The Stacks project

Lemma 38.5.8. Let $X \to S$ be a finite type morphism of schemes. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_ X$-module. Let $s \in S$ be a point. There exists an elementary étale neighbourhood $(S', s') \to (S, s)$ and étale morphisms $h_ i : Y_ i \to X_{S'}$, $i = 1, \ldots , n$ such that for each $i$ there exists a complete dévissage of $\mathcal{F}_ i/Y_ i/S'$ over $s'$, where $\mathcal{F}_ i$ is the pullback of $\mathcal{F}$ to $Y_ i$ and such that $X_ s = (X_{S'})_{s'} \subset \bigcup h_ i(Y_ i)$.

Proof. For every point $x \in X_ s$ we can find a diagram

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

of pointed schemes such that the horizontal arrows are elementary étale neighbourhoods and such that $g^*\mathcal{F}/X'/S'$ has a complete dévissage at $x'$. As $X \to S$ is of finite type the fibre $X_ s$ is quasi-compact, and since each $g : X' \to X$ as above is open we can cover $X_ s$ by a finite union of $g(X'_{s'})$. Thus we can find a finite family of such diagrams

\[ \vcenter { \xymatrix{ (X, x) \ar[d] & (X'_ i, x'_ i) \ar[l]^{g_ i} \ar[d] \\ (S, s) & (S'_ i, s'_ i) \ar[l] } } \quad i = 1, \ldots , n \]

such that $X_ s = \bigcup g_ i(X'_ i)$. Set $S' = S'_1 \times _ S \ldots \times _ S S'_ n$ and let $Y_ i = X_ i \times _{S'_ i} S'$ be the base change of $X'_ i$ to $S'$. By Lemma 38.5.3 we see that the pullback of $\mathcal{F}$ to $Y_ i$ has a complete dévissage over $s$ and we win. $\square$

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