On the approximation property of excellent rings
type: Item
instance of: scholarly article
title: On the approximation property of excellent rings
zbMATH Open document ID: 0614.13014
DOI: 10.1007/BF01405090
author: C. Rotthaus
published in: Inventiones Mathematicae
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143443
review text: Let A be a semilocal noetherian ring, with Jacobson radical I, whose generic formal fibres are geometrically regular. Suppose either that \({\mathbb{Q}}\subseteq A\) or that, for each minimal \(prime\quad g_ i\) of A, the fraction field of \(A/g_ i\) has a separating transcendence basis over a perfect subfield \(P_ i\) such that \(\prod P_ i\subset A.\quad U\sin g\) a theorem of \textit{M. Artin} [Contemp. Math. 13, 223-227 (1982; Zbl 0528.13021)], it is shown that \(\hat A,\) the I-adic completion of A, is a direct limit of generically smooth A-algebras. It is deduced that, for A an excellent henselian ring containing \({\mathbb{Q}}\), if a finite system of polynomials over A has a zero in \(\hat A\) then it has a zero A.
Mathematics Subject Classification ID: 13J15
Mathematics Subject Classification ID: 13H05
Mathematics Subject Classification ID: 14B15
zbMATH DE Number: 3993740
zbMATH Keywords: excellent semilocal ring
zbMATH Keywords: approximation property
zbMATH Keywords: adic completion
reviewed by: C. P. L. Rhodes
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3993740