TENSOR FUNCTORS AND FINITE REPRESENTATION TYPE

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, le t F(L) be the tensor functor (L X K-):Mod(A) --> Mod(L X(K) A), X bar arrow pointing right (L X(K) X). This functor is always faithful. We p rove that if for any extension L/K the functor FL is essentially surje ctive (i.e. each Y is-an-element-of Mod(L X(K) A) is isomorphic to som e F(L)(X) with X is-an-element-of Mod(A)), then A is of finite represe ntation type. The converse is not generally true. However, A is of fin ite representation type if and only if for each separable extension L/ K, F(L) is essentially surjective.