Extensions and biextensions of locally constant group schemes, tori and abelian schemes

Let S be a scheme. We compute explicitly the group of homomorphisms, the S-sheaf of homomorphisms, the group of extensions, and the S-sheaf of extensions involving locally constant S-group schemes, abelian S-schemes, and S-tori. Using the obtained results, we study the categories of biextensions involving these geometrical objets. In particular, we prove that if G_i (for i=1,2,3) is an extension of an abelian S-scheme A_i by an S-torus T_i, the category of biextensions of (G_1,G_2) by G_3 is equivalent to the category of biextensions of the underlying abelian S-schemes (A_1,A_2) by the underlying S-torus T_3.