Description of Complexes in the derived category

CLAUDIA CHAIO, ALFREDO GONZALEZ CHAIO E ISABEL PRATTI

Mexico DF

Congreso; XXIII Coloquio Latinoamericano de Álgebra; 2019

Let A be a finite dimensional algebra over an algebraically closed field. Wedenote by mod A the finitely generated module category and by proj A the fullsubcategory of mod A of the finitely generated projective A-modules.The categories Cn(proj A) of complexes of fixed size were defined and studied in[BSZ]. We say that a category Cn(proj A) is representation-finite if there are a finitenumber of classes of isomorphic indecomposable complexes in Cn(proj A). Moreover, in [CPS], the authors introduced a knitting technique to build the AuslanderReiten quiver of the category of complexes of fixed size.In this work we make use of the mentioned knitting technique to obtain specific Auslander-Reiten triangles in the bounded derived category. To show thisprocess, we consider two different families of finite dimensional algebras over analgebraically closed field. We describe the complexes that belong to the mouth ofnon-homogeneous tubes in the Auslander-Reiten quiver of their bounded derivedcategory, whenever this algebras are either derived equivalent to hereditary algebras of type Aen or Den. In case the algebras are discrete, we describe the complexesin the mouth of components of type ZA∞ of the Auslander-Reiten quiver of theirbounded derived category.