The Witt ring of a space of orderings

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 ) X\, = \,(X,\,G) is any space of orderings with Witt ring W(X), and X → Z X\, \to \,Z is any continuous function, then g is represented by an element of W(X) if and only if Σ σ ∈ V g ( σ ) ≡ 0 mod | V | {\Sigma _{\sigma \, \in \,V}}g(\sigma )\, \equiv \,0\,\bmod \,\left | V \right | holds for all finite fans V ⊆ X 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 → ± 1 g:\,X\, \to \, \pm 1 by elements of G.