# Do we have $\underline{\operatorname{Ext}}^i_\text{fppf}(\mathbb{G}_a,\mathbb{G}_m)=0$ for $i>0$?

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:

• Dear @LSpice, do you mind explaining why you changed the \textsf to \mathsf? Apr 13 at 16:49
• 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. Apr 13 at 16:52

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.