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

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.