Lemma 78.15.12. In Situation 78.15.3 assume in addition that $s, t$ are flat and locally of finite presentation and that $U$ is affine. Then there exists an affine scheme $U'$, an 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. The proof of this lemma is literally the same as the proof of Lemma 78.15.11 except that “strong splitting” needs to be replaced by “splitting” (2 times) and that the reference to Lemma 78.15.8 needs to be replaced by a reference to Lemma 78.15.9. $\square$

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