Vol. 2, No. 7, 2008

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The Frobenius structure of local cohomology

Florian Enescu and Melvin Hochster

Vol. 2 (2008), No. 7, 721–754

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.

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
Received: 27 July 2007
Revised: 15 July 2008
Accepted: 26 August 2008
Published: 23 November 2008
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