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

DOI:
https://doi.org/10.1090/S0002-9947-01-02643-5

Published electronically:
January 18, 2001

MathSciNet review:
1828602

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Abstract | References | Similar Articles | Additional Information

Let 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 commutes with localization and, if is local, with completion, under the additional hypothesis that the tight closure of zero in the injective hull of the residue field of every local ring of is equal to the finitistic tight closure of zero in . 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 -module is introduced and studied. This theory gives rise to an ideal of 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 in general, and shown to equal the test ideal under the hypothesis that in every local ring of .

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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:
https://doi.org/10.1090/S0002-9947-01-02643-5

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