The Stacks project

Lemma 37.38.8. Let $f : X \to S$ be a smooth morphism of schemes. Then there exists an étale covering $\{ U_ i \to X\} _{i \in I}$ such that $U_ i \to S$ factors as $U_ i \to V_ i \to S$ where $V_ i \to S$ is étale and $U_ i \to V_ i$ is a smooth morphism of affine schemes, which has a section, and has geometrically connected fibres.

Proof. Let $s \in S$. By Varieties, Lemma 33.25.6 the set of closed points $x \in X_ s$ such that $\kappa (x)/\kappa (s)$ is separable is dense in $X_ s$. Thus it suffices to construct an étale morphism $U \to X$ with $x$ in the image such that $U \to S$ factors in the manner described in the lemma. To do this, choose an immersion $Z \to X$ passing through $x$ such that $Z \to S$ is étale (Lemma 37.38.5). After replacing $S$ by $Z$ and $X$ by $Z \times _ S X$ we see that we may assume $X \to S$ has a section $\sigma : S \to X$ with $\sigma (s) = x$. Then we can first replace $S$ by an affine open neighbourhood of $s$ and next replace $X$ by an affine open neighbourhood of $x$. Then finally, we consider the subset $X^0 \subset X$ of Section 37.29. By Lemmas 37.29.6 and 37.29.4 this is a retrocompact open subscheme containing $\sigma $ such that the fibres $X^0 \to S$ are geometrically connected. If $X^0$ is not affine, then we choose an affine open $U \subset X^0$ containing $x$. Since $X^0 \to S$ is smooth, the image of $U$ is open. Choose an affine open neighbourhood $V \subset S$ of $s$ contained in $\sigma ^{-1}(U)$ and in the image of $U \to S$. Finally, the reader sees that $U \cap f^{-1}(V) \to V$ has all the desired properties. For example $U \cap f^{-1}(V)$ is equal to $U \times _ S V$ is affine as a fibre product of affine schemes. Also, the geometric fibres of $U \cap f^{-1}(V) \to V$ are nonempty open subschemes of the irreducible fibres of $X^0 \to S$ and hence connected. Some details omitted. $\square$

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