Graded algebras: theoretical and computational aspects

In this dissertation we study by a computational approach Hilbert functions and minimal graded free resolutions of finitely generated graded modules over two significant graded K-algebras, K being a field. More precisely, if E is the exterior algebra of a finite dimensional K-vector space and F is a finitely generated graded free E-module with a homogeneous basis, we characterize the Hilbert functions of graded E-modules of the type F/M, with M graded submodule of F, via the unique lexicographic submodule of F having the same Hilbert function as M. Furthermore, we study projective and injective resolutions over E. In particular, we give upper bounds for the graded Betti numbers and the graded Bass numbers of classes of E-modules. Moreover, we give a criterion to determine the extremal Betti numbers of a special class of monomial ideals of a standard polynomial ring S known as the t-spread strongly stable ideals, where t is an integer >=0. We are able to find a complete numerical characterization (positions as well as values) for the case t=0 and t=1. Instead, for the case t=2 we determine the structure of the t-spread strongly stable ideals with the maximal number of extremal Betti numbers. The approach to these topics is mainly computational because of the algorithmic nature of the topic themselves. Finally, we present some packages in order to work and manipulate specific objects in both contexts.