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 x 3 y 3 ∈ ( x 4 , y 4 , z 4 ) ∗ x^3y^3 \in (x^4,y^4,z^4)^* in K [ x , y , z ] / ( x 7 + y 7 − z 7 ) \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 K \mathbb {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 ] / ( x 7 + y 7 − z 7 , u x 4 + v y 4 + w z 4 + x 3 y 3 ) (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 0 and 1 1 .