Prime divisors, asymptotic R-sequences and unmixed local rings
type: Item
instance of: scholarly article
title: Prime divisors, asymptotic R-sequences and unmixed local rings
zbMATH Open document ID: 0601.13009
DOI: 10.1016/0021-8693(85)90092-4
author: Daniel Katz
published in: Journal of Algebra
publication date: 1985-01-01T00:00:00Z
review text: Let (R,M) be a local ring and \(R^*\) be its completion. The author introduces the notion of asymptotic R-sequence as follows: Elements \(x_ 1,...x_ d\) form an asymptotic R-sequence iff \(x_{i+1}\not\in \cup \{P| P\in \tilde A^*(x_ 1,...,x_ i)\},\quad 1\leq i\leq d-1,\) where: \(\tilde A^*(I)=\cap_{j}\{A^*(J)| \quad (\exists n) I^ nR\leq JR\leq \overline{I^ nR}\}\) with: \(A^*(J)=Ass R/J^ n\), \(n\gg 0\), \(\overline{I^ nR} =\) integral closure of \(I^ n\). The main results of this paper are the following: (i) The length of any maximal asymptotic R-sequence characterizes the minimal depth among prime divisors of zero in R. - (ii) R is unmixed iff each system of parameters forms an asymptotic R-sequence. - (iii) R is unmixed iff \(S=\sum_{n}| \cup_{k}(I^ n:M^ k))t^ n\) is a finite module over the Rees ring of I, for all ideals I with analytic spread less than dim R.
Mathematics Subject Classification ID: 13H99
Mathematics Subject Classification ID: 13A05
zbMATH DE Number: 3968735
zbMATH Keywords: unmixed local rings
zbMATH Keywords: maximal asymptotic R-sequence
zbMATH Keywords: minimal depth
zbMATH Keywords: system of parameters
reviewed by: Carla Massaza
MaRDI profile type: Publication
full work available at URL: 0021-8693(85)90092-4
OpenAlex ID: W2057867311
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description: scientific article; zbMATH DE number 3968735