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Let $S$ be a scheme over a field $k$, and let $G$ be a reductive group scheme over $S$. Let us call it trivial, if it is a pull-back of a group scheme over $k$ via the structure morphism $S\to k$. Is it always true that $G$ becomes trivial after a certain etale base change $S'\to S$? I am willing to assume that $S$ is smooth if needed.

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The question looks natural, but it would be interesting to know a little more about the background or the implications. – Jim Humphreys Apr 23 '11 at 14:56
If you look at SGA 3, Prop XIX.6.1, it is essentially shown that any reductive group over any base is split after a finite etale base change. So if you allow extensions of scalars of $k$ as well, then what you need follows. – Keerthi Madapusi Pera Apr 23 '11 at 15:11
Sorry, it appears as if the Proposition does not guarantee you a finite etale base change, though I would be surprised if this weren't true. – Keerthi Madapusi Pera Apr 23 '11 at 15:13
I have been informed by the esteemed BCnrd that my previous comment was too optimistic. In fact, in his paper 'Groups over $\mathbb{Z}$', B. Gross has shown the existence of non-split simply-connected semi-simple groups over $\mathbb{Q}$ that have reductive models over $\mathbb{Z}$. In particular, these models can only split after a non-finite etale base-change over $\mathbb{Z}$. – Keerthi Madapusi Pera Apr 23 '11 at 21:54
Answering the question of Jim, if the statement is correct, then reductive group schemes over $S$ are classified by $H^1_{et}(S, Aut_G)$, right? – Roman Fedorov Apr 25 '11 at 17:26
up vote 4 down vote accepted

Reductive groups schemes over $S$ are classified by $H^1_{fpqc}(S,Aut_G)$, see SGA 3 Exp. XXIV.

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You can indeed replace $fpqc$ by $et$, see SGA 3 Exp. XXIV Cor. 1.18 – Victor Petrov Apr 29 '11 at 11:26
Note that SGA3 has its own definition of reductive. The geometric fibers have to be connected. – Wilberd van der Kallen Apr 29 '11 at 15:55

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