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Zbl 0948.18001
Osborne, M.Scott
Basic homological algebra. (English)
[B] Graduate Texts in Mathematics. 196. New York, NY: Springer. x, 395 p. DM 98.00; öS 716.00; sFr. 89.50; \sterling 34.00; \$ 49.85 (2000). ISBN 0-387-98934-X/hbk

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). \par 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). \par 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. \par Each chapter contains a reasonable selection of exercises. Solution outlines for many of them are given at the end of the book. \par 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''). \par All in all, this book is a very welcome addition to the literature.
[T.W.Hungerford (Cleveland)]

MSC 2000:: *18-01 Textbooks (category theory)
18Gxx Homological algebra

Keywords: dimension theory; abstract homological algebra; Künneth theorems; Mitchell-Freyd theorem; Noether correspondences; set theoretic technicalities

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