The Stacks project

94.14.7 Variant on principal homogeneous spaces

Let $S$ be a scheme. Let $B = S$. Let $G$ be a group scheme over $B = S$. In this setting we can define a full subcategory $G\textit{-Principal-Schemes} \subset G\textit{-Principal}$ whose objects are pairs $(U, X)$ where $U$ is an object of $(\mathit{Sch}/S)_{fppf}$ and $X \to U$ is a principal homogeneous $G$-space over $U$ which is representable, i.e., a scheme.

It is in general not the case that $G\textit{-Principal-Schemes}$ is a stack in groupoids over $(\mathit{Sch}/S)_{fppf}$. The reason is that in general there really do exist principal homogeneous spaces which are not schemes, hence descent for objects will not be satisfied in general.

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Your email address will not be published.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