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 $(\mathsf{Sch}/k)_\text{fppf}$.)
In a previous question, we saw that $\underline{\operatorname{Ext}}^1(\mathbb{G}_a,\mathbb{G}_m)\neq 0$. Now I wonder about the formal analog. (Both questions were previously only one. But we got a great answer to half of the question, so I thought it was ok to split it in two.)
Is it true that the Ext sheaves $\underline{\operatorname{Ext}}^i(\widehat{\mathbb{G}}_a,\mathbb{G}_m)$ vanish for $i>0$?
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. Perhaps there's some homological dimension argument in here?
Reference:
- [BB] L. Barbieri-Viale, A. Bertapelle - Sharp de Rham Realization
