The Stacks project

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

  1. $f$ is locally of finite presentation,

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

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

Then there exists a commutative diagram of pointed schemes

\[ \xymatrix{ (X, x) \ar[d] & (X', x') \ar[l]^ g \ar[d] \\ (Y, y) & (Y', y') \ar[l] } \]

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. $\square$

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