I'm in particular interested in understanding Grothendieck's argument for this in SGA 1 (page 232 in <http://arxiv.org/pdf/math/0206203v2.pdf>)


Let $G$ be $\text{GL}\_n$ over a scheme $S$ for some integer $n$, and let $P/S$ be a principal $G$-bundle. Then we know that there is an fpqc morphism $S'\rightarrow S$ such that $P' := P\times\_S S'$ is isomorphic to $G' := G\times\_S S'$ over $S'$.


He claims in his proof that the statement follows from noting that $G(T) = \text{Aut}(\mathcal{O}\_T^n)$ (for any $S$-scheme $T$), and that fpqc morphisms are morphisms of effective descent in the category of locally free $\mathcal{O}\_T$-modules of rank $n$.


I guess his argument must rely on something special about automorphism group schemes which I'm missing. I admit I haven't read the entirety of his prior chapters on descent. Everything I know about descent pretty much comes from reading the stacks project.


I'd like to know how Grothendieck envisaged his argument would go, though I'd also appreciate any other argument proving this fact or relevant references.


thanks


* will



why are principal GL(n)-bundles (Zariski-)locally trivial?
I would argue using the associated fiber bundle construction in Exp. XI just after corollary 4.3 (a few pages back from your spot). For any $GL\_n$-torsor $P$, you may use the action of $GL\_n$ on $\mathbb{G}\_a^n$ to construct a canonical vector bundle $E = P \times^{GL\_n} \mathbb{G}\_{a,S}^n$, and there is a canonical locally free sheaf $\mathcal{F}$ such that $E = \mathbb{V}(\mathcal{F}) = \operatorname{Spec}\_S \operatorname{Sym}\_{\mathcal{O}\_S}(\mathcal{F})$. The structures $E$ and $\mathcal{F}$ can be constructed on $S$ by descent from the corresponding trivial objects on $S'$. Furthermore, the locally free sheaf $\mathcal{F}$ is Zariski-locally trivial.


By unwinding the definition of $E$, we see that there is a canonical isomorphism $P \cong \underline{\operatorname{Isom}}\_{S-vect}(\mathbb{G}\_{a,S}^n, E)$. Indeed, sections of $P$ are in canonical bijection with trivializations of $E$. Using the inverse of the $\mathbb{V}$ equivalence, we have an isomorphism $P \cong \underline{\operatorname{Isom}}\_{\mathcal{O}\_S-mod}(\mathcal{O}^{\oplus n}\_S, \mathcal{F})$. Thus, a Zariski-local trivialization of $\mathcal{F}$ induces a Zariski-local trivialization of $P$.


