REPRESENTATION THEORY AND RELATED TOPICS SEMINAR

ABSTRACT: Given a not-necessarily commutative ring with unit and an additive subcategory of the category of right modules, one can consider complexes of modules in the subcategory and the corresponding homotopy category. Sometimes, these homotopy categories are the first step in studying other (algebraic) homotopy categories, such us those associated to a scheme. To study these categories, one can use results from the category of modules or the category of complexes. In the first part of the talk, we will see how some results of homotopy categories of complexes extend to homotopy categories of N-complexes, for a natural number N greater than or equal to 2, using some techniques from module categories, such us the deconstruction of a class of modules.

Another approximation is to use other methods for studying homotopy categories, like those coming from triangulated categories. In some cases, the results obtained in homotopy categories have some consequences in the category of modules. In the second part of the talk, we will see how to prove the existence of Gorenstein-projective precovers for some specific rings using this approach.