Журнал обобщенной теории лжи и ее приложений

Homological Adventures Exploring Homological Algebra

Abstract

Soren Patwadi

Homological algebra is a branch of mathematics that delves into the study of structures through the lens of algebraic objects called homology and cohomology. It's a powerful tool with applications in various fields including algebraic topology, algebraic geometry, and representation theory. In this article, we embark on a journey through the fascinating realm of homological algebra, exploring its fundamental concepts, techniques, and applications. Homological algebra emerged from the study of algebraic topology in the early century, with pioneers like Emmy Noether and Saunders Mac Lane laying its groundwork. Central to homological algebra are chain complexes, sequences of abelian groups or modules connected by homomorphisms, which capture the essence of cycles, boundaries, and exact sequences. The notion of homology arises from these complexes, providing a measure of "holes" or "voids" in mathematical structures, while cohomology offers a dual perspective, focusing on mappings and boundaries.