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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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