The Stacks project

Lemma 38.10.9. Let $f : X \to S$ be a morphism which is locally of finite presentation. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module of finite type. If $x \in X$ and $\mathcal{F}$ is flat at $x$ over $S$, then $\mathcal{F}_ x$ is an $\mathcal{O}_{X, x}$-module of finite presentation.

Proof. Let $s = f(x)$. By Proposition 38.10.3 there exists an elementary ├ętale neighbourhood $(S', s') \to (S, s)$ such that the pullback of $\mathcal{F}$ to $X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'})$ is of finite presentation in a neighbourhood of the point $x' \in X_{s'} = X_ s$ corresponding to $x$. The ring map

\[ \mathcal{O}_{X, x} \longrightarrow \mathcal{O}_{X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'}), x'} = \mathcal{O}_{X \times _ S S', x'} \]

is flat and local as a localization of an ├ętale ring map. Hence $\mathcal{F}_ x$ is of finite presentation over $\mathcal{O}_{X, x}$ by descent, see Algebra, Lemma 10.83.2 (and also that a flat local ring map is faithfully flat, see Algebra, Lemma 10.39.17). $\square$

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