Generic flatness for strongly Noetherian algebras
type: Item
instance of: scholarly article
title: Generic flatness for strongly Noetherian algebras
zbMATH Open document ID: 0958.16024
author: Lance W. Small
author: J. J. Zhang
author: Michael Artin
published in: Journal of Algebra
publication date: 2001-04-03T00:00:00Z
review text: The central result of this paper is the following theorem: Let \(R\) be a commutative domain which is affine over a field or an excellent Dedekind domain. Let \(A\) be a strongly right Noetherian \(R\)-algebra. Then every finitely generated right \(A\)-module is generically flat as an \(R\)-module. Here, ``strongly right Noetherian'' means that \(A\otimes_RS\) is right Noetherian for every commutative Noetherian \(R\)-algebra \(S\); and an \(R\)-module \(M\) is ``generically flat'' if there exists \(0\neq r\in R\) such that \(M[r^{-1}]\) is a flat \(R[r^{-1}]\)-module. The latter concept was introduced by Grothendieck in the 60s, in the setting of a commutative \(R\)-algebra \(A\), and has been much studied since then for various classes of noncommutative Noetherian algebras \(A\), in part because of its relevance for the Jacobson condition and the noncommutative Nullstellensatz -- see, for example, Chapter 9 of [\textit{J. C. McConnell} and \textit{J. C. Robson}, Noncommutative Noetherian rings, Wiley, Chichester (1987; Zbl 0644.16008)] and the references cited there. These topics are also revisited in the present paper, which also discusses a number of interesting examples and counterexamples, and which develops methods which may well prove useful in other contexts.
Mathematics Subject Classification ID: 16P40
Mathematics Subject Classification ID: 16D40
Mathematics Subject Classification ID: 16W50
Mathematics Subject Classification ID: 16P10
Mathematics Subject Classification ID: 16S38
zbMATH DE Number: 1400110
zbMATH Keywords: graded rings
zbMATH Keywords: excellent Dedekind domains
zbMATH Keywords: strongly right Noetherian algebras
zbMATH Keywords: finitely generated right modules
zbMATH Keywords: generically flat modules
zbMATH Keywords: noncommutative Noetherian algebras
zbMATH Keywords: noncommutative Nullstellensatz
reviewed by: Kenneth A. Brown
MaRDI profile type: Publication
full work available at URL: jabr.1999.7997
OpenAlex ID: W2015390802
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DOI: 10.1006/JABR.1999.7997
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description: scientific article; zbMATH DE number 1400110