Noetherian domains inside a homomorphic image of a completion
type: Item
instance of: scholarly article
title: Noetherian domains inside a homomorphic image of a completion
zbMATH Open document ID: 0944.13009
author: C. Rotthaus
author: Sylvia Wiegand
author: William J. Heinzer
published in: Journal of Algebra
publication date: 2000-09-12T00:00:00Z
review text: Given a field \(k\) and and a field \(L\) of finite type over \(k\), a general problem is to describe the local Noetherian domains \(A\) between \(k\) and \(L\), such that \(A\) has fraction field \(L\) and residue field \(k\). It could be shown that every \(A\) is realizable as the intersection of \(L\) with a homomorphic image of the \(\mathfrak{m}\)-adic completion of a local Noetherian domain (of maximal ideal \(\mathfrak{m}\)) essentially of finite type over \(k\) and having \(L\) as fraction field. However it is not true in general that such intersection is Noetherian. In the paper under review, motivated by the above problem, the authors give the following criterion for the intersection domain to be Noetherian: Let \(R\) be a Noetherian integral domain with fraction field \(L\). Let \(x\) be a nonzero nonunit of \(R\) and let \(R^*\) denote the \((x)\)-adic completion of \(R\). Suppose \(I\) is an ideal of \(R^*\) with the property that \(P \cap R=(0)\) for each \(P\in \text{Ass}(R^*/I)\). Then \(R\to (R^*/I)[1/x]\) is flat if and only if \(L\cap (R^*/I)\) is Noetherian and is realizable as a localization of a subring of \(R[1/x]\). The authors also give several examples where flatness holds, and study some regularity properties related to the above setting.
reviewed by: Philippe Elbaz-Vincent
Mathematics Subject Classification ID: 13E05
Mathematics Subject Classification ID: 13J10
Mathematics Subject Classification ID: 13G05
Mathematics Subject Classification ID: 13B30
Mathematics Subject Classification ID: 13B10
Mathematics Subject Classification ID: 13B35
zbMATH DE Number: 1309132
zbMATH Keywords: Noetherian domain
zbMATH Keywords: Noetherianness
zbMATH Keywords: flatness condition
zbMATH Keywords: \(I\)-adic completion
zbMATH Keywords: regular morphism
zbMATH Keywords: fraction field
zbMATH Keywords: residue field
Wikidata QID: Q123151428
MaRDI profile type: Publication
full work available at URL: jabr.1998.7763
OpenAlex ID: W1989775711
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DOI: 10.1006/JABR.1998.7763
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description: scientific article; zbMATH DE number 1309132