The Stacks project

Lemma 58.5.4. Let $X$ be a scheme. Given $U, V$ finite étale over $X$ there exists a scheme $W$ finite étale over $X$ such that

\[ \mathop{\mathrm{Mor}}\nolimits _ X(X, W) = \mathop{\mathrm{Mor}}\nolimits _ X(U, V) \]

and such that the same remains true after any base change.

Proof. By More on Morphisms, Lemma 37.68.4 there exists a scheme $W$ representing $\mathit{Mor}_ X(U, V)$. (Use that an étale morphism is locally quasi-finite by Morphisms, Lemmas 29.36.6 and that a finite morphism is separated.) This scheme clearly satisfies the formula after any base change. To finish the proof we have to show that $W \to X$ is finite étale. This we may do after replacing $X$ by the members of an étale covering (Descent, Lemmas 35.23.23 and 35.23.6). Thus by Étale Morphisms, Lemma 41.18.3 we may assume that $U = \coprod _{i = 1, \ldots , n} X$ and $V = \coprod _{j = 1, \ldots , m} X$. In this case $W = \coprod _{\alpha : \{ 1, \ldots , n\} \to \{ 1, \ldots , m\} } X$ by inspection (details omitted) and the proof is complete. $\square$

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