The Stacks project

Lemma 29.43.18. Let $f : X \to S$ be a universally closed morphism. Let $\mathcal{L}$ be an $f$-ample invertible $\mathcal{O}_ X$-module. Let $s \in \Gamma (X, \mathcal{L})$. Then $X_ s \to S$ is an affine morphism.

Proof. The question is local on $S$ (Lemma 29.11.3) hence we may assume $S$ is affine. By Lemma 29.43.17 we can write $X = \text{Proj}(A)$ where $A$ is a graded ring and $s$ corresponds to $f \in A_1$ and $X_ s = D_+(f)$ (Properties, Lemma 28.26.9) which proves the lemma by construction of $\text{Proj}(A)$, see Constructions, Section 27.8. $\square$

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