Finiteness Dimension and Bass Numbers of Generalized Local Cohomology Modules

Abstract

Let R be a commutative Noetherian ring, a an ideal of R, and M , N two nonzero finitely generated R-modules. Let t be a non-negative integer. It is shown that dim Supp H i a (M, N ) ≤ 1 for all i < t if and only if there exists an ideal b of R such that dim R/b ≤ 1 and H i a (M, N ) ∼ = H i b (M, N ) for all i < t. As a consequence all Bass numbers and all Betti numbers of generalized local cohomology modules H i a (M, N ) are finite for all i < t, provided that the projective dimension pd(M ) is finite.