The Stacks project

Remark 39.11.5. Let $(G, m)$ be a group scheme over the scheme $S$. In this situation we have the following natural types of questions:

  1. If $X \to S$ is a pseudo $G$-torsor and $X \to S$ is surjective, then is $X$ necessarily a $G$-torsor?

  2. Is every $\underline{G}$-torsor on $(\mathit{Sch}/S)_{fppf}$ representable? In other words, does every $\underline{G}$-torsor come from a fppf $G$-torsor?

  3. Is every $G$-torsor an fppf (resp. smooth, resp. ├ętale, resp. Zariski) torsor?

In general the answers to these questions is no. To get a positive answer we need to impose additional conditions on $G \to S$. For example: If $S$ is the spectrum of a field, then the answer to (1) is yes because then $\{ X \to S\} $ is a fpqc covering trivializing $X$. If $G \to S$ is affine, then the answer to (2) is yes (this follows from Descent, Lemma 35.37.1). If $G = \text{GL}_{n, S}$ then the answer to (3) is yes and in fact any $\text{GL}_{n, S}$-torsor is locally trivial (this follows from Descent, Lemma 35.7.6).

Comments (2)

Comment #7722 by Anonymous on

For the two instances of (insert future reference here), perhaps you intend Lemma 35.37.1 for affine and Lemma 35.7.6 for .

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Your email address will not be published.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 049C. Beware of the difference between the letter 'O' and the digit '0'.