Homological Methods in Non-Commutative Algebra

Abstract

Homological methods play a central role in the structural analysis of non-commutative algebras by providing powerful invariants and categorical frameworks to investigate modules, extensions, and deformation phenomena. This paper examines the application of homological algebra techniques—including projective and injective resolutions, derived functors such as Ext and Tor, and homological dimensions—to the study of non-commutative rings and graded algebras. Particular emphasis is placed on derived categories, Hochschild (co)homology, and their relevance to representation theory and non-commutative geometry. The work further explores Artin–Schelter regularity, Koszul duality, and deformation theory as fundamental tools for understanding structural and geometric properties of non-commutative spaces. By synthesizing classical homological constructions with modern categorical approaches, the study highlights how homological invariants facilitate classification, equivalence detection, and the resolution of algebraic singularities in non-commutative settings.