The Stacks project

Lemma 29.43.17. Let $f : X \to S$ be a universally closed morphism. Let $\mathcal{L}$ be an $f$-ample invertible $\mathcal{O}_ X$-module. Then the canonical morphism

\[ r : X \longrightarrow \underline{\text{Proj}}_ S \left( \bigoplus \nolimits _{d \geq 0} f_*\mathcal{L}^{\otimes d} \right) \]

of Lemma 29.37.4 is an isomorphism.

Proof. Observe that $f$ is quasi-compact because the existence of an $f$-ample invertible module forces $f$ to be quasi-compact. By the lemma cited the morphism $r$ is an open immersion. On the other hand, the image of $r$ is closed by Lemma 29.41.7 (the target of $r$ is separated over $S$ by Constructions, Lemma 27.16.9). Finally, the image of $r$ is dense by Properties, Lemma 28.26.11 (here we also use that it was shown in the proof of Lemma 29.37.4 that the morphism $r$ over affine opens of $S$ is given by the canonical morphism of Properties, Lemma 28.26.9). Thus we conclude that $r$ is a surjective open immersion, i.e., an isomorphism. $\square$

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