On ordinary and symbolic powers of prime ideals
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.
scientific article; zbMATH DE number 3957256
scientific article