Rings with approximation property
type: Item
instance of: scholarly article
title: Rings with approximation property
zbMATH Open document ID: 0702.13007
DOI: 10.1007/BF01446905
author: C. Rotthaus
published in: Mathematische Annalen
publication date: 1990-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164697
review text: Let A be a noetherian semilocal ring which satisfies the Artin approximation property. It is well known that A is henselian, universally catenary and that the formal fibres of A are geometrically normal. The present paper proves that the formal fibres of A are geometrically regular; and hence, that A is an excellent ring. [The converse, that every henselian excellent semilocal ring which contains the field of rational numbers has the Artin approximation property, has already been proved by the author in Invent. Math. 88, 39-63 (1987; Zbl 0614.13014).] The idea of the proof is to find ``enough'' equations for describing the singularity of the localization \(\hat A_ P\) where \(\hat A\) is the completion of A and P is a prime ideal in the singular locus of A. The main tool in the proof is the theorem (due to the author and Artin) that if R is an excellent discrete valuation ring, then the formal power series ring \(RX_ 1,...,X_ n\) is a direct limit of smooth R- algebras.
Mathematics Subject Classification ID: 13B40
Mathematics Subject Classification ID: 14B12
Mathematics Subject Classification ID: 13J15
zbMATH DE Number: 4150348
zbMATH Keywords: henselian ring
zbMATH Keywords: Néron desingularization
zbMATH Keywords: Artin approximation property
zbMATH Keywords: excellent ring
zbMATH Keywords: singularity of the localization
zbMATH Keywords: formal power series ring
reviewed by: Andrew Kustin
MaRDI profile type: Publication
cites work: Algebraic approximation of structures over complete local rings
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cites work: On the approximation property of excellent rings
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description: scientific article; zbMATH DE number 4150348