On the finiteness properties of local cohomology modules for regular local rings

Abstract

Let $\mathfrak{a}$ denote an ideal in a regular local (Noetherian) ring $R$ and let $N$ be a finitely generated $R$-module with support in $V(\mathfrak{a})$. The purpose of this paper is to show that all homomorphic images of the $R$-modules $\Ext^j_R(N, H^i_{\frak a}(R))$ have only finitely many associated primes, for all $i,j\ge 0$, whenever $\dim R\le 4$ or $\dim R/\mathfrak{a}\le 3$ and $R$ contains a field. In addition, we show that if $\dim R=5$ and $R$ contains a field, then the $R$-modules $\Ext^j_R(N, H^i_{\frak a}(R))$ have only finitely many associated primes, for all $i,j\ge 0$.

13 pages, To appear in Tokyo Journal of Mathematics