The Stacks project

Lemma 37.46.3. Let $f : X \to S$ be a morphism of schemes which is locally of finite presentation and flat with geometrically reduced fibres. Then there exists an étale covering $\{ X_ i \to X\} _{i \in I}$ such that $X_ i \to S$ factors as $X_ i \to S_ i \to S$ where $S_ i \to S$ is étale and $X_ i \to S_ i$ is flat of finite presentation with geometrically connected and geometrically reduced fibres.

Proof. Pick a point $x \in X$ with image $s \in S$. We will produce a diagram

\[ \xymatrix{ X' \ar[r] \ar[rd] & S' \times _ S X \ar[r] \ar[d] & X \ar[d] \\ & S' \ar[r] & S } \]

and points $s' \in S'$, $x' \in X'$, $y \in S' \times _ S X$ such that $x'$ maps to $x$, $(S', s') \to (S, s)$ is an étale neighbourhood, $(X', x') \to (S' \times _ S X, y)$ is an étale neighbourhood1, and $X' \to S'$ has geometrically connected fibres. If we can do this for every $x \in X$, then the lemma follows (with members of the covering given by the collection of étale morphisms $X' \to X$ so produced). The first step is the replace $X$ and $S$ by affine open neighbourhoods of $x$ and $s$ which reduces us to the case that $X$ and $S$ are affine (and hence $f$ of finite presentation).

Choose a separable algebraic extension $\overline{k}$ of $\kappa (s)$. Denote $X_{\overline{k}}$ the base change of $X_ s$. Choose a point $\overline{x}$ in $X_{\overline{k}}$ mapping to $x \in X_ s$. Choose a connected quasi-compact open neighbourhood $\overline{V} \subset X_{\overline{k}}$ of $\overline{x}$. (This is possible because any scheme locally of finite type over a field is locally connected as a locally Noetherian topological space.) By Varieties, Lemma 33.7.9 we can find a finite separable extension $k'/\kappa (s)$ and a quasi-compact open $V' \subset X_{k'}$ whose base change is $\overline{V}$. In particular $V'$ is geometrically connected over $k'$, see Varieties, Lemma 33.7.7. By Lemma 37.35.2 we can find an étale neighbourhood $(S', s') \to (S, s)$ such that $\kappa (s')$ is isomorphic to $k'$ as an extension of $\kappa (s)$. Denote $x' \in (S' \times _ S X)_{s'} = X_{k'}$ the image of $\overline{x}$. Thus after replacing $(S, s)$ by $(S', s')$ and $(X, x)$ by $(S' \times _ S X, x')$ we reduce to the case handled in the next paragrah.

Assume there is a quasi-compact open $V \subset X_ s$ which contains $x$ and is geometrically irreducible. Then we can apply Lemma 37.46.2 to find an affine étale neighbourhood $(S', s') \to (S, s)$ and a quasi-compact open $X' \subset S' \times _ S X$ such that $X' \to S'$ has geometrically connected fibres and such that $X'$ contains a point mapping to $x$. This finishes the proof. $\square$

[1] The proof actually gives an open $X' \subset S' \times _ S X$.

Comments (1)

Comment #8738 by Yicheng Zhou on

I guess that in the first sentence of the last paragraph should be only "geometrically connected"?

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 0EY5. Beware of the difference between the letter 'O' and the digit '0'.