Walter Ferrer Santos

We introduce the notion of Hopf sheaf on an abelian variety and prove the existence of a contravariant equivalence between the category of faithful commutative Hopf sheaves on a given abelian variety A, and the category of affine extensions of A, generalizing in this manner the well-known \op-equivalence between commutative Hopf algebras and affine group schemes. 

We develop a representation theory for the Hopf sheaves and prove that if

q_H : G_H |-> A is the affine extension associated to H (a Hopf sheaf on A), then the category of q-coherent H--comodules is equivalent to the category of representations of q_H. 

Our exposition will center in the algebraic aspects of the theory with special emphasis in the monoidality of the constructions.