The Stacks project

Lemma 79.15.9. In Situation 79.15.3 assume in addition that $s, t$ are flat and locally of finite presentation. Then there exists a scheme $U'$, a separated étale morphism $U' \to U$, and a point $u' \in U'$ lying over $u$ with $\kappa (u) = \kappa (u')$ such that the restriction $R' = R|_{U'}$ of $R$ to $U'$ is split over $u'$.

Proof. This follows from the construction of $U'$ in the proof of Lemma 79.15.6 because in this case $U' = (R_ s/U, e)_{fin}$ is a scheme separated over $U$ by Lemmas 79.12.14 and 79.12.15. $\square$

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