The Stacks project

Lemma 76.4.1. Let $S$ be a scheme. Let $X \to Y$ be a finite type morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_ X$-module. Let $y \in |Y|$ be a point. There exists an étale morphism $(Y', y') \to (Y, y)$ with $Y'$ an affine scheme and étale morphisms $h_ i : W_ i \to X_{Y'}$, $i = 1, \ldots , n$ such that for each $i$ there exists a complete dévissage of $\mathcal{F}_ i/W_ i/Y'$ over $y'$, where $\mathcal{F}_ i$ is the pullback of $\mathcal{F}$ to $W_ i$ and such that $|(X_{Y'})_{y'}| \subset \bigcup h_ i(W_ i)$.

Proof. The question is étale local on $Y$ hence we may assume $Y$ is an affine scheme. Then $X$ is quasi-compact, hence we can choose an affine scheme $X'$ and a surjective étale morphism $X' \to X$. Then we may apply More on Flatness, Lemma 38.5.8 to $X' \to Y$, $(X' \to Y)^*\mathcal{F}$, and $y$ to get what we want. $\square$


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