Almost split sequences for complexes of fixed size

Let A be an additive k-category, k a commutative artinian ring and n>1. We denote by Cn(A) the category of complexes X=(Xi,dXi)i∈Z in A with Xi=0 if i∉{1,…,n}. We see that Cn(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 Cn(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 Cn(ProjΛ).