A dimension theorem for real spectra

Abstract

In this note, the author proves, in the context of excellent rings, two results on chains of specializations in the real spectrum and some corollaries about real dimension. The first result (Theorem I) is the following: let α0 be a point of SpecrA of dimension d0 in the closure of a constructible set S of real dimension d at α0; then there exists a chain of strict specializations of exactly d points αd→⋯→α0 entirely contained in S, specializing to α0. This theorem is in fact a full generalization, in that context, of the "curve selection lemma'' (which corresponds to the existence of α1 in the case d0=1). The second result (2.1) says that if A is an excellent local domain, any ordering of the fraction field inducing an ordering in the residue field extends to a "formal branch'' (that is, an irreducible component of the completion Â). Both results are simultaneously proved by induction on the Krull dimension of A [see also the author, C. R. Acad. Sci. Paris Sér. I Math. 302 (1986), no. 2, 67–69].