J0 the kernel of restriction to the center of a Sylow p-sub*
*group. Then
J = J0.
Our proof is a generalization of a simplified version of the proof of Carlso*
*n's theorem
given by Evens and Feshbach [3].
________________________________*
The author is very grateful for the support and hospitality of the Sonderfor*
*schungsbereich 170 "Ge-
ometrie und Analysis," Universit"at G"ottingen, for the period during which thi*
*s research was carried out.
1
2 Elementary abelian subgroups
For the convenience of the reader, we begin this section with a proof of the es*
*sential
ingredient, which is a theorem of Feshbach ([4], Theorem 2.4; see also Priddy a*
*nd Wilk-
erson [5], Theorem II). We then show how to use Feshbach's theorem to express e*
*lements
as restrictions of transfers from centralizers.
Let E be an elementary abelian p-group of order pr, and k an algebraically c*
*losed
field of characteristic p. Then modulo the radical, H*(E; k) is a polynomial r*
*ing on r
generators in degree one if p = 2 and degree two if p is odd. It can be viewed *
*as the ring
of polynomial functions on the affine space VE = k Fp E ~=Ar(k). The element
Y
oeE = fi(x);
06=x2H1(E;Fp)
namely the product of the Bocksteins of the degree one elements defined over Fp*
*, may be
thought of as a polynomial function whose zeros VE (oeE ) consist of the union *
*of the proper
subspaces of VE defined over Fp; in other words, the subspaces corresponding to*
* the proper
subgroups of E. It is clear from the definition that oeE is fixed by all automo*
*rphisms of E.
If o is an automorphism of E, consider the ideal Io in H*(E; k) generated by*
* elements
of the form x - o(x). The zeros of Io form a linear subspace VEo VE , where Eo*
* is the
set of fixed points in E of o. In particular, if o acts non-trivially, this is *
*contained in the
zeros of oeE , and so by Hilbert's Nullstellensatz, some power of oeE lies in I*
*o. So we obtain
an equation of the form
Xt
oesE= aj(bj - o(bj)):
j=1
P t
Setting at+1 = - j=1ajo(bj) and bt+1 = 1, we have
t+1X t+1X
ajbj = oesE; ajo(bj) = 0:
j=1 j=1
Now suppose W is a group acting faithfully as automorphisms on E. For each n*
*on-
identity element o 2 W , we may obtain such elements aj(o) and bj(o). If we fo*
*rm the
product of the index sets, and form the products of a's, and products of b's an*
*d the product
of the s's, we obtain elements aj and bj such that
(
X oes o = 1
ajo(bj) = 0E o 6= 1:
P
Thus oesEis a linear combination of transfers ajTr1;W(bj). Now the transfers*
* form an
ideal in the ring of invariants H*(E; k)W , and H*(E; k) is finitely generated *
*as a module
over H*(E; k)W , so we may apply the following (well known) ring theoretic lemm*
*a.
Lemma 2.1 Let R S be commutative rings, with S finitely generated as an R-m*
*odule.
If I is an ideal in R and a is an element of R with the property that a lies in*
* the ideal in
S generated by I, then some power of a lies in I.
2
P
Proof Let S be generated by x1; : :;:xn as an R-module, and let a:xi = jaijx*
*j with
aij2 I. Let A be the matrix (aij), and x the column vector formed by x1; : :;:*
*xn, so
that (a:1 - A):x = 0. Denote by B the transposed matrix of cofactors of a:1 - A*
*, so that
B(a:1 - A) = det(a:1 - A):1. Then
det(a:1 - A):x = B(a:1 - A):x = 0
and since the identity element of S is an R-linear combination of the xi, it fo*
*llows that
det(a:1 - A) = 0. This equation shows that an 2 I. 2
Applying this lemma with R = H*(E; k)W , S = H*(E; k), x = oesEand I the ima*
*ge
of the transfer map, we deduce the following theorem. The final part about rest*
*rictions is
guaranteed by replacing ff by ff:oeE if necessary.
Theorem 2.2 (Feshbach) Suppose that W is a group acting faithfully as automo*
*rphisms
on an elementary abelian p-group E, and let oeE 2 H*(E; k) be as above. Then t*
*here is
an element ff 2 H*(E; k) with the property that Tr1;W(ff) is a power of oeE , a*
*nd ff may be
taken to restrict to zero on every proper subgroup of E.
