TRANSFER AND CHERN CLASSES FOR EXTRASPECIAL p-GROUPS

DAVID JOHN GREEN AND PHAM ANH MINH

Abstract. In the cohomology ring of an extraspecial p-group, the subring generated by Chern classes and transfers is studied. This subring is strictly larger than the Chern subring, but still not the whole cohomology ring, even modulo nilradical. A formula is obtained relating Chern classes to transfers.

Introduction

Methods to determine the cohomology ring of a finite group almost always presuppose that the cohomology of the Sylow p-subgroups is known. Calculating the cohomology ring of a p-group is however a delicate and difficult task. The extraspecial p-groups of exponent p are in some sense the minimal difficult cases: minimal because their proper quotients are all elementary abelian, and their automorphism groups are very large. For this reason, many papers have been written, investigating their cohomology. Developments up till 1991 are surveyed in the paper [BC92]. In particular, M. Tezuka and N. Yagita obtained the prime ideal spectrum of the cohomology ring.

The usual method to calculate the cohomology of a p-group is to write the group as an extension, and solve the associated Lyndon–Hochschild–Serre spectral sequence. For the extraspecial p-groups however, such spectral sequences are intractable, and one is forced to look for other techniques. Now, standard constructions such as transfer (or corestriction) from subgroups and taking Chern classes of group representations provide us with a large number of cohomology classes. So many in fact, that for any p-group the classes provided by these two constructions generate a subring that has the same prime ideal spectrum as the cohomology ring (see [GL96]). In this paper, we study this subring in the case of the extraspecial p-groups, and ask whether it is the whole cohomology ring.

Actually, these constructions yield very few odd-dimensional classes, and so it is rather more realistic to ask whether we obtain the whole cohomology ring modulo nilradical. At least for mod-p cohomology, Proposition 9.4 answers this question too in the negative. For integral cohomology however, the problem remains open, and the significance of Corollary 8.2 is that this subring is the biggest studied to date in the cohomology ring of an extraspecial p-group.

This last result is proved using cohomology classes which we denote χ_{r,ϕ}. They are constructed in a manner foreshadowed in [Min95]. Take a product of Chern classes for the group of order p^{2n−1}. By inflation and then corestriction, obtain a cohomology class χ_{r,ϕ} for the group of order p^{2n+1}. In Theorem 5.2, an elegant formula is obtained relating the Chern classes and the χ_{r,ϕ}, and in Theorem 7.2 we show that the p^{th} power of any Chern class or any χ_{r,ϕ} lies in the subring generated by top Chern classes: this is the subring Tezuka and Yagita used to obtain the spectrum of the cohomology ring.

Date: 15 November 1996.

1991 Mathematics Subject Classification. Primay 20J06; Secondary 20D15, 55R40.

The first author was supported by the Deutsche Forschungsgemeinschaft Schwerpunktprogramm “Algorithmische Zahlentheorie und Algebra”.

The second author held a DAAD fellowship.

We are very grateful to Bruno Kahn for interesting discussions; and to Hélène Esnault and Eckhart Viehweg, who arranged for the second author to visit Essen.

1. A relation between Dickson invariants

We shall assume that the reader is familiar with the Dickson invariants: such familiarity may be acquired by consulting Benson’s book [Ben93], for example.

Let V be an m-dimension 𝔽_{p}-vector space, and let 0 ≤ r ≤ m − 1. We shall write D_{r}(V) for the Dickson invariant in degree p^{m} − p^{r} in S(V), or just D_{m,r} if V is clear from the context. Recall that, for an indeterminate X, we have the equation

V (V; X) = X^{p^{m}} + ∑_{r=0}^{m−1}(−1)^{m−r}D_{r}(V)X^{p^{r}} , (1.1)

where V (V; X) in S(V)[X] is defined by

V (V; X) = ∏_{ϕ∈V}(X − ϕ) . (1.2)

The polynomial V (V; X) is called the Mui invariant, as it is the most important of a family of invariants for subgroups of GL_{m+1} studied by Mui [Mui75]. Note that V (V; X) is 𝔽_{p}-linear as a function of X. By convention, we define D_{m,r} = 0 if r < 0, and D_{m,m} = 1.

Lemma 1.1. Let V be an m-dimensional 𝔽_{p}-vector space, let U be a proper subspace of V, and let s be the codimension of U in V. Let Hyp(V, U) denote the set of all hyperplanes in V which contain U: this set is clearly nonempty. Then

∑_{W∈Hyp(V,U)} V (W; X) = V (U; X)^{p^{s−1}} , (1.3)

and it follows that

(1.4)  ∑_{W∈Hyp(V,U)} D_{r−1}(W) = D_{r−s}(U)^{p^{s−1}} for all 1 ≤ r ≤ m.

Proof. Let T be a complementary subspace of U in V, so dim(T ) = s. The elements of Hyp(V, U) are those subspaces W of V such that W ∩T is a hyperplane in T and W = U ⊕ (W ∩ T ). So choosing such a T induces a bijection between Hyp(V, U) and the projective space ℙT^{∗} of lines in the dual space of T .

