The structure and homological properties of generalized standard Auslander–Reiten components

Abstract

We describe the structure and homological properties of arbitrary generalized standard Auslander–Reiten components of artin algebras. In particular, we prove that for all but finitely many indecomposable modules in such components the Euler characteristic is defined and nonnegative. Further, we provide a handy criterion for an infinite Auslander–Reiten component of an artin algebra to be generalized standard. We solve also the long standing open problem concerning the structure of artin algebras admitting a separating family of Auslander–Reiten components.