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?
PS: What about the sheaf of the preceding blog post? Here G is formally étale.
[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
F(Spec(R)) = {r in R | r^{p^n} = 0 for some n > 0}.
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.]
Johan, by Artin’s theorem if $G$ is represented by a smooth group scheme (no affine hypotheses) then such a torsor is an algebraic space, and in particular smooth as such, so it admits sections etale-locally on the base. A general formally smooth group sheaf is quite beyond this case, but anyway the affine condition can be weakened. Doesn’t Grothendieck address exactly your general question in section 11 of Brauer III, even comparing higher cohomologies in the commutative case? (My copy is in the office, so I cannot check on it right now.) My vague recollection is that he made no representability hypotheses, but that some aspect of his proofs made no sense to me without representability (perhaps I was just exhausted at the time).
Yes, I was just trying to be careful; too careful in fact. Namely, it is true that if G is a group in algebraic spaces over S and T is an fppf G-torsor, then T is an algebraic space over S, see Lemma Tag 04U1. In particular, if G is smooth over S, then T is an algebraic space smooth over S and hence T is also a torsor for the étale topology. Thanks!
[Well, maybe to apply Artin’s theorem one might need to impose a quasi-separatedness or quasi-compactness hypothesis on $G$, though it would all be moot if Brauer III solves the general case.]
Johan, I checked in Brauer III, and there (in section 11) Grothendieck considers just the case of $G$ that satisfies a mild variant on formal smoothness along with the condition of being “representable near the identity”, and most importantly he requires $G$ to be abelian. Bhargav’s example violates both of Grothendieck’s hypotheses.
OK, thanks Brian! I checked out the reference Brauer III. I agree that Bhargav’s example satisfies neither condition (L) nor (R). [[By the way, it seems to me there is a typo in the formulation of (R) and the morphism U –> \bar{X} should be surjective, not U —> X. Right?]] The condition (L) seems quite a bit stronger than formal smoothness, doesn’t it?
Also, Remark (11.8) part 3) of the reference tells us that if we’re looking at torsors, i.e., just H^1 we may consider noncommutative sheaves with properties (R) and (L) and we get the same result.
Thanks again.
Johan, I only skimmed section 11 this morning, so I missed Remark 11.8 part (3) though I had tried looking for some comment about the non-commutative aspect. Good to hear he did address it in there. I agree that (L) is quite a bit stronger than formal smoothness (I wrote “mild variant” above because I was thinking that in the representable case it is the same and (R) seems barely more interesting than representability). You’re probably right about the type in (R), but my recollection is that I was only able to understand the proof assuming representability and not the weaker condition (R), so I never really thought about (R) too closely.