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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On the commutation of the test ideal with localization and completion


Authors: Gennady Lyubeznik and Karen E. Smith
Journal: Trans. Amer. Math. Soc. 353 (2001), 3149-3180
MSC (1991): Primary 13A35; Secondary 13C99
Published electronically: January 18, 2001
MathSciNet review: 1828602
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Abstract:

Let $R$ be a reduced ring that is essentially of finite type over an excellent regular local ring of prime characteristic. Then it is shown that the test ideal of $R$ commutes with localization and, if $R$ is local, with completion, under the additional hypothesis that the tight closure of zero in the injective hull $E$ of the residue field of every local ring of $R$ is equal to the finitistic tight closure of zero in $E$. It is conjectured that this latter condition holds for all local rings of prime characteristic; it is proved here for all Cohen-Macaulay singularities with at most isolated non-Gorenstein singularities, and in general for all isolated singularities. In order to prove the result on the commutation of the test ideal with localization and completion, a ring of Frobenius operators associated to each $R$-module is introduced and studied. This theory gives rise to an ideal of $R$ which defines the non-strongly F-regular locus, and which commutes with localization and completion. This ideal is conjectured to be the test ideal of $R$ in general, and shown to equal the test ideal under the hypothesis that $0_E^*=0_E^{fg*}$in every local ring of $R$.


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Additional Information

Gennady Lyubeznik
Affiliation: Department of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
Email: gennady@math.umn.edu

Karen E. Smith
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
Email: kesmith@math.lsa.umich.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-01-02643-5
PII: S 0002-9947(01)02643-5
Keywords: Tight closure, test ideal, localisation, Frobenius action
Received by editor(s): January 4, 1999
Received by editor(s) in revised form: July 12, 1999, and March 25, 2000
Published electronically: January 18, 2001
Additional Notes: Both authors were supported by the National Science Foundation
Article copyright: © Copyright 2001 American Mathematical Society