Lemma 77.4.6. Let S be a scheme. Let f : X \to Y be a morphism of algebraic spaces over S. Let \mathcal{F} be a quasi-coherent sheaf on X. Let y \in |Y|. Set F = f^{-1}(\{ y\} ) \subset |X|. Assume that
f is of finite type,
\mathcal{F} is of finite type, and
\mathcal{F} is flat over Y at all x \in F.
Then there exists an étale morphism (Y', y') \to (Y, y) where Y' is a scheme and a commutative diagram of algebraic spaces
\xymatrix{ X \ar[d] & X' \ar[l]^ g \ar[d] \\ Y & \mathop{\mathrm{Spec}}(\mathcal{O}_{Y', y'}) \ar[l] }
such that X' \to X \times _ Y \mathop{\mathrm{Spec}}(\mathcal{O}_{Y', y'}) is étale, |X'_{y'}| \to F is surjective, X' is affine, and \Gamma (X', g^*\mathcal{F}) is a free \mathcal{O}_{Y', y'}-module.
Proof.
Choose an étale morphism (Y', y') \to (Y, y) where Y' is an affine scheme. Then X \times _ Y Y' is quasi-compact. Choose an affine scheme X' and a surjective étale morphism X' \to X \times _ Y Y'. Picture
\xymatrix{ X \ar[d] & X' \ar[l]^ g \ar[d] \\ Y & Y' \ar[l] }
Then \mathcal{F}' = g^*\mathcal{F} is flat over Y' at all points of X'_{y'}, see Morphisms of Spaces, Lemma 67.31.3. Hence we can apply the lemma in the case of schemes (More on Flatness, Lemma 38.12.11) to the morphism X' \to Y', the quasi-coherent sheaf g^*\mathcal{F}, and the point y'. This gives an étale morphism (Y'', y'') \to (Y', y') and a commutative diagram
\xymatrix{ X \ar[d] & X' \ar[l]^ g \ar[d] & X'' \ar[l]^{g'} \ar[d] \\ Y & Y' \ar[l] & \mathop{\mathrm{Spec}}(\mathcal{O}_{Y'', y''}) \ar[l] }
To get what we want we take (Y'', y'') \to (Y, y) and g \circ g' : X'' \to X.
\square
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