Grade functions and two classical theorems
type: Item
instance of: scholarly article
title: Grade functions and two classical theorems
zbMATH Open document ID: 0557.13002
DOI: 10.1016/0022-4049(85)90040-4
published in: Journal of Pure and Applied Algebra
publication date: 1985-01-01T00:00:00Z
review text: 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\((P/p)=1\}.\)
Mathematics Subject Classification ID: 13A15
Mathematics Subject Classification ID: 13H10
zbMATH DE Number: 3887817
zbMATH Keywords: grade functions
zbMATH Keywords: subset of Spec
zbMATH Keywords: asymptotic sequences
zbMATH Keywords: essential sequences
zbMATH Keywords: R-sequences
zbMATH Keywords: Noetherian ring
zbMATH Keywords: A-primes
zbMATH Keywords: A-sequence
zbMATH Keywords: A-grade
zbMATH Keywords: Cohen-Macaulay
author: Louis J. Ratliff
MaRDI profile type: Publication
full work available at URL: 0022-4049(85)90040-4
OpenAlex ID: W2082842567
cites work: Asymptotic Stability of Ass(M/I n M)
cites work: Five theorems on Macaulay rings
cites work: Q5603302
cites work: Q4293528
cites work: Asymptotic prime divisors
cites work: On the asymptotic cograde of an ideal
cites work: Essential sequences
cites work: On asymptotic prime divisors
cites work: Asymptotic sequences
cites work: Essential sequences over an ideal and essential cograde
cites work: Q3237214
cites work: Q3244153
cites work: A Note on General Ideals
description: scientific article; zbMATH DE number 3887817