On the Derived Categories and Quasitilted Algebras

Abstract

In this paper, we present some results on the bounded derived category of Artin algebras, and in particular on the indecomposable objects in these categories, using homological properties. Given a complex X*, we consider the set \(J_{X^*}=\{i \in \mathbb{Z}\, |\, H^i(X^*)\neq 0\}\) and we define the application \(l(X^*)=\text{max}J_{X^*}-\text{min}J_{X^*}+1\). We give relationships between some homological properties of an algebra and the respective application l. On the other hand, using homological properties again, we determine two subcategories of the bounded derived category of an algebra, which turn out to be the bounded derived categories of quasi-tilted algebras. As a consequence of these results we obtain new characterizations of quasi-tilted and quasi-tilted gentle algebras.