Direct limits of finite spaces of orderings

Abstract

Denote by Wthe Witt ring of (X, G) and by W{ the Witt ring of (Xi9 Gt). W is the subring of the ring-theoretic inverse limit of the W{ consisting of compatible families with bounded anisotropic dimension. The inverse limit topology on G (resp. W) is the weakest such that the function a: G -• {± 1} (resp. a: W -* Z) is continuous for all a e Xc. Further if a e X, then the following are equivalent : (i) a e Xc, (ii) a : W-+ Z is continuous and (iii) a: G -» {±1} is continuous. Thus Xc depends only on the topology on W(resp. G) and not on the particular of (X, G) as a direct limit of finite spaces. Also, the set y : = {Y: Y is a finite subspace of X and Y c Xc} forms a directed system and (X, G) is the direct limit of this system. This is the standard presentation of (A, G) as a direct limit of finite spaces. The stability index of (X, G) is the supremum of the stability indices of the spaces in the set -V. This is finite iff the following two conditions hold: (i) Xc, with the topology induced from X, is discrete and (ii) X is the Stone-Cech compactification of Xc. If the stability index of (X, G) is finite then a function 0: Xc -> Z with finite image which satisfies the congruence Tta^v^is) = 0 mod \V for all finite fans Ve ^represents an element of W. In the presence of certain finiteness conditions (e.g., finite stability will do) (X, G) is built up canonically from singleton spaces exactly as in the finite case, using sums and group extensions, except that now infinite sums are required. However, not all direct limits of finite spaces of orderings are built up in this way. Since each finite space of orderings is realized as the space of orderings of a Pythagorean field, one can ask if the same is true for arbitrary spaces