Let τ_{1}, …, τ_{s} be a basis for T . Observe that V (U; τ_{1}), …, V (U; τ_{s}), all homogeneous of degree p^{n−s}, are algebraically independent over 𝔽_{p}: for the image of V (U; τ_{i}) in S(V/U) ≅ S(T ) is τ_{i}^{p^{n−s}}.

Denote by \underline{λ} an s-tuple (λ_{1}, …, λ_{s}) in 𝔽_{p}^{s}. Given ϕ ∈ ℙT^{∗}, say that \underline{λ} belongs to ker(ϕ) if ∑_{i=1}^{s} λ_{i}τ_{i} does. Using Eqn. (1.2) and the linearity of V (U; −), we obtain

(1.5)  ∑_{W∈Hyp(V,U)} V (W; X) = ∑_{ϕ∈ℙT ^{∗}} ∏_{t∈ker(ϕ)} V (U; X − t)

= ∑_{ϕ} ∏_{\underline{λ}∈ker(ϕ)} (V (U; X) − ∑_{i=1}^{s} λ_{i}V (U; τ_{i})) . (1.6)

Consider the expression (1.6). It is a polynomial in V (U; X), V (U; τ_{1}), …, V (U; τ_{s}), and as such is homogeneous of degree p^{s−1}. Treat V (U; X) as the main variable: then there is no constant term. Moreover, the coefficients are polynomials in V (U; τ_{1}), …, V (U; τ_{s}) which are invariant under the natural action of GL_{s}(𝔽_{p}).

So the coefficients must be polynomials in the Dickson invariants in these s variables: but there are no such polynomials in positive degree less than p^{s−1}.

Therefore the coefficient in (1.6) of V (U; X)^{j} is zero except when j = p^{s−1}. This coefficient is the size of ℙT^{∗}, congruent to 1 modulo p.

2. Chern classes and extraspecial p-groups

The integral (group) cohomology of the unitary group U(n) is a polynomial algebra with n generators. A unitary representation of a finite group G pulls these generators back to the integral cohomology of G; the images of the generators are the Chern classes of the representation. We refer the reader to [Tho86] for more information about Chern classes and their properties.

Let p be an odd prime. For n ≥ 1, denote by P = P_{n} the extraspecial p-group of order p^{2n+1} and exponent p. This fits into a central extension

1 → Z → P_{n} → E_{n} → 1 , (2.1)

where Z = Z(P) is cyclic of order p, and E = E_{n} is elementary abelian of prank 2n. We may identify Z with 𝔽_{p} and view E as a 2n-dimensional 𝔽_{p}-vector space: then the commutator map on P induces a nondegenerate symplectic bilinear form E ⊗_{𝔽p} E → Z.

Ignoring the p^{2n} linear characters, the remaining irreducible characters of P all have degree p^{n}, and are distinguished by their restrictions to Z. Pick a nontrivial linear character γ of Z, and define ρ_{γ} to be the unique irreducible representation of P whose restriction to Z has character p^{n}γ. The other degree p^{n} representations are then ψ^{j}(ρ_{γ}) for 2 ≤ j ≤ p − 1.

Since E is the abelianization of P, Chern classes of one-dimensional representations lie in the image of inflation from E to P. The Chern classes of ψ^{j}(ρ_{γ}) are scalar multiples of those of ρ_{γ}.

Pick an embedding of 𝔽_{p} in ℂ^{×}: for each elementary abelian p-group A, this allows us to identify Hom(A, ℂ^{×}) with the dual space A^{∗}.

The maximal elementary abelian subgroups M of P all have p-rank n + 1, and are permuted transitively by the automorphism group of P. Each contains Z, and the map M ↦ I = M/Z induces a bijection between the M and the maximal totally isotropic subspaces I of E. Note that the dual I^{∗} is naturally a subspace of M^{∗}.

View γ as an element of H^{2}(Z, ℤ), since this isomorphic to Hom(Z, ℂ^{×}). The restriction of ρ_{γ} to M decomposes as the direct sum of all representations whose restriction to Z has character γ. The first Chern classes of these summands are all the classes in H^{2}(M, ℤ) whose restriction to Z is γ; pick γ̄ to be one of these first Chern classes, and observe that the kernel in H^{2}(M, ℤ) ≅ M^{∗} of restriction to Z is I^{∗}. From the definition of V, the total Chern class of ρ_{γ} therefore restricts to M as V (I^{∗}; 1 + γ̄), independent of the choice of γ̄. Define ζ_{n} ∈ H^{2p^{n}}(P, ℤ) and, for 0 ≤ r ≤ n − 1, define κ_{r} = κ_{n,r} in H^{2(p^{n}−p^{r})}(P_{n}, ℤ) by

ζ_{n} = c_{p^{n}}(ρ_{γ}) κ_{n,r} = (−1)^{n−r}c_{p^{n}−p^{r}}(ρ_{γ}) . (2.2)

Then these are the only non-nilpotent Chern classes of ρ_{γ}, and

Res_{M}^{P} ζ_{n} = V (I^{∗}; γ̄) Res_{M}^{P}(κ_{r}) = D_{r}(I^{∗}) . (2.3)

It is useful, and sensible, to define P_{0} to be Z and ζ_{0} to be γ. Also to define κ_{n,n} to be 1 ∈ H^{0}(P, ℤ).

