The Stacks Project


Tag 06E0

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

Lemma 70.9.3. In Situation 70.9.2 the composition morphism $c : R \times_{s, U, t} R \to R$ is flat and universally open. In Situation 70.9.1 the group law $m : G \times_k G \to G$ is flat and universally open.

Proof. The composition is isomorphic to the projection map $\text{pr}_1 : R \times_{t, U, t} R \to R$ by Diagram (70.3.0.2). The projection is flat as a base change of the flat morphism $t$ and open by Morphisms of Spaces, Lemma 58.6.6. The second assertion follows immediately from the first because $m$ matches $c$ in (70.9.2.1). $\square$

    The code snippet corresponding to this tag is a part of the file spaces-more-groupoids.tex and is located in lines 662–672 (see updates for more information).

    \begin{lemma}
    \label{lemma-groupoid-on-field-open-multiplication}
    In
    Situation \ref{situation-groupoid-on-field}
    the composition morphism $c : R \times_{s, U, t} R \to R$ is flat and
    universally open.
    In
    Situation \ref{situation-group-over-field}
    the group law $m : G \times_k G \to G$ is flat and
    universally open.
    \end{lemma}
    
    \begin{proof}
    The composition is isomorphic to the projection map
    $\text{pr}_1 : R \times_{t, U, t} R \to R$ by
    Diagram (\ref{equation-pull}).
    The projection is flat as a base change of the flat morphism $t$
    and open by
    Morphisms of Spaces,
    Lemma \ref{spaces-morphisms-lemma-space-over-field-universally-open}.
    The second assertion follows immediately from the first because
    $m$ matches $c$ in (\ref{equation-groupoid-from-group}).
    \end{proof}

    Comments (0)

    There are no comments yet for this tag.

    Add a comment on tag 06E0

    Your email address will not be published. Required fields are marked.

    In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the lower-right corner).

    All contributions are licensed under the GNU Free Documentation License.




    In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following box. So in case this where tag 0321 you just have to write 0321. Beware of the difference between the letter 'O' and the digit 0.

    This captcha seems more appropriate than the usual illegible gibberish, right?