An application of generalized Matlis duality for quasi-\F-modules to the Artinianness of local cohomology modules

TL;DR

This paper explores the use of generalized Matlis duality for quasi- ext{F}-modules to analyze the Artinianness of local cohomology modules, providing new insights and examples in the context of ext{F}-modules and local duality.

Contribution

It introduces a novel application of generalized Matlis duality to quasi- ext{F}-modules and offers new examples of non-Artinian local cohomology modules, enhancing understanding of their structure.

Findings

Describes generalized Matlis duals for certain quasi- ext{F}-modules.

Provides examples of non-Artinian local cohomology modules.

Offers a new perspective on Hartshorne's counterexample related to Grothendieck's conjecture.

Abstract

We use a result of Hellus about generalized local duality to describe some generalized Matlis duals for certain quasi-\F-modules. Furthermore, we apply this description to obtain examples for non-artinian local cohomology modules by the theory of \F-modules. In particular, we get a new view on Hartshorne's counterexample for a conjecture by Grothendieck about the finiteness of $Hom_R(R/I,H^i_I(R))$ for a noetherian local Ring $R$ and an ideal $I \subseteq R$.