Bass Numbers of Local Cohomology Modules with Respect to an Ideal

Abstract

Let \({\left( {R,\mathfrak{m}} \right)}\) be a commutative Noetherian local ring and let \(\mathfrak{a}\) be an ideal of R. We give some inequalities between the Bass numbers of an R–module and those of its local cohomology modules with respect to \(\mathfrak{a}\). As an application of these inequalities, we recover results of Delfino-Marley and Kawasaki by showing that for a minimax R-module M and for any non-negative integer i, the Bass numbers of the ith local cohomology module \({\text{H}}^{i}_{\mathfrak{a}} {\left( M \right)}\) are finite if one of the following holds: (a) \(R \mathord{\left/ {\vphantom {R \mathfrak{a}}} \right. \kern-\nulldelimiterspace} \mathfrak{a} = 1\), (b) \(\mathfrak{a}\) is a principal ideal.