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 $\kappa (s') \subset \kappa (y')$ 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 37.43.2 and 37.43.3. 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. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 057S. Beware of the difference between the letter 'O' and the digit '0'.