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

摘要

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 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).