On the Cohomology of Galois Groups Determined by Witt Rings

Department of Mathematics, University of Wisconsin, Madison, Wisconsin, 53706, f1E-mail: {adem,dikran}@math.wisc.eduf1; Department of Mathematics, University of Western Ontario, London, Ontario, Canada, N6A 5B7f2E-mail: minac@uwo.caf2
Advances in Mathematics 08/1998; DOI: 10.1006/aima.1999.1847

ABSTRACT Let F denote a field of characteristic different from two. In this paper we describe the mod 2 cohomology of a Galois group F (called the W-group of F) which is known to essentially characterize the Witt ring WF of anisotropic quadratic modules over F. We show that H*(F, 2) contains the mod 2 Galois cohomology of F and that its structure will reflect important properties of the field. We construct a space XF endowed with an action of an elementary abelian group E such that the computation of the cohomology of F reduces to calculating the equivariant cohomology H*E(XF, 2). For the case of a field which is not formally real this amounts to computing the cohomology of an explicit Euclidean space form, an object which is interesting in its own right. We provide a number of examples and a substantial combinatorial computation for the cohomology of the universal W-groups.

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