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Transactions of the American Mathematical Society

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Test ideals in non- $ \mathbb{Q}$-Gorenstein rings


Author: Karl Schwede
Journal: Trans. Amer. Math. Soc. 363 (2011), 5925-5941
MSC (2010): Primary 13A35, 14F18, 14B05
DOI: https://doi.org/10.1090/S0002-9947-2011-05297-9
Published electronically: June 3, 2011
MathSciNet review: 2817415
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Abstract: Suppose that $ X = \operatorname{Spec}R$ is an $ F$-finite normal variety in characteristic $ p > 0$. In this paper we show that the big test ideal $ \tau_b(R) = \widetilde{\tau(R)}$ is equal to $ \sum_{\Delta} \tau(R; \Delta)$, where the sum is over $ \Delta$ such that $ K_X + \Delta$ is $ \mathbb{Q}$-Cartier. This affirmatively answers a question asked by various people, including Blickle, Lazarsfeld, K. Lee and K. Smith. Furthermore, we have a version of this result in the case that $ R$ is not even necessarily normal.


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Additional Information

Karl Schwede
Affiliation: Department of Mathematics, The Pennsylvania State University, 318C McAllister Building, University Park, Pennsylvania 16802
Email: schwede@math.psu.edu

DOI: https://doi.org/10.1090/S0002-9947-2011-05297-9
Keywords: Tight closure, test ideal, $\mathbb{Q}$-Gorenstein, log $\mathbb{Q}$-Gorenstein, multiplier ideal, $F$-singularities
Received by editor(s): June 24, 2009
Received by editor(s) in revised form: November 30, 2009
Published electronically: June 3, 2011
Additional Notes: The author was partially supported by a National Science Foundation postdoctoral fellowship and by RTG grant number 0502170.
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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