Cones

Claus Michael Ringel

Dedicated to the Memory of Maurice Auslander

ABSTRACT. A cone is the full translation subquiver of
ZA_\infty given by the predecessors of some fixed 
vertex on the boundary. We consider an abelian category
A with Auslander-Reiten sequences and we denote its
Auslander-Reiten quiver by \Gamma. We are going to
present a criterion for an indecomposable object X in
A in order that the predecessors of X in \Gamma form
a cone. As a consequence, we will see that for suitable
graded algebras, any stable Auslander-Reiten component
which contains a Koszul module is of the form ZA_\infty.