## Tag `06E1`

Chapter 70: More on Groupoids in Spaces > Section 70.9: Properties of groups over fields and groupoids on fields

Lemma 70.9.5. In Situation 70.9.2. Let $k \subset k'$ be a field extension, $U' = \mathop{\mathrm{Spec}}(k')$ and let $(U', R', s', t', c')$ be the restriction of $(U, R, s, t, c)$ via $U' \to U$. In the defining diagram $$ \xymatrix{ R' \ar[d] \ar[r] \ar@/_3pc/[dd]_{t'} \ar@/^1pc/[rr]^{s'} \ar@{..>}[rd] & R \times_{s, U} U' \ar[r] \ar[d] & U' \ar[d] \\ U' \times_{U, t} R \ar[d] \ar[r] & R \ar[r]^s \ar[d]_t & U \\ U' \ar[r] & U } $$ all the morphisms are surjective, flat, and universally open. The dotted arrow $R' \to R$ is in addition affine.

Proof.The morphism $U' \to U$ equals $\mathop{\mathrm{Spec}}(k') \to \mathop{\mathrm{Spec}}(k)$, hence is affine, surjective and flat. The morphisms $s, t : R \to U$ and the morphism $U' \to U$ are universally open by Morphisms, Lemma 28.22.4. Since $R$ is not empty and $U$ is the spectrum of a field the morphisms $s, t : R \to U$ are surjective and flat. Then you conclude by using Morphisms of Spaces, Lemmas 58.5.5, 58.5.4, 58.6.4, 58.20.5, 58.20.4, 58.30.4, and 58.30.3. $\square$

The code snippet corresponding to this tag is a part of the file `spaces-more-groupoids.tex` and is located in lines 719–740 (see updates for more information).

```
\begin{lemma}
\label{lemma-restrict-groupoid-on-field}
In
Situation \ref{situation-groupoid-on-field}.
Let $k \subset k'$ be a field extension, $U' = \Spec(k')$
and let $(U', R', s', t', c')$ be the restriction of
$(U, R, s, t, c)$ via $U' \to U$. In the defining diagram
$$
\xymatrix{
R' \ar[d] \ar[r] \ar@/_3pc/[dd]_{t'} \ar@/^1pc/[rr]^{s'} \ar@{..>}[rd] &
R \times_{s, U} U' \ar[r] \ar[d] &
U' \ar[d] \\
U' \times_{U, t} R \ar[d] \ar[r] &
R \ar[r]^s \ar[d]_t &
U \\
U' \ar[r] &
U
}
$$
all the morphisms are surjective, flat, and universally open.
The dotted arrow $R' \to R$ is in addition affine.
\end{lemma}
\begin{proof}
The morphism $U' \to U$ equals $\Spec(k') \to \Spec(k)$,
hence is affine, surjective and flat. The morphisms $s, t : R \to U$
and the morphism $U' \to U$ are universally open by
Morphisms, Lemma \ref{morphisms-lemma-scheme-over-field-universally-open}.
Since $R$ is not empty and $U$ is the spectrum of a field the morphisms
$s, t : R \to U$ are surjective and flat. Then you conclude by using
Morphisms of Spaces, Lemmas
\ref{spaces-morphisms-lemma-base-change-surjective},
\ref{spaces-morphisms-lemma-composition-surjective},
\ref{spaces-morphisms-lemma-composition-open},
\ref{spaces-morphisms-lemma-base-change-affine},
\ref{spaces-morphisms-lemma-composition-affine},
\ref{spaces-morphisms-lemma-base-change-flat}, and
\ref{spaces-morphisms-lemma-composition-flat}.
\end{proof}
```

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