Some results on Artinian cofinite top local cohomology modules

Abstract

Let $(R,\mathfrak {m})$ be a Noetherian local ring, I be an ideal of R, and M be a finitely generated R-module such that ${\text {H}}_I^t(M)$ is Artinian and I-cofinite, where $t={\text {cd}}\,(I,M)$. In this paper, we give some equivalent conditions for the property

$$\begin{aligned} {\text {Ann}}\,_R\left( 0:_{{\text {H}}_I^t (M)} \mathfrak {p}\right) =\mathfrak {p}~\text {for all prime ideals }~ \mathfrak {p}\supseteq {\text {Ann}}\,_R{\text {H}}_I^t(M).(*) \end{aligned}$$

Also, we show that if ${\text {H}}_I^t(M)$ satisfies the property $(*)$, then ${\text {H}}_I^t(M)\cong {\text {H}}_{\mathfrak {m}}^t(M/N)$ for some submodule N of M with ${\text {dim}}\,(M/N)=t$.