Abelian equals A-finite for Anderson A-modules

Anderson introduced t -modules as higher dimensional analogs of Drinfeld modules. Attached to such a t -module, there are its t -motive and its dual t-motive. The t -module gets the attribute “abelian” when the t -motive is a finitely generated module, and the attribute “ t -finite” when the dual t -motive is a finitely generated module. The main theorem of this article is the affirmative answer to the long standing question whether these two attributes are equivalent. The proof relies on an invariant of the t -module and a condition for that invariant which is necessary and sufficient for both being abelian and being t -finite. We further show that this invariant also provides the information whether the t-module is pure or not. Moreover, we conclude that also over general coefficient rings A, i.e. for Anderson A-modules, the attributes of being abelian and being A-finite are equivalent.