Tensor functors and finite representation type

Abstract

Let A be a finite-dimensonal algebra over an infinite field K and Mod( A) be the category of all (left) A modules. For each extension L K , let F L be the tensor functor ( L⊗ K −):Mod( A)→Mod( L⊗ K A), X→( L⊗ K X). This functor is always faithful. We prove that if for any extension L K the functor F L is essentially surjective (i.e. each YϵMod( L⊗ K A) is isomorphic to some F L ( X) with XϵMod( A)), then A is of finite representation type. The converse is not generally true. However, A is of finite representation type if and only if for each separable extension L K , F L is essentially surjective.