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

Question

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

Answer 1

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.