Note on Asymptotic Prime Divisors, Analytic Spreads and the Altitude Formula

Abstract

Abstract.The main result gives several characterizations of Noethenan domains that satisfy the altitude formula in terms of analytic spreads and asymptotic prime divisors.1. Introduction. In [7], I proved several results concerning the prime divisors ( = associated primes) of /" and the integral closure of /", where / is an ideal in a Noetherian ring R and n is large. These results were considerably sharpened in [1] and [2]. In particular, in [1], Brodmann showed that all large powers of /" have the same prime divisors, and in [7, Theorem 2.5], I showed that the integral closures of all large powers of /" have the same prime divisors. Using this last result, McAdam proved,in [3, Theorem 3], the very interesting result that if / C P are ideals in a Noetherian domain R that satisfies the altitude formula and if P is prime, then P is a prime divisor of the integral closures of all large powers of /" if and only if (height P) equals the analytic spread of IRP. Our main results, Theorems 1 and 2, were suggested by McAdam's result and the related result [6, Theorem 2.29], and they characterize Noetherian domains that satisfy the altitude formula in terms of these asymptotic prime divisors and analytic spreads. Then, after proving several corollaries of these theorems, we close by showing certain additional rings (besides those in [3, Theorem 6]) have the property that a prime ideal is a prime divisor of all large powers of an ideal / if and only if it is a prime divisor of the integral