Char $0$ example of fppf surjectivity without etale surjectivity

Question

If we have a surjective morphism of representable fppf sheaves over the fppf site over a field，it may not be surjective as etale sheaves. For example, consider $p$ power from $G_m$ to $G_m$ when the base field has char $p$ nonzero.

Do we know any such example in char 0 with finite type assumption？

Answer 1

Let me write as an answer the claim that I was trying to make in the above comments. I believe the following is correct:

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NB: All sites in the following are the 'big' versions of the sites or, if your fppf site is the small fppf site, below is discussing the stronger claim that you still get an etale sheaf with underlying category the same as that of the small fppf site but with the etale topology. You can get even stronger claims (I think), like having a lot of these "for all $X$-schemes $Y$" conditions go away in a lot of situations if you only care about getting a sheaf on the small etale site.

In fact, if you want to know that a short exact sequence of fppf sheaves on $\mathrm{Spec}(k)$ is actually short exact on the small etale of $\mathrm{Spec}(k)$ when $k$ is characteristic $0$ (or perfect), then this is actually fairly easy.

Namely, suppose that $k$ is arbitrary and that we have a short exact sequence of etale sheaves

$$0\to \mathcal{F}_1\to\mathcal{F}_2\xrightarrow{f}\mathcal{F}_3\to 0$$

of sheaves on the fppf site of $\mathrm{Spec}(k)$. We claim that the map $\mathcal{F}_2(\overline{k})\to \mathcal{F}_3(\overline{k})$ is surjective. Indeed, if $z\in\mathcal{F}_3(\overline{k})$ then we know there is an fppf map $X\to\mathrm{Spec}(\overline{k})$ and some $x\in \mathcal{F}_2(X)$ such that $f(x)=z\mid_X$. But, note that $X$ has an $\overline{k}$-point $y$ since it's finite type. Note then that

$$f(x\mid_y)=(z\mid_X)\mid_y=z$$

so that, in fact, $z$ is in the image of $f$ on $\overline{k}$.

Now, to show that our sequence is short exact on the small etale site of $\mathrm{Spec}(k)$ we know that it suffices to check surjectivity of $\mathcal{F}_2\to\mathcal{F}_3$ on stalks. This amounts to checking that $\mathcal{F}_2(k^\mathrm{sep})\to\mathcal{F}_3(k^\mathrm{sep})$ is surjective. Since $\mathrm{char}(k)=0$ $k^\mathrm{sep}=\overline{k}$ and we're done.

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Let us make the following claim:

Claim 1: Let $X$ be a scheme and let $$0\to\mathcal{F}_1\to\mathcal{F}_2\xrightarrow{f}\mathcal{F}_3\to 0\qquad (\ast)$$ be a short exact sequence of fppf sheaves of groups on $X$. Suppose that the natural map $$H^1_\text{et}(Y,\mathcal{F}_1)\to H^1_\text{fppf}(Y,\mathcal{F}_1)$$ is a bijection for all $X$-schemes $Y$. Then, $(\ast)$ is a short exact sequence of etale sheaves on $X$.

Proof: Let $U\to X$ be an $X$-scheme and let $z\in \mathcal{F}_3(U)$. Then, the functor

$$K:\mathsf{Sch}_U\to \mathsf{Set}$$ given by

$$K(T)=\{x\in\mathcal{F}_2(T):f(x)=z\mid_T\}$$

is a $\mathcal{F}_1$-torsor for the fppf site on $U$. By assumption, we have that every $\mathcal{F}_1$-torsor for the fppf site on $U$ is actually split etale locally. Thus, $K$ actually obtains a section etale locally on $U$. This shows that $f$ is etale locally surjective, and so $(\ast)$ is a short exact sequence on the etale site. $\blacksquare$

Recall then we have the following observation of Grothendieck:

Observation 1(Grothendieck): Let $U$ be any scheme and let $\mathcal{G}$ be a smooth commtuative group scheme on $U$. Then, the natural map $H^i_\text{et}(U,\mathcal{G})\to H^i_\text{fppf}(U,\mathcal{G})$ is an isomorphism for all $i\geqslant 0$.

Proof:$\text{ }$ This is Le Groupe de Brauer III, Theorem 11.7. $\blacksquare$

If you don't want to assume abelian, you can upgrade this easily to the following:

Observation 2: Let $U$ be a scheme and let $\mathcal{G}$ be a smooth group scheme on $U$ such that every fppf torsor of $\mathcal{G}$ is representable. Then, the natural map $H^1_\text{et}(U,\mathcal{G})\to H^1_\text{fppf}(U,\mathcal{G})$ is bijective.

Remark: The condition that every fppf torsor of $\mathcal{G}$ is representable is true fairly generally. For example, if:

• $\mathcal{G}$ is affine (note that this is preserved by base change!)
• $\dim U\leqslant 1$ (and $\mathcal{G}$ is separated).
• $\mathcal{G}$ is an abelian scheme and $U$ is regular.

as some examples (see Theorem 4.3 of Milne's book on etale cohomology).

Proof (Observation 2): Let $\mathcal{F}$ be an fppf torsor for $\mathcal{G}$. By assumption, we know that $\mathcal{F}$ is represented by some $U$-scheme $\mathcal{T}$. Since $\mathcal{T}$ is a $\mathcal{G}$-torsor and $\mathcal{G}$ is smooth, we know that $\mathcal{T}$ is smooth. Thus, we know that it has a section etale locally on $U$ and so is split locally. The conclusion follows. $\blacksquare$

Combining all of this we deduce the following:

Claim 2: Let $X$ be a scheme and let $\mathcal{G}$ be a smooth group scheme on $X$ for which every $X$-scheme $Y$ any fppf torsor of $\mathcal{G}_Y$ is representable (or $\mathcal{G}$ is abelian). Then, any short exact sequence of fppf group sheaves on $X$ $$0\to\mathcal{G}\to \mathcal{F}_2\to\mathcal{F}_3\to 0$$ is actually a short exact sequence of etale sheaves on $X$.

Note though the following well-known result of Cartier:

Theorem(Cartier): Let $k$ be a field of characteristic $0$ and let $G/k$ be a finite type group scheme. Then, $G$ is smooth.

Proof: See Tag047N. $\blacksquare$

Thus:

Conclusion: Let $k$ be a field of characteristic $0$ and let $X$ be a $k$-scheme. Let $G$ be a finite type $k$ scheme and set $\mathcal{G}:=G_X$. Then, if for every $X$-scheme $Y$ every fppf torsor for $\mathcal{G}_Y$ is representable (or $\mathcal{G}$ is affine) then any short exact sequence of fppf sheaves on $X$ $$0\to\mathcal{G}\to\mathcal{F}_2\to\mathcal{F}_3\to0$$ is a short exact sequence of etale sheaves.

I think this shows that there is no example of a sequence $$0\to\mathcal{F}_1\to\mathcal{F}_2\to\mathcal{F}_3\to 0$$ of fppf sheaves on $\mathrm{Spec}(k)$ where $k$ is a field of characteristic $0$ such that $\mathcal{F}_1$ is representable by a finite type group scheme $G$ that is either affine or an abelian group (or, more generally, an abelian group or every $k$-scheme $Y$ we have that every fppf torsor for $G_Y$ is representable).