On the arithmetic of tight closure
Abstract
We provide a negative answer to an old question in tight closure theory by showing that the containment x3y3 ∈ (x4,y4,z4)* in K[x,y,z]/(x7+y7-z7) 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) ⊂ K[x,y,z,u,v,w]/(x7+y7-z7, ux4+vy4+wz4+x3y3) has then the property that the cohomological dimension fluctuates arithmetically between 0 and 1.
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