Offenes Unterschema/Affin und Erweiterungsideal/Endlich erzeugt/Fakt/en

Let ${}R$ be a noetherian ring and  ${}U=D({\mathfrak {a}})\subseteq \operatorname {Spec} {\left(R\right)}$  an open subset.

Then the following hold.

1. ${}U$ is an affine scheme if and only if  ${}{\mathfrak {a}}\Gamma (U,{\mathcal {O}}_{X})=(1)$. 
2. If this holds, and  ${}q_{1}f_{1}+\cdots +q_{n}f_{n}=1$  with  ${}f_{1},\ldots ,f_{n}\in {\mathfrak {a}}$  and  ${}q_{i}\in \Gamma (U,{\mathcal {O}}_{X})$,  then  ${}\Gamma (U,{\mathcal {O}}_{X})=R[q_{1},\ldots ,q_{n}]$.  In particular, the ring of global sections over ${}U$ is finitely generated over ${}R$.