3. The new classes

Since every group homomorphism P → 𝔽_{p} factors through E, we may identify E^{∗} with the 𝔽_{p}-vector space Hom(P, 𝔽_{p}). Each maximal subgroup H ≤ P induces a one-dimensional subspace of Hom(P, 𝔽_{p}), namely the subspace generated by any ϕ with ker(ϕ) = H. Consequently there is a natural bijection between the set of maximal subgroups of P and the projective space ℙE^{∗}.

We are interested in the corestriction map from maximal subgroups to P. Let ϕ ∈ ℙE^{∗}, and let H = ker(ϕ) be the corresponding maximal subgroup of P. Then H/Z is a (2n − 1)-dimensional 𝔽_{p}-vector space, carrying a symplectic bilinear form with one-dimensional kernel. The centre Z_{H} of H is elementary abelian of p-rank 2, and Z_{H}/Z is the kernel of the form on H/Z. The maximal elementary abelian subgroups of H all have p-rank n + 1 and all contain Z_{H}. They are permuted transitively by the automorphism group of H.

Definition 3.1. For n ≥ 1, for 0 ≤ r ≤ n − 1 and for ϕ ∈ ℙE_{n}^{∗}, the class χ_{r,ϕ} in H^{∗}(P_{n}, ℤ) with degree 2(p^{n} − p^{r}) is defined as follows. Set H = ker(ϕ) and pick a rank one subgroup A ≠ Z of Z_{H}. Choosing such an A induces a split epimorphism H → H/A ≅ P_{n−1}. Set

χ_{r,ϕ} = Cor_{H}^{P_{n}} Inf_{P_{n−1}}^{H}(κ_{n−1,r}ζ_{n−1}^{p−1}) . (3.1)

Lemma 3.2. The class χ_{r,ϕ} is well-defined. That is, it does not depend on the choice of A ≤ Z_{H}.

Proof. The inner automorphisms of P_{n} permute transitively all such subgroups A ≤ Z_{H}, always fixing Z pointwise and therefore sending the ρ_{γ} for one P_{n−1} to the ρ_{γ} for the other. Since H is normal in P_{n} and corestriction commutes with conjugation, the result follows.

4. Restriction and the new classes

In this section, we study restrictions of the classes χ_{r,ϕ}. We start however with a preparatory lemma.

Lemma 4.1. Let V be an 𝔽_{p}-vector space, and let U be a hyperplane in V. Pick v in V \ U. Then the element V (U; v)^{p−1} of S(V) is invariant under all transformations of V which preserve U. In particular, it is independent of the choice of v.

Proof. Any transformation which preserves U acts on the coset space V/U as multiplication by some scalar in 𝔽_{p}^{×}. But, from its definition, V (U; v) is invariant under all transformations which preserve U and the coset v + U.

Definition 4.2. In the situation of Lemma 4.1, we denote by V (U, V) the invariant V (U; v)^{p−1}.

Lemma 4.3. Let V be an 𝔽_{p}-vector space, and let U be a subspace with codimension two in V. Then

∑_{W} V (U, W) = 0 , (4.1)

where the sum is over all hyperplanes W in V which themselves contain U as a hyperplane.

Proof. Let v, w be a basis for a subspace of V complementary to U. Then

(4.2)  ∑_{W} V (U, W) = ∑_{[λ: μ]∈𝔽pℙ} V (U; λv + μw)^{p−1}

= ∑_{[λ: μ]} (λV (U; v) + μV (U; w))^{p−1} . (4.3)

This is an invariant of GL_{2}(𝔽_{p}) acting on the rank 2 polynomial algebra generated by V (U; v) and V (U; w). Its degree in the generators is however p − 1, and there are no invariants other than zero in this degree.

We now investigate the image of χ_{r,ϕ} under restriction to each maximal elementary abelian M ≤ P. If M is contained in H = ker(ϕ), then Z_{H} is contained in M. Define I_{ϕ} to be the quotient M/Z_{H}. Then I_{ϕ}^{∗} is a hyperplane in I^{∗}, which is itself a hyperplane in M^{∗}. Concretely, I_{ϕ}^{∗} is the annihilator in I^{∗} of Z_{H}/Z and also the annihilator in M^{∗} of Z_{H}.

