Sections in the bounded derived category of piecewise hereditary algebras

Abstract

Let [Formula: see text] be a finite-dimensional algebra over an algebraically closed field. We study sections in the Auslander–Reiten quiver of the categories of complexes of fixed size and in the Auslander–Reiten quiver of the bounded derived category. One of the aims of this work is to show where the indecomposable [Formula: see text]-modules can be found in the Auslander–Reiten quiver of the derived category, whenever [Formula: see text] is a piecewise hereditary algebra. We also prove that we can obtain a transjective component of the bounded derived category from a section. As an application of the results of sections, we find a bound on the strong global dimension of some piecewise hereditary algebras of tree type. Moreover, we also determine the strong global dimension of some algebras taking into account their ordinary quivers with relations.