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On the arithmetic of tight closure

Authors: Holger Brenner and Mordechai Katzman
Journal: J. Amer. Math. Soc. 19 (2006), 659-672
MSC (2000): Primary 13A35; Secondary 11A41, 14H60
Published electronically: December 22, 2005
MathSciNet review: 2220102
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Abstract: We provide a negative answer to an old question in tight closure theory by showing that the containment $ x^3y^3 \in (x^4,y^4,z^4)^*$ in $ \mathbb{K}[x,y,z]/(x^7+y^7-z^7)$ holds for infinitely many but not for almost all prime characteristics of the field $ \mathbb{K}$. This proves that tight closure exhibits a strong dependence on the arithmetic of the prime characteristic. The ideal $ (x,y,z) \subset \mathbb{K}[x,y,z,u,v,w]/(x^7+y^7-z^7, ux^4+vy^4+wz^4+x^3y^3)$ has then the property that the cohomological dimension fluctuates arithmetically between 0 and $ 1$.

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Additional Information

Holger Brenner
Affiliation: Department of Pure Mathematics, University of Sheffield, Hicks Building, Sheffield S3 7RH, United Kingdom

Mordechai Katzman
Affiliation: Department of Pure Mathematics, University of Sheffield, Hicks Building, Sheffield S3 7RH, United Kingdom

Keywords: Tight closure, dependence on prime numbers, cohomological dimension, semistable bundles.
Received by editor(s): December 3, 2004
Published electronically: December 22, 2005
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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