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

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 ${}R$.

Proof, Write another proof