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.
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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)
From the Back Cover
With a wealth of examples as well as abundant applications to Algebra, this is a must-read work: a clearly written, easy-to-follow guide to Homological Algebra. The author provides a treatment of Homological Algebra which approaches the subject in terms of its origins in algebraic topology. In this brand new edition the text has been fully updated and revised throughout and new material on sheaves and abelian categories has been added. Applications include the following: * to rings -- Lazard's theorem that flat modules are direct limits of free modules, Hilbert's Syzygy Theorem, Quillen-Suslin's solution of Serre's problem about projectives over polynomial rings, Serre-Auslander-Buchsbaum characterization of regular local rings (and a sketch of unique factorization); * to groups -- Schur-Zassenhaus, Gaschutz's theorem on outer automorphisms of finite p-groups, Schur multiplier, cotorsion groups; * to sheaves -- sheaf cohomology, Cech cohomology, discussion of Riemann-Roch Theorem over compact Riemann surfaces. Learning Homological Algebra is a two-stage affair. Firstly, one must learn the language of Ext and Tor, and what this describes. Secondly, one must be able to compute these things using a separate language: that of spectral sequences. The basic properties of spectral sequences are developed using exact couples. All is done in the context of bicomplexes, for almost all applications of spectral sequences involve indices. Applications include Grothendieck spectral sequences, change of rings, Lyndon-Hochschild-Serre sequence, and theorems of Leray and Cartan computing sheaf cohomology. Joseph Rotman is Professor Emeritus of Mathematics at the University of Illinois at Urbana-Champaign. He is the author of numerous successful textbooks, including Advanced Modern Algebra (Prentice-Hall 2002), Galois Theory, 2nd Edition (Springer 1998) A First Course in Abstract Algebra (Prentice-Hall 1996), Introduction to the Theory of Groups, 4th Edition (Springer 1995), and Introduction to Algebraic Topology (Springer 1988).
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11 of 12 people found the following review helpful
5.0 out of 5 stars
A good introduction, easier than Weibel, more detailed in places
February 26, 2011
By Justin Curry
Format:Paperback|Amazon Verified Purchase
I have no idea why this book has earned the scorn of other reviewers. The book is largely error-free and includes many nice examples. The review titled "Publish or Perish..." levies a confusing charge as Rotman is an emeritus professor and no longer needs to scramble in the academic rat race.
I got Rotman's book before getting Weibel's classic on Homological Algebra and have no regrets. In particular, the sheaf theory section does a very nice development of the etale espace approach to sheaves and connects this with the more standard development in terms of pre-sheaves and sheafification.
It is granted that this book is not meant to be deemed a classic, but is pragmatic and unpretentious. Rotman points out implications of definitions that in most classic and "elegant" texts readers are supposed to gleam for him or herself. As another case in point, Rotman quotes extensively from an internet forum post providing an intuitive introduction to Riemann-Roch over Riemann surfaces. There are other quirks that come from quoting texts of historical mathematical importance, which I find charming and appropriate coming from a senior professor.
There is also more review of basic algebra than in Weibel, which may prove useful to the neophyte. Lastly, the current discounted price on Amazon (~$26) is nearly $20 cheaper than Amazon's price of Weibel.
I got Rotman's book before getting Weibel's classic on Homological Algebra and have no regrets. In particular, the sheaf theory section does a very nice development of the etale espace approach to sheaves and connects this with the more standard development in terms of pre-sheaves and sheafification.
It is granted that this book is not meant to be deemed a classic, but is pragmatic and unpretentious. Rotman points out implications of definitions that in most classic and "elegant" texts readers are supposed to gleam for him or herself. As another case in point, Rotman quotes extensively from an internet forum post providing an intuitive introduction to Riemann-Roch over Riemann surfaces. There are other quirks that come from quoting texts of historical mathematical importance, which I find charming and appropriate coming from a senior professor.
There is also more review of basic algebra than in Weibel, which may prove useful to the neophyte. Lastly, the current discounted price on Amazon (~$26) is nearly $20 cheaper than Amazon's price of Weibel.
12 of 21 people found the following review helpful
By P. Huling
Format:Paperback
Simply put, this book could have some real purpose for someone wanting a gentle introduction into homological algebra if not for one huge blunder. Rotman does do a good job at motivating a lot of the topics and not becoming too longwinded, but there is also an unfortunate fatal flaw in this book as well. This book contains far too many errors to be acceptable. While I acknowledge that all books will contain errors, this amount is beyond a level that should have been allowed to be printed without correction. Some are simple typos that will not affect the average reader. Others, however, will make this book not cater well to its target audience. The pace of this book is too slow as to make it a necessary resource, as Weibel's book of the same title or Kenneth Brown's "Cohomology of Groups" are far more rigorous and complete. This books aim seems to be aimed thus at the graduate level or possibly a mathematician from a different field. However, this audience will be in for a chore. Many mistakes lead to incorrect proofs and even worse incorrect proposition and theorem statements. When trying to understand the functorality of certain constructions, for instance, it is crucial that the reader understand exactly how things work. The mixing up of rings and modules often leaves statements paradoxical. The advanced reader will have no problem finding and fixing these errors, but for those not comfortable in this area of mathematics, this may be a huge challenge. This book may be helpful to some as a secondary resource as it does work out some simpler results that many books (e.g. the ones mentioned above) take for granted. I would not recommend this book for any other reason though.
I will be fair and say that if this book were to receive a major editing job removing most of the errors that it could be a very useful introduction. However, until such a revision is produced, buy a better book.
I will be fair and say that if this book were to receive a major editing job removing most of the errors that it could be a very useful introduction. However, until such a revision is produced, buy a better book.
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