On the integral closures of certain ideals generated by regular sequences
type: Item
instance of: scholarly article
title: On the integral closures of certain ideals generated by regular sequences
zbMATH Open document ID: 0959.13005
DOI: 10.1016/S0022-4049(99)00127-9
published in: Journal of Pure and Applied Algebra
publication date: 2001-04-23T00:00:00Z
review text: The goal of the paper under review is to introduce a method for computing the integral closures of certain ideals generated by regular sequences which are generically good. More precisely, let \(A=kX,Y,Z\) be the formal power series ring over a field \(k\), and \(\alpha\), \(\alpha'\), \(\beta\), \(\beta'\), \(\gamma\), \(\gamma'\) be arbitrary positive integers. Set \(a=Z^{\gamma+\gamma'}-X^{\alpha'}Y^{\beta'}\), \(b=X^{\alpha+\alpha'}-Y^{\beta}Z^{\gamma'}\), and \(c=Y^{\beta+\beta'}-X^{\alpha}Z^{\gamma}\). Let \(J=(a,b)A\) and \(I\) be the ideal of \(A\) generated by \(a\), \(b\) and the set of the \(X^iZ^jc\) with \(i,j\geq 0\) and \(i/\alpha'+j/\gamma'\geq 1\). Then, the author shows that the integral closure \(\overline{J^n}\) is equal to \(J^{n-1}I\) for all \(n\geq 1\) and the Rees algebra \(R(\overline{J})\) is a Cohen-Macaulay ring. The key results for the proof are: The associated primes in \(A\) of \(A/J\) are \((a,b,c)A\) and \((X,Y)A\), and \(A/J^nI\) is Cohen-Macaulay for all \(n\geq 0\).
reviewed by: Philippe Elbaz-Vincent
Mathematics Subject Classification ID: 13B22
Mathematics Subject Classification ID: 13A30
Mathematics Subject Classification ID: 13H10
zbMATH DE Number: 1517308
zbMATH Keywords: integral closure
zbMATH Keywords: regular sequences
zbMATH Keywords: Rees algebras
zbMATH Keywords: Cohen-Macaulay ring
zbMATH Keywords: formal power series ring
author: Kohji Nishida
MaRDI profile type: Publication
cites work: Symbolic algebras of monomial primes.
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full work available at URL: s0022-4049(99)00127-9
OpenAlex ID: W2087738214
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description: scientific article; zbMATH DE number 1517308