The Stacks project

Lemma 99.3.5. Let $P$ be a property of morphisms of algebraic spaces as above. Let $\tau \in \{ {\acute{e}tale}, smooth, syntomic, fppf\} $. Let $\mathcal{X} \to \mathcal{Y}$ and $\mathcal{Y} \to \mathcal{Z}$ be morphisms of algebraic stacks representable by algebraic spaces. Assume

  1. $\mathcal{X} \to \mathcal{Y}$ is surjective and ├ętale, smooth, syntomic, or flat and locally of finite presentation,

  2. the composition has $P$, and

  3. $P$ is local on the source in the $\tau $ topology.

Then $\mathcal{Y} \to \mathcal{Z}$ has property $P$.

Proof. Let $Z$ be a scheme and let $Z \to \mathcal{Z}$ be a morphism. Set $X = \mathcal{X} \times _\mathcal {Z} Z$, $Y = \mathcal{Y} \times _\mathcal {Z} Z$. By (1) $\{ X \to Y\} $ is a $\tau $ covering of algebraic spaces and by (2) $X \to Z$ has property $P$. By (3) this implies that $Y \to Z$ has property $P$ and we win. $\square$

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