Extensions of Picard stacks and their homological interpretation

TL;DR

This paper introduces extensions of strictly commutative Picard stacks over a site and relates their classification to a specific cohomology group, providing a homological perspective on their structure.

Contribution

It defines new operations on Picard stacks and establishes a homological interpretation of their extensions via Ext^1 groups.

Findings

Extensions are parametrized by Ext^1([P],[Q])

Defined pull-back, push-down, and sum operations on extensions

Homological interpretation links extensions to cohomology groups

Abstract

Let S be a site. We introduce the notion of extensions of strictly commutative Picard S-stacks. We define the pull-back, the push-down, and the sum of such extensions and we compute their homological interpretation: if P and Q are two strictly commutative Picard S-stacks, the equivalence classes of extensions of P by Q are parametrized by the cohomology group Ext^1([P],[Q]), where [P] and [Q] are the complex associated to P and Q respectively.