A local homology theory for linearly compact modules

TL;DR

This paper develops a dual local homology theory for linearly compact modules, establishing properties and dualities with local cohomology, and generalizing key results in the field.

Contribution

It introduces a new local homology framework for linearly compact modules, dual to Grothendieck's local cohomology, with foundational properties and duality relations.

Findings

Proves noetherianness and vanishing properties of local homology modules.

Establishes a duality between local homology and local cohomology modules.

Generalizes classical results in local cohomology to semi-discrete linearly compact modules.

Abstract

We introduce a local homology theory for linearly compact modules which is in some sense dual to the local cohomology theory of A. Grothendieck. Some basic properties such as the noetherianness, the vanishing and non-vanishing of local homology modules of linearly compact modules are proved. A duality theory between local homology and local cohomology modules of linearly compact modules is developed by using Matlis duality and Macdonald duality. As consequences of the duality theorem we obtain some generalizations of well-known results in the theory of local cohomology for semi-discrete linearly compact modules.