Some duality and equivalence results

TL;DR

This paper explores properties of the Matlis dual of certain local cohomology modules in relative Cohen-Macaulay rings, establishing duality results that generalize classical theorems like the Local Duality Theorem.

Contribution

It introduces new duality and equivalence results involving the Matlis dual of local cohomology modules, extending known theorems to broader classes of rings.

Findings

Matlis duals behave like canonical modules over Cohen-Macaulay rings

Established duality and equivalence results involving these modules

Generalized the Local Duality Theorem to relative Cohen-Macaulay rings

Abstract

Let $(R,\fm)$ be a relative Cohen-Macaulay local ring with respect to an ideal $\fa$ of $R$ and set $c:=\h\fa$. In this paper, we investigate some properties of the Matlis dual $\H_{\fa}^c(R)^{\vee}$ of the $R$-module $\H_{\fa}^c(R)$ and we show that such modules treat like canonical modules over Cohen-Macaulay local rings. Also, we provide some duality and equivalence results with respect to the module $\H_{\fa}^c(R)^{\vee}$ and so these results lead to achieve generalizations of some known results, such as the Local Duality Theorem, which have been provided over a Cohen-Macaulay local ring which admits a canonical module.