On ideals whose adic and symbolic topologies are linearly equivalent
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.
scientific article; zbMATH DE number 4051769
scientific article