The Stacks project

Lemma 58.31.9. Let $S$ be a quasi-compact and quasi-separated integral normal scheme with generic point $\eta $. Let $f : X \to S$ be a quasi-compact and quasi-separated smooth morphism with geometrically connected fibres. Let $\sigma : S \to X$ be a section of $f$. Let $Z \to X_\eta $ be a finite étale Galois cover (Section 58.7) with group $G$ of order invertible on $S$ such that $Z$ has a $\kappa (\eta )$-rational point mapping to $\sigma (\eta )$. Then there exists a finite étale Galois cover $Y \to X$ with group $G$ whose restriction to $X_\eta $ is $Z$.

Proof. If $S$ is Noetherian, then this is the result of Lemma 58.31.8. The general case follows from this by a standard limit argument. We strongly urge the reader to skip the proof.

We can write $S = \mathop{\mathrm{lim}}\nolimits S_ i$ as a directed limit of a system of schemes with affine transition morphisms and with $S_ i$ of finite type over $\mathbf{Z}$, see Limits, Proposition 32.5.4. For each $i$ let $S \to S'_ i \to S_ i$ be the normalization of $S_ i$ in $S$, see Morphisms, Section 29.53. Combining Algebra, Proposition 10.162.16 Morphisms, Lemmas 29.53.15 and 29.53.13 we conclude that $S'_ i$ is of finite type over $\mathbf{Z}$, finite over $S_ i$, and that $S'_ i$ is an integral normal scheme such that $S \to S'_ i$ is dominant. By Morphisms, Lemma 29.53.5 we obtain transition morphisms $S'_{i'} \to S'_ i$ compatible with the transition morphisms $S_{i'} \to S_ i$ and with the morphisms with source $S$. We claim that $S = \mathop{\mathrm{lim}}\nolimits S'_ i$. Proof of claim omitted (hint: look on affine opens over a chosen affine open in $S_ i$ for some $i$ to translate this into a straightforward algebra problem). We conclude that we may write $S = \mathop{\mathrm{lim}}\nolimits S_ i$ as a directed limit of a system of normal integral schemes $S_ i$ with affine transition morphisms and with $S_ i$ of finite type over $\mathbf{Z}$.

For some $i$ we can find a smooth morphism $X_ i \to S_ i$ of finite presentation whose base change to $S$ is $X \to S$. See Limits, Lemmas 32.10.1 and 32.8.9. After increasing $i$ we may assume the section $\sigma $ lifts to a section $\sigma _ i : S_ i \to X_ i$ (by the equivalence of categories in Limits, Lemma 32.10.1). We may replace $X_ i$ by the open subscheme $X_ i^0$ of it studied in More on Morphisms, Section 37.29 since the image of $X \to X_ i$ clearly maps into it (openness by More on Morphisms, Lemma 37.29.6). Thus we may assume the fibres of $X_ i \to S_ i$ are geometrically connected. After increasing $i$ we may assume $|G|$ is invertible on $S_ i$. Let $\eta _ i \in S_ i$ be the generic point. Since $X_\eta $ is the limit of the schemes $X_{i, \eta _ i}$ we can use the exact same arguments to descent $Z \to X_\eta $ to some finite étale Galois cover $Z_ i \to X_{i, \eta _ i}$ after possibly increasing $i$. See Lemma 58.14.1. After possibly increasing $i$ once more we may assume $Z_ i$ has a $\kappa (\eta _ i)$-rational point mapping to $\sigma _ i(\eta _ i)$. Then we apply the lemma in the Noetherian case and we pullback to $X$ to conclude. $\square$

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