Proposition 4.4. With the above notation, we have

(4.4) Res_{M}(χ_{r,ϕ}) = \begin{cases}−D_{r}(I_{ϕ}^{∗})V (I_{ϕ}^{∗}, I^{∗}) & if M ≤ ker(ϕ) \\ 0 & otherwise. \end{cases}

Proof. Since χ_{r,ϕ} was defined as a corestriction, we use the Mackey formula to determine its restriction to M. Both M and H are normal in P, and corestriction to M from any proper subgroup is the zero map, at least in positive degree. This proves the result when M is not contained in H. So we may now assume that M ≤ H.

As in the definition of χ_{r,ϕ}, choose a cyclic subgroup A of Z_{H} such that Z_{H} = A × Z. Then M/A is a maximal elementary abelian subgroup of H/A ≅ P_{n−1}. Note that (M/A)^{∗} is a hyperplane in M^{∗}, and itself contains I_{ϕ}^{∗} as a hyperplane: namely, the annihilator of Z. Then

(4.5)  Res_{M}(χ_{r,ϕ}) = ∑_{g∈P_{n}/H} g^{∗} Res_{M}^{H} Inf_{P_{n−1}}^{H} κ_{n−1,r}ζ_{n−1}^{p−1}

= ∑_{g∈P_{n}/H} D_{r}(I_{ϕ}^{∗})V (I_{ϕ}^{∗}, g^{∗}(M/A)^{∗}) . (4.6)

Now, there were p possible choices for A, each of which yields a different (M/A)^{∗}. The possible (M/A)^{∗} are exactly those hyperplanes in M^{∗} which contain I_{ϕ}^{∗} but are not equal to I^{∗}. There are permuted faithfully and transitively by P_{n}/H. The result therefore follows from Eqn. (4.6) by Lemma 4.3.

5. Describing Chern classes in terms of corestrictions

The Chern classes κ_{n,r} restrict to Z as zero. A theorem of Carlson [Eve91, §10.2] says that any such class has some power which is a sum of corestrictions from proper subgroups. In this section we shall derive a formula for κ_{n,r} in terms of the corestrictions χ_{r,ϕ} and the image of inflation from E. First however, we recall a well-known fact about Dickson invariants.

Lemma 5.1. Let V be an 𝔽_{p}-vector space, and U a hyperplane in V. Then

V (V; X) = V (U; X)^{p} − V (U; X)V (U, V) , (5.1)

and so, for 0 ≤ r ≤ n − 1,

D_{r}(V) = D_{r−1}(U)^{p} + D_{r}(U)V (U, V) . (5.2)

Here, recall that D_{−1}(U) is zero.

Let M_{0} be a maximal elementary abelian subgroup of P, and denote by J_{0} the quotient P/M_{0}, itself elementary abelian of rank n. Then J_{0}^{∗} is the annihilator in E^{∗} of M_{0}. In addition, we may view J_{0}^{∗} as a subspace of H^{2}(E, ℤ).

Theorem 5.2. For all 0 ≤ r ≤ n − 1, the degree 2(p^{n} − p^{r}) class ν_{r,M_{0}} in H^{∗}(P, ℤ) defined by

ν_{r,M_{0}} = κ_{n,r} − Inf_{E}^{P} D_{r}(J_{0}^{∗}) + ∑_{ϕ∈ℙJ_{0}^{∗}} χ_{r,ϕ} (5.3)

is nilpotent.

Proof. By a theorem of Quillen [Eve91, Cor. 8.3.4], a class in the mod-p cohomology ring of a finite group is nilpotent if and only if the restriction to every elementary abelian p-subgroup is. Now, a class in integral cohomology reduces to zero in modp cohomology only if it is in the ideal generated by p. But in positive degree, all such classes are nilpotent. Hence it suffices to prove that the restriction to every elementary abelian subgroup is zero.

Let M be a maximal elementary abelian subgroup of P. As before, define I = M/Z, and I_{ϕ} = M/(Z(ker ϕ)) for ϕ ∈ ℙE^{∗}. Define S to be the annihilator in J_{0}^{∗} of M, recalling that J_{0}^{∗} may be viewed as a space of homomorphisms P → 𝔽_{p}. Then S = {ϕ ∈ E^{∗} | M, M_{0} ∈ ker(ϕ)}.

If S = {0}, then Res_{M}(χ_{r,ϕ}) = 0 for every ϕ ∈ J_{0}^{∗}, and both κ_{n,r} and Inf_{E}^{P} D_{r}(J_{0}^{∗}) restrict to M as D_{r}(I^{∗}). So Res_{M}(ν_{r,M_{0}}) = 0 as required.

