Notes on local cohomology and duality

TL;DR

This paper explores local cohomology and duality in Gorenstein domains, providing formulas, duality theorems, and generalizations that extend classical results to broader classes of ideals and modules.

Contribution

It introduces a formula for the Matlis dual of injective hulls, proves a dual version of Hartshorne-Lichtenbaum vanishing, and generalizes local duality to cohomologically complete intersection ideals.

Findings

Derived a formula for the Matlis dual of injective hulls of prime ideals.

Proved a dual version of Hartshorne-Lichtenbaum vanishing theorem.

Generalized local duality to cohomologically complete intersection ideals.

Abstract

We provide a formula (see Theorem 1.5) for the Matlis dual of the injective hull of $R/\mathfrak{p}$ where $\mathfrak p$ is a one dimensional prime ideal in a local complete Gorenstein domain $(R,\mathfrak{m})$. This is related to results of Enochs and Xu (see [4] and [3]). We prove a certain 'dual' version of the Hartshorne-Lichtenbaum vanishing (see Theorem 2.2). There is a generalization of local duality to cohomologically complete intersection ideals $I$ in the sense that for $I=\mathfrak{m}$ we get back the classical Local Duality Theorem. We determine the exact class of modules to which a characterization of cohomologically complete intersection from [6] generalizes naturally (see Theorem 4.4).