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On the Cohomology of Galois Groups Determined by Witt Rings

✍ Scribed by Alejandro Adem; Dikran B Karagueuzian; Ján Mináč

Publisher
Elsevier Science
Year
1999
Tongue
English
Weight
494 KB
Volume
148
Category
Article
ISSN
0001-8708

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✦ Synopsis

Let F denote a field of characteristic different from two. In this paper we describe the mod 2 cohomology of a Galois group G 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*(G F , 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 X F endowed with an action of an elementary abelian group E such that the computation of the cohomology of G F reduces to calculating the equivariant cohomology H* E (X F , F 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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