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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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The graded Witt ring and Galois cohomology. II
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by J贸n Kr. Arason, Richard Elman and Bill Jacob PDF
Trans. Amer. Math. Soc. 314 (1989), 745-780 Request permission

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 鈥檆lose鈥 to $WF$ or $H_q^{\ast } K$ is 鈥檆lose鈥 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
  • © Copyright 1989 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 314 (1989), 745-780
  • MSC: Primary 11E04; Secondary 11E81, 12G05, 18F25, 19G12
  • DOI: https://doi.org/10.1090/S0002-9947-1989-0964897-9
  • MathSciNet review: 964897