Shapes of Auslander-Reiten Triangles
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Shapes of Auslander-Reiten Triangles ˆ ´ Giraldo3 Edson Ribeiro Alvares1 · Sonia Maria Fernandes2 · Hernan Received: 10 December 2018 / Accepted: 12 November 2019 / © Springer Nature B.V. 2019
Abstract Giraldo and Merklen classified the irreducible morphisms in the bounded derived categories of finite dimensional algebras in three classes. The Auslander-Reiten triangles in these categories are made of irreducible morphisms and we classify these triangles in terms of Giraldo and Merklen’s classes. As a byproduct this yields an explicit description of the cone of any irreducible morphism. For tilted algebras this applies to a constructive description of the transjective components of the Auslander-Reiten quiver. Keywords Representation theory of algebras · Auslander-Reiten triangles · irreducible morphisms Mathematics Subject Classification (2010) 16G10 · 16G70
1 Introduction Auslander-Reiten theory is a fundamental tool to understand module categories of artin algebras (see [4, 5]). Its main features are the Auslander-Reiten sequences, the irreducible morphisms, and the Auslander-Reiten quiver. Later Happel [9, 11] adapted this theory to bounded derived categories of finite dimensional algebras with finite global dimension. In particular, he described the associated Auslander-Reiten triangles, irreducible morphisms, and Auslander-Reiten quivers in the case of hereditary algebras. Presented by: Christof Geiss Edson Ribeiro Alvares
[email protected]; [email protected] Sˆonia Maria Fernandes [email protected] Hern´an Giraldo [email protected] 1
Centro Polit´ecnico, Departamento de Matem´atica, Universidade Federal do Paran´a, CP019081, Jardim das Americas, Curitiba-PR, 81531-990, Brazil
2
Departamento de Matem´atica, Universidade Federal de Vic¸osa, Vic¸osa, Brazil
3
Instituto de Matem´aticas, Universidad de Antioquia, Calle 67 No. 53-108, Medell´ın, Colombia
E.R. Alvares et al.
Bautista-Souto Salorio studied in [6] the Auslander-Reiten sequences for complexes. They characterized the irreducible morphisms between bounded complexes of projectives in case of a self-injective artin algebra. Giraldo and Merklen [8] studied irreducible morphisms in the category of complexes over an abelian Krull-Schmidt category. As an application, they characterized irreducible morphisms in the bounded derived category of a finite dimensional algebra. They showed that an irreducible morphism can be separated in three types of morphisms: smonic morphism, sepic morphism and sirreducible morphism. However, although irreducible morphisms are better understood, a complete description of the morphisms and the cone in the Auslander-Reiten triangle are not known. u v w Given an Auslander-Reiten triangle X → Y → Z → X[1], we determine the type (smonic, sepic or sirreducible) of v according to the type of u. We go one step further with this characterization. We give a complete description of the cone in each case and show a special shape of the Auslander-Reiten triangle for each one of them. Note that a complete understanding of the co
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