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

of Lemma 29.37.4 is an isomorphism.

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.
$\square$

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)