Product Description
The landscape of homological algebra has evolved over the past half-century into a fundamental tool for the working mathematician. This book provides a unified account of homological algebra as it exists today. The historical connection with topology, regular local rings, and semi-simple Lie algebras is also described. The first half of the book takes as its subject the canonical topics in homological algebra: derived functors, Tor and Ext, projective dimensions and spectral sequences. Homology of group and Lie algebras illustrate these topics. Intermingled are less canonical topics, such as the derived inverse limit functor lim1, local cohomology, Galois cohomology, and affine Lie algebras. The last part of the book covers less traditional topics that are a vital part of the modern homological toolkit:simplicial methods, Hochschild and cyclic homology, derived categories and total derived functors. ... Read more

Customer Reviews (5)

Review of homological algebra - Weibel
This is one of the worst books I have read. I am reviewing it because I have seen the other reviews and think they are entirely misleading. The book makes endless definitions without explaining motivation, use or history of any of the subjects. There is no context and the book makes the subject pointless. For example the definition of Tor gives no clues as to why it is named Tor. As the root is Torsion and is clearly explained in MacLane (a fantastic start, forget comments about it being dated) which immediately gives a great insight why not let the reader know. Then there is no motivation for it's use and the definition gives a definition from a projective resolution which is great if you already know all about Tor and projective resolutions. The whole book goes on in this style. There is a list of references at the back but as you go through nothing, no comments on derivation, motivation, application or anything. The book is great as a reference on how not to write notes and how to demotivate readers. I would sell mine secondhand but I couldn't live with the guilt.

Excellent Text
This is an excellent book on Homological Algebra. I am surprised at how often I find myself turning to it.

Long Live the Homoligarchy!
This is by far the most accessible and well-written Homological Algebra book I've dealt with.I'm an analyst at heart, so the subject does not in any way come easy to me.However, the exposition in this book greatly shortened the learning curve, and Dr. Weibel provided a plethora of understandable examples and accessible exercises, all of which greatly aided my understanding of the subject.

Deeeep!
Now Homological Algebra is not a simple subject that can be picked up with only a minimal background.The subject is deep and permeates many different braches of mathematics.If you are thinking of buying this text, then I have two suggestions.First, just buy it.For the price, you will not find another book that is as up to date or as rich in content.Second, ask yourself if you are really ready for this text?What I would do is use Amazon's "Search Inside" in order to check out the first two or three pages of chapter one.If you feel at home with the material presented there, then just buy it.You will not be upset.Well, you may get upset-this book covers a lot of material and at times can be rather overwhelming.Let me stress that the level of comfort with the preliminary material is analogous to a tenured professors level of comfort with, say, Calculus.That is, very comfortable.With respect to the actual text, I am hard pressed to find any negative words.Weibel has set the modern standard and this text is fast becoming the standard from which all other texts will have to model themselves after.The material is presented in small palatable chunks that can be consumed at any time.Just make sure you have a new pencil and couple notebooks handy when you read this text.

Great homological algebra book!
This book was absolutely priceless for me while I was doing my qualifying exams and dissertation. Excellently written, with modern notation and applications and a clearly-exposited writing style. I wish all "hard" math books were written like this. Thanks Prof. Weibel ... Read more

Product Description

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.

Customer Reviews (1)

Publish or perish... but at least proofread the thing!
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. ... Read more

Product Description
This classic book provides a broad introduction to homological algebra, including a comprehensive set of exercises. Since publication of the first edition homological algebra has found a large number of applications in many different fields. Today, it is a truly indispensable tool in fields ranging from finite and infinite group theory to representation theory, number theory, algebraic topology and sheaf theory. In this new edition, the authors have selected a number of different topics and describe some of the main applications and results to illustrate the range and depths of these developments. The background assumes little more than knowledge of the algebraic theories groups and of vector spaces over a field. ... Read more

Customer Reviews (2)

Buying madness
I am a retired professor of mathematics. When I retired I got rid of my library and now I am back to rebuying many of the books I rid myself of, including the current one under review. Hilton and Stammbach's book is quite useful to learn homology from. (I'm not an expert in homology although I have published research in homology,) I prefer MacLane's Homology if it can be found.

