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 characteristic 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 C_{G}(K) are not conjugate 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 from subgroups in H, and J′ be the ideal in H^{∗}(G, k) given by the intersection of the kernels of restriction to subgroups in K. Then the ideals J and J′ have the same radical, \sqrt{J} = \sqrt{J′}.

The theorem of Carlson is the special case of the above theorem in which H is 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 J′ 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 characteristic p. Let J be the ideal in H^{∗}(G, k) spanned by the transfers from subgroups of index divisible by p, and J′ the kernel of restriction to the center of a Sylow p-subgroup. Then \sqrt{J} = \sqrt{J′}.

Our proof is a generalization of a simplified version of the proof of Carlson’s theorem given by Evens and Feshbach [3].

∗The author is very grateful for the support and hospitality of the Sonderforschungsbereich 170 “Geometrie und Analysis,” Universität Göttingen, for the period during which this research was carried out.

2 Elementary abelian subgroups

For the convenience of the reader, we begin this section with a proof of the essential ingredient, which is a theorem of Feshbach ([4], Theorem 2.4; see also Priddy and Wilkerson [5], Theorem II). We then show how to use Feshbach’s theorem to express elements as restrictions of transfers from centralizers.

Let E be an elementary abelian p-group of order p^{r}, and k an algebraically closed field of characteristic p. Then modulo the radical, H^{∗}(E, k) is a polynomial ring 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 V_{E} = k ⊗_{F_{p}} E ≅ A^{r}(k). The element

σ_{E} = ∏_{0≠x∈H^{1}(E,F_{p})} β(x),

namely the product of the Bocksteins of the degree one elements defined over F_{p}, may be thought of as a polynomial function whose zeros V_{E}(σ_{E}) consist of the union of the proper subspaces of V_{E} defined over F_{p}; in other words, the subspaces corresponding to the proper subgroups of E. It is clear from the definition that σ_{E} is fixed by all automorphisms of E.

If τ is an automorphism of E, consider the ideal I_{τ} in H^{∗}(E, k) generated by elements of the form x − τ(x). The zeros of I_{τ} form a linear subspace V_{E^{τ}} ⊆ V_{E}, where E^{τ} is the set of fixed points in E of τ. In particular, if τ acts non-trivially, this is contained in the zeros of σ_{E}, and so by Hilbert’s Nullstellensatz, some power of σ_{E} lies in I_{τ}. So we obtain an equation of the form

σ_{E}^{s} = ∑_{j=1}^{t} a_{j}(b_{j} − τ(b_{j})).

Setting a_{t+1} = − ∑_{j=1}^{t} a_{j}τ(b_{j}) and b_{t+1} = 1, we have

∑_{j=1}^{t+1} a_{j}b_{j} = σ_{E}^{s},  ∑_{j=1}^{t+1} a_{j}τ(b_{j}) = 0.

Now suppose W is a group acting faithfully as automorphisms on E. For each nonidentity element τ ∈ W, we may obtain such elements a_{j}(τ) and b_{j}(τ). If we form the product of the index sets, and form the products of a’s, and products of b’s and the product of the s’s, we obtain elements a_{j} and b_{j} such that

∑ a_{j}τ(b_{j}) = \{ \begin{matrix}σ_{E}^{s} & τ = 1 \\ 0 & τ ≠ 1. \end{matrix}

Thus σ_{E}^{s} is a linear combination of transfers ∑ a_{j}Tr_{1,W} (b_{j}). 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 lemma.

Lemma 2.1 Let R ⊆ S be commutative rings, with S finitely generated as an R-module. 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.

