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Let $k$ be a characteristic zero field and consider the category $(\mathsf{Sch}/k)_\text{fppf}$ of schemes over $k$ with the fppf topology. I know that $\underline{\operatorname{Hom}}(\mathbb{G}_a,\mathbb{G}_m)=\widehat{\mathbb{G}}_a$. (Here we see all the groups as abelian sheaves on $(\mathsf{Sch}/k)_\text{fppf}$.)

Is it true that the Ext sheaves $\underline{\operatorname{Ext}}^i(\mathbb{G}_a,\mathbb{G}_m)$ vanish for $i>0$?

I know that [Br] shows that the Ext groups $\operatorname{Ext}^i(\mathbb{G}_a,\mathbb{G}_m)$ vanish for $i>0$. But I don't know how to deduce what I want from this. (This is very close to the problem in Vanishing of $\text{Ext}^2$ sheaf from abelian variety to multiplicative group.)

Reference:

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  • $\begingroup$ Dear @LSpice, do you mind explaining why you changed the \textsf to \mathsf? $\endgroup$
    – Gabriel
    Commented Apr 13, 2023 at 16:49
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    $\begingroup$ Re, because \textsf is semantically for use in text and \mathsf is semantically for use in math. Of course in MathJax (as opposed to TeX) the distinction is otiose, and may even result in identical rendering (I don't know), but I figured it was better to be semantically correct while I was adding the links. But, if you don't like it, then feel free to revert (although hopefully leaving the links), and please accept my apologies for an unwanted edit. $\endgroup$
    – LSpice
    Commented Apr 13, 2023 at 16:52

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This is false, see Remark 2.2.16 of Rosengarten - Tate Duality In Positive Dimension Over Function Fields in which a nontrivial local extension of $\mathbb{G}_a$ by $\mathbb{G}_m$ is constructed. However, the same paper shows that the first Ext-sheaf vanishes in positive characteristic.

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    $\begingroup$ Your link is anchored at p. 34. Because my browser doesn't obey that directive, I don't know whether that links to physical or logical page 34, but the actual remark seems to be logical page 29 = physical page 31. $\endgroup$
    – LSpice
    Commented Apr 13, 2023 at 22:43
  • $\begingroup$ Wow that's surprising. Do you know something about the higher ext sheaves or about the case of the formal additive group? $\endgroup$
    – Gabriel
    Commented Apr 14, 2023 at 10:06
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    $\begingroup$ Something sounds off. There is a general rule of thumb that anything which holds in enough positive characteristics holds in characteristic 0. What about this result is allowed to violate that rule of thumb? $\endgroup$
    – Theo Johnson-Freyd
    Commented Apr 14, 2023 at 12:29
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    $\begingroup$ @TheoJohnson-Freyd Actually, reading the counterexample a little more carefully, the characteristic zero assumption seems to be used in an essential way when showing that if an extension splits fppf locally then it splits etale locally $\endgroup$
    – Exit path
    Commented Apr 14, 2023 at 17:30
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    $\begingroup$ @LSpice Agreed! Your comment is why I responded again to Theo Johnson-Freyd amending the reason I thought the characteristic zero case was different. The existence of these homomorphisms can't on its own explain the discrepancy for the reason you point out $\endgroup$
    – Exit path
    Commented Apr 14, 2023 at 23:29

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