The Stacks project

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

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

  2. a commutative diagram

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

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

such that the following properties hold

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

  2. $i$ is a closed immersion of finite presentation,

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

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

  5. the extension $\kappa (y')/\kappa (s')$ is purely transcendental,

  6. $h$ 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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