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

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

2010-07-27

arxiv.org/abs/1007.4658v2

Mathematics

Algebraic Geometry

Accepted for publication in the Proceedings of the London Mathematical Society. 54 pages

Scientific paper

10.1112/plms/pds002

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.

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