Extending a valuation centered in a local domain to the formal completion
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.
Cite
@article{arxiv.1007.4658,
title = {Extending a valuation centered in a local domain to the formal completion},
author = {F. J. Herrera Govantes and M. A. Olalla Acosta and M. Spivakovsky and B. Teissier},
journal= {arXiv preprint arXiv:1007.4658},
year = {2012}
}
Comments
Accepted for publication in the Proceedings of the London Mathematical Society. 54 pages