Irreducible morphisms, the Gabriel-valued quiver and colocalizations for coalgebras

Given a basicK-coalgebraC, we study the left Gabriel-valued quiver(QC,dC)ofCby means of irreducible morphisms between indecomposable injective leftC-comodules and by means of the powersradmof the radicalradof the categoryC-injof the socle-finite injective leftC-comodules. Connections between the valued quiver(QC,dC)ofCand the valued quiver(QC¯,dC¯)of a colocalization coalgebra quotientfE:C→C¯ofCare established.

The Structure of Stable Components

Let A be an artin algebra. Let ( be a component of the stable Auslander- Reiten quiver of A. If is periodic, then the structure of G is known. Here, we are going to consider the case when is non-periodic: we will show that is isomorphic to Z with a valued quiver. In particular, there is no cyclic path in .