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 objects. 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. © 2008 Springer-Verlag.