The Stacks project

The classifying stack of a group scheme or group algebraic space.

Lemma 94.15.4. Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $G$ be a group algebraic space over $B$. Then the stacks in groupoids

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

are all canonically equivalent. If $G \to B$ is flat and locally of finite presentation, then these are also equivalent to $G\textit{-Principal}$.

Proof. The equivalence $G\textit{-Torsors} \to \mathcal{G}/\mathcal{B}\textit{-Torsors}$ is given in Lemma 94.14.10. The equivalence $[B/G] \to [[B/G]]$ is given in Proposition 94.15.3. Unwinding the definition of $[[B/G]]$ given in Section 94.15 we see that $[[B//G]] = G\textit{-Torsors}$.

Finally, assume $G \to B$ is flat and locally of finite presentation. To show that the natural functor $G\textit{-Torsors} \to G\textit{-Principal}$ is an equivalence it suffices to show that for a scheme $U$ over $B$ a principal homogeneous $G_ U$-space $X \to U$ is fppf locally trivial. By our definition of principal homogeneous spaces (Groupoids in Spaces, Definition 77.9.3) there exists an fpqc covering $\{ U_ i \to U\} $ such that $U_ i \times _ U X \cong G \times _ B U_ i$ as algebraic spaces over $U_ i$. This implies that $X \to U$ is surjective, flat, and locally of finite presentation, see Descent on Spaces, Lemmas 73.11.6, 73.11.13, and 73.11.10. Choose a scheme $W$ and a surjective ├ętale morphism $W \to X$. Then it follows from what we just said that $\{ W \to U\} $ is an fppf covering such that $X_ W \to W$ has a section. Hence $X$ is an fppf $G_ U$-torsor. $\square$

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