Cohomological dimension and top local cohomology modules

Abstract

Let $R$ be a Noetherian ring, $I$ an ideal of $R$ and $M$ an $R$-module. In this paper, we first determine a condition under which a given integer $t$ is a lower bound for the cohomological dimension $\operatorname {cd}(I,M)$, and use this to conclude that non-catenary Noetherian domains contain prime ideals that are not set-theoretic complete intersection. We also show the existence of a descending chain of ideals with successive diminishing cohomological dimensions. We then resolve the Artinianness of top local cohomology modules over local unique factorization domains of Krull dimension at most three, and obtain several related results on the top local cohomology modules for much more general cases.