Dimensão global forte e complexidade na categoria derivada

Abstract

We introduce in this thesis a definition of complexity in the derived category of bounded above complexes of modules over a finite dimensional k-algebra. One of our result shows a relationship between the complexity of indecomposable objects and the notion of strong global dimension. More specifically, we prove that the existence of an indecomposable object in the category derived bounded above whose complexity is not zero is a sufficient condition for corresponding algebra being of infinite strong global dimension. We also investigate the existence of a relationship between the global dimension and the strong global dimension of shod algebras (Coelho and Lanzilotta, 1999). Our motivation came from characterization of quasitilted algebras (Happel, Reiten and Smalo, 1996) by its strong global dimension, given by D. Happel and D. Zacharia (2008), and from the fact that shod algebras are a generalization of quasitilted algebras. Our conclusion was that there is not in general a characterization of shod algebras in terms of its strong global dimension. This conclusion comes from the fact that we showed that for each integer d > 2 there exists a strictly shod algebra whose strong global dimension is d.