We may therefore assume that the 𝔽_{p}-vector space S has positive dimension s, which implies that the number of elements of the projective space ℙS is congruent to one modulo p. Denote by R the subspace Res_{M} J_{0}^{∗} of I^{∗}, and observe that S is the kernel of the quotient map J_{0}^{∗} → R. Now, S contains by Proposition 4.4 the set of those ϕ ∈ J_{0}^{∗} such that Res_{M}(χ_{r,ϕ}) is nonzero. Hence

(5.4)  Res_{M} ( ∑_{ϕ∈ℙJ_{0}^{∗}} χ_{r,ϕ}) = − ∑_{ϕ∈ℙS} D_{r}(I_{ϕ}^{∗})V (I_{ϕ}^{∗}, I^{∗}) by Proposition 4.4

= ∑_{ϕ∈ℙS}(D_{r−1}(I_{ϕ}^{∗})^{p} − D_{r}(I^{∗})) by Lemma 5.1 (5.5)

= D_{r−s}(R)^{p^{s}} − D_{r}(I^{∗}) by Lemma 1.1 (5.6)

= Res_{M}(Inf_{E}^{P} D_{r}(J_{0}^{∗}) − κ_{n,r}) . (5.7)

Therefore Res_{M}(ν_{r,M_{0}}) = 0, as desired.

6. Taking p^{th} powers of Chern classes

In the next section we shall show that the p^{th} powers of the new classes χ_{r,ϕ} lie in the image of inflation from E, at least modulo nilradical. In preparation for this, we shall here prove the same result for the p^{th} powers of the Chern classes κ_{n,r}.

Recall that E carries a nondegenerate symplectic form, say (,). Pick a symplectic basis A_{1}, …, A_{n}, B_{1}, …, B_{n} for E: so A_{i} ⊥ A_{j}, B_{i} ⊥ B_{j} and (A_{i}, B_{j}) = δ_{ij}. Let α_{1}, …, α_{n}, β_{1}, …, β_{n} be the corresponding dual basis for E^{∗}, which we recall may be identified with H^{2}(E, ℤ).

Definition 6.1. For r ≥ 1, define z_{n}^{(r)} in S(E^{∗}) by

z_{n}^{(i)} = ∑_{i=1}^{n}(α_{i}β_{i}^{p^{r}} − α_{i}^{p^{r}}β_{i}) . (6.1)

By the work of Carlisle and Kropholler on symplectic invariants (see [Ben93, §8.3]), the z_{n}^{(i)} are invariant under symplectic transformations of E. In particular, this means that they do not depend upon the choice of symplectic basis. Tezuka and Yagita proved that, at least for p odd, z_{n}^{(1)}, …, z_{n}^{(n)} is a regular sequence in S(E^{∗}); and that the ideal they generate contains every z_{n}^{(i)} and is the intersection with S(E^{∗}) of the kernel of inflation from H^{∗}(E, ℤ) to H^{∗}(P, ℤ). (See [BC92, Prop. 8.2 and §10].)

Lemma 6.2. Let x_{1}, …, x_{m} be a regular sequence in a commutative ring R, and suppose that elements a_{1}, …, a_{m} of R satisfy

∑_{i=1}^{m} a_{i}x_{i} = 0 . (6.2)

Then each a_{i} lies in the ideal generated by x_{1}, …, x̂_{i}, …, x_{m}.

Proof. We note that the case m = 1 is trivial, and proceed by induction on m. The product a_{m}x_{m} lies in the ideal generated by x_{1}, …, x_{m−1}, and therefore so does a_{m} by regularity: say a_{m} = ∑_{i=1}^{m−1} b_{i}x_{i}. Defining a′_{i} = a_{i} + b_{i}x_{m} for 1 ≤ i ≤ m − 1, we have ∑_{i=1}^{m−1} a′_{i}x_{i} = 0. Thus we have reduced to the case of m − 1, and the result follows by induction.

Corollary 6.3. Suppose that f ∈ S(E^{∗}) lies in the kernel of Inf_{E}^{P}. Then there exist elements f_{1}, …, f_{n} of S(E^{∗}) such that f = ∑_{i=1}^{n} f_{i}z_{n}^{(i)}, and the images Inf_{E}^{P}(f_{i}) of the f_{i} under inflation depend only on f, not on the choice of the f_{i}.

Proposition 6.4. There are unique classes g_{0}, …, g_{n−1} in H^{∗}(P, ℤ) such that there exist f_{0}, …, f_{n−1} in S(E^{∗}) which satisfy both g_{i} = Inf_{E}^{P}(f_{i}) and

z_{n}^{(n+1)} + ∑_{i=0}^{n−1}(−1)^{n−i}z_{n}^{(i+1)}f_{i} = 0 . (6.3)

Each g_{i} has degree 2(p^{n+1} − p^{i+1}), and κ_{n,r}^{p} − g_{r} is nilpotent for all 0 ≤ r ≤ n − 1.

Proof. Existence and uniqueness of the g_{i} follows from Corollary 6.3. We once again demonstrate nilpotence by proving that restriction to every maximal elementary abelian subgroup M is zero.

