On the arithmetic of tight closure

Brenner, Holger¹ ; Katzman, Mordechai¹

¹ Department of Pure Mathematics, University of Sheffield, Hicks Building, Sheffield S3 7RH, United Kingdom

Journal of the American Mathematical Society, Tome 19 (2006) no. 3, pp. 659-672

Résumé

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 $1$.

Détail

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

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