Excellent rings, Henselian rings and the approximation property
type: Item
instance of: scholarly article
title: Excellent rings, Henselian rings and the approximation property
zbMATH Open document ID: 0881.13009
author: C. Rotthaus
published in: Rocky Mountain Journal of Mathematics
publication date: 1997-10-27T00:00:00Z
full work available at URL: CONT27-1.html
review text: The author gives an overview on excellent rings, Henselian rings and the property of approximation. As an application she describes a solution of the Bass-Quillen conjecture obtained by Popescu and Spivakovsky. After M. Artin's approximation theorem, the central result in the theory of excellent Henselian rings was proved by \textit{D. Popescu} [Nagoya Math. J. 100, 97-126 (1985; Zbl 0561.14008) and 104, 85-115 (1986; Zbl 0592.14014)]: An excellent Henselian ring has the property of (Artin) approximation. This is a consequence of Popescu's more general result: A morphism between two Noetherian rings is regular if and only if it is a filtered inductive limit of smooth finite type morphisms. At least after \textit{M. Spivakovsky}'s paper [``Smoothing of ring homomorphisms, approximation theorems and the Bass-Quillen conjecture'' (preprint 1993)] Popescu's result and the validity of his proof was established in the mathematical community. It seems that the author completely ignores this fact and knows only her own proof for a special case which appeared 1987. From this point of view the paper suggests a wrong picture.
Mathematics Subject Classification ID: 13F40
Mathematics Subject Classification ID: 13J15
Mathematics Subject Classification ID: 14B12
Mathematics Subject Classification ID: 13J10
zbMATH DE Number: 1051538
zbMATH Keywords: excellent rings
zbMATH Keywords: Henselian rings
zbMATH Keywords: approximation
reviewed by: Gerhard Pfister
MaRDI profile type: Publication
OpenAlex ID: W1980683761
cites work: Algebraic approximation of structures over complete local rings
cites work: Algebraic structure of power series rings
cites work: On Schanuel's conjectures
cites work: The Approximation Property for Some 5-Dimensional Henselian Rings
cites work: Über den regulären Ort in ausgezeichneten Ringen
cites work: Q4143433
cites work: Resolution of singularities of an algebraic variety over a field of characteristic zero. I
cites work: Generic local structure of the morphisms in commutative algebra
cites work: On the Bass-Quillen conjecture concerning projective modules over polynomial rings
cites work: Q3992979
cites work: Q5735274
cites work: General Néron desingularization based on the idea of Popescu
cites work: General Néron desingularization
cites work: General Néron desingularization and approximation
cites work: Projective modules over polynomial rings
cites work: Anneaux locaux henséliens
cites work: Komplettierung semilokaler quasiausgezeichneter Ringe
cites work: Zur Komplettierung ausgezeichneter Ringe
cites work: On the approximation property of excellent rings
cites work: Rings with approximation property
cites work: Q4124942
cites work: A few theorems on completion of excellent rings
DOI: 10.1216/RMJM/1181071964
description: scientific article; zbMATH DE number 1051538