Filter Regular Sequences and Generalized Local Cohomology Modules#

Abstract

Let 𝔞 be an ideal of a commutative Noetherian ring R and let M, N be finitely generated R-modules. We prove that whenever n is a positive integer such that i. (N) has a finitely many associated prime ideals; and, ii. (M, (N)) is finitely generated for all i = 1, 2,…, n − 1 then the set of associated prime ideals of generalized local cohomology module (M, N) is finite. As a consequence, we provide some sufficient conditions for finiteness of Ass R (M, N). Also, we show that if M has finite projective dimension d then (M, N) ≅ for any positive integer n and any 𝔞-filter regular sequence a 1,…, a n on N.