Remark Priddy and Wilkerson observe that this theorem may be interpreted as sa*
*ying
that the localization H*(E; k)[oe-1E] is a projective kW -module. We shall not *
*make use of
this observation.
In fact, Priddy and Wilkerson, in the last proposition of their paper, write*
* down an
explicit element ff in the case W = GL(E) (which is, of course, sufficient), wi*
*th s = r (the
rank of E). We find the above argument more enlightening.
We can make this theorem more powerful by using the Evens norm map [2] as fo*
*llows.
If E is an elementary abelian p-subgroup of G, and C = CG (E) is its centralize*
*r, let
|C : E| = pa:h with h coprime to p. Then given any element x 2 H*(E; k), the Ma*
*ckey
double coset formula shows that
a
resC;EnormE;C(1 + x) = (1 + x)|C:E|= 1 + hxp + . .:.
So in particular xpa is a restriction from C. Furthermore, if x is assumed to *
*restrict to
zero on every proper subgroup of E, then for E0 a subgroup of C not containing *
*E we
have
resC;E0normE;C(1 + x) = 1:
Now, raising elements to the power of pa is a ring homomorphism on H*(E; k) whi*
*ch
commutes with the action of W , and hence with Tr1;W. This means that we may as*
*sume
that the element ff in the above theorem is a restriction from C, say ff = resC*
*;E(j),
j 2 H*(C; k). Then setting N = NG (E), the normalizer of E in G, we have
oesE= resN;ETrC;N(j) = resG;ETrC;G(j):
In this last step, we have used the fact that if we expand out the restriction *
*from G to
E of TrC;G(j) using the Mackey double coset formula, the only terms which are n*
*on-zero
3
come from the normalizer. The remaining terms involve restricting to subgroups *
*of C not
containing E, which gives zero. So we have proved the following strengthened fo*
*rm of the
above theorem; the statement about the form of s is easy to see from the above *
*proof.
Theorem 2.3 Suppose that E is an elementary abelian p-subgroup of G and let o*
*eE 2
H*(E; k) be as above. Then some power of oeE is a restriction from G of a trans*
*fer from
C = CG (E),
oesE= resG;ETrC;G(j):
The number s in this formula may be taken to be a multiple of any given integer*
*, and j
may be taken to restrict to zero on every proper subgroup of E.
Now suppose that y is an element of H*(G; k) with the property that resG;E0(*
*y) = 0
for all proper subgroups E0of E. In this situation, one may apply a lemma of Qu*
*illen and
Venkov.
Lemma 2.4 (Quillen and Venkov [7]) Suppose that P is a p-group, and P 0is a*
* sub-
group of index p. Let x 2 H1(P; Fp) ~= Hom (P; Z=p) be a corresponding degree *
*one
cohomology element. If i is an element of H*(P; k) with resP;P0(i) = 0 then the*
*re is an
element i02 H*(P; k) with i2 = fi(x)i0.
By the hypothesis on y, we may apply this lemma to resG;E(y) for each subgro*
*up E0
of index p in E. We find that resG;E(ym ) = oeE :u, for some m > 0 and u 2 H*(*
*E; k).
We can clearly also demand that resE;E0(u) = 0 for every proper subgroup E0 of *
*E, for
example by making u divisible by oeE . Now if g 2 N then conjugation by g is tr*
*ivial on the
restriction of ym , so we have oeE :g(u) = oeE :u. Since oeE is not a zero divi*
*sor in H*(E; k),
this implies that u is invariant under the action of N. So by the Evens norm ar*
*gument
given above, some power of u is a restriction from G. So we have a formula of t*
*he form
resG;E(ymt) = oetEresG;E(z)
with z 2 H*(G; k). Now by the above theorem, there is an element j 2 H*(C; k) *
*with
resG;ETrC;G(j) = oestE. Then
resG;E(ymst) = oestEresG;E(zs) = resG;E(TrC;G(j)) resG;E(zs) = resG;E(TrC;G(j r*
*esG;C(zs)))
and so
resG;E(ymst - TrC;G(j resG;C(zs))) = 0:
To summarize, we have proved the following.
Theorem 2.5 If E is an elementary abelian p-subgroup of G with centralizer C *
*= CG (E),
and y is an element of H*(G; k) with the property that resG;E0(y) = 0 for all p*
*roper
subgroups E0 of E, then there exists r > 0 and x 2 H*(C; k) such that
(i) resG;E(yr - TrC;G(x)) = 0, and
(ii) x restricts to zero on all proper subgroups of E.