Now, each z_{n}^{(j)} is a symplectic invariant. Hence, for any symplectic transformation σ of E, Corollary 6.3 says that Eqn. (6.3) still holds if each f_{i} is replaced by σ^{∗}(f_{i}). Consequently, σ^{∗} fixes each g_{i}. As any linear transformation of a maximal totally isotropic subspace I may be extended to a symplectic transformation of E, it follows that each Res_{M}(g_{i}) is a polynomial in the Dickson invariants for I^{∗}. Since the symplectic transformations of E permute the I transitively, Res_{M}(g_{i}) is the same polynomial for each M. Finally, we see by comparing degrees that for each i there is a scalar λ_{i} ∈ 𝔽_{p} such that Res_{M}(g_{i}) = λ_{i}D_{i}(I^{∗})^{p} for all M.

To establish the result from here, we have to prove that each λ_{i} is 1. First we shall prove that

ϕ^{p^{n+1}} + ∑_{i=0}^{n−1}(−1)^{n−i}ϕ^{p^{i+1}}g_{i} = 0 for all ϕ ∈ E^{∗} ⊆ H^{2}(P, ℤ). (6.4)

Since the g_{i} are invariants for the symplectic group, it suffices to prove this for one nonzero ϕ, say β_{1}. Recall that the inflation of z_{n}^{(j)} is zero, and observe that differentiating z_{n}^{(j)} with respect to α_{1} yields β_{1}^{p^{j}}. Consequently, differentiating both sides of Eqn. (6.3) with respect to α_{1} and then inflating yields Eqn. (6.4) with ϕ replaced by β_{1}. This establishes Eqn. (6.4).

Now restrict to any one maximal elementary abelian M ≤ P and take p^{th} roots to obtain

ϕ^{p^{n}} + ∑_{i=0}^{n−1}(−1)^{n−i}ϕ^{p^{i}}λ_{i}D_{i}(I^{∗})^{p} = 0 for all ϕ ∈ I^{∗}. (6.5)

This equation also holds for all ϕ ∈ I^{∗} with each λ_{i} replaced by 1. Therefore each λ_{i} must be 1, or else taking the difference of the two left hand sides would yield a polynomial of degree less than p^{n}, with too many roots in an integral domain.

7. Taking p^{th} powers of the new classes

There are h_{1}, …, h_{n−1} in S(E_{n−1}^{∗}) such that z_{n−1}^{(n)} = ∑_{i=1}^{n−1} h_{i}z_{n−1}^{(i)}, by Corollary 6.3. Now, S(E_{n−1}^{∗}) embeds in S(E_{n}^{∗}), and z_{n}^{(i)} = z_{n−1}^{(i)} + z′_{1}^{(i)}. Here z′_{1}^{(i)} signifies z_{1}^{(i)} as a function of α_{n}, β_{n}, that is α_{n}β_{n}^{p^{i}} − α_{n}^{p^{i}}β_{n}. So we have

z_{n}^{(n)} = z′_{1}^{(n)} − ∑_{i=1}^{n−1} h_{i}z′_{1}^{(i)} + ∑_{i=1}^{n−1} h_{i}z_{n}^{(i)} . (7.1)

As z′_{1}^{(i)} is divisible by β_{n}, there is a unique η ∈ S(E^{∗}) such that

z_{n}^{(n)} = β_{n}η + ∑_{i=1}^{n−1} h_{i}z_{n}^{(i)} . (7.2)

We may consider η to be an element of H^{2p^{n}}(E, ℤ).

Lemma 7.1. For every ϕ ∈ ℙE^{∗}, the cohomology class χ_{n−1,ϕ}^{p} lies in the image of Inf_{E}^{P} modulo nilpotent elements.

Proof. As the symplectic group permutes ℙE^{∗} transitively, it is enough to prove the lemma for one ϕ. We shall show that χ_{n−1,β_{n}}^{p} − Inf_{E}^{P}(η) is nilpotent. Write χ for χ_{n−1,β_{n}}.

Once more, we prove that the restriction of χ^{p} − Inf_{E}^{P}(η) to every maximal elementary abelian M ≤ P is zero. The restriction of χ^{p} to each M is known by Proposition 4.4. If the image of β_{n} in I^{∗} is nonzero, then it follows from Eqn. (7.2) that Res_{M} Inf^{P} (η) is zero. Henceforth we may assume that M lies in the kernel of β_{n}, which is to say that M contains A_{n} and lies in E_{n−1}×⟨A_{n}⟩. Then Res_{M} χ^{p} = −V (I_{β_{n}}^{∗}; α_{n})^{p} by Proposition 4.4.

Differentiating Eqn. (7.2) with respect to β_{n}, then inflating to P and finally restricting to M, we have

(7.3) −α_{n}^{p^{n}} = Res_{M} Inf^{P} (η) − ∑_{i=1}^{n−1} Res_{M} Inf^{P} (h_{i})α_{n}^{p^{i}} .

