Let S be a scheme. Suppose that G is a sheaf of groups on (Sch\/S)_{fppf}. What kind of conditions guarantee that any fppf torsor is actually an étale torsor? I know that if G is representable by an affine smooth group scheme this is OK. So this suggests looking at formally smooth sheaves G. Is there a counter example?<\/p>\n
PS: What about the sheaf of the preceding blog post? Here G is formally étale.<\/p>\n
[Edit Tuesday January 04, 2011. Bhargav just send me the following example of a formally smooth G with an fppf torsor which is not an etale torsor. Let S = Spec(k) where k is separably closed but not algebraically closed of characteristic p > 0. Let F be the sheaf which for a k-algebra R gives<\/p>\n
F(Spec(R)) = {r in R | r^{p^n} = 0 for some n > 0}.<\/p>\n
In other words F is the colimit of the sheaves alpha_{p^n} of p^n roots of zero. Allowing arbitrary p-power roots of 0 gives formal smoothness. The injective map alpha_p -> F gives fppf F-torsors that are non-trivial because H^1_{fppf}(Spec(k), alpha_p) -> H^1_{fppf}(Spec(k), F) is injective. And H^1_{fppf}(k, alpha_p) is non-zero by non-perfectness of k. But H^1_{etale}(Spec(k), F) = 0 since S has no connected etale covers.]<\/p>\n","protected":false},"excerpt":{"rendered":"
Let S be a scheme. Suppose that G is a sheaf of groups on (Sch\/S)_{fppf}. What kind of conditions guarantee that any fppf torsor is actually an étale torsor? I know that if G is representable by an affine smooth … Continue reading