The Stacks project

Proof. The equivalence $G\textit{-Torsors} \to \mathcal{G}/\mathcal{B}\textit{-Torsors}$ is given in Lemma 95.14.10. The equivalence $[B/G] \to [[B/G]]$ is given in Proposition 95.15.3. Unwinding the definition of $[[B/G]]$ given in Section 95.15 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 78.9.3) 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 74.11.6, 74.11.13, and 74.11.10. Choose a scheme $W$ and a surjective étale 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. $\square$