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Basic homological algebra. (English) Zbl 0948.18001
Graduate Texts in Mathematics. 196. New York, NY: Springer. x, 395 p. DM 98.00; öS 716.00; sFr. 89.50; £34.00; $ 49.85 (2000).

This book is suitable for a one-quarter introduction that includes both Ext and Tor, but also presents a variety of other topics that provide several options for a longer course (one semester or two quarters).

Its nine chapters are: 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 (including injective envelopes, universal coefficients, and the Künneth theorems).

There are four appendices: A. GCDs, LCMs, PIDs, and UFDs; B. The ring of entire functions; C. The Mitchell-Freyd theorem; D. Noether correspondences in abelian categories.

Each chapter contains a reasonable selection of exercises. Solution outlines for many of them are given at the end of the book.

The author is more careful than is customary with set theoretic technicalities, which is very helpful. The general approach is abstract, without being impenetrable for the novice. Nevertheless, the novice is assumed to be well-prepared: the book’s prerequisite is a (strong) graduate algebra course and its intended audience is second or third year graduate students in algebra, algebraic topology, or other fields that use homological algebra. Subject to that caveat, the author’s style is both readable and entertaining (e.g. his description of a theorem “whose hypotheses seem almost dopey” or an axiom that is “a pain in the neck to enforce”, as well as a section titled “Cheating with projectives”).

All in all, this book is a very welcome addition to the literature.

18-01Textbooks (category theory)
18GxxHomological algebra (category theory)