Irreducible morphisms in the bounded derived category

Abstract

We study irreducible morphisms in the bounded derived category of finitely generated modules over an Artin algebra Λ , denoted D b ( Λ mod ) , by means of the underlying category of complexes showing that, in fact, we can restrict to the study of certain subcategories of finite complexes. We prove that as in the case of modules there are no irreducible morphisms from X to X if X is an indecomposable complex. In case Λ is a selfinjective Artin algebra we show that for every irreducible morphism f in C b ( Λ proj ) either f j is split monomorphism for all j ∈ Z or split epimorphism, for all j ∈ Z . Moreover, we prove that all the non-trivial components of the Auslander–Reiten quiver of C b ( Λ proj ) are of the form Z A ∞ .