# The Stacks Project

## Tag: 04S0

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

The corresponding content:

Lemma 57.11.9. Assumptions and notation as in Situation 57.11.3. 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 quasi-split over $u'$.

Proof. The proof of this lemma is literally the same as the proof of Lemma 57.11.8 except that ''splitting'' needs to be replaced by ''quasi-splitting'' (2 times) and that the reference to Lemma 57.11.6. needs to be replaced by a reference to Lemma 57.11.7. $\square$

\begin{lemma}
\label{lemma-quasi-splitting-affine-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
and that $U$ is affine.
Then there exists an affine scheme $U'$, an \'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}
The proof of this lemma is literally the same as the proof of
Lemma \ref{lemma-splitting-affine-scheme}
except that splitting'' needs to be replaced by quasi-splitting''
(2 times) and that the reference to
Lemma \ref{lemma-splitting-scheme}.
needs to be replaced by a reference to
Lemma \ref{lemma-quasi-splitting-scheme}.
\end{proof}


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