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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, $U' = \mathop{\rm Spec}(k')$ 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 $\mathop{\rm Spec}(k') \to \mathop{\rm Spec}(k)$, 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, 58.29.4, and 58.29.3. $\square$

    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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