The lifting of determinantal prime ideals
type: Item
instance of: scholarly article
title: The lifting of determinantal prime ideals
zbMATH Open document ID: 0876.13003
DOI: 10.1007/BF02567967
author: Trung Ngo Viet
published in: Manuscripta Mathematica
publication date: 1997-04-13T00:00:00Z
full work available at URL: https://eudml.org/doc/156242
review text: One main result (theorem 1.1) of this paper is inspired by the generic perfection of determinantal ideals in the sense of Hochster and Eagon. It shows that a perfect prime ideal \(I_r(A)\) generated by the maximal minors of an \(r\times s\) matrix \(A= (a_{ij})\) of elements \(a_{ij}\) in a noetherian ring \(R\), \(r\leq s\), has the equality between symbolic and ordinary powers if the grade of the ideals generated by the low order minors of the matrix is large. More precisely: If \({\mathfrak p}=I_r(A)\) is prime with \(\text{grade } {\mathfrak p} =s-r+1\) and if \(\text{grade} (I_t(A)) \geq(r-t+1) \cdot (s-r) +2\) for \(t=1, \dots, r-1\), then the form ring \(\text{gr}_{\mathfrak p} (R)\) of \({\mathfrak p}\) is a domain and \({\mathfrak p}^{(n)} ={\mathfrak p}^n\) for all \(n\geq 1\). For Cohen-Macaulay rings \(R\) this result was also proved by Eisenbud and Huneke but with other methods. The proof of theorem 1.1 is based on the possibility to transfer the property of \(\text{gr}_{\mathfrak p} (R)\) being a domain in (weakly) lifting prime ideals. The second main result is concerned with the question whether one can lift a homogeneous determinantal ideal to a determinantal prime ideal with nice properties by a regular sequence of linear forms. The statement (theorem 1.2 and 4.4) is as follows: Let \(R\) be a finitely generated graded algebra over a field which is a domain and let \(A\) be an \(r\times s\) matrix of homogeneous elements of \(R\) such that all 2-minors of \(A\) are homogeneous, \(r\leq s\). If \({\mathfrak p}=I_r(A)\) is a prime ideal of \(R\) with \(\text{grade} {\mathfrak p}= s-r+1\), then \({\mathfrak p}\) can be lifted by a regular sequence of linear forms to a prime ideal \(I\) generated by the maximal minors of a matrix of homogeneous elements of a finitely generated algebra \(S\) such that \(\text{gr}_I(S)\) is a domain and \(I^{(n)} =I^n\) for all \(n\geq 1\).
Mathematics Subject Classification ID: 13A15
Mathematics Subject Classification ID: 13C40
zbMATH DE Number: 980724
zbMATH Keywords: symbolic powers
zbMATH Keywords: lifting
zbMATH Keywords: generic perfection of determinantal ideals
zbMATH Keywords: grade
zbMATH Keywords: form ring
reviewed by: Manfred Herrmann
MaRDI profile type: Publication
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OpenAlex ID: W2026266150
description: scientific article; zbMATH DE number 980724