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Parameter-test-ideals of Cohen–Macaulay rings

Part of: Commutative algebra: Homological methods Computational aspects and applications General commutative ring theory

Published online by Cambridge University Press:  01 July 2008

Mordechai Katzman
Mordechai Katzman*
Affiliation:
Department of Pure Mathematics, University of Sheffield, Hicks Building, Sheffield S3 7RH, UK (email: M.Katzman@sheffield.ac.uk)
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Abstract

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We describe an algorithm for computing parameter-test-ideals in certain local Cohen–Macaulay rings. The algorithm is based on the study of a Frobenius map on the injective hull of the residue field of the ring and on the application of Sharp’s notion of ‘special ideals’. Our techniques also provide an algorithm for computing indices of nilpotency of Frobenius actions on top local cohomology modules of the ring and on the injective hull of its residue field. The study of nilpotent elements on injective hulls of residue fields also yields a great simplification of the proof of the celebrated result in the article Generators of D-modules in positive characteristic (J. Alvarez-Montaner, M. Blickle and G. Lyubeznik, Math. Res. Lett. 12 (2005), 459–473).

Keywords

tight closuretest-idealFrobenius maplocal cohomology

MSC classification

Secondary: 13A35: Characteristic $p$ methods (Frobenius endomorphism) and reduction to characteristic $p$; tight closure 13D45: Local cohomology 13P99: None of the above, but in this section
Type
Research Article
Information
Compositio Mathematica , Volume 144 , Issue 4 , July 2008 , pp. 933 - 948
Copyright
Copyright © Foundation Compositio Mathematica 2008