The Stacks project

Lemma 37.63.2. For a morphism of schemes $f : X \to Y$, the following are equivalent:

  1. $f$ is ind-quasi-affine,

  2. for every affine open subscheme $V \subset Y$ and every quasi-compact open subscheme $U \subset f^{-1}(V)$, the induced morphism $U \to V$ is quasi-affine.

  3. for some cover $\{ V_ j \} _{j \in J}$ of $Y$ by quasi-compact and quasi-separated open subschemes $V_ j \subset Y$, every $j \in J$, and every quasi-compact open subscheme $U \subset f^{-1}(V_ j)$, the induced morphism $U \to V_ j$ is quasi-affine.

  4. for every quasi-compact and quasi-separated open subscheme $V \subset Y$ and every quasi-compact open subscheme $U \subset f^{-1}(V)$, the induced morphism $U \to V$ is quasi-affine.

In particular, the property of being an ind-quasi-affine morphism is Zariski local on the base.

Proof. The equivalence (1) $\Leftrightarrow $ (2) follows from the definitions and Morphisms, Lemma 29.13.3. For (2) $\Rightarrow $ (4), let $U$ and $V$ be as in (4). By Schemes, Lemma 26.21.14, the induced morphism $U \to V$ is quasi-compact. Thus, for every affine open $V' \subset V$, the fiber product $V' \times _ V U$ is quasi-compact, so, by (2), the induced map $V' \times _ V U \to V'$ is quasi-affine. Thus, $U \to V$ is also quasi-affine by Morphisms, Lemma 29.13.3. This argument also gives (3) $\Rightarrow $ (4): indeed, keeping the same notation, those affine opens $V' \subset V$ that lie in one of the $V_ j$ cover $V$, so one needs to argue that the quasi-compact map $V' \times _ V U \to V'$ is quasi-affine. However, by (3), the composition $V' \times _ V U \to V' \to V_ j$ is quasi-affine and, by Schemes, Lemma 26.21.13, the map $V' \to V_ j$ is quasi-separated. Thus, $V' \times _ V U \to V'$ is quasi-affine by Morphisms, Lemma 29.13.8. The final implications (4) $\Rightarrow $ (2) and (4) $\Rightarrow $ (3) are evident. $\square$


Comments (0)

There are also:

  • 2 comment(s) on Section 37.63: Ind-quasi-affine morphisms

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Your email address will not be published.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.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0F1U. Beware of the difference between the letter 'O' and the digit '0'.