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.

Key Words:

• Local cohomology modules
• Associated primes
• Filter regular sequences

Acknowledgment

The first author was partially supported by a grant from Institute for Studies in Theoretical Physics and Mathematics (IPM) Iran (No. 81130021).

Notes

^#Communicated by H. Bruns.