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:

- [Br] L. Breen - Extensions of Abelian Sheaves and Eilenberg–Maclane Algebras

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