Biextensions of Picard stacks and their homological interpretation

Details Biextensions of Picard stacks and their homological interpretation Biextensions of Picard stacks and their homological interpretation

2011-03-03

arxiv.org/abs/1103.0723v1

Mathematics

Algebraic Geometry

37 pages

Scientific paper

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.

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