On asymptotic prime divisors

Abstract

Several results are proved concerning the set A*(I) = {P E Spec R; P is a prime divisor of the integral closure (Γ) a of /' for all large /}, where / is an ideal in a Noetherian ring R. Among these are: if P is a prime divisor of (Γ) a for some i > 1, then P is a prime divisor of (I") a for all n > i\ a characterization of Cohen-Macaulay rings and of altitude two local UFDs in terms of A*(I); and, some results on the relationship of A*(I) to A*(IS) with S a flat Λ-algebra and to A*((I + z)/z) with z a minimal prime ideal in R.

Asymptotic growth of algebras associated to powers of ideals

“…The multiplicity of a finitely generated R-module M will be denoted by e(M ). Theorem 1.1 below is inspired by results of MacAdam [23], Ratliff [25], Katz [19], Schenzel [27] and others on the asymptotic associated primes of ideals of small analytic spread. Part (a) of Theorem 1.1 follows from Theorem 4.1 of Katz's paper [19] and Theorem 5.6 of Schenzel's paper [27].…”

Δ-closures of ideals and rings

“…Q.E.D. 7. The A-closure of P-algebras In this section it is shown that the A-closure analogues of several standard results concerning the integral closure of P-algebras are valid.…”

A characterization of Cohen-Macaulay modules and local cohomology

“…. , n. In [15], L.J. Ratliff, Jr., introduced the interesting set of associated primes * (I ) := Ass R R/(I n ) a for large n, and he showed that this finite set has some nice properties in the theory of asymptotic prime divisors.…”

“…In the case N = R, A * a (I, N ) is the asymptotic primes * (I ) of I introduced by L.J. Ratliff, Jr., [15].…”