On the associated primes of generalized local cohomology modules

In [8], Huneke conjectured that if M is a finitely generated R-module, then the set of associated primes of H i a (M) is finite. Singh [15] provides a counter example for this conjecture. However, it is known that the conjecture is true in many situations. For example, in [11] it is shown that if R is local and dimR/a = 1, then for a finitely generated R-module M , the set AssR(H i a (M)) is finite for all i ≥ 0. Also, Brodmann and Lashgari [2] showed that the first non-finitely generated local cohomology module of a finitely generated R-module has only finitely many associated primes. Also, see [10] and [4] for a far reaching generalizations of this result. The following generalization of local cohomology theory is due to Herzog [7] (see also [17]). The generalized local cohomology functor H i a (., .) is defined by