Real Algebra

Summary

We consider general properties of the real spectrum of a commutative ring with unit. In Sections 1 and 2 we collect the basic facts, and for more information we refer to [B-C-R] and [Kn-Schd]. In Section 3 valuation theory enters the scene. It is of fundamental importance for the whole work. As an application we obtain in Section 4 the first results on going-up and going-down in the real spectrum. In Section 5 we present the notion and basic properties of abstract semialgebraic functions on constructible sets of the real spectrum; this is done in two equivalent ways which have both their advantages. These functions are used in Section 6 to construct cylindrical decompositions with respect to systems of polynomials. Finally, in Section 7 we introduce real strict localizations, which are the real analogues of the strict localizations used in etale cohomology.