On the arithmetic of tight closure

Original · EN

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.