On the associated primes of the generalized d -local cohomology modules

The first part of the paper is concerned to relationship between the sets of associated primes of the generalized d-local cohomology modules and the ordinary gen eralized local cohomology modules. Assume that R is a commutative Noetherian local ring, M and N are finitely generated R-modules and d,t are two integers. We prove that AssHt d(M,N) = ∪I∈Φ AssHt I(M,N) whenever Hi d(M,N) = 0 for all i < t and Φ = {I : I is an ideal of R with dimR/I ≤ d}. In the second part of the paper, we give some information about the non-vanishing of the generalized d-local cohomology modules. To be more precise, we prove that Hi d(M,R) ̸= 0 if and only if i = n−d whenever R is a Gorenstein ring of dimension n and pdR(M) < ∞. This result leads to an example which shows that AssHn−d d (M,R) is not necessarily a finite set.