The Stacks project

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. $\square$