Lemma 101.34.1. Let \mathcal{P} be a property of morphisms of algebraic spaces which is étale-smooth local on the source-and-target. Let f : \mathcal{X} \to \mathcal{Y} be a DM morphism of algebraic stacks. Consider commutative diagrams
\xymatrix{ U \ar[d]_ a \ar[r]_ h & V \ar[d]^ b \\ \mathcal{X} \ar[r]^ f & \mathcal{Y} }
where U and V are algebraic spaces, V \to \mathcal{Y} is smooth, and U \to \mathcal{X} \times _\mathcal {Y} V is étale. The following are equivalent
for any diagram as above the morphism h has property \mathcal{P}, and
for some diagram as above with a : U \to \mathcal{X} surjective the morphism h has property \mathcal{P}.
If \mathcal{X} and \mathcal{Y} are representable by algebraic spaces, then this is also equivalent to f (as a morphism of algebraic spaces) having property \mathcal{P}. If \mathcal{P} is also preserved under any base change, and fppf local on the base, then for morphisms f which are representable by algebraic spaces this is also equivalent to f having property \mathcal{P} in the sense of Properties of Stacks, Section 100.3.
Proof.
Let us prove the implication (1) \Rightarrow (2). Pick an algebraic space V and a surjective and smooth morphism V \to \mathcal{Y}. As f is DM there exists a scheme U and a surjective étale morphism U \to V \times _\mathcal {Y} \mathcal{X}, see Lemma 101.21.7. Thus we see that (2) holds. Note that U \to \mathcal{X} is surjective and smooth as well, as a composition of the base change \mathcal{X} \times _\mathcal {Y} V \to \mathcal{X} and the chosen map U \to \mathcal{X} \times _\mathcal {Y} V. Hence we obtain a diagram as in (1). Thus if (1) holds, then h : U \to V has property \mathcal{P}, which means that (2) holds as U \to \mathcal{X} is surjective.
Conversely, assume (2) holds and let U, V, a, b, h be as in (2). Next, let U', V', a', b', h' be any diagram as in (1). Picture
\xymatrix{ U \ar[d] \ar[r]_ h & V \ar[d] \\ \mathcal{X} \ar[r]^ f & \mathcal{Y} } \quad \quad \xymatrix{ U' \ar[d] \ar[r]_{h'} & V' \ar[d] \\ \mathcal{X} \ar[r]^ f & \mathcal{Y} }
To show that (2) implies (1) we have to prove that h' has \mathcal{P}. To do this consider the commutative diagram
\xymatrix{ U \ar[d]^ h & U \times _\mathcal {X} U' \ar[l] \ar[d]^{(h, h')} \ar[r] & U' \ar[d]^{h'} \\ V & V \times _\mathcal {Y} V' \ar[l] \ar[r] & V' }
of algebraic spaces. Note that the horizontal arrows are smooth as base changes of the smooth morphisms V \to \mathcal{Y}, V' \to \mathcal{Y}, U \to \mathcal{X}, and U' \to \mathcal{X}. Note that the squares
\xymatrix{ U \ar[d] & U \times _\mathcal {X} U' \ar[l] \ar[d] & U \times _\mathcal {X} U' \ar[d] \ar[r] & U' \ar[d] \\ V \times _\mathcal {Y} \mathcal{X} & V \times _\mathcal {Y} U' \ar[l] & U \times _\mathcal {Y} V' \ar[r] & \mathcal{X} \times _\mathcal {Y} V' }
are cartesian, hence the vertical arrows are étale by our assumptions on U', V', a', b', h' and U, V, a, b, h. Since \mathcal{P} is smooth local on the target by Descent on Spaces, Lemma 74.21.2 part (2) we see that the base change t : U \times _\mathcal {Y} V' \to V \times _\mathcal {Y} V' of h has \mathcal{P}. Since \mathcal{P} is étale local on the source by Descent on Spaces, Lemma 74.21.2 part (1) and s : U \times _\mathcal {X} U' \to U \times _\mathcal {Y} V' is étale, we see the morphism (h, h') = t \circ s has \mathcal{P}. Consider the diagram
\xymatrix{ U \times _\mathcal {X} U' \ar[r]_{(h, h')} \ar[d] & V \times _\mathcal {Y} V' \ar[d] \\ U' \ar[r]^{h'} & V' }
The left vertical arrow is surjective, the right vertical arrow is smooth, and the induced morphism
U \times _\mathcal {X} U' \longrightarrow U' \times _{V'} (V \times _\mathcal {Y} V') = V \times _\mathcal {Y} U'
is étale as seen above. Hence by Descent on Spaces, Definition 74.21.1 part (3) we conclude that h' has \mathcal{P}. This finishes the proof of the equivalence of (1) and (2).
If \mathcal{X} and \mathcal{Y} are representable, then Descent on Spaces, Lemma 74.21.3 applies which shows that (1) and (2) are equivalent to f having \mathcal{P}.
Finally, suppose f is representable, and U, V, a, b, h are as in part (2) of the lemma, and that \mathcal{P} is preserved under arbitrary base change. We have to show that for any scheme Z and morphism Z \to \mathcal{X} the base change Z \times _\mathcal {Y} \mathcal{X} \to Z has property \mathcal{P}. Consider the diagram
\xymatrix{ Z \times _\mathcal {Y} U \ar[d] \ar[r] & Z \times _\mathcal {Y} V \ar[d] \\ Z \times _\mathcal {Y} \mathcal{X} \ar[r] & Z }
Note that the top horizontal arrow is a base change of h and hence has property \mathcal{P}. The left vertical arrow is surjective, the induced morphism
Z \times _\mathcal {Y} U \longrightarrow (Z \times _\mathcal {Y} \mathcal{X}) \times _ Z (Z \times _\mathcal {Y} V)
is étale, and the right vertical arrow is smooth. Thus Descent on Spaces, Lemma 74.21.3 kicks in and shows that Z \times _\mathcal {Y} \mathcal{X} \to Z has property \mathcal{P}.
\square
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