  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.
