Associated primes of graded components of generalized local cohomology modules

摘要

We show that the ith generalized local cohomology module of finitely generated graded modules M and N over a standard positive graded commutative Noetherian ring R, with respect to the irrelevant ideal R+, is itself graded; all its graded components are finitely generated modules over R-0, the component of R of degree zero. Also we prove that the nth component of H-R+(i) (M, N)(n) of this generalized local cohomology module H-R+(i) (M, N) is zero for all n 0. Moreover, in this article, we study the asymptotic behaviour of ASS(R0) (H-R+(i) (M, N)(n)) as n -> -infinity in the following cases: (i) i is the least integer j for which H-R+(i) (M, N) is not finitely generated. (ii) The base ring R-0 is local of dimension at most one and for all i >= 0.