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.
Categories: math.AC