Lemma 109.51.1. Let $S$ be a scheme. Let $G$ be a group scheme over $S$. The stack $G\textit{-Principal}$ classifying principal homogeneous $G$-spaces (see Examples of Stacks, Subsection 94.14.5) and the stack $G\textit{-Torsors}$ classifying fppf $G$-torsors (see Examples of Stacks, Subsection 94.14.8) are not equivalent in general.

Proof. The discussion above shows that the functor $G\textit{-Torsors} \to G\textit{-Principal}$ isn't essentially surjective in general. $\square$

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