Introduction to Homological Algebra (Pure and Applied Mathematics, No. 85)
Joseph J. Rotman | 1979-07-12 00:00:00 | Academic Press | 376 | Mathematics
With a wealth of examples as well as abundant applications to algebra, this is a must-read work: an easy-to-follow, step-by-step 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.
Reviews
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.
Download this book!
Free Ebooks Download