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