Almost split sequences for complexes of fixed size

Let be an additive k-category, k a commutative artinian ring and n>1. We denote by the category of complexes in with Xi=0 if i {1,…,n}. We see that is endowed with a natural exact structure and its global dimension is at most n−1. In case is a dualizing category, we prove that 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 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 Cn(ProjΛ).