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⊆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∈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.