Spaces of Orderings IV

Abstract

A major goal of this paper is to give a proof of the following isotropy criterion: Let X = (X,G) be a space of orderings in the terminology of [9] or [10], and let f be a form defined over G.Then f is anisotropic over X if and only if f is anisotropic over some finite subspace of X.This is the content of Theorem 1.4, and generalizes [1, Corollary 3.4]. Moreover, in view of the known structure of finite spaces (see [9]), this has, essentially, the strength of [2, Satz 3.9] or [12, Theorem 8.12]. The technique used to prove this criterion is roughly patterned on that of [6], and yields some interesting by-products: An interesting invariant of a space of orderings called the chain length is introduced (Definition 1.1) and spaces of orderings with finite chain length are classified (Theorem 1.6).