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

We consider the Fermat cubic  $R=K[X,Y,Z]/(X^3+Y^3+Z^3)$,  the ideal  $I=(X,Y)$  and the element $Z^2$. We claim that in positive characteristic $\ne 3$ the element $Z^2$ does belong to the tight closure of $I$. Equivalently, the open subset

$D(X,Y)\subseteq \mathrm{Spec}(R[S,T]/(XS+YT+Z^2))$

is not affine. The element $Z^2$ defines the cohomology class

$c=\frac{Z^2}{XY}\in H^1(D(X,Y),{𝒪}_X)$

and its Frobenius pull-backs are

$F^{e\ast }(c)=\frac{Z^{2q}}{X^q{Y}^q}\in H^1(D(X,Y),{𝒪}_X).$

This cohomology module has a $\mathbb{Z}$-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 $0$ in positive degree (this is related to the fact that the corresponding projective curve is elliptic). Therefore for any homogeneous element  $t\in R$  of positive degree we have  $t{F}^{e\ast }(c)=0$  and so $Z^2$ belongs to the tight closure.

From this it follows also that in characteristic $0$ the element $Z^2$ belongs to the solid closure, because affineness is an open property in an arithmetic family.