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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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On the commutation of the test ideal with localization and completion
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by Gennady Lyubeznik and Karen E. Smith PDF
Trans. Amer. Math. Soc. 353 (2001), 3149-3180 Request permission

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
  • MR Author ID: 117320
  • Email: gennady@math.umn.edu
  • Karen E. Smith
  • Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
  • MR Author ID: 343614
  • Email: kesmith@math.lsa.umich.edu
  • 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
  • © Copyright 2001 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 353 (2001), 3149-3180
  • MSC (1991): Primary 13A35; Secondary 13C99
  • DOI: https://doi.org/10.1090/S0002-9947-01-02643-5
  • MathSciNet review: 1828602