Bass Numbers of Local Cohomology Modules

Abstract

Let $A$ be a regular local ring of positive characteristic. This paper is concerned with the local cohomology modules of $A$ itself, but with respect to an arbitrary ideal of $A$. The results include that all the Bass numbers of all such local cohomology modules are finite, that each such local cohomology module has finite set of associated prime ideals, and that, whenever such a local cohomology module is Artinian, then it must be injective. (This last result had been proved earlier by Hartshorne and Speiser under the additional assumptions that $A$ is complete and contains its residue field which is perfect.) The paper ends with some low-dimensional evidence related to questions about whether the analogous statements for regular local rings of characteristic $0$ are true.