Lemma 95.14.10 (04UT)—The Stacks project

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Table of contents
Part 7: Algebraic Stacks
Chapter 95: Examples of Stacks
Section 95.14: Classifying torsors
Lemma 95.14.10 (cite)

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Lemma 95.14.10. 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. $\square$

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