Noethersches Schema/Affin/Kohomologisches Kriterium/en/Fakt

Let ${}X$ denote a noetherian scheme. Then the following properties are equivalent.

1. ${}X$ is an affine scheme.
2. For every quasicoherent sheaf ${}{\mathcal {F}}$ on ${}X$ and all ${}i\geq 1$ we have ${}H^{i}(X,{\mathcal {F}})=0$.
3. For every coherent ideal sheaf ${}{\mathcal {I}}$ on ${}X$ we have ${}H^{1}(X,{\mathcal {I}})=0$.

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