The structure and homological properties of generalized standard Auslander-Reiten components
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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.
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Artin algebra, Auslander-Reiten quiver, generalized standard component, tilted algebra, quasitilted algebra, Euler form, generically tame algebra