Lemma 76.44.1. Let \mathcal{P} be a property of morphisms of schemes which is étale local on the target. Let S be a scheme. Let f : X \to Y be a representable morphism of algebraic spaces over S. Consider commutative diagrams
\xymatrix{ X \times _ Y V \ar[d] \ar[r] & V \ar[d] \\ X \ar[r]^ f & Y }
where V is a scheme and V \to Y is étale. The following are equivalent
for any diagram as above the projection X \times _ Y V \to V has property \mathcal{P}, and
for some diagram as above with V \to Y surjective the projection X \times _ Y V \to V 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}.
Proof.
Let us prove the equivalence of (1) and (2). The implication (1) \Rightarrow (2) is immediate. Assume
\xymatrix{ X \times _ Y V \ar[d] \ar[r] & V \ar[d] \\ X \ar[r]^ f & Y } \quad \quad \xymatrix{ X \times _ Y V' \ar[d] \ar[r] & V' \ar[d] \\ X \ar[r]^ f & Y }
are two diagrams as in the lemma. Assume V \to Y is surjective and X \times _ Y V \to V has property \mathcal{P}. To show that (2) implies (1) we have to prove that X \times _ Y V' \to V' has \mathcal{P}. To do this consider the diagram
\xymatrix{ X \times _ Y V \ar[d] & (X \times _ Y V) \times _ X (X \times _ Y V') \ar[l] \ar[d] \ar[r] & X \times _ Y V' \ar[d] \\ V & V \times _ Y V' \ar[l] \ar[r] & V' }
By our assumption that \mathcal{P} is étale local on the source, we see that \mathcal{P} is preserved under étale base change, see Descent, Lemma 35.22.2. Hence if the left vertical arrow has \mathcal{P} the so does the middle vertical arrow. Since U \times _ X U' \to U' is surjective and étale (hence defines an étale covering of U') this implies (as \mathcal{P} is assumed local for the étale topology on the target) that the left vertical arrow has \mathcal{P}.
If X and Y are representable, then we can take \text{id}_ Y : Y \to Y as our étale covering to see the final statement of the lemma is true.
\square
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