Extensions of Abelian Schemes and the Additive Group

TL;DR

This paper studies the extension sheaves of abelian schemes and the additive group, providing new proofs and descriptions of key extension groups in algebraic geometry.

Contribution

It offers a generalized proof of the Barsotti--Weil formula, proves the vanishing of a specific Ext sheaf, and describes an extension group in characteristic zero.

Findings

Generalized proof of the Barsotti--Weil formula

Vanishing of ^2(A, \u211d_m) for abelian schemes

Description of ^1(_a, _m) in characteristic zero

Abstract

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.