Valuations and henselization

Abstract

We study the extension of valuations centered in a local domain to its henselization. We prove that a valuation $\nu$ centered in a local domain R uniquely determines a minimal prime $H(\nu )$ of the henselization $R^h$ of R and an extension of $\nu$ centered in $R^h/H(\nu )$, which has the same value group as $\nu$. Using the integrality and functoriality of henselization, this is equivalent to the fact that the henselization of a valuation ring is a valuation ring with the same value group, which is a fundamental result in the theory of valued fields. We present here a more constructive approach to this result and some consequences of this approach. Our method, which assumes neither that R is noetherian nor that it is integrally closed, is to reduce the problem to the extension of the valuation to a quotient of a standard étale local R-algebra and in that situation to draw valuative consequences from the observation that the Newton–Hensel algorithm for constructing roots of polynomials produces sequences that are, for any valuation centered in R, pseudo–convergent in the sense of Ostrowski. We then apply this method to the study of the approximation of elements of the henselization of a valued field by elements of the field and give a characterization of the henselian property of a local domain $(R,m_R)$ in terms of the limits of certain pseudo–convergent sequences of elements of $m_R$ for a valuation centered in it. Another consequence of our work is to establish in full generality a bijective correspondence between the minimal primes of the henselization of a local domain R and the connected components of the Riemann–Zariski space of valuations centered in R.