Categorical methods in graded ring theory

Let G be a group, R a G-graded ring and X a right G-set . We study functors between categories of modules graded by G-sets, continuing the work of [M]. As an application we obtain generalizations of Cohen-Montgomery Duality Theorems by categorical methods. Then we study when some functors introduced in [M] (which generalize some functors ocurring in [D1], [D2] and [NRV]) are separable. Finally we obtain an application to the study of the weak dimension of a group graded ring

Weak dimension of group-graded rings

We study the weak dimension of a group-graded ring using methods developed in [B1], [Q] and [R]. We prove that if R is a G-graded ring with G locally finite and the order of every subgroup of G is invertible in R, then the graded weak dimension of R is equal to the ungraded one

Finite groups in integral group rings

Notes used for a course held in 2016 in the School of Advances in Group Theory and Applications, for some lectures given in 2018 for the students of the Master in Mathematics of the Vrije Universiteit Brussels and a course for master and Ph.D. students at the Universidade de S\~ao Paulo. We revise some problems on the study of finite subgroups of the group of units of integral group rings of finite groups and some techniques to attack them.Comment: Revised after course in S\~ao Paul