This is the analogue of More on Flatness, Section 38.12.
Proposition 77.5.1. 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 $x \in |X|$ with image $y \in |Y|$. Assume that
$f$ is locally of finite presentation,
$\mathcal{F}$ is of finite presentation, and
$\mathcal{F}$ is flat at $x$ over $Y$.
Then there exists a commutative diagram of pointed schemes
whose horizontal arrows are étale such that $X'$, $Y'$ are affine and such that $\Gamma (X', g^*\mathcal{F})$ is a projective $\Gamma (Y', \mathcal{O}_{Y'})$-module.
Proof. As formulated this proposition immediately reduces to the case of schemes, which is More on Flatness, Proposition 38.12.4. $\square$
Lemma 77.5.2. 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 presentation,
$\mathcal{F}$ is of finite presentation, and
$\mathcal{F}$ is flat over $Y$ at all $x \in F$.
Then there exists a commutative diagram of algebraic spaces
such that $h$ and $g$ are étale, there is a point $y' \in |Y'|$ mapping to $y$, we have $F \subset g(|X'|)$, the algebraic spaces $X'$, $Y'$ are affine, and $\Gamma (X', g^*\mathcal{F})$ is a projective $\Gamma (Y', \mathcal{O}_{Y'})$-module.
Proof. As formulated this lemma immediately reduces to the case of schemes, which is More on Flatness, Lemma 38.12.5. $\square$
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