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
-
![{\displaystyle {}D(X,Y)\subseteq \operatorname {Spec} {\left(R[S,T]/(XS+YT-Z)\right)}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/22a2adff12d669705e0acf2bc5555d5901de846d)
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
-
![{\displaystyle {}X^{3}+Y^{3}=(XS+YT)^{3}=X^{3}S^{3}+3X^{2}S^{2}YT+3XSY^{2}T^{2}+Y^{3}T^{3}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f80bc4887533c6879d08a7a00a445c7e26094f2)
or, equivalently,
-
![{\displaystyle {}X^{3}{\left(S^{3}-1\right)}+3X^{2}YS^{2}T+3XY^{2}ST^{2}+Y^{3}{\left(T^{3}-1\right)}=0\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ad625a4228cfd57fb5b28305309d5f063d9a43e)
We write this as
![{\displaystyle {}{\begin{aligned}X^{3}{\left(S^{3}-1\right)}&=-3X^{2}YS^{2}T-3XY^{2}ST^{2}-Y^{3}{\left(T^{3}-1\right)}\\&=Y{\left(-3X^{2}S^{2}T-3XYST^{2}-Y^{2}{\left(T^{3}-1\right)}\right)},\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b4ef1dca8f4708b37a0fcd731494deda5e6d299)
which yields on
the rational function
-
![{\displaystyle {}Q={\frac {S^{3}-1}{Y}}={\frac {-3X^{2}S^{2}T-3XYST^{2}-Y^{2}{\left(T^{3}-1\right)}}{X^{3}}}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0bfebf545d6d004095793fcda597736be9382614)
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.