The Stacks project

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

  1. $f$ is of finite type,

  2. $\mathcal{F}$ is of finite type, and

  3. $\mathcal{F}$ 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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