Locally Dynkin quivers and hereditary coalgebras whose left comodules are direct sums of finite dimensional comodules

Let C be an indecomposable hereditary K-coalgebra, where K is an algebraically closed field. We prove that every left C-comodule is a direct sum of finite dimensional C-comodules if and only if C is comodule Morita equivalent (see [19]) with a path K-coalgebra KQ(op), where Q is a pure semisimple locally Dynkin quiver, that is, Q is either a finite quiver whose underlying graph is any of the Dynkin diagrams A(n), n greater than or equal to 1, D-n, n greater than or equal to 4, E-6, E-7, E-8, or Q is any of the infinite quivers A(infinity)((s)), (infinity)A(infinity)((S)), D-infinity((s)), with s greater than or equal to 0, shown in Sec. 2. In particular, we get in Corollaries 2.5 and 2.6 a K-coalgebra analogue of Gabriel's theorem [11] characterising representation-finite hereditary K-algebras (see also [2, Sec. VIII.5]).