## 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)

## 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Your email address will not be published.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.

There are no comments yet for this tag.