Basic Homological Algebra

This book is intended for one-quarter or one semester-courses in homological algebra. The aim is to cover Ext and Tor early, and still have enough material for a one-semester course going off in any of several possible directions. The prerequisite for this book is a graduate algebra course. The book can also be used for independent study.

Review

“Each chapter contains a reasonable selection of exercises. … its intended audience is second or third year graduate students in algebra, algebraic topology, or other fields that use homological algebra. … the author’s style is both readable and entertaining … . All in all, this book is a very welcome addition to the literature.” (T.W.Hungerford, zbMATH 0948.18001, 2022)

“The book is well written. We find here many examples. Each chapter is followed by exercises, and at the end of the book there are outline solutions to some of them. … I especially appreciated the lively style of the book; compared with some other books on homological algebra, one has here the good feeling that one understands why a notion is defined in this way,that one can easily remember at least the structure of the theory, and that one is quickly able to find necessary details. The prerequisite for this book is a graduate course on algebra, but one get quite far with a modest knowledge of algebra. The book can be strongly recommended as a textbook for a course on homological algebra.”

EMS Newsletter, June 2001

Table of Contents

1 Categories

2 Modules

3 Ext and Tor

4 Dimension Theory

5 Change of Rings

6 Derived Functors

7 Abstract Homological Algebra

8 Colimits and Tor

9 Odds and Ends

A GCDs, LCMs, PIDs, and UFDs

B The Ring of Entire Functions

C The Mitchell-Freyd Theorem and Cheating in Abelian Categories

D Noether Correspondences in Abelian Categories

Solution Outlines