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

Abstract

Let S be a locally noetherian scheme and consider two extensions G₁ and G₂ of abelian S-schemes by S-tori. In this note we prove that the fppf-sheaf $\mathbf{Cor}{\mathbf{r}}_S(G_1,G_2)$ of divisorial correspondences between G₁ and G₂ 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 $\mathbf{Pi}{\mathbf{c}}_{G/S}.$

Dedicated to M. Raynaud.

Keywords:

• Group schemes
• divisorial correspondences

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 14F22
• 16H05