The Stacks project

Example 39.23.1. Let $k$ be a field. Let $U = \text{GL}_{2, k}$. Let $B \subset \text{GL}_2$ be the closed subgroup scheme of upper triangular matrices. Then the quotient sheaf $\text{GL}_{2, k}/B$ (in the Zariski, étale or fppf topology, see Definition 39.20.1) is representable by the projective line: $\mathbf{P}^1 = \text{GL}_{2, k}/B$. (Details omitted.) On the other hand, the ring of invariant functions in this case is just $k$. Note that in this case the morphisms $s, t : R = \text{GL}_{2, k} \times _ k B \to \text{GL}_{2, k} = U$ are smooth of relative dimension $3$.

Comments (0)

There are also:

  • 6 comment(s) on Section 39.23: Finite flat groupoids, affine case

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Your email address will not be published.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 03BG. Beware of the difference between the letter 'O' and the digit '0'.



View Example 39.23.1 as pdf