4
3 Proof of the main theorem
We are now in a position to prove Theorem 1.1. First, we remark that if k0is an*
* algebraic
closure of k, then H*(G; k0) = k0k H*(G; k), and the ideals J and J0 defined ov*
*er k0
are obtained by extension of scalars from the corresponding ideals defined over*
* k. So it
suffices to prove the theorem in case k = k0 is algebraically closed, which we *
*now assume.
We may also assume without loss of generality that if H 2 H then so is every co*
*njugate
and every subgroup of H, because this does not alter the sum of the images of t*
*ransfer.
Next, we recall a theorem of Quillen [6], which states that an element of co*
*homology
H*(G; k) is nilpotent if and only if its restriction to every elementary abelia*
*n p-subgroup
of G is nilpotent. It follows that we may replace K by the collection K0 of el*
*ementary
abelianpp-subgroups_E G with the Sylow p-subgroups of CG (E) not in H, and the*
* ideal
J0 is unaffected by this (although J0 may get larger).
After this is done, we claim that J J0. Namely, we claim that for any H 2 *
*H,
E 2 K0 and i 2 H*(H; k), we have resG;ETrH;G(i) = 0. To prove this, we look at*
* the
double cosets of H and E in G, and examine the corresponding Mackey formula. Th*
*is is
the same as looking at the orbits of E on the cosets of H. The hypothesis that *
*the Sylow
p-subgroups of CG (E) are not in H ensures that each orbit of CG (E) on the cos*
*ets of H
has length divisible by p. So there are two types of double cosets: those corre*
*sponding to
non-trivial orbits of E, and those corresponding to fixed points of E. For the *
*non-trivial
orbits of E, the Mackey formula tells us to transfer from a proper subgroup of *
*E, so we
get zero. The centralizer CG (E) acts on the cosets corresponding to the fixed*
* points of
E, and the contributions from double cosets in the same CG (E)-orbit are equal.*
* Since
the orbits have length divisible by p, and we are in characteristic p, the tota*
*l contribution
from each CG (E)-orbit of fixed points is therefore zero, and we are done.
Conversely, we must show that if y 2 J0 then some power of y is in J. The pl*
*an is to
proceed in steps involving replacing y by a power of y, and subtracting from y *
*transfers
from subgroups in H, until y has zero restriction to every elementary abelian s*
*ubgroup
of G, at which stage Quillen's theorem tells us that y is nilpotent, and we are*
* done. We
proceed by induction on the elementary abelian subgroups E of G, being sure to *
*treat all
proper subgroups of E before treating E. If E is in K0, then the restriction of*
* y to E is
already zero, so we are done. If E is not in K0, then a Sylow p-subgroup of C =*
* CG (E) is
in H. Now transfer is surjective from a Sylow p-subgroup of C to C, so by Theor*
*em 2.5,
we may replace y by a power of y, then subtract off a transfer from an element *
*of H,
so that the new y restricts to zero on E. Using the double coset formula and p*
*art (ii)
of Theorem 2.5, we see that y still restricts to zero on the previously treated*
* elementary
abelian subgroups. This completes the inductive step, and hence the proof of t*
*he main
theorem.
References
[1]J. F. Carlson. Varieties and transfers. J. Pure Appl. Algebra 44 (1987), 99*
*-105.
5
[2]L. Evens. A generalization of the transfer map in the cohomology of groups. *
*Trans.
A.M.S. 108 (1963), 54-65.
[3]L. Evens and M. Feshbach. Carlson's theorem on varieties and transfer. J. Pu*
*re Appl.
Algebra 57 (1989), 39-45.
[4]M. Feshbach. p-subgroups of compact Lie groups and torsion of infinite heigh*
*t in
H*(BG), II. Mich. Math. J. 29 (1982), 299-306.
[5]S. Priddy and C. Wilkerson. Hilbert's theorem 90 and the Segal Conjecture fo*
*r ele-
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[6]D. G. Quillen. The spectrum of an equivariant cohomology ring, I, II. Ann. o*
*f Math.
94 (1971), 549-572; 573-602.
[7]D. G. Quillen and B. B. Venkov. Cohomology of finite groups and elementary a*
*belian
subgroups. Topology 11 (1972), 317-318.
6