By Proposition 6.4 (for P_{n−1}) and Eqn. (1.1), the choice of the h_{i} implies that

α_{n}^{p^{n}} − ∑_{i=1}^{n−1} Res_{M} Inf^{E}(h_{i})α_{n}^{p^{i}} = V (I_{β_{n}}^{∗}; α_{n})^{p} , (7.4)

whence the result follows.

Theorem 7.2. For all 0 ≤ r ≤ n − 1 and all ϕ ∈ ℙE^{∗}, the class χ_{r,ϕ}^{p} lies in Im(Inf_{E}^{P}) + \sqrt{0}.

Proof. By a result of Evens [Eve68], Steenrod operations commute with corestriction. Therefore, at least in mod-p cohomology, taking p^{th} powers also commutes with corestriction. Hence Lemma 7.1 and Proposition 6.4 (applied to P_{n−1}) yield the result.

8. Linear independence of corestrictions

We have been looking at three subrings of the cohomology of the extraspecial pgroups. Firstly the subring generated by top Chern classes, then the Chern subring, and then the subring generated by top Chern classes and the corestrictions χ_{r,ϕ}. In [Gre96], it is shown that top Chern classes generate a strictly smaller subring than the Chern subring, even modulo nilradical. Theorem 5.2 demonstrates that (at least modulo nilradical) the Chern subring is contained in the subring generated by top Chern classes and the χ_{r,ϕ}. This containment is now shown to be strict.

Proposition 8.1. Let E′′ be a nondegenerate, codimension 2 subspace of E. Exactly p + 1 elements ϕ ∈ ℙE^{∗} satisfy E′′ ⊆ ker(ϕ). When n = 2, no non-trivial 𝔽_{p}-linear combination of these χ_{n−1,ϕ} lies in Im(Inf_{E}^{P}) + \sqrt{0}.

Corollary 8.2. Assume n = 2. Modulo nilradical, the Chern subring is strictly contained in the subring generated by top Chern classes and the χ_{r,ϕ}.

Proof of Corollary. Modulo nilradical, the degree 2(p^{2} − p) part of the Chern subring consists of the image of inflation together with one extra class, κ_{1}.

Let E′ be the orthogonal complement of E′′. Pick a symplectic basis for E such that E′ has basis A_{1}, B_{1} and E′′ has basis A_{2}, B_{2}. Note that each maximal isotropic subspace I of E satisfies either I = (I ∩ E′) ⊕ (I ∩ E′′) or (I ∩ E′) = (I ∩ E′′) = 0.

Lemma 8.3. Consider the x ∈ S(E^{∗}) whose restriction to S(I^{∗}) is zero for every maximal totally isotropic I ⊆ E satisfying (I ∩ E′) = (I ∩ E′′) = 0. These x form the ideal generated by z_{2}^{(1)} and γ_{2} = D_{1}(E′^{∗}) − D_{1}(E′′^{∗}).

Proof. Recall that z_{2}^{(1)}, z_{2}^{(2)} form a regular sequence in S(E^{∗}), and generate the joint kernel of restriction to every S(I^{∗}). Observe that (α_{1}β_{1}^{p} − α_{1}^{p}β_{1})γ_{2} ∈ z_{2}^{(2)} + (z_{2}^{(1)}). Moreover, γ_{2} has zero restriction to S(I^{∗}) if and only if I satisfies (I ∩ E′) = (I ∩ E′′) = 0, whereas (α_{1}β_{1}^{p} − α_{1}^{p}β_{1}) restricts to zero if and only if I satisfies I = (I ∩ E′) ⊕ (I ∩ E′′). So the ideal in question is the ideal of x such that (α_{1}β_{1}^{p} − α_{1}^{p}β_{1})x has zero restriction to every S(I^{∗}).

Proof of Proposition 8.1. Let ϕ ∈ ℙE^{∗} have E′′ ⊆ ker(ϕ), and let H be the corresponding maximal subgroup of P. For any maximal elementary abelian M ≤ H, have that I = M/Z satisfies I = (I ∩ E′) ⊕ (I ∩ E′′). By Proposition 4.4 it follows that Res_{M}(χ_{1,ϕ}) = 0 for all such ϕ and all M satisfying (I ∩ E′) = (I ∩ E′′) = 0. Invoking Lemma 8.3 and comparing degrees, the only element of Im(Inf_{E}^{P}) that comes into consideration is γ_{2} itself.

For no two of these ϕ do the corresponding maximal subgroups H share a common maximal elementary abelian subgroup. Applying Proposition 4.4 again, these χ_{1,ϕ} are therefore 𝔽_{p}-linearly independent. Moreover, no nonzero restriction of a χ_{1,ϕ} is a scalar multiple of the restriction of γ_{2}.

Remark 8.4. Proposition 8.1 and Corollary 8.2 can be proved in the same way for general n ≥ 2, using the results of [Gre96].

9. Chern classes and transfer not sufficient

Calculating cohomology rings of p-groups would be easier if there were a list of constructions, such as transfer and Chern classes, which together always yielded a set of generators for the cohomology ring. Being able to construct the cohomology ring modulo nilradical would be an important first step. This gave rise to the socalled \overline{Ch}-conjecture, related to a construction of Moselle [Mos89]. We provide a counterexample to that conjecture.

