Homological interpretation of extensions and biextensions of 1-motives

TL;DR

This paper establishes a homological framework for understanding extensions and biextensions of 1-motives over a separably closed field, linking geometric notions with cohomological groups.

Contribution

It introduces geometric definitions of extensions and biextensions of 1-motives and proves their homological interpretations via Ext groups.

Findings

Biext^0(K_1,K_2;K_3) is isomorphic to Ext^0(K_1⊗K_2,K_3)

Biext^1(K_1,K_2;K_3) is isomorphic to Ext^1(K_1⊗K_2,K_3)

Provides a cohomological perspective on biextensions of 1-motives.

Abstract

Let k be a separably closed field. Let K_i=[A_i \to B_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 cohomology group Ext^0(K_1 \otimes K_2,K_3), 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 cohomology group Ext^1(K_1 \otimes K_2,K_3).