## 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