Homological interpretation of extensions and biextensions of 1-motives

Let k be a separably closed field. Let K i=[A i→ uiB i] (for i=1, 2, 3) be three 1-motives defined over k. We define the geometrical notions of extension of K 1 by K 3 and of biextension of (K 1, K 2) by K 3. We then compute the homological interpretation of these new geometrical notions: namely, the group Biext 0(K 1, K 2;K 3) of automorphisms of any biextension of (K 1, K 2) by K 3 is canonically isomorphic to the group Ext0(K1⊗LK2,K3), and the group Biext 1(K 1, K 2;K 3) of isomorphism classes of biextensions of (K 1, K 2) by K 3 is canonically isomorphic to the group Ext 1(K 1⊗L{double-struck}K 2,K 3). © 2012 Elsevier Inc