On the cofiniteness of local cohomology modules in dimension < 2

Abstract

In this note, we prove the cofiniteness of local cohomology modules $H^i_I(N)$ with respect to $I$ for all $i\lt t$ and the finiteness of $(0:_{H^t_I(N)}I)$ and ${\rm Ext}^1_R(R/I,H^t_I(N))$ provided that ${\rm Ext}^i_R(R/I,N)$ is finitely generated for all $i\le t+1$ and $H^i_I(N)$ is in dimension $\lt 2$ for all $i \lt t$, where $t\ge 1$ is an integer (here, $N$ is not necessarily finitely generated over $R$). This extends the results of Bahmanpour-Naghipour [5, Theorem 2.6], Aghapournahr-Bahmanpour [2, Theorem 3.4], Bahmanpour-Naghipour-Sedghi [4, Theorem 2.8] by a different proof method.