Algebra & Number Theory

The Frobenius structure of local cohomology

Florian Enescu and Melvin Hochster

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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.

Article information

Algebra Number Theory, Volume 2, Number 7 (2008), 721-754.

Received: 27 July 2007
Revised: 15 July 2008
Accepted: 26 August 2008
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 13A35: Characteristic p methods (Frobenius endomorphism) and reduction to characteristic p; tight closure [See also 13B22]
Secondary: 13D45: Local cohomology [See also 14B15]

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


Enescu, Florian; Hochster, Melvin. The Frobenius structure of local cohomology. Algebra Number Theory 2 (2008), no. 7, 721--754. doi:10.2140/ant.2008.2.721.

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