The Stacks project

Lemma 99.7.1. 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 66.39.6 and 66.37.2. 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. $\square$


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