Showing $GL_n$ is a special algebraic group

Question

So there's this notion of a group scheme $G$ being 'special' if any principal $G$-bundle over a scheme $X$ (say defined in the etale topology) is also locally trivial in the Zariski topology. I would like to see why $GL_n$ is special in this sense. The few books I've seen mention this refer to other books to as their justification of this fact and the only 'proof' I've seen is in Milne's Etale Cohomology, but it uses many notions which I'm not familiar at all with. I've just started to look at stacks so I was hoping there a more accessible approach to show this?

Answer 1

Let $\pi : P \rightarrow X$ be a $GL(n)-$torsor and is locally trivial in the etale topology. We want to prove that it is locally trivial in the Zariski topology.Hence forth we denote $GL(n)$ by $G$ for convenience.

First let us construct the natural associated vector bundle. We just imitate the classical construction. Let $g \in GL(n)$ act on $P \times \mathbb{A}^n$ by $g.(x,y) = (x.g, g^{-1}.y)$, where $GL(n)$ acts on the right on $P$ and in a natural manner from the left on $\mathbb{A}^n$. Note that this action is free since the action is free on $P$. Let us look at the $GL(n)$ orbit of the action.

Claim : All GL(n) orbit on $P$ is contained in an open affine subset of $P$.

Proof of Claim : We know that for $p \in P$, we have $p.G = \pi^{-1}(\pi(p))$. Also note that $\pi$ is an affine map, since it is affine after etale base change. This is a statement that "affine morphism is local on the target". Now choose an open affine neighbourhood of $\pi(p)$, say $U_{\pi(p)}$ and let $U_p := \pi^{-1}(U_{\pi(p)})$. Since $\pi$ is affine, hence $U_p$ is affine and it clearly contains the orbit. Hence the claim.

Using the claim, we get that orbit of $GL(n)$ on $P \times \mathbb{A}^n$ is contained in an open affine subset since $\mathbb{A}^n$ is affine. Also note that the action is free. This allows us to form a quotient space say $E$ which has an obvious map to $X$ which comes from quotient of the $G-$ equivariant projection map $P \times \mathbb{A}^n \rightarrow P$.

Since $GL(n)$ is a smooth group scheme, $P$ is smooth over $X$. This follows from the following : Let $U \rightarrow X$ be etale cover such that $P \times_X U \rightarrow U$ is locally trivial. Since $GL(n)$ is smooth scheme, hence this is a smooth map. Thus we have the following situation $P \times_X U \rightarrow P$ is a smooth map and $P\times_X U \rightarrow U \rightarrow X$ is a smooth map, hence the map $P \rightarrow X$ is also smooth. This statement is known as "smoothness is etale local on the target"

It can be checked from the construction that $E$ is also etale locally trivial with fibers $\mathbb{A}^n$ and hence $E \rightarrow X$ is smooth affine. Let us assign a name $f : E \rightarrow X$.

Let $U_i \xrightarrow{\phi_i} X$ be etale cover such that for all $i$, we have $E \times_X U_i \rightarrow U_i$ is trivial. Thus we have $\phi^*(f_*\mathcal{O}_E) \cong \mathcal{O}_{U_i}[T_1,\dots T_n]$. Let $F_i = \oplus \mathcal{O}_{U_i}T_i$. Note that since $E$ is locally trivial for etale topology, we automatically have a descent data for $\lbrace F_i, \lbrace{U_i\phi_i} \rbrace \rbrace$(I have supressed the notation for coordinate transformations). Thus we have a zariski locally free sheaf $F$ on $X$, such that $\phi_i^*F \cong F_i$. We have $Spec(Sym(F_i)) \cong E \times_X U_i = \phi_i^*(E) \cong \phi_i^*(f_*\mathcal{O}_E)$. This implies that $Sym(F_i) \cong \phi_i^*(f_*\mathcal{O}_E)$. Thus we have a morphism of (effective)descent data and hence we have a map, infact an isomorphism $E \cong Spec(SymF)$(see 3).

This shows that $E$ is infact locally trivial in the Zariski topology. Now the rest should be clear from the answer here : https://mathoverflow.net/a/168004/58056

I will write it here for completeness. We have $P \cong \underline{Isom}(\mathbb{A}^n_S, E)$. Since $E$ is Zariksi locally trivial, we obtain that $P$ is locally trivial.

Here are some references for the descent arguments.

1. https://stacks.math.columbia.edu/tag/02L5 a lemma which says that the property of morphism being affine is local on the base for the fppf topology and hence also in the etale topology.

2. https://stacks.math.columbia.edu/tag/023B is the definition for the definition of descent and morphism of descent data for quasi-coherent sheaves.

3. https://stacks.math.columbia.edu/tag/023E says that the descent data is always effective and also implies that morphism of descent data gives a unique morphism for the quasi-coherent sheaves.

There might be some gaps in the argument. I do not know of a way to avoid all this terminology except maybe by following the line of argument given in the comment above(http://www-personal.umich.edu/~takumim/takumim_Spr14Thesis.pdf).