Tensor functors and finite representation type

AbstractLet 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 LK, let FL be the tensor functor (L⊗K−):Mod(A)→Mod(L⊗KA),X→(L⊗KX). This functor is always faithful. We prove that if for any extension LK the functor FL is essentially surjective (i.e. each YϵMod(L⊗KA) is isomorphic to some FL(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 LK, FL is essentially surjective