Projectively stable artin algebras: representation theory and quivers with relations

Abstract

A left artinian ring Λ is projectively stable if no non-zero morphism f: M → N of finitely generated left Λ-modules M,N having no non-zero projective direct summands factors through a projective module. Such rings have many good properties and, although the definition is given in terms of the category of modules, the structure of projectively stable left artinian rings allows a satisfying description: they are “built” from well-known left artinian rings, left hereditary and serial, by a pull back of rings construction. We show that representations of projectively stable artin algebras are also “built” from representations of well-studied artin algebras, hereditary and serial. Namely, we give a simple geometric construction that produces the Auslander-Reiten quiver or the ordinary quiver of a projectively stable algebra from those of an hereditary algebra and a serial algebra.