Good overview of the subject
This book, written by two of the leading experts in the area, is a soundexposition of a very abstract/abtruse subject. The logic is impeccable andthe organization nicely done. Algebraic topology is given a rigorousfoundation in this book and readers with a background in that subject willappreciate the discussion more. By far the best chapter in the book is theone on exact couples and spectral sequences as it gives proofs that wouldtake a lot of time to find in the original literature. A good book forreference and for detailed study. ... Read more

Product Description
Homological algebra was developed as an area of study almost 50 years ago, and many books on the subject exist. However, few, if any, of these books are written at a level appropriate for students approaching the subject for the first time.An Elementary Approach to Homological Algebra fills that void. Designed to meet the needs of beginning graduate students, it presents the material in a clear, easy-to-understand manner. Complete, detailed proofs make the material easy to follow, numerous worked examples help readers understand the concepts, and an abundance of exercises test and solidify their understanding.Often perceived as dry and abstract, homological algebra nonetheless has important applications in many important areas. The author highlights some of these, particularly several related to group theoretic problems, in the concluding chapter. Beyond making classical homological algebra accessible to students, the author's level of detail, while not exhaustive, also makes the book useful for self-study and as a reference for researchers. ... Read more

Customer Reviews (1)

Maybe not Elementary
The depth of knowledge required in order to fully appreciate the subject is indeed deep.One must possess a solid foundation in basic algebra at the level of Rotman or Hungerford.That is, you must have spent some serious time in the trenches sweating through a vast array of quality problems.Furthermore, it would be beneficial if the reader had a passing acquaintance with algebraic topology; say the later chapters of Munkres' text.Having this background will make your journey through this text more palatable and much smoother.In general, this is a very nice concise text and the author does an admirable job highlighting the core topics.In my opinion, the area in which this text is lacking is in the exercises-they are either trivial or they do not work towards extending the material-after all the sole purpose of including exercises is to test the reader's ability to "do" mathematics.Of course one could argue that because this is an "elementary" text, and hence necessarily "elementary" readers, that such problems are appropriate.But this brings me to my point, namely, Homological Algebra is not an elementary subject that should be introduced to pre-graduate level students and, to some extent, first or second year graduate students.Why?Simple, the normal run-of-the-mill student will not see the big picture.

With that said, I do recommend this text and I did enjoy reading this text.Again, for someone who has had some quality exposure to algebra I firmly believe you can make your way through the entire book and, by the end, begin to "see" the big picture of where and how Homological Algebra fits in. ... Read more

Product Description
This book is intended for one-quarter or one semester-courses in homological algebra. The aim is to cover Ext and Tor early and without distraction. It includes several further topics, which can be pursued independently of each other. Many of these, such as Lazard's theorem, long exact sequences in Abelian categories with no cheating, or the relation between Krull dimension and global dimension, are hard to find elsewhere. The intended audience is second or third year graduate students in algebra, algebraic topology, or any other field that uses homological algebra. ... Read more

Product Description
When this book was written, methods of algebraic topology had caused revolutions in the world of pure algebra. To clarify the advances that had been made, Cartan and Eilenberg tried to unify the fields and to construct the framework of a fully fledged theory. The invasion of algebra had occurred on three fronts through the construction of cohomology theories for groups, Lie algebras, and associative algebras. This book presents a single homology (and also cohomology) theory that embodies all three; a large number of results is thus established in a general framework. Subsequently, each of the three theories is singled out by a suitable specialization, and its specific properties are studied.

The starting point is the notion of a module over a ring. The primary operations are the tensor product of two modules and the groups of all homomorphisms of one module into another. From these, "higher order" derived of operations are obtained, which enjoy all the properties usually attributed to homology theories. This leads in a natural way to the study of "functors" and of their "derived functors."

This mathematical masterpiece will appeal to all mathematicians working in algebraic topology. ... Read more

Customer Reviews (3)

Old - Antiquated
Yes, this is a landmark book and yes this book has served many a fine mathematician but, let's be honest here, this book is out dated.Since the creation of this book there have been both advances within the field and the core terminology has been agreed upon.Why read a text, especially as an introduction, whose notation and basic terminology is extinct?Yes, it is nice to go back and read the works of the masters, the pioneers and wrap yourself up in a nice warm blanket of original and new thought.Whereas, if your trying to learn the subject matter for your own benefit or for research reasons, then I would not recommend this text.For those of you who are new to the field, my suggestion is to stay far away from this text.Try L.R. Vermani's text or Rotmans text instead.For those of you who are not new to the field then I am sure you already own your own copy, so enjoy.Finally, a word about the 3-stars.Sure this text deserves 5 stars for being well written, for being a landmark text and for many other reasons but, this texts hay day has gone and went.It is time to give up the old and bring in the up to date and more relavent texts such as Weibel's.In general, save your time and your money.

A landmark, that's it.
As the title of the series suggests, this is another "landmark" inbooks of mathematics. In fact, this was the only book on homological algebra for a certain period of time. I don't think this is a fungus, but with other modern good textbooks available (Weibel, for example) I am not inclined to give it "five stars".

