Lemma 100.3.2. Let $P$ be a property of morphisms of algebraic spaces as above. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks representable by algebraic spaces. The following are equivalent:
$f$ has $P$,
for every algebraic space $Z$ and morphism $Z \to \mathcal{Y}$ the morphism $Z \times _\mathcal {Y} \mathcal{X} \to Z$ has $P$.
Proof. The implication (2) $\Rightarrow $ (1) is immediate. Assume (1). Let $Z \to \mathcal{Y}$ be as in (2). Choose a scheme $U$ and a surjective étale morphism $U \to Z$. By assumption the morphism $U \times _\mathcal {Y} \mathcal{X} \to U$ has $P$. But the diagram
is cartesian, hence the right vertical arrow has $P$ as $\{ U \to Z\} $ is an fppf covering. $\square$
Your email address will not be published.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.
Comments (0)