The Stacks project

Lemma 28.18.3. Let $X$ be a scheme. Let $f \in \Gamma (X, \mathcal{O}_ X)$. Assume $X$ is quasi-compact and quasi-separated and assume that $X_ f$ is affine. Then the canonical morphism

\[ j : X \longrightarrow \mathop{\mathrm{Spec}}(\Gamma (X, \mathcal{O}_ X)) \]

from Schemes, Lemma 26.6.4 induces an isomorphism of $X_ f = j^{-1}(D(f))$ onto the standard affine open $D(f) \subset \mathop{\mathrm{Spec}}(\Gamma (X, \mathcal{O}_ X))$.

Proof. This is clear as $j$ induces an isomorphism of rings $\Gamma (X, \mathcal{O}_ X)_ f \to \mathcal{O}_ X(X_ f)$ by Lemma 28.17.1 above. $\square$

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