Lemma 38.3.2. Let $f : X \to S$ be morphism of schemes which is locally of finite type. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_ X$-module. Let $x \in X$ with image $s = f(x)$ in $S$. Set $\mathcal{F}_ s = \mathcal{F}|_{X_ s}$ and $n = \dim _ x(\text{Supp}(\mathcal{F}_ s))$. Then we can construct

elementary étale neighbourhoods $g : (X', x') \to (X, x)$, $e : (S', s') \to (S, s)$,

a commutative diagram

\[ \xymatrix{ X \ar[dd]_ f & X' \ar[dd] \ar[l]^ g & Z' \ar[l]^ i \ar[d]^\pi \\ & & Y' \ar[d]^ h \\ S & S' \ar[l]_ e & S' \ar@{=}[l] } \]a point $z' \in Z'$ with $i(z') = x'$, $y' = \pi (z')$, $h(y') = s'$,

a finite type quasi-coherent $\mathcal{O}_{Z'}$-module $\mathcal{G}$,

such that the following properties hold

$X'$, $Z'$, $Y'$, $S'$ are affine schemes,

$i$ is a closed immersion of finite presentation,

$i_*(\mathcal{G}) \cong g^*\mathcal{F}$,

$\pi $ is finite and $\pi ^{-1}(\{ y'\} ) = \{ z'\} $,

the extension $\kappa (s') \subset \kappa (y')$ is purely transcendental,

$h$ is smooth of relative dimension $n$ with geometrically integral fibres.

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