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

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 HR+i(M)n of this local cohomology module HR+i(M) is finitely generated over R0 and vanishes for n 0. In this paper, we want to understand the behavior of HR+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 R0=1, we will show that AssR0(HR+i(M)n) becomes constant when n becomes negatively large. When R0 is local, and dim R0=2, we will show that there exists an integer N such that either HR+i(M)n=(0) for all n