Extending valuations of local domains to complete local domains without changing the value group

Abstract

Let $(R,m,k)$ be an excellent local noetherian domain with field of fractions $K$. Let $$ \nu:K^*\twoheadrightarrow\Gamma $$ be a valuation centered at $R$ and let $R_\nu$ be the corresponding valuation ring of $K$, dominating $R$. Denote by $\widehat R$ 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 $\widehat R$ and $\nu$ by a suitable extension $\widehat\nu_-$ to $\frac{\widehat R}P$ for a suitably chosen prime ideal $P$, such that $$ P\cap R=(0). $$ In a previous article we gave a systematic description of all such extensions $\widehat\nu_-$ and defined the notion of tight extensions that are of particular interest for applications (see Herrera, Olalla, Spivakovsky and Teissier, Extending a valuation centered in a local domain to its formal completion, Proc. London Math. Soc. (3) 105 (2012) 571--621). If $\widehat\nu_-$ is a tight extension then its graded algebra is birational to that of $\nu$ (the converse is not known and might not be true). In particular, the value group of $\widehat\nu_-$ is $\Gamma$. The existence of tight extensions was conjectured by the last author (see Teissier, Valuations, deformations, and toric geometry, Fields Institute Communications, 33, 2003, 361-459). In the present paper we give a proof of Teissier's conjecture. An intended application of this result is an important step in two recent approaches to local uniformization in positive characteristic.