Some Applications and Recent Developments

The first section of this chapter describes how homological algebra arose by abstraction from algebraic topology and how it has contributed to the knowledge of topology. The other four sections describe applications of the methods and results of homological algebra to other parts of algebra. Most of the material presented in these four sections was not available when this text was first published. Since then homological algebra has indeed found a large number of applications in many different fields, ranging from finite and infinite group theory to representation theory, number theory, algebraic topology, and sheaf theory. Today it is a truly indispensable tool in all these fields. For the purpose of illustrating to the reader the range and depth of these developments, we have selected a number of different topics and describe some of the main applications and results. Naturally, the treatments are somewhat cursory, the intention being to give the flavor of the homological methods rather than the details of the arguments and results.