Tensor functors and finite representation type (Q1323659)

Let \(A\) be a finite-dimensional algebra over an infinite field \(K\). Given a field extension \(L\) of \(K\), one may consider the two algebras \(A\) over \(K\) and \(L \otimes_{K} A\) over \(L\). The tensor functor \(F_ L = L \otimes_ K - : \text{Mod} (A) \to \text{Mod} (L \otimes_ K A)\) from the category of all (left) \(A\)-modules to the category of all (left) \(L \otimes_ K A\)-modules, given by \(X \mapsto F_ L(X) = L \otimes_ K X\) is a faithful functor. The main result in the paper says that if for any extension \(L\) over \(K\) the functor \(F_ L\) is dense, then \(A\) is representation-finite. Moreover, the author shows that \(A\) is of finite- representation type if and only if for each separable extension \(L\) over \(K\), \(F_ L\) is dense.