On the Composition of Three Irreducible Morphisms in the Bounded Homotopy Category

Abstract

Let A be an artin algebra of finite global dimension. We study when the composition of three irreducible morphisms between indecomposable complexes in {K}^{b}(proj A) is a non-zero morphism in the fourth power of the radical. We apply such results to prove that the composition of three irreducible morphisms between indecomposable complexes in the bounded derived category of a gentle Nakayama algebra, not selfinjective, whose ordinary quiver is an oriented cycle, belongs to the fourth power of the radical if and only if it vanishes.