The Stacks project

Lemma 79.9.3. In Situation 79.9.2 the composition morphism $c : R \times _{s, U, t} R \to R$ is flat and universally open. In Situation 79.9.1 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 (79.3.0.2). The projection is flat as a base change of the flat morphism $t$ and open by Morphisms of Spaces, Lemma 67.6.6. The second assertion follows immediately from the first because $m$ matches $c$ in (79.9.2.1). $\square$

Comments (0)

Post a comment

Your email address will not be published.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 toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

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 input field. As a reminder, this is tag 06E0. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 79.9.3 as pdf