The Frobenius Structure of Local Cohomology

TL;DR

This paper investigates the Frobenius action on local cohomology modules in prime characteristic rings, identifying conditions under which these modules have finitely many Frobenius-invariant submodules, with implications for F-pure Gorenstein rings and face rings.

Contribution

It introduces the concept of anti-nilpotent Frobenius actions and characterizes rings with finitely many Frobenius-invariant submodules in local cohomology.

Findings

F-pure Gorenstein local rings have finitely many Frobenius-invariant submodules.

Face rings of finite simplicial complexes exhibit this property.

The lattice of Frobenius-invariant submodules satisfies the Ascending Chain Condition.

Abstract

Given a local ring of positive prime characteristic there is a natural Frobenius action on its local cohomology modules with support at its maximal ideal. In this paper we study the local rings for which the local cohomology modules have only finitely many submodules invariant under the Frobenius action. In particular we prove that F-pure Gorenstein local rings as well as the face ring of a finite simplicial complex localized or completed at its homogeneous maximal ideal have this property. We also introduce the notion of an anti-nilpotent Frobenius action on an Artinian module over a local ring and use it to study those rings for which the lattice of submodules of the local cohomology that are invariant under Frobenius satisfies the Ascending Chain Condition.