On ideals whose adic and symbolic topologies are linearly equivalent
type: Item
instance of: scholarly article
title: On ideals whose adic and symbolic topologies are linearly equivalent
zbMATH Open document ID: 0645.13005
DOI: 10.1016/0022-4049(87)90062-4
author: J. K. Verma
published in: Journal of Pure and Applied Algebra
publication date: 1987-01-01T00:00:00Z
review text: Let R be a Noetherian ring and I an arbitrary ideal. Much work has been done on the question: When are the topologies on R induced by the powers and the symbolic powers of I linearly equivalent? This question was raised by Hartshorne in 1970 and answered by P. Schenzel and J. L. Ratliff jun. for prime and primary ideals in 1985. The paper under review builds upon the work of D. Katz, S. McAdam, the above mentioned researchers, and others. In this paper such ideals are called s-ideals. They are characterized by the following two equivalent properties: I. Every u-essential prime divisor of I is a minimal prime divisor. II. If \(Q\in Ass(R/I^ n)\) for all large n and is not a minimal prime divisor, then \(\ell ((IR^*_ Q+z)/z)<\dim (R^*_ Q/z)\) for all \(z\in Ass(R^*_ Q/z)\). \((R^*_ Q\) is the Q-adic completion of \(R_ Q\) and \(\ell\) denotes analytic spread). These characterizations - especially the first one - are useful in studying s-ideals. Typical results obtained are: s-ideals remain s-ideals when extended to a non-degenerate or to a faithfully flat ring extension of R, the image of an s-ideal in R/z is an s-ideal for \(z\in Ass(R_ Q/z)\), if \(\sqrt{I}=p\) is prime, \(R_ M\) is unmixed for all maximal ideals \(M\supseteq I\), \(IR_ p\) is normal, and \(gr(IR_ p,R_ p)_{red}\) is a domain, then I is an s-ideal iff \(I^{(n)}=(I^ n)\) (integral closure) for all \(n\geq 1\) iff \(gr(IR,R)_{red}\) is a domain (a generalization of a result by Huneke about primes in universally catenarian Noetherian rings), and a short proof of Katz' characterization of locally unmixed rings in terms of s-ideals.
Mathematics Subject Classification ID: 13J10
Mathematics Subject Classification ID: 13A15
Mathematics Subject Classification ID: 13E05
zbMATH DE Number: 4051769
zbMATH Keywords: adic completion
zbMATH Keywords: linearly equivalent topologies
zbMATH Keywords: Noetherian ring
zbMATH Keywords: symbolic powers
zbMATH Keywords: s-ideals
zbMATH Keywords: essential prime divisor
zbMATH Keywords: analytic spread
reviewed by: Robert O. Robson
MaRDI profile type: Publication
full work available at URL: 0022-4049(87)90062-4
OpenAlex ID: W2044861075
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description: scientific article; zbMATH DE number 4051769