The Stacks project

Lemma 98.3.2. 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:

  1. $f$ has $P$,

  2. 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 ├ętale morphism $U \to Z$. By assumption the morphism $U \times _\mathcal {Y} \mathcal{X} \to U$ has $P$. But the diagram

\[ \xymatrix{ U \times _\mathcal {Y} \mathcal{X} \ar[d] \ar[r] & Z \times _\mathcal {Y} \mathcal{X} \ar[d] \\ U \ar[r] & Z } \]

is cartesian, hence the right vertical arrow has $P$ as $\{ U \to Z\} $ is an fppf covering. $\square$

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