Extensions of Abelian Schemes and the Additive Group

Gabriel Ribeiro, Zev Rosengarten

Published 2025-06-20Version 1

We compute extension sheaves of abelian schemes and of the additive group by the multiplicative group in the fppf topology. Our main results include a generalized and streamlined proof of the Barsotti--Weil formula, the vanishing of $\underline{\operatorname{Ext}}^2(A,\mathbb{G}_m)$ for an abelian scheme $A$ over a general base, and a description of $\underline{\operatorname{Ext}}^1(\mathbb{G}_a,\mathbb{G}_m)$ in characteristic zero.