On finiteness properties of local cohomology modules over Cohen–Macaulay local rings

Abstract

Let $A$ be a Cohen-Macaulay local ring which contains a field $k$, and let $I \subseteq A$ be an ideal generated by polynomials in a system of parameters of $A$ with coefficients in $k$. In this paper, we shall prove that all the Bass numbers of local cohomology modules are finite for all $j \in{\mathbb Z}$ provided that the residue field is separable over $k$. We also prove that the set of associated prime ideals of those is a finite set under the same hypothesis. Furthermore, we shall discuss finiteness properties of local cohomology modules over regular local rings.