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I have seen the following claim without proof in more than one paper, but it is sufficiently general that I suspect it is stated too strongly to be true:

Let $G$ be an affine group scheme (say, over a field of characteristic zero), let $X$ be a scheme (smooth over the same field), and let $P \to X$ be a $G_X$-torsor. If $Y$ is a scheme with a $G$-action, then the associated bundle $P \times^G Y$ is a scheme.

I can use fpqc descent to show that this is true when $Y$ is affine (or some other effective fpqc descent class), and I can use Zariski descent when $P$ is Zariski-locally trivial. In full generality, I suspect one can assemble known counterexamples of descent to falsify this claim, but I have been unable to do so.

Question: Is there a counterexample known? Failing that, is there a proof of the claim?

I'm somewhat more interested in the case where $G$ is connected, but here is a candidate that I don't know how to prove: Take $X$ to be a smooth curve with a nontrivial étale double cover $P$ (with $G$ constant of order 2), and set $Y$ to be Hironaka's 3-fold with involution (whose quotient sheaf is not a scheme).

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I don't know about counterexamples. But there are positive results in SGA 3 and in follow-up work by Raynaud. – Jason Starr Aug 5 '11 at 12:45
Thanks. I looked in SGA3, but didn't find anything this strong. I'll have a look at Raynaud. – S. Carnahan Aug 5 '11 at 15:47
up vote 7 down vote accepted

This is false; there is a counterexample (with $G$ a finite group of order $2$) in my notes on descent theory, subsection 4.4.2.

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Thank you. Somehow, I had missed that part before. – S. Carnahan Aug 8 '11 at 2:35

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