Some aspects of local cohomology theory - Holanda, Rafael Ferreira (2022)
This work is about some features of local cohomology theory. We develop a new tool called Mayer-Vietoris spectral sequence that allows us to study several local cohomology modules supported in different ideals, which led us to generalize or retrieve previous results of several authors and also produce new ones, especially in what concerns multigraded polynomial rings. We also deal with generalized Cohen-Macaulay modules and deficiency modules, providing relations between their Bass and Betti numbers in order to both generalize classical results and produce new ones as a case of the conjecture of Auslander and Reiten in a particular case. Finally, local cohomology is viewed as an important tool for the studying of the interplay between finiteness of homological dimensions and the vanishing of Ext modules.