Quasi-compact group schemes, Hopf sheaves, and their representations

TL;DR

This paper generalizes Tannaka Duality to affine extensions of abelian varieties, establishing a correspondence between their representation categories and categories of Hopf sheaves, thus extending classical affine group scheme theory.

Contribution

It introduces a duality framework for affine extensions of abelian varieties, linking their representations to Hopf sheaves, generalizing classical results for affine group schemes.

Findings

Characterization of categories of representations of affine extensions

Existence of an equivalence between affine extensions and Hopf sheaves

Representation categories are equivalent to categories of Hopf sheaf comodules

Abstract

We explore the notion of representation of an affine extension of an abelian variety -- such an extension is a faithfully flat affine morphism of $\Bbbk$-group schemes $q:G\to A$, where $A$ is an abelian variety. We characterize the categories that arise as the category of representations of an affine extension $q:G\to A$, generalizing the classical results of Tannaka Duality established for affine $\Bbbk$-group schemes (that is, when $A=\operatorname{Spec}(\Bbbk)$). We also prove the existence of a contravariant equivalence between the category of affine extensions of a given $A$ and the category of faithful commutative Hopf sheaves on $A$, generalizing in this manner the well-known op-equivalence between affine group schemes and commutative Hopf algebras. If $\mathcal H_q$ is the Hopf sheaf on $A$ associated to $q$, the category of representations of $q$ is equivalent to the category of $\mathcal H_q$-comodules.