Lemma 38.10.6. Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $s \in S$. Assume that
$f$ is of finite presentation,
$\mathcal{F}$ is of finite type, and
$\mathcal{F}$ is flat over $S$ at every point of the fibre $X_ s$.
Then there exists an elementary étale neighbourhood $(S', s') \to (S, s)$ and an open subscheme
\[ V \subset X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'}) \]
which contains the fibre $X_ s = X \times _ S s'$ such that the pullback of $\mathcal{F}$ to $V$ is an $\mathcal{O}_ V$-module of finite presentation and flat over $\mathcal{O}_{S', s'}$.
Proof.
For every point $x \in X_ s$ we can use Proposition 38.10.3 to find an elementary étale neighbourhood $(S_ x, s_ x) \to (S, s)$ and an open $V_ x \subset X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S_ x, s_ x})$ such that $x \in X_ s = X \times _ S s_ x$ is contained in $V_ x$ and such that the pullback of $\mathcal{F}$ to $V_ x$ is an $\mathcal{O}_{V_ x}$-module of finite presentation and flat over $\mathcal{O}_{S_ x, s_ x}$. In particular we may view the fibre $(V_ x)_{s_ x}$ as an open neighbourhood of $x$ in $X_ s$. Because $X_ s$ is quasi-compact we can find a finite number of points $x_1, \ldots , x_ n \in X_ s$ such that $X_ s$ is the union of the $(V_{x_ i})_{s_{x_ i}}$. Choose an elementary étale neighbourhood $(S' , s') \to (S, s)$ which dominates each of the neighbourhoods $(S_{x_ i}, s_{x_ i})$, see More on Morphisms, Lemma 37.35.4. Set $V = \bigcup V_ i$ where $V_ i$ is the inverse images of the open $V_{x_ i}$ via the morphism
\[ X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'}) \longrightarrow X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S_{x_ i}, s_{x_ i}}) \]
By construction $V$ contains $X_ s$ and by construction the pullback of $\mathcal{F}$ to $V$ is an $\mathcal{O}_ V$-module of finite presentation and flat over $\mathcal{O}_{S', s'}$.
$\square$
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