Ascending the divided and going-down properties by absolute flatness
type: Item
instance of: scholarly article
title: Ascending the divided and going-down properties by absolute flatness
zbMATH Open document ID: 1310.13014
author: Gabriel Picavet
published in: Arabian Journal of Mathematics
publication date: 2012-12-18T00:00:00Z
review text: An \textit{absolutely flat ring morphism} (a notion introduced by J. P. Olivier) is a flat ring morphism \(R\to S\) such that the codiagonal morphism \(S\otimes_R S \to S\) is flat. Examples are (strict) henselizations and étale morphisms. In the paper under review the author is primarily interested in showing that the ``divided ring'' and ``going-down ring'' properties ascend along absolutely flat ring morphisms. The author is motivated by some papers by M. G. Cinquegrani where mainly the ascent by henselization of the ``divided domain'' property is considered, and, as a corollary, by integral extensions. The author extends some of Cinquegrani's results to rings which may not be integral domains and absolutely flat morphisms. He also gives ascent results for the ``going-down ring'' property by using some recent results of \textit{A. Badawi} and \textit{D. E. Dobbs} [Commun. Algebra 29, 2805--2825 (2001; Zbl 1104.13301)]. As a by-product he obtains new results about the ascent of the ``going-down ring'' property by integral extensions. All the results are established under either a unibranchedness hypothesis or at least an irreducibility condition. The consideration of unibranchedness is necessary because, for instance, if \(R^h\) is divided, then \(R\) is unibranched and \(R\to R^h\) is absolutely flat.
reviewed by: Hamid Kulosman
Mathematics Subject Classification ID: 13B21
Mathematics Subject Classification ID: 13G05
Mathematics Subject Classification ID: 13A15
Mathematics Subject Classification ID: 13B40
Mathematics Subject Classification ID: 13C15
Mathematics Subject Classification ID: 14A05
zbMATH DE Number: 6116569
zbMATH Keywords: ``divided ring'' property
zbMATH Keywords: ``going-down ring'' property
zbMATH Keywords: absolutely flat ring morphisms
zbMATH Keywords: ascent by absolutely flat morphisms
zbMATH Keywords: integral extensions
zbMATH Keywords: henselizations
zbMATH Keywords: étale morphisms
MaRDI profile type: Publication
full work available at URL: s40065-012-0008-3
OpenAlex ID: W2000284438
Wikidata QID: Q59270827
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DOI: 10.1007/S40065-012-0008-3
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description: scientific article; zbMATH DE number 6116569