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

Beweis

We only give a sketch. (1). There always exists a natural scheme morphism

${\displaystyle U\longrightarrow \operatorname {Spec} {\left(\Gamma (U,{\mathcal {O}}_{X})\right)},}$

and ${\displaystyle {}U}$ is affine if and only if this morphism is an isomorphism. It is always an open embedding (because it is an isomorphism on the ${\displaystyle D(f)}$, ${\displaystyle {}f\in {\mathfrak {a}}}$), and the image is ${\displaystyle {}D({\mathfrak {a}}\Gamma (U,{\mathcal {O}}_{X}))}$. This is everything if and only if the extended ideal is the unit ideal.

(2). We write ${\displaystyle {}1=q_{1}f_{1}+\cdots +q_{n}f_{n}}$ and consider the natural morphism

${\displaystyle U\longrightarrow \operatorname {Spec} {\left(R[q_{1},\ldots ,q_{n}]\right)}}$

corresponding to the ring inclusion ${\displaystyle {}R[q_{1},\ldots ,q_{n}]\subseteq \Gamma (U,{\mathcal {O}}_{X})}$. This morphism is again an open embedding and its image is everything.