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 (y')/\kappa (s') is purely transcendental,
h is smooth of relative dimension n with geometrically integral fibres.
Proof.
Let V \subset S be an affine neighbourhood of s. Let U \subset f^{-1}(V) be an affine neighbourhood of x. Then it suffices to prove the lemma for f|_ U : U \to V and \mathcal{F}|_ U. Hence in the rest of the proof we assume that X and S are affine.
First, suppose that X_ s = \text{Supp}(\mathcal{F}_ s), in particular n = \dim _ x(X_ s). Apply More on Morphisms, Lemmas 37.47.2 and 37.47.3. This gives us a commutative diagram
\xymatrix{ X \ar[dd] & X' \ar[l]^ g \ar[d]^\pi \\ & Y' \ar[d]^ h \\ S & S' \ar[l]_ e }
and point x' \in X'. We set Z' = X', i = \text{id}, and \mathcal{G} = g^*\mathcal{F} to obtain a solution in this case.
In general choose a closed immersion Z \to X and a sheaf \mathcal{G} on Z as in Lemma 38.3.1. Applying the result of the previous paragraph to Z \to S and \mathcal{G} we obtain a diagram
\xymatrix{ X \ar[dd]_ f & Z \ar[l] \ar[dd]_{f|_ Z} & Z' \ar[l]^ g \ar[d]^\pi \\ & & Y' \ar[d]^ h \\ S & S \ar@{=}[l] & S' \ar[l]_ e }
and point z' \in Z' satisfying all the required properties. We will use Lemma 38.2.1 to embed Z' into a scheme étale over X. We cannot apply the lemma directly as we want X' to be a scheme over S'. Instead we consider the morphisms
\xymatrix{ Z' \ar[r] & Z \times _ S S' \ar[r] & X \times _ S S' }
The first morphism is étale by Morphisms, Lemma 29.36.18. The second is a closed immersion as a base change of a closed immersion. Finally, as X, S, S', Z, Z' are all affine we may apply Lemma 38.2.1 to get an étale morphism of affine schemes X' \to X \times _ S S' such that
Z' = (Z \times _ S S') \times _{(X \times _ S S')} X' = Z \times _ X X'.
As Z \to X is a closed immersion of finite presentation, so is Z' \to X'. Let x' \in X' be the point corresponding to z' \in Z'. Then the completed diagram
\xymatrix{ X \ar[dd] & X' \ar[dd] \ar[l] & Z' \ar[l]^ i \ar[d]^\pi \\ & & Y' \ar[d]^ h \\ S & S' \ar[l]_ e & S' \ar@{=}[l] }
is a solution of the original problem.
\square
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