$F$-stable submodules of top local cohomology modules of Gorenstein rings

TL;DR

This paper characterizes specific submodules of top local cohomology modules over Gorenstein rings using $F$-finite modules, providing an explicit description of the test ideal in terms of ideals in a regular ring.

Contribution

It offers a concrete description of $F$-stable submodules of top local cohomology modules in Gorenstein rings, linking them to test ideals via $F$-finite modules.

Findings

Explicit description of $F$-stable submodules in Gorenstein rings.

Connection between test ideals and ideals in regular rings.

Application of Lyubeznik's $F$-finite modules to local cohomology.

Abstract

This paper applies G. Lyubeznik's notion of $F$-finite modules to describe in a very down-to-earth manner certain annihilator submodules of some top local cohomology modules over Gorenstein rings. As a consequence we obtain an explicit description of the test ideal of Gorenstein rings in terms of ideals in a regular ring.