Dreidimensionale Standardquadrik/Nicht affin/Beispiel/en

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

${}R=K[x,y,u,v]/(xu-yv)\,.$

The ideal ${}{\mathfrak {p}}=(x,y)$ is a prime ideal in ${}R$ of height one. Hence the open subset ${}U=D(x,y)$ is the complement of an irreducible hypersurface. However, ${}U$ is not affine. For this we consider the closed subscheme

${}{\mathbb {A} }_{K}^{2}\cong Z=V(u,v)\subseteq \operatorname {Spec} {\left(R\right)}\,$

and ${}Z\cap U\subseteq U$. If ${}U$ were affine, then also the closed subscheme ${}Z\cap U\cong {\mathbb {A} }_{K}^{2}\setminus \{(0,0)\}$ would be affine, but this is not true, since the complement of the punctured plane has codimension ${}2$.