The Stacks project

Lemma 94.14.10. Let $B$ be an algebraic space over $S$. Let $G$ be a group algebraic space over $B$. Denote $\mathcal{G}$, resp. $\mathcal{B}$ the algebraic space $G$, resp. $B$ seen as a sheaf on $(\mathit{Sch}/S)_{fppf}$. The functor

\[ G\textit{-Torsors} \longrightarrow \mathcal{G}/\mathcal{B}\textit{-Torsors} \]

which associates to a triple $(U, b, X)$ the triple $(U, b, \mathcal{X})$ where $\mathcal{X}$ is $X$ viewed as a sheaf is an equivalence of stacks in groupoids over $(\mathit{Sch}/S)_{fppf}$.

Proof. We will use the result of Stacks, Lemma 8.4.8 to prove this. The functor is fully faithful since the category of algebraic spaces over $S$ is a full subcategory of the category of sheaves on $(\mathit{Sch}/S)_{fppf}$. Moreover, all objects (on both sides) are locally trivial torsors so condition (2) of the lemma referenced above holds. Hence the functor is an equivalence. $\square$

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Your email address will not be published.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