Lemma 67.22.1. Let \mathcal{P} be a property of morphisms of schemes which is étale local on the source-and-target. Let S be a scheme. Let f : X \to Y be a morphism of algebraic spaces over S. Consider commutative diagrams
\xymatrix{ U \ar[d]_ a \ar[r]_ h & V \ar[d]^ b \\ X \ar[r]^ f & Y }
where U and V are schemes and the vertical arrows are 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 X surjective the morphism h has property \mathcal{P}.
If X and Y are representable, then this is also equivalent to f (as a morphism of schemes) having property \mathcal{P}. If \mathcal{P} is also preserved under any base change, and fppf local on the base, then for representable morphisms f this is also equivalent to f having property \mathcal{P} in the sense of Section 67.3.
Proof.
Let us prove the equivalence of (1) and (2). The implication (1) \Rightarrow (2) is immediate (taking into account Spaces, Lemma 65.11.6). Assume
\xymatrix{ U \ar[d] \ar[r]_ h & V \ar[d] \\ X \ar[r]^ f & Y } \quad \quad \xymatrix{ U' \ar[d] \ar[r]_{h'} & V' \ar[d] \\ X \ar[r]^ f & Y }
are two diagrams as in the lemma. Assume U \to X is surjective and h has property \mathcal{P}. To show that (2) implies (1) we have to prove that h' has \mathcal{P}. To do this consider the diagram
\xymatrix{ U \ar[d]_ h & U \times _ X U' \ar[l] \ar[d]^{(h, h')} \ar[r] & U' \ar[d]^{h'} \\ V & V \times _ Y V' \ar[l] \ar[r] & V' }
By Descent, Lemma 35.32.5 we see that h has \mathcal{P} implies (h, h') has \mathcal{P} and since U \times _ X U' \to U' is surjective this implies (by the same lemma) that h' has \mathcal{P}.
If X and Y are representable, then Descent, Lemma 35.32.5 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 X the base change Z \times _ Y X \to Z has property \mathcal{P}. Consider the diagram
\xymatrix{ Z \times _ Y U \ar[d] \ar[r] & Z \times _ Y V \ar[d] \\ Z \times _ Y 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 étale and surjective and the right vertical arrow is étale. Thus Descent, Lemma 35.32.5 once again kicks in and shows that Z \times _ Y X \to Z has property \mathcal{P}.
\square
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