Associated primes of graded components of local cohomology modules

Abstract

The i i -th local cohomology module of a finitely generated graded module M M over a standard positively graded commutative Noetherian ring R R , with respect to the irrelevant ideal R + R_+ , is itself graded; all its graded components are finitely generated modules over R 0 R_0 , the component of R R of degree 0 0 . It is known that the n n -th component H R + i ( M ) n H^i_{R_+}(M)_n of this local cohomology module H R + i ( M ) H^i_{R_+}(M) is zero for all n >> 0 n>> 0 . This paper is concerned with the asymptotic behaviour of Ass R 0 ⁡ ( H R + i ( M ) n ) \operatorname {Ass}_{R_0}(H^i_{R_+}(M)_n) as n → − ∞ n \rightarrow -\infty . The smallest i i for which such study is interesting is the finiteness dimension f f of M M relative to R + R_+ , defined as the least integer j j for which H R + j ( M ) H^j_{R_+}(M) is not finitely generated. Brodmann and Hellus have shown that Ass R 0 ⁡ ( H R + f ( M ) n ) \operatorname {Ass}_{R_0}(H^f_{R_+}(M)_n) is constant for all n >> 0 n > > 0 (that is, in their terminology, Ass R 0 ⁡ ( H R + f ( M ) n ) \operatorname {Ass}_{R_0}(H^f_{R_+}(M)_n) is asymptotically stable for n → − ∞ n \rightarrow -\infty ). The first main aim of this paper is to identify the ultimate constant value (under the mild assumption that R R is a homomorphic image of a regular ring): our answer is precisely the set of contractions to R 0 R_0 of certain relevant primes of R 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 i > f . They noted that Singh’s study of a particular example (in which f = 2 f = 2 ) shows that Ass R 0 ⁡ ( H R + 3 ( R ) n ) \operatorname {Ass}_{R_0}(H^3_{R_+}(R)_n) need not be asymptotically stable for n → − ∞ n \rightarrow -\infty . The second main aim of this paper is to determine, for Singh’s example, Ass R 0 ⁡ ( H R + 3 ( R ) n ) \operatorname {Ass}_{R_0}(H^3_{R_+}(R)_n) quite precisely for every integer n n , and, thereby, answer one of the questions raised by Brodmann and Hellus.