Proof Let S be generated by x_{1}, …, x_{n} as an R-module, and let a.x_{i} = ∑_{j} a_{ij}x_{j} with a_{ij} ∈ I. Let A be the matrix (a_{ij}), and x the column vector formed by x_{1}, …, x_{n}, 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 x_{i}, it follows that det(a.1 − A) = 0. This equation shows that a^{n} ∈ I. □

Applying this lemma with R = H^{∗}(E, k)^{W}, S = H^{∗}(E, k), x = σ_{E}^{s} and I the image of the transfer map, we deduce the following theorem. The final part about restrictions is guaranteed by replacing α by α.σ_{E} if necessary.

Theorem 2.2 (Feshbach) Suppose that W is a group acting faithfully as automorphisms on an elementary abelian p-group E, and let σ_{E} ∈ H^{∗}(E, k) be as above. Then there is an element α ∈ H^{∗}(E, k) with the property that Tr_{1,W} (α) is a power of σ_{E}, and α 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 saying that the localization H^{∗}(E, k)[σ_{E}^{−1}] 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 α in the case W = GL(E) (which is, of course, sufficient), with 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 follows. If E is an elementary abelian p-subgroup of G, and C = C_{G}(E) is its centralizer, let |C : E| = p^{a}.h with h coprime to p. Then given any element x ∈ H^{∗}(E, k), the Mackey double coset formula shows that

res_{C,E}norm_{E,C}(1 + x) = (1 + x)^{|C:E|} = 1 + hx^{p^{a}} + · · · .

So in particular x^{p^{a}} is a restriction from C. Furthermore, if x is assumed to restrict to zero on every proper subgroup of E, then for E′ a subgroup of C not containing E we have

res_{C,E′}norm_{E,C}(1 + x) = 1.

Now, raising elements to the power of p^{a} is a ring homomorphism on H^{∗}(E, k) which commutes with the action of W, and hence with Tr_{1,W}. This means that we may assume that the element α in the above theorem is a restriction from C, say α = res_{C,E}(η), η ∈ H^{∗}(C, k). Then setting N = N_{G}(E), the normalizer of E in G, we have

σ_{E}^{s} = res_{N,E}Tr_{C,N}(η) = res_{G,E}Tr_{C,G}(η).

In this last step, we have used the fact that if we expand out the restriction from G to E of Tr_{C,G}(η) using the Mackey double coset formula, the only terms which are non-zero 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 form 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 σ_{E} ∈ H^{∗}(E, k) be as above. Then some power of σ_{E} is a restriction from G of a transfer from C = C_{G}(E),

σ_{E}^{s} = res_{G,E}Tr_{C,G}(η).

The number s in this formula may be taken to be a multiple of any given integer, and η 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 res_{G,E′}(y) = 0 for all proper subgroups E′ of E. In this situation, one may apply a lemma of Quillen and Venkov.

Lemma 2.4 (Quillen and Venkov [7]) Suppose that P is a p-group, and P′ is a subgroup of index p. Let x ∈ H^{1}(P, F_{p}) ≅ Hom(P, Z/p) be a corresponding degree one cohomology element. If ζ is an element of H^{∗}(P, k) with res_{P,P′}(ζ) = 0 then there is an element ζ′ ∈ H^{∗}(P, k) with ζ^{2} = β(x)ζ′.

By the hypothesis on y, we may apply this lemma to res_{G,E}(y) for each subgroup E′ of index p in E. We find that res_{G,E}(y^{m}) = σ_{E}.u, for some m > 0 and u ∈ H^{∗}(E, k). We can clearly also demand that res_{E,E′}(u) = 0 for every proper subgroup E′ of E, for example by making u divisible by σ_{E}. Now if g ∈ N then conjugation by g is trivial on the restriction of y^{m}, so we have σ_{E}.g(u) = σ_{E}.u. Since σ_{E} is not a zero divisor in H^{∗}(E, k), this implies that u is invariant under the action of N. So by the Evens norm argument given above, some power of u is a restriction from G. So we have a formula of the form

res_{G,E}(y^{mt}) = σ_{E}^{t}res_{G,E}(z)

with z ∈ H^{∗}(G, k). Now by the above theorem, there is an element η ∈ H^{∗}(C, k) with res_{G,E}Tr_{C,G}(η) = σ_{E}^{st}. Then

res_{G,E}(y^{mst}) = σ_{E}^{st} res_{G,E}(z^{s}) = res_{G,E}(Tr_{C,G}(η)) res_{G,E}(z^{s}) = res_{G,E}(Tr_{C,G}(η res_{G,C}(z^{s})))

and so

res_{G,E}(y^{mst} − Tr_{C,G}(η res_{G,C}(z^{s}))) = 0.

