Lemma 29.13.8. Suppose $g : X \to Y$ is a morphism of schemes over $S$. If $X$ is quasi-affine over $S$ and $Y$ is quasi-separated over $S$, then $g$ is quasi-affine. In particular, any morphism from a quasi-affine scheme to a quasi-separated scheme is quasi-affine.
Proof. The base change $X \times _ S Y \to Y$ is quasi-affine by Lemma 29.13.5. The morphism $X \to X \times _ S Y$ is a quasi-compact immersion as $Y \to S$ is quasi-separated, see Schemes, Lemma 26.21.11. A quasi-compact immersion is quasi-affine by Lemma 29.13.6 and the composition of quasi-affine morphisms is quasi-affine (see Lemma 29.13.4). Thus we win. $\square$
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Your email address will not be published.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).
All contributions are licensed under the GNU Free Documentation License.