The Auslander-Reiten quiver of a category of fixed size complexes

CHAIO, CLAUDIA, PRATTI ISABEL AND SOUTO SALORIO M.J.

Guanajuato

Conferencia; Advances in Representation Theory of Algebras; 2012

Joint work with Isabel Pratti and Souto Salorio María José. Let $A$ be an artin algebra. We denote by mod A the category of all the finitely generated right A-modules and by proj A the full subcategory of mod A consisting of all the finitely generated projective A-modules. We consider C_n(proj A) the full subcategory of C(mod A) whose objects are the complexes X = (X^i,d^i_x)_{i in Z} such that X^i = 0 if i is not in {1, ...,n} and X^i is projective if i is in {1, ...,n}. These categories were introduced by R. Bautista in [B] for n = 2 and later generalized in [BSZ] for n greater or equal to 2. The authors consider such categories in order to study the Auslander-Reiten triangles in bounded derived categories of finitely generated modules over an artin algebra.These categories are exact with enough projective and injective objects and they have finite global dimension. In this talk, we are going to show how to build the Auslander-Reiten quiver of C_n(proj A) for A a finite-dimensional algebra over an algebraically closed field k and n greater or equal to 2. We are also going to refer to sectional paths in such a category. In particular, we study some properties in C_n(proj H), for H a hereditary algebra. References: [B] R, Bautista. The category of morphisms between projective modules. Communications in Algebra 32 (11), (2004), 4303-4331. [BSZ] R, Bautista, M.J. Souto Salorio, R. Zuazu. Almost split sequences for complexes of fixed size. J. Algebra {287}, (2005), 140-168.