On ordinary and symbolic powers of prime ideals
type: Item
instance of: scholarly article
title: On ordinary and symbolic powers of prime ideals
zbMATH Open document ID: 0595.13002
DOI: 10.1016/0021-8693(86)90184-5
published in: Journal of Algebra
publication date: 1986-01-01T00:00:00Z
review text: Let P be a proper prime ideal of the (commutative) Noetherian ring R and let n be a positive integer. The n-th symbolic power of P, denoted by \(P^{(n)}\), is defined to be \(R\cap P^ nR_ P\), the P-primary component of \(P^ n\). Then \(P^ n=P^{(n)}\) precisely when \(P^ n\) is primary, or, equivalently when \(P^ n\) has no embedded prime. Results of \textit{M. Hochster} and \textit{J. A. Eagon} [Am. J. Math. 93, 1020-1058 (1971; Zbl 0244.13012)] led \textit{M. Hochster} [Math. Z. 133, 53-65 (1973; Zbl 0251.13012)] to consider the question of whether the two kinds of power of P being equal is related to R/P being Cohen-Macaulay, at least if R is a polynomial ring over a field. However Hochster (loc. cit.) established equivalent conditions for equality and used these to produce a polynomial ring R with a prime ideal P such that R/P is not Cohen- Macaulay but \(P^ n=P^{(n)}\) for all n. On the other hand an example on page 29 of the book by \textit{D. G. Northcott} [''Ideal theory'', (Cambridge 1953; Zbl 0052.268)] shows that R/P may be Cohen-Macaulay but \(P^ 2\neq P^{(2)}.\) The article under review shows that, under certain restrictions, the condition \(P^ 2=P^{(2)}\) is a condition on the quotient ring R/P. Specifically, the author shows that if k is a field of characteristic zero, S and S' are localizations of finitely generated k-algebras at regular closed points and P and P' are prime ideals of S and S' respectively such that S/P\(\cong S'/P'\), then \(P^ 2=P^{(2)}\) if and only if \((P')^ 2=(P')^{(2)}\). His proof is distinctly homological and uses properties of higher derivations and extensions of algebras.
Mathematics Subject Classification ID: 13A15
Mathematics Subject Classification ID: 13E05
Mathematics Subject Classification ID: 13H10
Mathematics Subject Classification ID: 13F20
zbMATH DE Number: 3957256
zbMATH Keywords: symbolic prime power of ideals
zbMATH Keywords: Noetherian ring
zbMATH Keywords: symbolic power
zbMATH Keywords: Cohen- Macaulay
zbMATH Keywords: higher derivations
zbMATH Keywords: extensions of algebras
author: Orlando E. Villamayor
reviewed by: John Clark
MaRDI profile type: Publication
full work available at URL: 0021-8693(86)90184-5
OpenAlex ID: W2131307083
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description: scientific article; zbMATH DE number 3957256