Presentation depth and the Lipman-Sathaye Jacobian theorem
type: Item
instance of: scholarly article
title: Presentation depth and the Lipman-Sathaye Jacobian theorem
zbMATH Open document ID: 1018.13003
author: Melvin Hochster
published in: Homology, Homotopy and Applications
publication date: 2002-08-01T00:00:00Z
full work available at URL: https://eudml.org/doc/50234
full work available at URL: volume4-2-2.htm
review text: A theorem of \textit{J. Lipman} and \textit{A. Sathaye} [Michigan Math. J. 28, 199-222 (1981; Zbl 0438.13019)] is generalized by weakening the hypotheses substantially: Let \(R\) be a Noetherian domain with fraction field \(K\). Assume that the \(S_2\)-locus is open in algebras essentially of finite type over \(R\) (which is always true if \(R\) is excellent). Let \(S\) be an extension algebra, essentially of finite type over \(R\) being torsion free and generically étale. Suppose that for every maximal ideal \(\mathfrak{m} \subset S\), \(\mathfrak{m} \cap R = \mathfrak p\), \(R_{\mathfrak p}\) is normal and \(S_{\mathfrak{m}}\) has a relatively \(S_2\) presentation over \(R_{\mathfrak p}\) (which is always true if \(R\) is normal and Cohen-Macaulay). Let \(L = K \otimes_R S\) and \(S^\prime\) be the integral closure of \(S\) in \(L\) and assume that \(S^\prime\) is a finite \(S\)-module (which is always true if \(R\) is excellent). Suppose that, for every height one prime ideal \(\mathfrak{q}\) of \(S^\prime\) the ring \(R_{\mathfrak{q} \cap R}\) is regular. Then \[ S^\prime\colon{_L J}_{S^\prime|R} \subseteq S\colon{_L J}_{S|R}. \] Here \(J_{S|R}\) is the Jacobian ideal of \(S\) over \(R\).
reviewed by: Gerhard Pfister
Mathematics Subject Classification ID: 13B02
Mathematics Subject Classification ID: 13B21
Mathematics Subject Classification ID: 13C15
zbMATH DE Number: 1770078
zbMATH Keywords: Jacobian theorem
zbMATH Keywords: algebra presentation
zbMATH Keywords: depth
zbMATH Keywords: integral extension
zbMATH Keywords: Jacobian ideal
MaRDI profile type: Publication
DOI: 10.4310/HHA.2002.V4.N2.A14
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description: scientific article; zbMATH DE number 1770078