Inseparable local uniformization
type: Item
instance of: scholarly article
title: Inseparable local uniformization
zbMATH Open document ID: 1276.14021
published in: Journal of Algebra
publication date: 2013-07-02T00:00:00Z
full work available at URL: https://arxiv.org/abs/0804.1554
review text: The paper under review deals with local uniformization of algebraic varieties, which can be considered as the local part of desingularization. The author proves that it always hold after a purely inseparable extension of the function field of the variety. More precisely, let \(X\) be an integral algebraic variety. Then \(X\) may be covered by a finite number of integral and regular varieties \(Y_{1},\dotsc,Y_{n}\) such that the extensions \(k(Y_{i})/k(X)\) are all purely inseparable. The proof goes by induction on the dimension and involves fibrating by curves. It is worth noticing that the proof of the induction step relies on inseparable local uniformization for some points on non-Archimedean \(k\)-analytic curves (type 1 and 4 in Berkovich's terminology; they are called terminal by the author). A decompletion procedure then allows to deduce a uniformization result for certains valuations on a curve over \(k^\circ\). The base case of the induction, which consists in local uniformization for Abhyankar valuations, is proved at the end of the paper. It involves techniques from logarithmic geometry.
reviewed by: Jérôme Poineau
Mathematics Subject Classification ID: 14E15
Mathematics Subject Classification ID: 14G20
Mathematics Subject Classification ID: 14G22
zbMATH DE Number: 6182877
zbMATH Keywords: inseparable local uniformization
zbMATH Keywords: desingularization
zbMATH Keywords: Berkovich spaces
author: Michael Temkin
MaRDI profile type: Publication
OpenAlex ID: W2962697396
arXiv ID: 0804.1554
cites work: Simultaneous Resolution for Algebraic Surfaces
cites work: Q4944999
cites work: Q3998951
cites work: Étale cohomology for non-Archimedean analytic spaces
cites work: Vanishing cycles for formal schemes
cites work: Smooth \(p\)-adic analytic spaces are locally contractible
cites work: Q3325851
cites work: Stable reduction and uniformization of abelian varieties. I
cites work: Formal and rigid geometry. I: Rigid spaces
cites work: Functoriality in resolution of singularities
cites work: Eine bemerkenswerte Eigenschaft der formellen Fasern affinoider Räume
cites work: Q3843687
cites work: Q5459656
cites work: Resolution of singularities of threefolds in positive characteristic. I: Reduction to local uniformization on Artin-Schreier and purely inseparable coverings
cites work: Resolution of singularities of threefolds in positive characteristic. II
cites work: Non-archimedean analytification of algebraic spaces
cites work: Smoothness, semi-stability and alterations
cites work: Schemes over $$ \mathbb{F}_1 $$
cites work: Every tamely ramified form of an open polydisc is trivial
cites work: Q5602063
cites work: Q3438063
cites work: Almost ring theory
cites work: Resolution of singularities of an algebraic variety over a field of characteristic zero. I
cites work: Logarithmic structures of Fontaine-Illusie. II
cites work: Log smooth deformation theory
cites work: Semistable reduction for overconvergentF-isocrystals, IV: local semistable reduction at nonmonomial valuations
cites work: Good formal structures for flat meromorphic connections, II: Excellent schemes
cites work: Abhyankar places admit local uniformization in any characteristic
cites work: Every place admits local uniformization in a finite extension of the function field
cites work: Toroidal embeddings. I
cites work: Elimination of ramification I: The generalized stability theorem
cites work: Fermeture intégrale et changement de base
cites work: Q4935967
cites work: Critères de platitude et de projectivité. Techniques de ''platification'' d'un module. (Criterial of flatness and projectivity. Technics of ''flatification of a module.)
cites work: On local properties of non-Archimedean analytic spaces
cites work: On local properties of non-Archimedean analytic spaces. II
cites work: Stable modification of relative curves
cites work: Desingularization of quasi-excellent schemes in characteristic zero
cites work: Relative Riemann-Zariski spaces
cites work: Functorial desingularization of quasi-excellent schemes in characteristic zero: the nonembedded case
cites work: Murphy's law in algebraic geometry: Badly-behaved deformation spaces
cites work: Homology of schemes
cites work: Reduction of the singularities of algebraic three dimensional varieties
DOI: 10.1016/J.JALGEBRA.2012.09.023
description: scientific article