Fermat-Kubik/z^2 in tight closure von (x,y)/Beispiel/en

We consider the Fermat cubic , the ideal and the element . We claim that in positive characteristic the element does belong to the tight closure of . Equivalently, the open subset

is not affine. The element defines the cohomology class

and its Frobenius pull-backs are

This cohomology module has a -graded structure (the degree is given by the difference of the degree of the numerator and the degree of the denominator) and, moreover, it is in positive degree (this is related to the fact that the corresponding projective curve is elliptic). Therefore for any homogeneous element of positive degree we have and so belongs to the tight closure.

From this it follows also that in characteristic the element belongs to the solid closure, because affineness is an open property in an arithmetic family.