On The Cohomological Dimension of Local Cohomology Modules

Abstract

Let $R$ be a Noetherian ring, $I$ an ideal of $R$ and $M$ an $R$-module with $\mathrm{cd}(I,M)=c$. In this article, we first show that there exists a descending chain of ideals $I=I_c⊋I_{c-1}⊋\cdots ⊋I_0$ of $R$ such that for each $0\le i\le c-1$, $\mathrm{cd}(I_i,M)=i$ and that the top local cohomology module $H_{I_i}^i(M)$ is not Artinian. We then give sufficient conditions for a non-negative integer $t$ to be a lower bound for $\mathrm{cd}(I,M)$ 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.

References in corpus (1)

• Cohomological and projective dimensions

Cited by in corpus (3)

• The finiteness dimension of modules and relative Cohen-Macaulayness
• A note on the non-Artinianness of top local cohomology modules
• Top Local Cohomology Modules Are Almost Always Non-Artinian