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 $\underline{\operatorname{Hom}}(\mathbb{G}_a,\mathbb{G}_m)=\widehat{\mathbb{G}}_a$ and so, by Cartier duality, $\underline{\operatorname{Hom}}(\widehat{\mathbb{G}}_a,\mathbb{G}_m)=\mathbb{G}_a$. (Here we see all the groups as abelian sheaves on $(\mathsf{Sch}/k)_\text{fppf}$.)

Is it true that the Ext sheaves $\underline{\operatorname{Ext}}^i(\mathbb{G}_a,\mathbb{G}_m)$ and $\underline{\operatorname{Ext}}^i(\widehat{\mathbb{G}}_a,\mathbb{G}_m)$ 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.)

Also, I know that [BB, Lemma A.4.6] proves that $\underline{\operatorname{Ext}}^1(\widehat{\mathbb{G}}_a,\mathbb{G}_m)=0$ but I don't know how to deal with higher extensions.

References:

• [Br] L. Breen - Extensions of Abelian Sheaves and Eilenberg–Maclane Algebras
• [BB] L. Barbieri-Viale, A. Bertapelle - Sharp de Rham Realization

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

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

Is it true that the Ext sheaves $\underline{\operatorname{Ext}}^i(\mathbb{G}_a,\mathbb{G}_m)$ and $\underline{\operatorname{Ext}}^i(\widehat{\mathbb{G}}_a,\mathbb{G}_m)$ 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)

Also, I know that [BB, Lemma A.4.6] proves that $\underline{\operatorname{Ext}}^1(\widehat{\mathbb{G}}_a,\mathbb{G}_m)=0$ but I don't know how to deal with higher extensions.

References:

• [Br] L. Breen - Extensions of Abelian Sheaves and Eilenberg-MacLane Algebras
• [BB] L. Barbieri-Viale, A. Bertapelle - Sharp de Rham Realization

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 $\underline{\operatorname{Hom}}(\mathbb{G}_a,\mathbb{G}_m)=\widehat{\mathbb{G}}_a$ and so, by Cartier duality, $\underline{\operatorname{Hom}}(\widehat{\mathbb{G}}_a,\mathbb{G}_m)=\mathbb{G}_a$. (Here we see all the groups as abelian sheaves on $(\mathsf{Sch}/k)_\text{fppf}$.)

Is it true that the Ext sheaves $\underline{\operatorname{Ext}}^i(\mathbb{G}_a,\mathbb{G}_m)$ and $\underline{\operatorname{Ext}}^i(\widehat{\mathbb{G}}_a,\mathbb{G}_m)$ 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.)

Also, I know that [BB, Lemma A.4.6] proves that $\underline{\operatorname{Ext}}^1(\widehat{\mathbb{G}}_a,\mathbb{G}_m)=0$ but I don't know how to deal with higher extensions.

References:

• [Br] L. Breen - Extensions of Abelian Sheaves and Eilenberg–Maclane Algebras
• [BB] L. Barbieri-Viale, A. Bertapelle - Sharp de Rham Realization