Fermat equation/x^2 in (y,z)^*/Forcing algebra/Several characteristics/Example

Let ${}K$ be a field and consider the Fermat ring

${}R=K[X,Y,Z]/{\left(X^{d}+Y^{d}+Z^{d}\right)}\,$

together with the ideal ${}I=(X,Y)$ and ${}f=Z^{2}$. For ${}d\geq 3$ we have ${}Z^{2}\notin (X,Y)$. This element is however in the tight closure ${}(X,Y)^{*}$ of the ideal in positive characteristic (assume that the characteristic ${}p$ does not divide ${}d$) and is therefore also in characteristic ${}0$ inside the tight closure and inside the solid closure. Hence the open subset

${}D(X,Y)\subseteq \operatorname {Spec} {\left(K[X,Y,Z,S,T]/{\left(X^{d}+Y^{d}+Z^{d},SX+TY-Z^{2}\right)}\right)}\,$

is not an affine scheme. In positive characteristic, ${}Z^{2}$ is also contained in the plus closure ${}(X,Y)^{+}$ and therefore this open subset contains punctured surfaces (the spectrum of the forcing algebra contains two-dimensional closed subschemes which meet the exceptional fiber ${}V(X,Y)$ in only one point; the ideal ${}(X,Y)$ has superheight two in the forcing algebra). In characteristic zero however, due to remark the superheight is one and therefore by fact the algebra ${}\Gamma ({D(X,Y),{\mathcal {O}}}_{B})$ is not finitely generated. For ${}K=\mathbb {C}$ and ${}d=3$ one can also show that ${}D(X,Y)_{\mathbb {C} }$ is, considered as a complex space, a Stein space.