Specialization chains of real valuation rings

Abstract

The present paper is a continuation of the study of real places in function fields initiated by the author in [ 11. Indeed we prove here the existence of specialization chains of real valuation rings with prescribed centers, ranks, dimensions, and value groups, solving completely a problem answered partially in [2]. A similar result has been obtained by Brocker and Schiilting [4] by means of Hironaka’s desingularization theorem. Here the proof is based on Noether’s normalization theorem and is quite elementary. There are (at least) two reasons which make this elementary proof worthwhile: first, historically, valuation theory was the starting point for desingularization; second, in real algebraic geometry, and in the theory of real spectra (which I use in this paper), real valuation rings are basic tools (for instance, for dimension theory or in the definition of the sheaf of semialgebraic functions). Thus it is nice to have at hand a strong result concerning the existence of real valuation rings with the least possible effort. Also, our approach is based on selecting curves, which once more coincides with a classical point of view of valuation theory. As pointed out above, I use the language of real spectra. This makes some statements slightly different from those of [l]. Also, Corollary 2.5 is more general than Theorem 5.1 of [ 11. However, we refer to [l] for some proofs which remain valid almost word for word. 1. Let