# The Stacks Project

## Tag: 04RY

This tag has label spaces-more-groupoids-lemma-quasi-splitting-scheme and it points to

The corresponding content:

Lemma 56.11.7. Assumptions and notation as in Situation 56.11.3. 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 quasi-split over $u'$.

Proof. This follows from the construction of $U'$ in the proof of Lemma 56.11.5 because in this case $U' = (R_s/U, e)_{fin}$ is a scheme separated over $U$ by Lemmas 56.8.14 and 56.8.15. $\square$

\begin{lemma}
\label{lemma-quasi-splitting-scheme}
Assumptions and notation as in
Situation \ref{situation-etale-localize-quasi}.
Assume in addition that $s, t$ are flat and locally of finite presentation.
Then there exists a scheme $U'$, a separated \'etale 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 quasi-split over $u'$.
\end{lemma}

\begin{proof}
This follows from the construction of $U'$ in the proof of
Lemma \ref{lemma-quasi-splitting-general}
because in this case $U' = (R_s/U, e)_{fin}$ is a scheme separated
over $U$ by
Lemmas \ref{lemma-finite-separated-flat-locally-finite-presentation} and
\ref{lemma-finite-plus-section}.
\end{proof}


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