Grade functions and two classical theorems
During the past few years there have been dozens of papers in the literature concerning asymptotic sequences and essential sequences, and of course there are many papers in the literature concerning R-sequences. The new papers have shown that most of the basic properties of R- sequences, associated primes, and Cohen-Macaulay rings have valid analogues for asymptotic (resp., essential) sequences, asymptotic (resp., essential) prime divisors, and quasi-unmixed (resp., unmixed) Noetherian ring so there are (at least) three parallel theories: the standard, asymptotic, and essential, and results valid in one theory have analogous results which are often valid in the other theories. - The goal of the present paper is to study sequences and associated primes abstractly. This is done by specifying for each ideal I in a Noetherian ring R a finite subset A(I) of Spec(R); the primes in A(I) and called the A-primes of I. If the sets A(I) satisfy certain natural conditions, then it is shown that the concept of an A-sequence in R and of the A-grade of I can be defined in such a way that many of the basic properties of R-sequences and classical grade have valid analogues in the A-theory. Several examples (some new and some known) of suitable choices for the set A(I) are mentioned, and then, as an application of the A-theory it is shown that the A-theory version of the following two classical theorems hold: (a) if I is an ideal in \(F[X_ 1,...,X_ n]\) such that I can be generated by \(h=height(I)\) elements, then every prime divisor of \(I^ k\) has height h for all \(k\geq 1\). (b) if p is a prime ideal in a Noetherian ring R such that \(R_ p\) is Cohen-Macaulay, then \(R_ p\) is Cohen- Macaulay for all but finitely many ideals P in \(\{P\in Spec(R);\) \(p\subset P\) and height\((P/p)=1\}.\)
scientific article; zbMATH DE number 3887817