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

Let $R$ be a noetherian ring and $U = D (𝔞) \subseteq Spec (R)$ an open subset.

Then the following hold.

$U$ is an affine scheme if and only if $𝔞 Γ (U, 𝒪_{X}) = (1)$ .

If this holds, and $q_{1} f_{1} + \dots + q_{n} f_{n} = 1$ with $f_{1}, \dots, f_{n} \in 𝔞$ and $q_{i} \in Γ (U, 𝒪_{X})$ , then $Γ (U, 𝒪_{X}) = R [q_{1}, \dots, q_{n}]$ . In particular, the ring of global sections over $U$ is finitely generated over $R$ .