Biextensions of Picard stacks and their homological interpretation

TL;DR

This paper introduces a 2-category framework for biextensions of strictly commutative Picard stacks and links their classification to specific cohomology groups, providing a homological interpretation.

Contribution

It defines the structure and operations of biextensions of Picard stacks and establishes their classification via Ext cohomology groups, offering a new homological perspective.

Findings

Biextensions are parametrized by Ext^1 cohomology groups.

Isomorphism classes of automorphisms correspond to Ext^0 groups.

Automorphisms of automorphisms are described by Ext^{-1} groups.

Abstract

Let S be a site. We introduce the 2-category of biextensions of strictly commutative Picard S-stacks. We define the pull-back, the push-down, and the sum of such biextensions and we compute their homological interpretation: if P,Q and G are strictly commutative Picard S-stacks, the equivalence classes of biextensions of (P,Q) by G are parametrized by the cohomology group Ext^1([P] {\otimes} [Q] ,[G]), the isomorphism classes of arrows from such a biextension to itself are parametrized by the cohomology group Ext^0([P]{\otimes} [Q] ,[G]) and the automorphisms of an arrow from such a biextension to itself are parametrized by the cohomology group Ext^{-1}([P]{\otimes}[Q] ,[G]), where [P],[Q] and [G] are the complex associated to P,Q and G respectively.