Showing $GL_n$ is a special algebraic group

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?

• This seems to answer your question mathoverflow.net/a/168004/58056 – random123 Nov 30 '18 at 15:32
• @random123 Do you happen to understand that answer? I don't understand a lot of it unfortunately and perhaps an answer to my question just requires more background work. – Fromage Nov 30 '18 at 16:30
• I may be wrong here but the answer doesnot seem to use too much of machinery. Could you be more specific about what is unclear in the proof there? – random123 Nov 30 '18 at 16:48
• I am not sure if this came up in your google search but perhaps this is a better reference than the one in the comment above. Page 30 of www-personal.umich.edu/~takumim/takumim_Spr14Thesis.pdf – random123 Nov 30 '18 at 17:11
• It probab;y doesn't use that much but I have just started looking at stacks etc. so it's all new to me. The 2nd reference is more complicated imo. In terms of what I don't understand in the overflow answer are: the construction of $E= P \times^{GL_N} \mathbb G$ so what does this product mean, how are we getting the locally free sheaf $\mathcal F$ (it's discussed in the comments but I don't know what effective descent means in that context). Also i'm using the etale topology not the fpqc one, so how would that change things? – Fromage Nov 30 '18 at 20:11

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).

• Hey thanks alot for writing up as an answer, I had look at the descent stuff and I'll probably go over it in depth later on but for now I can see the outline of the argument. Just one question though: all the stuff about the orbits being contained in an affine subset and smoothness, were they just needed to construct $E$? If so I read in other places that the construction of $E$ is essentially just replacing the etale local copies of $GL_n$ in the principal bundle with copies of $\mathbb A^n$ and then using the same glueing data. So do we really need to do all that extra work to define $E$? – Fromage Dec 7 '18 at 11:54
• @Fromage I myself got to understand this a bit better while writing this answer. I guess you are right, but then one will have to argue that these new trivial vector bundles one the etale open subsets of the etale cover of $X$ glue together to form a scheme over $X$. That seems like a clean way to do it. The construction I did is somewhat standard in differential geometry. It seemed as a good idea at the time of writing the answer to follows the constructions from differential geometry. I guess it is a bit messy this way. – random123 Dec 7 '18 at 12:08
• Ah okay, it was still useful to see that idea though and it's nice to know it comes from diff geom. Thanks once again for going through this so thoroughly with me, it has been really helpful! – Fromage Dec 7 '18 at 14:34
• @Fromage Sure! You are welcome. – random123 Dec 7 '18 at 15:14