## 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 $1$.## References

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## Bibliographic Information

**Holger Brenner**- Affiliation: Department of Pure Mathematics, University of Sheffield, Hicks Building, Sheffield S3 7RH, United Kingdom
- MR Author ID: 322383
- Email: H.Brenner@sheffield.ac.uk
**Mordechai Katzman**- Affiliation: Department of Pure Mathematics, University of Sheffield, Hicks Building, Sheffield S3 7RH, United Kingdom
- Email: M.Katzman@sheffield.ac.uk
- Received by editor(s): December 3, 2004
- Published electronically: December 22, 2005
- © Copyright 2005
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: J. Amer. Math. Soc.
**19**(2006), 659-672 - MSC (2000): Primary 13A35; Secondary 11A41, 14H60
- DOI: https://doi.org/10.1090/S0894-0347-05-00514-X
- MathSciNet review: 2220102