The i-th local cohomology module of a ﬁnitely generated graded module M over a standard positively graded commutative Noetherian ring R, with respect to the irrelevant ideal R+ , is itself graded; all its graded components are ﬁnitely generated modules over R0 , the component of R of degree 0. It is known that the n-th component Hi R+ (M )n of this local cohomology module Hi R+ (M ) is zero for all n >> 0. This paper is concerned with the asymptotic behaviour of AssR 0 (Hi R+ (M )n ) as n → −∞.
The smallest i for which such study is interesting is the ﬁniteness dimension f of M relative to R+ , deﬁned as the least integer j for which Hj R+ (M ) is not ﬁnitely generated. Brodmann and Hellus have shown that AssR 0 (H f R+ (M )n ) is constant for all n << 0 (that is, in their terminology, AssR 0 (H f R+ (M )n ) is asymptotically stable for n → −∞). The ﬁrst main aim of this paper is to identify the ultimate constant value (under the mild assumption that R is a homomorphic image of a regular ring): our answer is precisely the set of contractions to R0 of certain relevant primes of R whose existence is conﬁrmed by Grothendieck’s Finiteness Theorem for local cohomology.
Brodmann and Hellus raised various questions about such asymptotic behaviour when i > f . They noted that Singh’s study of a particular example (in which f = 2) shows that AssR 0 (H3 R+ (R)n ) need not be asymptotically stable for n → −∞. The second main aim of this paper is to determine, for Singh’s example, AssR 0 (H3 R+ (R)n ) quite precisely for every integer n, and, thereby, answer one of the questions raised by Brodmann and Hellus.