Almost split sequences for complexes of fixed size

• Published in: Journal of algebra 2005-05, Vol.287 (1), p.140-168
• Main Authors: Bautista, Raymundo, Souto Salorio, Maria Jose, Zuazua, Rita
• Format: Article
• Language: English
• ISSN: 0021-8693; 1090-266X
• DOI: 10.1016/j.jalgebra.2005.01.032

Let A be an additive k-category, k a commutative artinian ring and n > 1 . We denote by C n ( A ) the category of complexes X = ( X i , d X i ) i ∈ Z in A with X i = 0 if i ∉ { 1 , … , n } . We see that C n ( A ) is endowed with a natural exact structure and its global dimension is at most n − 1 . In case A is a dualizing category, we prove that C n ( A ) has almost split sequences in the sense of [P. Dräxler, I. Reiten, S.O. Smalø, Ø. Solberg, Exact categories and vector space categories, with an appendix by B. Keller, Trans. Amer. Math. Soc. 351 (2) (1999) 647–682] or [R. Bautista, The category of morphisms between projective modules, Comm. Algebra 32 (11) (2004) 4303–4331]. If A is the category of finitely generated projective Λ-modules ( Λ an Artin algebra), we prove that the ends of an almost split sequence are related by an Auslander–Reiten translation functor which is defined in the most general category C n ( Proj Λ ) .