The Stacks project

Remark 39.7.5. Warning: The result of Lemma 39.7.4 does not mean that every irreducible component of $G/k$ is geometrically irreducible. For example the group scheme $\mu _{3, \mathbf{Q}} = \mathop{\mathrm{Spec}}(\mathbf{Q}[x]/(x^3 - 1))$ over $\mathbf{Q}$ has two irreducible components corresponding to the factorization $x^3 - 1 = (x - 1)(x^2 + x + 1)$. The first factor corresponds to the irreducible component passing through the identity, and the second irreducible component is not geometrically irreducible over $\mathop{\mathrm{Spec}}(\mathbf{Q})$.

Comments (0)

There are also:

  • 7 comment(s) on Section 39.7: Properties of group schemes over a field

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Your email address will not be published.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 04L9. Beware of the difference between the letter 'O' and the digit '0'.



View Remark 39.7.5 as pdf