Fermat-Kubik/z nicht 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 $Z$ . We claim that for characteristic $\neq 3$ the element $Z$ does not belong to the solid closure of $I$ . Equivalently, the open subset

D (X, Y) \subseteq Spec (R [S, T] / (X S + Y T - Z))

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

X^{3} + Y^{3} = (X S + Y T)^{3} = X^{3} S^{3} + 3 X^{2} S^{2} Y T + 3 X S Y^{2} T^{2} + Y^{3} T^{3}

or, equivalently,

X^{3} (S^{3} - 1) + 3 X^{2} Y S^{2} T + 3 X Y^{2} S T^{2} + Y^{3} (T^{3} - 1) = 0 .

We write this as

\begin{matrix} X^{3} (S^{3} - 1) & = - 3 X^{2} Y S^{2} T - 3 X Y^{2} S T^{2} - Y^{3} (T^{3} - 1) \\ = Y (- 3 X^{2} S^{2} T - 3 X Y S T^{2} - Y^{2} (T^{3} - 1)), \end{matrix}

which yields on $D (X, Y)$ the rational function

Q = \frac{S^{3} - 1}{Y} = \frac{- 3 X^{2} S^{2} T - 3 X Y S T^{2} - Y^{2} (T^{3} - 1)}{X^{3}} .

This shows that $S^{3} - 1 = Q Y$ belongs to the extended ideal. Similarly, one can show that also the other coefficients $3 S^{2} T, 3 S T^{2}, T^{3} - 1$ belong to the extended ideal. Therefore in characteristic different from $3$ , the extended ideal is the unit ideal.