Introduction to Homological Algebra

Graduate mathematics students will find this book an easy-to-follow, step-by-step guide to the subject. Rotman’s book gives a treatment of homological algebra which approaches the subject in terms of its origins in algebraic topology. In this new edition the book has been updated and revised throughout and new material on sheaves and cup products has been added. The author has also included material about homotopical algebra, alias K-theory. Learning homological algebra is a two-stage affair. First, one must learn the language of Ext and Tor. Second, one must be able to compute these things with spectral sequences. Here is a work that combines the two.

From the reviews of the second edition:"Joseph J. Rotman is a renowned textbook author in contemporary mathematics. Over the past four decades, he has published numerous successful texts of introductory character, mainly in the field of modern abstract algebra and its related disciplines. … Now, in the current second edition, the author has reworked the original text considerably.  While the first edition covered exclusively aspects of the homological algebra of groups, rings, and modules, that is, topics from its first period of development, the new edition includes some additional material from the second period, together with numerous other, more recent results from the homological algebra of groups, rings, and modules. The new edition has almost doubled in size and represents a substantial updating of the classic original. … All together, a popular classic has been turned into a new, much more topical and comprehensive textbook on homological algebra, with all the great features that once distinguished the original, very much to the belief [of its] new generation of readers." (Werner Kleinert, Zentralblatt)"The new expanded second edition … attempts to cover more ground, basically going from the (concrete) category of modules over a given ring, as in the first edition, to an abelian category and to treat the important example of the category of sheaves on a topological space. … the exercise at the end of every section, plenty of examples and motivation for the many new concepts set this book apart and make it an ideal textbook for a course on the subject." (Felipe Zaldivar, MAA Online, December, 2008)"This is the second edition of Rotman’s introduction to the more classical aspects of homological algebra … . The book is mainly concerned with homological algebra in module categories … . The book is full of illustrative examples and exercises. It contains many references for further study and also to original sources. All this makes Rotman’s book very convenient for beginners in homological algebra as well as a reference book." (Fernando Muro, Mathematical Reviews, Issue 2009 i)