On the finiteness properties of associated primes of generalized local cohomology modules

Abstract

Let R be a commutative Noetherian ring, a an ideal of R and N an R-module. We prove that, for every finitely generated R-module of finite projective dimension t, the elements in the support of generalized local cohomology module H nþt a ðM; NÞ of height n are finite for all n b 0. This implies that, if R is a d-dimensional local ring, then H dþtÀ1 a ðM; NÞ has finite support for arbitrary R, a and N. In addition, for a non-negative integer n, we show that if M and N are arbitrary finitely generated R-modules such that the R-modules H i a ðNÞ and H i a ðM; NÞ have finite support for all i < n, then Ass H n a ðM; NÞ is finite.