The Stacks project

Lemma 37.45.3. 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 37.34.2 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 37.45.2 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 $X' \subset S' \times _ S X$.

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