On The Cohomological Dimension of Local Cohomology Modules

Let $R$ be a Noetherian ring, $I$ an ideal of $R$ and $M$ an $R$-module with $\operatorname{cd}(I,M)=c$. In this article, we first show that there exists a descending chain of ideals $I=I_c\supsetneq I_{c-1}\supsetneq \cdots \supsetneq I_0$ of $R$ such that for each $0\leq i\leq c-1$, $\operatorname{cd}(I_i,M)=i$ and that the top local cohomology module $\operatorname{H}^i_{I_i}(M)$ is not Artinian. We then give sufficient conditions for a non-negative integer $t$ to be a lower bound for $\operatorname{cd}(I,M)$ and use this to conclude that in non-catenary Noetherian local integral domains, there exist prime ideals that are not set theoretic complete intersection. Finally, we set conditions which determine whether or not a top local cohomology module is Artinian.