Definition 9.1. For a finite group G and a prime p, the subring \overline{Ch}(G) of H^{∗}(G, 𝔽_{p}) is defined recursively as the subring generated by Chern classes for G, together with the images under corestriction of \overline{Ch}(H) for proper subgroups H of G.

Remark 9.2. For p = 2, one should use Stiefel–Whitney rather than Chern classes to get the largest possible subring.

Remark 9.3. Inclusion of \overline{Ch}(G) in H^{∗}(G, 𝔽_{p}) always induces an isomorphism of varieties: see [GL96].

Proposition 9.4. Let G be the extraspecial 3-group of order 27 and exponent 3. Then \overline{Ch}(G)/\sqrt{0} is strictly contained in H^{∗}(G, 𝔽_{p})/\sqrt{0}.

Proof. Take a symplectic basis A, B for E = E_{1}. We may view A, B as elements of P = P_{1}. We shall consider restrictions to the four cyclic subgroups generated by A, AB, AB^{2}, B respectively. For each of these four cyclic groups K, denote by ξ the class in H^{1}(K, 𝔽_{p}) ≅ Hom(K, 𝔽_{p}) taking the generator to 1. Let x ∈ H^{2} be the Bockstein of ξ.

For p-groups H_{1} < H_{2}, recall that Cor_{H_{1}}^{H_{2}} Res_{H_{1}}^{H_{2}} is zero in mod-p cohomology. In particular, Cor_{H_{1}}^{H_{2}} is zero whenever H_{2} is elementary abelian. Using the Mackey formula, it follows that Res_{K}^{P} Cor_{H}^{P} is zero for any H < P and for K any of the four cyclic subgroups.

Let α, β be the dual basis for E^{∗}. These may be viewed as elements of H^{1}(P, 𝔽_{p}). Let a, b ∈ H^{2} be their Bocksteins. Then a, b are also the first Chern classes of the corresponding representations, and all first Chern classes are in their span. Note that the restrictions of a to the four cyclics are x, x, x, 0 respectively, whereas the restrictions of b are 0, x, −x, x respectively.

We refer the reader to [Lea92] for the fact that αβ = 0 in H^{2}(P, 𝔽_{p}), and for an account of Massey triple products in this context. Define Y to be the Massey product ⟨α, α, β⟩ ∈ H^{2}(P, 𝔽_{p}). By naturality, this has zero restriction to the cyclics generated by A, B, whereas its restriction to the subgroup generated by AB is ⟨ξ, ξ, ξ⟩. This is −x. Hence Y lies outside \overline{Ch}(P), even modulo nilradical.

Remark 9.5. We conclude from the proof of Proposition 9.4 that any list of constructions that was always guaranteed to provide enough generators modulo nilradical for mod-p cohomology would have to include Massey products.

Remark 9.6. Strictly, Proposition 9.4 is only a counterexample to the mod-p version of the \overline{Ch}-conjecture. One may however define in the same way a subring \overline{Ch}(G) of H^{∗}(G, ℤ), and it is an open question whether containment modulo nilradical is strict in this case.

References

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[BC92] D. J. Benson and J. F. Carlson. The cohomology of extraspecial groups. Bull. London Math. Soc. 24 (1992), 209–235. Erratum: Bull. London Math. Soc. 25 (1993), 498.

[Eve68] L. Evens. Steenrod operations and transfer. Proc. Amer. Math. Soc. 19 (1968), 1387– 1388.

[Eve91] L. Evens. The Cohomology of Groups. (Oxford Univ. Press, 1991).

[Gre96] D. J. Green. “Chern classes and extraspecial groups of order p^{5}.” Submitted for publication.

[GL96] D. J. Green and I. J. Leary. “The spectrum of the Chern subring.” Submitted for publication.

[Lea92] I. J. Leary. The mod-p cohomology rings of some p-groups. Math. Proc. Cambridge Philos. Soc. 112 (1992), 63–75.

[Mos89] B. Moselle. “Calculations in the cohomology of finite groups.” Unpublished essay. Univ. of Cambridge, 1989.

[Min95] P. A. Minh. Transfer map and Hochschild–Serre spectral sequences. J. Pure Appl. Algebra 104 (1995), 89–95.

[Mui75] H. Mui. Modular invariant theory and cohomology algebras of symmetric groups. J. Fac. Sci. Univ. Tokyo Sect. 1A Math. 22 (1975), 316–369.

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Inst. f. Exp. Math., Ellernstr. 29, D–45326 Essen, Germany

   E-mail address: david@exp-math.uni-essen.de

Fachbereich 6 Mathematik, Universität Essen, D–45117 Essen, Germany

Current address: Department of Mathematics, University of Hue, Dai hoc Tong hop Hue, Hue, Vietnam