An Introduction to Homological Algebra (Universitext) 2nd Edition by Joseph J. Rotman (PDF)

Description

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.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐A highly useful text for studying homological algebra. It was technically recommended but not required for my class, but this text has been excellent for filling in additional details that the professor hasn’t covered entirely in proofs and whatnot.It is print-on-demand, however; my text was actually printed *after* I ordered it. I don’t mind that, but it is clearly a cheaper printing method, since my book has been wearing out quite rapidly—though part of that is also down to how much I’ve been using it.

⭐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.

⭐The authors explanation of the material is excellent. I have no headaches learning this.

⭐clear exposition and very comprehensive

⭐This book is an excellent textbook for learning homological algebra and seeing the connections to other areas of mathematics, including algebraic topology, commutative algebra, algebraic geometry, and others. I started reading this book as an undergraduate, and I found Rotman’s unpretentious style to be very approachable. I still came back to this book when I studied H.A. in graduate school.One thing I would like to emphasize: this book is a great place to start learning the basics of category theory. It defines the concepts very carefully and immediately follows up each definition with an example in R-Mod or Sets, which are the most familiar categories for those new to the subject. When it comes to the category theory material, I think that Rotman strikes a masterful balance between content and examples. There isn’t a huge laundry list of examples, just one or two concrete examples to bring a definition down to earth. Also, he does not define dual notions by saying “just reverse all arrows”, e.g. both products and coproducts are carefully defined and motivated. I really appreciate this.I also don’t agree with the other review that says this book is full of errors. I haven’t noticed any glaring omissions or typos after having read most of it.

⭐If you’re concerned about errors in this book just go to the author’s page at the University of Illinois at Urbana-Champaign and download the correction sheet. Contrary to complaints in that negative review about errors, there aren’t really many mistakes and the book is so good that it’s well worth taking five or ten minutes to pencil in the corrections. Get the errata at math.uiuc.edu/~rotman.

⭐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.

⭐La première édition de ce livre (datant de 1979) était déjà une excellente introduction à l’algèbre homologique, ayant à la fois des similitudes et des différences avec le fameux “Homological Algebra” de Cartan et Eilenberg. Cette seconde édition est aussi claire, mais son contenu est incomparablement plus riche. La grande différence est que le livre ne se cantonne plus à la catégorie des modules, mais, en se plaçant dans le cadre plus général des catégories abéliennes, traite aussi des faisceaux, dans l’esprit de l’article célèbre de Grothendieck “Sur quelques points d’algèbre homologique” paru dans le Tohoku Math. Journal en 1957.La première partie du livre (qui couvre les chapitre 1 à 4) est à peu près inchangée par rapport à la première édition. L’introduction du chapitre 1 explique, notamment au plan étymologique, ce qu’on doit entendre par “homologie”. Les chapitres 2 à 4 passent en revue les foncteurs classiques dans les catégories de modules que sont Hom et le produit tensoriel, puis les types de modules qui rendent ces foncteurs “exacts” (modules projectifs, injectifs et plats), enfin les types d’anneaux qui jouent un rôle particulier vis-à-vis de ces foncteurs (anneaux semi-simples, réguliers au sens de von Neumann, héréditaires (en particulier, les anneaux de Dedekind, mais définis et étudiés dans le cas commutatif seulement), semi-héréditaires (en particulier les anneaux de Prüfer), etc. La localisation des anneaux commutatifs est également étudiée, et enfin les anneaux de polynômes avec le théorème de Quillen-Suslin.Commence alors, avec le chapitre 5, la seconde partie du livre. Rotman explique son objet dès la courte introduction de ce chapitre: “Nous projetons d’utiliser l’Algèbre homologique pour démontrer des résultats concernant les modules, les groupes, et les faisceaux. Un contexte commun pour discuter de ces sujets est celui des catégories abéliennes, ou plus généralement des catégories de complexes.” On peut mentionner quelques points saillants de ce programme: au chapitre 6, la cohomologie des faisceaux et le théorème de Riemann-Roch (version élémentaire, pas celle de Grothendieck!), les foncteurs dérivés Tor et Ext au chapitre 7, au chapitre 8 le théorème des Syzygies de Hilbert, et au chapitre 9 des points liés à la cohomologie des groupes. Le chapitre 10 (et dernier du livre) est consacré aux suites spectrales et se termine avec les théorèmes de Künneth.Le contenu du livre ne recouvre pas complètement (loin s’en faut) celui de “Homological Algebra” de Cartan et Eilenberg. Pour compléter sa lecture, mais dans une tout autre direction, le lecteur pourra également consulter “Categories and Sheaves” de Kashiwara et Schapira.Un superbe livre sur l’Algèbre homologique!Merci beaucoup, le livre est en parfait état.À une prochaine fois.A. Chakhar.ホモロジー代数については，和書では共立出版から出ている『層とホモロジー代数』があるのでそちらを読まれればよいかと思います．このRotmanの本も有名なので読んでみました．非形式的な背景や応用が丁寧に説明されていてよかったのですが，誤植やミスが極めて多く，閉口させられました．so good thanks for delivered.

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