Open subscheme/Affine and extended ideal/Finitely generated/Fact/Proof

Proof

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

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

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

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

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

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

To fact