A Beginner's Guide to Homological Algebra: A Comprehensive Introduction for Students

TL;DR

This paper offers an accessible introduction to homological algebra, covering core concepts like chain complexes, homology, resolutions, and their connections to category theory for advanced students.

Contribution

It provides a comprehensive beginner's guide to homological algebra, emphasizing foundational topics and their relevance to category theory, suitable for students starting in the field.

Findings

Introduces chain complexes and homology modules.

Explains projective resolutions and Tor functors.

Connects homological algebra to category theory.

Abstract

Homological algebra is often understood as the translator between the world of topology and algebra. However, this branch of mathematics is worth studying by itself, given that it provides fascinating perspectives about other disciplines, most notably, category theory. In this paper, we seek to provide an introductory guide for advanced students of mathematics and related specialties seeking to get started on homological algebra, covering the necessary central topics to later delve deeper into more complex aspects of this field and beyond. This work starts by presenting the notion of chain complexes of algebraic structures, and then moves into exploring homology modules and chain homotopies. Next, we provide an overview of projective resolutions and conclude by entering the world of category theory by looking at Tor functors.