  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.
