论文标题
同源代数的初学者指南：针对学生的全面介绍
A Beginner's Guide to Homological Algebra: A Comprehensive Introduction for Students
论文作者
论文摘要
同源代数通常被理解为拓扑结构和代数之间的翻译。但是，鉴于它提供了有关其他学科的迷人观点，最值得注意的是类别理论，因此数学的这个分支本身值得研究。在本文中，我们试图为试图开始使用同源代数的数学和相关专业的高级学生提供介绍性指南，涵盖了必要的中心主题，以便深入研究该领域及其他地区更复杂的方面。这项工作首先介绍代数结构的链复合物的概念，然后转向探索同源模块和链同位素。接下来，我们提供了投射决议的概述，并通过查看TOR函数来进入类别理论的世界。
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.
