# A note on divisorial correspondences of extensions of abelian schemes by tori

## TL;DR

This paper proves the representability of the sheaf of divisorial correspondences between extensions of abelian schemes by tori and explores how line bundles induce group homomorphisms in this context.

## Contribution

It establishes the representability of the sheaf of divisorial correspondences and links line bundles on extensions to group homomorphisms, advancing understanding of these geometric structures.

## Key findings

- Divisorial correspondence sheaf is representable.
- Line bundles induce group homomorphisms.
- Extension structures relate to Picard schemes.

## Abstract

Let S be a locally noetherian scheme and consider two extensions G_1 and G_2 of abelian S-schemes by S-tori. In this note we prove that the fppf-sheaf Corr _S(G_1,G_2) of divisorial correspondences between G_1 and G_2 is representable. Moreover, using divisorial correspondences, we show that line bundles on an extension G of an abelian scheme by a torus define group homomorphisms between G and Pic_{ G/S}.