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

TL;DR

This paper investigates the structure and homological properties of generalized standard Auslander-Reiten components in artin algebras, providing criteria for their characterization and solving a longstanding open problem.

Contribution

It introduces a criterion for identifying generalized standard components and addresses the structure of artin algebras with separating families of such components.

Findings

Most indecomposable modules in these components have nonnegative Euler characteristic.

A criterion is established for infinite Auslander-Reiten components to be generalized standard.

The structure of artin algebras with separating families of components is characterized.

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.