Graded local cohomology modules and their associated primes: the Cohen–Macaulay case

Abstract

It is well-known that the ith local cohomology of a finitely generated R-module M over a positively graded commutative Noetherian ring R, associated to the irrelevant ideal R + is graded. Furthermore, for every integer n, the nth component H R + i ( M) n of this local cohomology module H R + i ( M) is finitely generated over R 0 and vanishes for n⪢0. In this paper, we want to understand the behavior of H R + i ( M) n for n⪡0 in the case where R is a Cohen–Macaulay ring and M is a Cohen–Macaulay R-module. When dim R 0=1 , we will show that Ass R 0 ( H R + i ( M) n ) becomes constant when n becomes negatively large. When R 0 is local, and dim R 0=2 , we will show that there exists an integer N such that either H R + i ( M) n =(0) for all n< N or, H R + i ( M) n ≠(0) for all n< N.