Some Additive Galois Cohomology Rings

ABSTRACT

Let p ≥ 3 be a prime. We consider the cyclotomic extension Z _(p)[ζ _{p ²} ] | Z _(p), with Galois group G = (Z/p ²)*. Since this extension is wildly ramified, the Z _(p) G-module Z _(p)[ζ _{p ²} ] is not projective. We calculate its cohomology ring H*(G,Z _(p)[ζ _{p ²} ];Z _(p)), carrying the cup product induced by the ring structure of Z _(p)[ζ _{p ²} ]. Formulated in a somewhat greater generality, our results also apply to certain Lubin-Tate extensions.

Key Words:

• Galois cohomology
• Twisted group rings

AMS Mathematics Subject Classification:

• 11R33
• 16S35

ACKNOWLEDGMENTS

We would like to thank G. Nebe for the method of reduction by blockwise decomposition. We would like to thank B. Keller for the argument that shows that Yoneda product and cup product coincide. We would like to thank several colleagues for comments and suggestions.

The first author would like to thank P. Littelmann for the kind hospitality in Strasbourg, where the first part of this work was done, and where he received support from the EU TMR-network ‘Algebraic Lie Representation', grant no. ERB FMRX-CT97-0100. Another part of the work was financed by a DFG research grant.

Communicated by A. Prestel.