Relative homological dimensions and controllable extensions - Primolan, Roger Ramirez (2023)
In 1956, Hochschild developed an homological theory for extensions of associative algebras. His theory went dormant for the next decades, but recent results related it to the Finitistic Dimension Conjecture: a 60 years old central conjecture for homological theory of finite dimensional algebras. In this work we will present a panoramic view on the relations between Relative Homological Algebra and the Finitistic Dimension Conjecture. We define and analyse a new class of extensions, the controllable extensions. We proved that this class preserves much of the properties of classical homological dimensions to the relative realm, in particular we are able to compute the relative global dimension of some extensions. We also translated some results of homological algebra to relative homological algebra, in particular we obtained a generalization of the homological behaviour of path algebras.