The Stacks project

Lemma 86.18.7. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of locally Noetherian formal algebraic spaces which is locally of finite type. Let $\{ g_ i : Y_ i \to Y\} $ be a family of morphisms of formal algebraic spaces which are representable by algebraic spaces and ├ętale such that $\coprod g_ i$ is surjective. Then $f$ is rig-surjective if and only if each $f_ i : X \times _ Y Y_ i \to Y_ i$ is rig-surjective.

Proof. Namely, if $f$ is rig-surjective, so is any base change (Lemma 86.18.4). Conversely, if all $f_ i$ are rig-surjective, so is $\coprod f_ i : \coprod X \times _ Y Y_ i \to \coprod Y_ i$. By Lemma 86.18.6 the morphism $\coprod g_ i : \coprod Y_ i \to Y$ is rig-surjective. Hence $\coprod X \times _ Y Y_ i \to Y$ is rig-surjective (Lemma 86.18.3). Since this morphism factors through $X \to Y$ we see that $X \to Y$ is rig-surjective by Lemma 86.18.5. $\square$


Comments (0)


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Your email address will not be published.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 0AQV. Beware of the difference between the letter 'O' and the digit '0'.