Valuations and henselization
Abstract
We study the extension of valuations centered in a local domain to its henseliza-tion. We prove that a valuation ν centered in a local domain R uniquely determines a minimal prime H(ν) of the henselization R h of R and an extension of ν centered in R h /H(ν), which has the same value group as ν. 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 always 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.
Related
BibTeX
@article{paramio2026valuations,
title = {{Valuations and henselization}},
author = {Ana Belén de Felipe Paramio and Bernard H Teissier},
journal = {Mathematische Annalen},
year = {2026},
month = oct,
doi = {10.1007/s0028-020-01970-1},
eprint = {1903.10793},
archivePrefix = {arXiv},
url = {https://hal.science/hal-02077611v2},
note = {CC BY-NC 4.0},
}
RIS
TY - JOUR
TI - Valuations and henselization
AU - Ana Belén de Felipe Paramio
AU - Bernard H Teissier
PY - 2026
DA - 2026/10/08
JO - Mathematische Annalen
DO - 10.1007/s0028-020-01970-1
UR - https://hal.science/hal-02077611v2
AB - We study the extension of valuations centered in a local domain to its henseliza-tion. We prove that a valuation ν centered in a local domain R uniquely determines a minimal prime H(ν) of the henselization R h of R and an extension of ν centered in R h /H(ν), which has the same value group as ν. 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 always 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.
SN - 0025-5831
SN - 1432-1807
ER -