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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 Witt ring of a space of orderings
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by Murray Marshall PDF
Trans. Amer. Math. Soc. 258 (1980), 505-521 Request permission

Abstract:

The theory of “space of orderings” generalizes the reduced theory of quadratic forms over fields (or, more generally, over semilocal rings). The category of spaces of orderings is equivalent to a certain category of “abstract Witt rings". In the particular case of the space of orderings of a formally real field K, the corresponding abstract Witt ring is just the reduced Witt ring of K. In this paper it is proved that if $X = (X, G)$ is any space of orderings with Witt ring W(X), and $X \to Z$ is any continuous function, then g is represented by an element of W(X) if and only if ${\Sigma _{\sigma \in V}}g(\sigma ) \equiv 0 \bmod \left | V \right |$ holds for all finite fans $V \subseteq X$. This generalizes a recent field theoretic result of Becker and Bröcker. Following the proof of this, applications are given to the computation of the stability index of X, and to the representation of continuous functions $g: X \to \pm 1$ by elements of G.
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Additional Information
  • © Copyright 1980 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 258 (1980), 505-521
  • MSC: Primary 10C05; Secondary 12D15
  • DOI: https://doi.org/10.1090/S0002-9947-1980-0558187-8
  • MathSciNet review: 558187