The Stacks project

95.14.11 Variant on torsors in fppf topology

Let $S$ be a scheme. Let $B = S$. Let $G$ be a group scheme over $B = S$. In this setting we can define a full subcategory $G\textit{-Torsors-Schemes} \subset G\textit{-Torsors}$ whose objects are pairs $(U, X)$ where $U$ is an object of $(\mathit{Sch}/S)_{fppf}$ and $X \to U$ is an fppf $G$-torsor over $U$ which is representable, i.e., a scheme.

It is in general not the case that $G\textit{-Torsors-Schemes}$ is a stack in groupoids over $(\mathit{Sch}/S)_{fppf}$. The reason is that in general there really do exist fppf $G$-torsors which are not schemes, hence descent for objects will not be satisfied in general.