Biextensions of 1-Motives by 1-Motives

Abstract

Let S be a scheme and let Gi (i = 1; 2; 3) be an extension of an abelian
S-scheme Ai by a S-torus Yi(1): We ¯rst sketch the proof of the following result:
In the topos Tfppf , the category of biextensions of (G1;G2) by G3 is equivalent to
the category of biextensions of (A1;A2) by Y3(1). Using this result, we de¯ne the
notion of biextension of 1-motives by 1-motives. If M(S) denotes what should be
the Tannakian category generated by 1-motives over S (in a geometrical sense),
as a candidate for the morphisms of M(S) from the tensor product of two 1-
motives M1 ­M2 to another 1-motive M3; we propose the isomorphism classes of
biextensions of (M1;M2) by M3.