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ISSN 1088-6834(e) ISSN 0894-0347(p)

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
Posted: December 22, 2005
MathSciNet review: 2220102
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Abstract | References | Similar Articles | Additional Information

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 .

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