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

We consider the Fermat cubic , the ideal and the element . We claim that for characteristic the element does not belong to the solid closure of . Equivalently, the open subset

is affine. For this we show that the extended ideal inside the ring of global sections is the unit ideal. First of all we get the equation

or, equivalently,

We write this as

which yields on the rational function

This shows that belongs to the extended ideal. Similarly, one can show that also the other coefficients belong to the extended ideal. Therefore in characteristic different from , the extended ideal is the unit ideal.