Local cohomology modules and their properties

Abstract

UDC 512.5

Let $(R, \mathfrak{m})$ be a complete Noetherian local ring and let $M$ be a generalized Cohen-Macaulay $R$-module of dimension $d \geq 2.$
We show that
$$
D \left(H_{\mathfrak{m}}^d\Big(D \big(H_{\mathfrak{m}}^d
(D_{\mathfrak{m}}(M))\big)\Big)\right) \approx D_{\mathfrak{m}} (M),
$$
where $D = {\rm Hom} (-, E)$ and $D_{\mathfrak{m}} (-)$ is the ideal transform functor.
Also, assuming that $I$ is a proper ideal of a local ring $R$, we obtain some results on the finiteness of Bass numbers, cofinitness, and cominimaxness of local cohomology modules with respect to $I.$