Fermatgleichung/x^2 in (y,z)^*/Erzwingende Algebra/Verschiedene Charakteristiken/Beispiel/en

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

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

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 Spec (K [X, Y, Z, S, T] / (X^{d} + Y^{d} + Z^{d}, S X + T Y - Z^{2}))

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 Fakt the superheight is one and therefore by Fakt the algebra $Γ ({D (X, Y), 𝒪}_{B})$ is not finitely generated. For $K = ℂ$ and $d = 3$ one can also show that $D (X, Y)_{ℂ}$ is, considered as a complex space, a Stein space.