2002/09/25 by Markus P. Brodmann, Mordechai Katzman, Rodney Y. Sharp
Mathematics · #math.AC #math.AG #msc:13D45 #msc:13E05 #msc:13A02 #msc:13P10
published as Transactions of the AMS, 354 (2002), pp. 4261-4283
arxiv created 2002/09/25 · arxiv updated 2009/11/30
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. This paper is concerned with the asymptotic behaviour of \AssR0(HiR+(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 HjR+(M) is not finitely generated. Brodmann and Hellus have shown that \AssR0(HfR+(M)n) is constant for all n < < 0 (that is, in their terminology, \AssR0(HfR+(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 \AssR0(H3R+(R)n) need not be asymptotically stable for n → -∞. The second main aim of this paper is to determine, for Singh's example, \AssR0(H3R+(R)n) quite precisely for every integer n, and, thereby, answer one of the questions raised by Brodmann and Hellus.