G-principal bundles on formal disc

Question

How can I prove that the $G$-fiber principal bundles on $\operatorname{Spec} \mathbb{C}[[t]]$ and $\operatorname{Spec} \mathbb{C}((t))$ are trivial when $G$ is an algebraic linear group on $\mathbb{C}$? There are some characterization or theorem which became trivial or easy in this case? I would be glad of any kind of hints.

Answer 1

Let $P \to \operatorname{Spf} \mathbb{C}[[t]]$ be a (Zariski-)locally trivial $G$-torsor for a linear algebraic group $G$ over $\mathbb{C}$. To show that $P$ is trivial, it suffices to exhibit a section $\operatorname{Spf} \mathbb{C}[[t]] \to P$. Since $\mathbb{C}$ is algebraically closed, there is a rational point $\operatorname{Spec} \mathbb{C} \to P$. It remains to extend this section over $\operatorname{Spec} \mathbb{C}$ to an infinitesimal neighborhood $\operatorname{Spf} \mathbb{C}[[t]]$. A map $\operatorname{Spf} \mathbb{C}[[t]] \to P$ is a compatible system of maps $\operatorname{Spec} \mathbb{C}[t]/t^n \to P$. Starting with $\operatorname{Spec} \mathbb{C} \to P$, this is the lifting problem posed by formal smoothness, which can be solved since $G$ is smooth.

For $\operatorname{Spf} \mathbb{C}((t))$, I don't think the statement is true: consider the nontrivial $\mu_n$-torsor $\operatorname{Spf} \mathbb{C}((t^{1/n})) \to \operatorname{Spf} \mathbb{C}((t))$ for $n > 1$.