Essential sequences over an ideal and essential cograde

Abstract

In [-6] the concepts of essential prime divisors, essential sequences, and the essential grade of an ideal I in a Noetherian ring were introduced and therein it was shown that these concepts are an excellent analogue of, respectively, associated primes, R-sequences, and the standard grade of I, in the classical theory, and also of, respectively, asymptotic prime divisors, asymptotic sequences, and the asymptotic grade of I, in the asymptotic theory. It turns out that essential sequences and essential grade yield useful information concerning all the prime divisors of zero in the completion of a local ring. Thus these concepts should prove useful in many future research papers. Asymptotic prime divisors, asymptotic sequences, and asymptotic grade have recently been useful in several research papers. And in [5] it was shown that asymptotic sequences over an ideal I in a Noetherian ring R and the asymptotic cograde of I (when R is local) have some useful properties, and several bounds on this cograde were established in [5]. The main purpose of this paper is to show that, similarly, essential sequences over I behave nicely when passing to certain rings related to R and that the essential cograde of I is well defined (when R is local) and satisfies certain rather natural inequalities. Section 2 contains the definitions and a list of the basic facts concerning essential prime divisors, essential sequences, and essential grade that are needed in the remainder of the paper. It also contains several results showing that essential sequences over an ideal I in a Noetherian ring R behave nicely with respect to passing to certain rings related to R. In Section 3 it is shown that the essential cograde of I is well defined when R is local and likewise behaves nicely when passing to the same type of related rings. In Section 4 we briefly