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The
Frobenius structure of local cohomology
Florian Enescu and Melvin Hochster |
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Vol. 2 (2008), No. 7, 721–754
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Abstract |
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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. |
Keywords
local cohomology, Frobenius action, Frobenius functor,
F-pure ring, Gorenstein ring, antinilpotent module, tight
closure, face ring, FH-finite ring, finite FH-length
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Mathematical Subject Classification 2000
Primary: 13A35
Secondary: 13D45
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Milestones
Received: 27 July 2007
Revised: 15 July 2008
Accepted: 26 August 2008
Published: 23 November 2008
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Authors |
| Florian Enescu | |
| Department of Mathematics and
Statistics Georgia State University 750 COE, 7th floor, 30 Pryor Street Atlanta, GA 30303-3083 United States |
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| Melvin Hochster | |
| Department of Mathematics University of Michigan 2074 East Hall 530 Church Street Ann Arbor, MI 48109-1043 United States |
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