The Stacks project

95.14.7 Variant on principal homogeneous spaces

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 $G\textit{-Principal-Schemes} \subset G\textit{-Principal}$ whose objects are pairs $(U, X)$ where $U$ is an object of $(\mathit{Sch}/S)_{fppf}$ and $X \to U$ is a principal homogeneous $G$-space over $U$ which is representable, i.e., a scheme.

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