To summarize, we have proved the following.

Theorem 2.5 If E is an elementary abelian p-subgroup of G with centralizer C = C_{G}(E), and y is an element of H^{∗}(G, k) with the property that res_{G,E′}(y) = 0 for all proper subgroups E′ of E, then there exists r > 0 and x ∈ H^{∗}(C, k) such that

(i) res_{G,E}(y^{r} − Tr_{C,G}(x)) = 0, and

(ii) x restricts to zero on all proper subgroups of E.

3 Proof of the main theorem

We are now in a position to prove Theorem 1.1. First, we remark that if k′ is an algebraic closure of k, then H^{∗}(G, k′) = k′ ⊗_{k} H^{∗}(G, k), and the ideals J and J′ defined over k′ are obtained by extension of scalars from the corresponding ideals defined over k. So it suffices to prove the theorem in case k = k′ is algebraically closed, which we now assume. We may also assume without loss of generality that if H ∈ H then so is every conjugate and every subgroup of H, because this does not alter the sum of the images of transfer.

Next, we recall a theorem of Quillen [6], which states that an element of cohomology H^{∗}(G, k) is nilpotent if and only if its restriction to every elementary abelian p-subgroup of G is nilpotent. It follows that we may replace K by the collection K′ of elementary abelian p-subgroups E ≤ G with the Sylow p-subgroups of C_{G}(E) not in H, and the ideal \sqrt{J′} is unaffected by this (although J′ may get larger).

After this is done, we claim that J ≤ J′. Namely, we claim that for any H ∈ H, E ∈ K′ and ζ ∈ H^{∗}(H, k), we have res_{G,E}Tr_{H,G}(ζ) = 0. To prove this, we look at the double cosets of H and E in G, and examine the corresponding Mackey formula. This is the same as looking at the orbits of E on the cosets of H. The hypothesis that the Sylow p-subgroups of C_{G}(E) are not in H ensures that each orbit of C_{G}(E) on the cosets of H has length divisible by p. So there are two types of double cosets: those corresponding 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 C_{G}(E) acts on the cosets corresponding to the fixed points of E, and the contributions from double cosets in the same C_{G}(E)-orbit are equal. Since the orbits have length divisible by p, and we are in characteristic p, the total contribution from each C_{G}(E)-orbit of fixed points is therefore zero, and we are done.

Conversely, we must show that if y ∈ J′ then some power of y is in J. The plan 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 subgroup 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 K′, then the restriction of y to E is already zero, so we are done. If E is not in K′, then a Sylow p-subgroup of C = C_{G}(E) is in H. Now transfer is surjective from a Sylow p-subgroup of C to C, so by Theorem 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 part (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 the main theorem.

References

[1] J. F. Carlson. Varieties and transfers. J. Pure Appl. Algebra 44 (1987), 99–105.

[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. Pure Appl. Algebra 57 (1989), 39–45.

[4] M. Feshbach. p-subgroups of compact Lie groups and torsion of infinite height 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 for elementary abelian p-groups. Amer. J. Math. 107 (1985), 775–785.

[6] D. G. Quillen. The spectrum of an equivariant cohomology ring, I, II. Ann. of Math. 94 (1971), 549–572; 573–602.

[7] D. G. Quillen and B. B. Venkov. Cohomology of finite groups and elementary abelian subgroups. Topology 11 (1972), 317–318.