Lemma 67.37.4. Let S be a scheme. Let f : X \to Y be a morphism of algebraic spaces over S. The following are equivalent:
f is smooth,
for every x \in |X| the morphism f is smooth at x,
for every scheme Z and any morphism Z \to Y the morphism Z \times _ Y X \to Z is smooth,
for every affine scheme Z and any morphism Z \to Y the morphism Z \times _ Y X \to Z is smooth,
there exists a scheme V and a surjective étale morphism V \to Y such that V \times _ Y X \to V is a smooth morphism,
there exists a scheme U and a surjective étale morphism \varphi : U \to X such that the composition f \circ \varphi is smooth,
for every commutative diagram
\xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y }where U, V are schemes and the vertical arrows are étale the top horizontal arrow is smooth,
there exists a commutative diagram
\xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y }where U, V are schemes, the vertical arrows are étale, and U \to X is surjective such that the top horizontal arrow is smooth, and
there exist Zariski coverings Y = \bigcup _{i \in I} Y_ i, and f^{-1}(Y_ i) = \bigcup X_{ij} such that each morphism X_{ij} \to Y_ i is smooth.
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