The Stacks project

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

  1. $f$ is of finite presentation,

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

  3. $\mathcal{F}$ is flat over $Y$ at all $x \in F$.

Then there exists a commutative diagram of algebraic spaces

\[ \xymatrix{ X \ar[d] & X' \ar[l]^ g \ar[d] \\ Y & Y' \ar[l]_ h } \]

such that $h$ and $g$ are ├ętale, 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 immmediately reduces to the case of schemes, which is More on Flatness, Lemma 38.12.5. $\square$

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