A going-down theorem for real spectra

Abstract

In this note, the author gives a proof of the following going-down theorem: If ψ:A→B is a regular map between Noetherian rings with A excellent, any chain of specialization in Spec r A comes from a chain of the same length in Spec r B with corresponding points of the same height. This theorem, trivially well known for Zariski spectra, has been originally proved by M. Coste and M. F. Coste-Roy for real spectra [in Ordered fields and real algebraic geometry (San Francisco, CA, 1981), 27–59, Contemp. Math., 8, Amer. Math. Soc., Providence, RI, 1982] with the extra assumption that ψ is finitely presented. The motivation for dropping that hypothesis is to be able to work in formal and analytic algebras, in which context the author has already contributed to make real algebra and geometry work pretty well. The counterexample of Coste and Coste-Roy to the real going-down with no assumption on ψ was simply the projection on the x-axis of the parabola x=y 2. The second part of the paper under review is devoted to the study of this example as the dimension increases, in the framework of convergent power series: If ψ is the map R{t}→R{x 1 ,⋯,x n } given by ψ(t)=∑x 2 i, the author shows that ψ has more and more regularity properties as n increases, without being regular (it clearly does not satisfy real going-down). More precisely, the generic fiber of ψ is geometrically regular and connected when n≥2, and the special fiber is a geometrically isolated singularity of codimension 1, multiplicity 2, normal for n≥3 and factorial for n≥5.