The Stacks project

Lemma 100.7.1. Let $\mathcal{P}$ be a property of schemes which is local in the smooth topology, see Descent, Definition 35.15.1. Let $\mathcal{X}$ be an algebraic stack. The following are equivalent

  1. for some scheme $U$ and some surjective smooth morphism $U \to \mathcal{X}$ the scheme $U$ has property $\mathcal{P}$,

  2. for every scheme $U$ and every smooth morphism $U \to \mathcal{X}$ the scheme $U$ has property $\mathcal{P}$,

  3. for some algebraic space $U$ and some surjective smooth morphism $U \to \mathcal{X}$ the algebraic space $U$ has property $\mathcal{P}$, and

  4. for every algebraic space $U$ and every smooth morphism $U \to \mathcal{X}$ the algebraic space $U$ has property $\mathcal{P}$.

If $\mathcal{X}$ is a scheme this is equivalent to $\mathcal{P}(U)$. If $\mathcal{X}$ is an algebraic space this is equivalent to $X$ having property $\mathcal{P}$.

Proof. Let $U \to \mathcal{X}$ surjective and smooth with $U$ an algebraic space. Let $V \to \mathcal{X}$ be a smooth morphism with $V$ an algebraic space. Choose schemes $U'$ and $V'$ and surjective étale morphisms $U' \to U$ and $V' \to V$. Finally, choose a scheme $W$ and a surjective étale morphism $W \to V' \times _\mathcal {X} U'$. Then $W \to V'$ and $W \to U'$ are smooth morphisms of schemes as compositions of étale and smooth morphisms of algebraic spaces, see Morphisms of Spaces, Lemmas 67.39.6 and 67.37.2. Moreover, $W \to V'$ is surjective as $U' \to \mathcal{X}$ is surjective. Hence, we have

\[ \mathcal{P}(U) \Leftrightarrow \mathcal{P}(U') \Rightarrow \mathcal{P}(W) \Rightarrow \mathcal{P}(V') \Leftrightarrow \mathcal{P}(V) \]

where the equivalences are by definition of property $\mathcal{P}$ for algebraic spaces, and the two implications come from Descent, Definition 35.15.1. This proves (3) $\Rightarrow $ (4).

The implications (2) $\Rightarrow $ (1), (1) $\Rightarrow $ (3), and (4) $\Rightarrow $ (2) are immediate. $\square$

Comments (0)

There are also:

  • 2 comment(s) on Section 100.7: Properties of algebraic stacks defined by properties of schemes

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Your email address will not be published.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 04YF. Beware of the difference between the letter 'O' and the digit '0'.