The Stacks project

Lemma 100.27.13. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks which is surjective, flat, and locally of finite presentation. Then for every scheme $U$ and object $y$ of $\mathcal{Y}$ over $U$ there exists an fppf covering $\{ U_ i \to U\} $ and objects $x_ i$ of $\mathcal{X}$ over $U_ i$ such that $f(x_ i) \cong y|_{U_ i}$ in $\mathcal{Y}_{U_ i}$.

Proof. We may think of $y$ as a morphism $U \to \mathcal{Y}$. By Properties of Stacks, Lemma 99.5.3 and Lemmas 100.27.3 and 100.25.3 we see that $\mathcal{X} \times _\mathcal {Y} U \to U$ is surjective, flat, and locally of finite presentation. Let $V$ be a scheme and let $V \to \mathcal{X} \times _\mathcal {Y} U$ smooth and surjective. Then $V \to \mathcal{X} \times _\mathcal {Y} U$ is also surjective, flat, and locally of finite presentation (see Morphisms of Spaces, Lemmas 66.37.7 and 66.37.5). Hence also $V \to U$ is surjective, flat, and locally of finite presentation, see Properties of Stacks, Lemma 99.5.2 and Lemmas 100.27.2, and 100.25.2. Hence $\{ V \to U\} $ is the desired fppf covering and $x : V \to \mathcal{X}$ is the desired object. $\square$

Comments (0)

There are also:

  • 2 comment(s) on Section 100.27: Morphisms of finite presentation

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Your email address will not be published.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 06QA. Beware of the difference between the letter 'O' and the digit '0'.