The Frobenius structure of local cohomology

Enescu, Florian; Hochster, Melvin

doi:10.2140/ant.2008.2.721

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The Frobenius structure of local cohomology

Florian Enescu and Melvin Hochster

Vol. 2 (2008), No. 7, 721–754

DOI: 10.2140/ant.2008.2.721

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 antinilpotent 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.

Keywords

local cohomology, Frobenius action, Frobenius functor, F-pure ring, Gorenstein ring, antinilpotent module, tight closure, face ring, FH-finite ring, finite FH-length

Mathematical Subject Classification 2000

Primary: 13A35

Secondary: 13D45

Milestones

Received: 27 July 2007

Revised: 15 July 2008

Accepted: 26 August 2008

Published: 23 November 2008

Authors

Florian Enescu
Department of Mathematics and Statistics Georgia State University 750 COE, 7th floor, 30 Pryor Street Atlanta, GA 30303-3083 United States

Melvin Hochster
Department of Mathematics University of Michigan 2074 East Hall 530 Church Street Ann Arbor, MI 48109-1043 United States

Vol. 2, No. 7, 2008