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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The graded Witt ring and Galois cohomology. II


Authors: Jón Kr. Arason, Richard Elman and Bill Jacob
Journal: Trans. Amer. Math. Soc. 314 (1989), 745-780
MSC: Primary 11E04; Secondary 11E81, 12G05, 18F25, 19G12
MathSciNet review: 964897
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Abstract: A primary problem in the theory of quadratic forms over a field $ F$ of characteristic different from two is to prove that the rings $ H_q^\ast F$ and $ GWF$ are isomorphic. Here $ H_q^\ast F = {H^\ast }(\operatorname{Gal}({F_q}/F),{\mathbf{Z}}/2{\mathbf{Z}}))$, where $ {F_q}$ is the quadratic closure of $ F$, and $ GWF$ is the graded Witt ring associated to the fundamental ideal of even dimensional forms in the Witt ring $ WF$ of $ F$. In this paper, we assume we are given a field extension $ K$ of $ F$ such that $ WK$ is 'close' to $ WF$ or $ H_q^{\ast} K$ is 'close' to $ H_q^\ast F$. A method is developed to obtain information about these graded rings over $ F$ and its $ 2$-extensions from information about the corresponding graded ring of $ K$. This relative theory extends and includes the previously developed absolute case where $ K = {F_q}$. Applications are also given to show that $ H_q^\ast F$ and $ GWF$ are isomorphic for a collection of fields arising naturally from the theory of abstract Witt rings.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1989-0964897-9
PII: S 0002-9947(1989)0964897-9
Article copyright: © Copyright 1989 American Mathematical Society