The Witt ring of a space of orderings

Author:
Murray Marshall

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

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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 is any space of orderings with Witt ring *W*(*X*), and is any continuous function, then *g* is represented by an element of *W*(*X*) if and only if holds for all finite fans . 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 by elements of *G*.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1980-0558187-8

Keywords:
Orderings,
Boolean space,
reduced Witt ring,
abstract Witt ring

Article copyright:
© Copyright 1980
American Mathematical Society