The Stacks project

Remark 39.11.5. Let $(G, m)$ be a group scheme over the scheme $S$. In this situation we have the following natural types of questions:

1. If $X \to S$ is a pseudo $G$-torsor and $X \to S$ is surjective, then is $X$ necessarily a $G$-torsor?

2. Is every $\underline{G}$-torsor on $(\mathit{Sch}/S)_{fppf}$ representable? In other words, does every $\underline{G}$-torsor come from a fppf $G$-torsor?

3. Is every $G$-torsor an fppf (resp. smooth, resp. étale, resp. Zariski) torsor?

In general the answers to these questions is no. To get a positive answer we need to impose additional conditions on $G \to S$. For example: If $S$ is the spectrum of a field, then the answer to (1) is yes because then $\{ X \to S\}$ is a fpqc covering trivializing $X$. If $G \to S$ is affine, then the answer to (2) is yes (this follows from Descent, Lemma 35.37.1). If $G = \text{GL}_{n, S}$ then the answer to (3) is yes and in fact any $\text{GL}_{n, S}$-torsor is locally trivial (this follows from Descent, Lemma 35.7.6).