TRANSFER AND CHERN CLASSES FOR EXTRASPECIAL p-GROUPS

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 χr,φ. They are constructed in a manner foreshadowed in [Min95]. Take a product of Chern classes for the group of order p. By inflation and then corestriction, obtain a cohomology class χr,φ for the group of order p . In Theorem 5.2, an elegant formula is obtained relating the Chern classes and the χr,φ, and in Theorem 7.2 we show that the p power of any Chern class or any χr,φ lies in the subring Date: 3 March 1997. 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. 1 2 D. J. GREEN AND P. A. MINH generated by top Chern classes: this is the subring Tezuka and Yagita used to obtain the spectrum of the cohomology ring. 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 Fp-vector space, and let 0 ≤ r ≤ m − 1. We shall write Dr(V) for the Dickson invariant in degree p − p in S(V), or just Dm,r if V is clear from the context. Recall that, for an indeterminate X , we have the equation (1.1) V (V ;X) = Xpm + m−1