On the arithmetic of tight closure
Commutative Algebra
2007-05-23 v2 Algebraic Geometry
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 K[x,y,z]/(x^7+y^7-z^7) holds for infinitely many but not for almost all prime characteristics of the field K. This proves that tight closure exhibits a strong dependence on the arithmetic of the prime characteristic. The ideal (x,y,z) \subset 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.
Keywords
Cite
@article{arxiv.math/0412033,
title = {On the arithmetic of tight closure},
author = {Holger Brenner and Mordechai Katzman},
journal= {arXiv preprint arXiv:math/0412033},
year = {2007}
}