On the arithmetic of tight closure

Brenner, Holger; Katzman, Mordechai

doi:10.1090/S0894-0347-05-00514-X

On the arithmetic of tight closure
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by Holger Brenner and Mordechai Katzman

J. Amer. Math. Soc. 19 (2006), 659-672

DOI: https://doi.org/10.1090/S0894-0347-05-00514-X

Published electronically: December 22, 2005

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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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