Irreducible morphisms and Auslander-Reiten triangles in the stable category of modules over a repetitive algebra take three canonical shapes, and every such triangle is induced by an Auslander-Reiten sequence.
Shapes of Auslander-Reiten Triangles
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abstract
Our main theorem classifies the Auslander-Reiten triangles according to properties of the morphisms involved. As a consequence, we are able to compute the mapping cone of an irreducible morphism. We finish by showing a technique for constructing the connecting component of the derived category of any tilted algebra. In particular we obtain a technique for constructing the derived category of any tilted algebra of finite representation type.
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On irreducible morphisms and Auslander-Reiten triangles in the stable category of modules over repetitive algebras
Irreducible morphisms and Auslander-Reiten triangles in the stable category of modules over a repetitive algebra take three canonical shapes, and every such triangle is induced by an Auslander-Reiten sequence.