Extending a valuation centered in a local domain to the formal completion

TL;DR

This paper systematically describes how valuations centered in a local domain extend to its formal completion, especially under the assumption that the domain is excellent, aiding applications in algebra and singularity theory.

Contribution

It provides a comprehensive classification of valuation extensions to the formal completion of an excellent local domain, highlighting those relevant for applications.

Findings

Systematic description of valuation extensions to formal completions.

Identification of classes of extensions with particular applications.

Assumption that R is excellent is crucial for the results.

Abstract

Let (R; m; k) be a local noetherian domain with field of fractions K and R_v a valuation ring, dominating R (not necessarily birationally). Let v|K be the restriction of v to K; by definition, v|K is centered at R. Let \hat{R} denote the m-adic completion of R. In the applications of valuation theory to commutative algebra and the study of singularities, one is often induced to replace R by its m-adic completion \hat{R} and v by a suitable extension \hat{v} to \hat{R}/P for a suitably chosen prime ideal P, such that P \cap R = (0). The purpose of this paper is to give, assuming that R is excellent, a systematic description of all such extensions \hat{v} and to identify certain classes of extensions which are of particular interest for applications.