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](https://mathoverflow.net/questions/247161/vanishing-of-textext2-sheaf-from-abelian-variety-to-multiplicative-group).)


Reference:


* [Br] L. Breen - [Extensions of Abelian Sheaves and Eilenberg–Maclane Algebras](https://doi.org/10.1007/BF01389887)



Do we have $\underline{\operatorname{Ext}}^i_\text{fppf}(\mathbb{G}_a,\mathbb{G}_m)=0$ for $i>0$?
This is false, see Remark 2.2.16 of [Rosengarten - Tate Duality In Positive Dimension Over Function Fields](https://arxiv.org/pdf/1805.00522.pdf#page34) 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.


