Scalar Extension of Abelian and Tannakian Categories

Abstract

We introduce and develop the notion of scalar extension for abelian categories. Given a field extension F'/F, to every F-linear abelian category A satisfying a suitable finiteness condition we associate an F'-linear abelian category A' and an exact F-linear functor t: A --> A'. This functor is universal among F-linear right exact functors with target an F'-linear abelian category. We discuss various basic properties of this concept, among others compatibilities with multilinear endofunctors such as tensor products, and the permanence of favourable properties of the functors and categories involved. We obtain the notion of scalar extension for Tannakian categories, which allows us to deduce consequences for the algebraic monodromy groups of Tannakian categories.