The Stacks project

Lemma 79.15.7 (Existence of quasi-splitting). In Situation 79.15.4 there exists an algebraic space $U'$, an étale morphism $U' \to U$, and a point $u' : \mathop{\mathrm{Spec}}(\kappa (u)) \to U'$ lying over $u : \mathop{\mathrm{Spec}}(\kappa (u)) \to U$ such that the restriction $R' = R|_{U'}$ of $R$ to $U'$ is quasi-split over $u'$.

Proof. Let $f : (U', Z_{univ}, s', t', c') \to (U, R, s, t, c)$ be as constructed in Lemma 79.14.1. Recall that $R' = R \times _{(U \times _ S U)} (U' \times _ S U')$. Thus we get a morphism $(f, t', s') : Z_{univ} \to R'$ of groupoids in algebraic spaces

\[ (U', Z_{univ}, s', t', c') \to (U', R', s', t', c') \]

(by abuse of notation we indicate the morphisms in the two groupoids by the same symbols). Now, as $Z_{univ} \subset R \times _{s, U, g} U'$ is open and $R' \to R \times _{s, U, g} U'$ is étale (as a base change of $U' \to U$) we see that $Z_{univ} \to R'$ is an open immersion. By construction the morphisms $s', t' : Z_{univ} \to U'$ are finite. It remains to find the point $u'$ of $U'$.

We think of $u$ as a morphism $\mathop{\mathrm{Spec}}(\kappa (u)) \to U$ as in the statement of the lemma. Set $F_ u = R \times _{s, U} \mathop{\mathrm{Spec}}(\kappa (u))$. The morphism $F_ u \to \mathop{\mathrm{Spec}}(\kappa (u))$ is quasi-finite at $e(u)$ by assumption. Hence we can find a decomposition into open and closed subschemes

\[ F_ u = Z_ u \amalg Rest \]

for some scheme $Z_ u$ finite over $\kappa (u)$ whose support is $e(u)$. Hence by the construction of $U'$ in Section 79.14 $(u, Z_ u)$ defines a $\mathop{\mathrm{Spec}}(\kappa (u))$-valued point $u'$ of $U'$. To finish the proof we have to show that $e'(u') \in Z_{univ}$ which is clear. $\square$

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