Stable Local Cohomology

TL;DR

This paper introduces stable local cohomology for modules over Gorenstein rings, extending classical local cohomology by using complete injective resolutions to connect with Gorenstein injective modules.

Contribution

It defines a new stable local cohomology functor that aligns with classical theory and establishes a Gorenstein injective approximation when only one local cohomology module is non-zero.

Findings

Stable local cohomology behaves similarly to classical local cohomology.

When only one local cohomology module is non-zero, the stable version provides a Gorenstein injective approximation.

The new functor maps to the stable category of Gorenstein injective modules.

Abstract

Let $R$ be a Gorenstein local ring, $\frak{a}$ an ideal in $R$, and $M$ an $R$-module. The local cohomology of $M$ supported at $\frak{a}$ can be computed by applying the $\frak{a}$-torsion functor to an injective resolution of $M$. Since $R$ is Gorenstein, $M$ has a complete injective resolution, so it is natural to ask what one gets by applying the $\frak{a}$-torsion functor to it. Following this lead, we define stable local cohomology for modules with complete injective resolutions. This gives a functor to the stable category of Gorenstein injective modules. We show that in many ways this behaves like the usual local cohomology functor. Our main result is that when there is only one non-zero local cohomology module, there is a strong connection between that module and the stable local cohomology module; in fact, the latter gives a Gorenstein injective approximation of the former.