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.
$\square$

## Post a comment

Your email address will not be published.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)