Valuations and henselization

We study the extension of valuations centered in a local domain to its henselization. We prove that a valuation. centered in a local domain R uniquely determines a minimal prime H(.) of the henselization Rh of R and an extension of. centered in Rh/H(.), which has the same value group as.. 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 etale 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 ofOstrowski. We then apply thismethod 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, mR) in terms of the limits of certain pseudo-convergent sequences of elements of mR 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 RiemannZariski space of valuations centered in R.