H. Cartan and S. Eilenberg are the authors
I believe this is volume #19 of the Princeton Math. Ser.entitled Homological Algebra. The authors are H. Cartan and S. Eilenberg. H. Cartan is one of E. Cartan's sons. ... Read more

Product Description
This book, the first printing of which was published as volume 38 of the Encyclopaedia of Mathematical Sciences, presents a modern approach to homological algebra, based on the systematic use of the terminology and ideas of derived categories and derived functors. The book contains applications of homological algebra to the theory of sheaves on topological spaces, to Hodge theory, and to the theory of modules over rings of algebraic differential operators (algebraic D-modules). The authors Gelfand and Manin explain all the main ideas of the theory of derived categories. Both authors are well-known researchers and the second, Manin, is famous for his work in algebraic geometry and mathematical physics. The book is an excellent reference for graduate students and researchers in mathematics and also for physicists who use methods from algebraic geometry and algebraic topology ... Read more

Product Description

This modern approach to homological algebra by two leading writers in the field is based on the systematic use of the language and ideas of derived categories and derived functors. It describes relations with standard cohomology theory and provides complete proofs. Coverage also presents basic concepts and results of homotopical algebra. This second edition contains numerous corrections.

Customer Reviews (1)

A brilliant work of mathematics
Homological algebra is one of those subjects that in order to understand, you need to know already.Category theory wouldn't hurt either, nor some algebraic geometry and algebraic topology.Unfortunately, you need to know homological algebra to do some of these things as well.The great strength of Gelfand and Manin's work is that it ties together examples from all of these areas and coherently integrates them into some of the best mathematical prose I've ever read.The book is recent enough that its authors write from a position of vast perspective on fifty years of research, and the subject as they present it is about as up-to-date as possible, yet cleanly developed and not overwhelming.Unlike many books whose subject matter was influenced by modern algebraic geometry, this one does not merely pay lip service to standard references on its vast prerequisites, but systematically develops them (specifically, the ideas of category theory and abelian categories) in an entire, large chapter.

The book's only tangible drawback is the presence of errors, despite the revision.The previous edition was said to be riddled with them, and the authors have indeed brought the count down to a nearly respectable level, with those remaining relatively minor.The remaining errors are more jarring than confusing, however, and this is not a sticking point.

Finally, I would like to emphasize that neither this book nor any other is suitable for beginners in homological algebra.This is an aspect of the field, and its remedy is to study the applications, algebraic geometry and algebraic topology most of all.The ideas of homological algebra are derived not from first principles but from mathematicians' experiences doing mathematics, and both the subject matter and the many excellent examples in the book will resonate more with a student whose knowledge they cast in a new light. ... Read more

Product Description

This second volume of this text covers the classical aspects of the theory of groups and their representations. It also offers a general introduction to the modern theory of representations including the representations of quivers and finite partially ordered sets and their applications to finite dimensional algebras. It reviews key recent developments in the theory of special ring classes including Frobenius, quasi-Frobenius, and others.

Product Description
About the book…

In honor of Edgar Enochs and his venerable contributions to a broad range of topics in Algebra, top researchers from around the world gathered at Auburn University to report on their latest work and exchange ideas on some of today's foremost research topics. This carefully edited volume presents the refereed papers of the participants of these talks along with contributions from other veteran researchers who were unable to attend.

These papers reflect many of the current topics in Abelian Groups, Commutative Algebra, Commutative Rings, Group Theory, Homological Algebra, Lie Algebras, and Module Theory. Accessible even to beginning mathematicians, many of these articles suggest problems and programs for future study. This volume is an outstanding addition to the literature and a valuable handbook for beginning as well as seasoned researchers in Algebra.

about the editors…

H. PAT GOETERS completed his undergraduate studies in mathematics and computer science at Southern Connecticut State University and received his Ph.D. in 1984 from the University of Connecticut under the supervision of William J. Wickless. After spending one year in a post-doctoral position in Wesleyan University under the tutelage of James D. Reid, Goeters was invited for a tenure track position in Auburn University by Ulrich F. Albrecht. Soon afterwards, William Ullery and Overtoun Jenda were hired, and so began a lively Algebra group.


OVERTOUN M. G. JENDA received his bachelor's degree in Mathematics from Chancellor College, the University of Malawi. He moved to the U.S. 1977 to pursue graduate studies at University of Kentucky, earning his Ph.D. in 1981 under the supervision of Professor Edgar Enochs. He then returned to Chancellor College, where he was a lecturer (assistant professor) for three years. He moved to the University of Botswana for another three-year stint as a lecturer before moving back to the University of Kentucky as a visiting assistant professor in 1987. In 1988, he joined the Algebra research group at Auburn University. ... Read more

Product Description
Packed with contributions from international experts, Commutative Algebra: Geometric, Homological, Combinatorial, and Computational Aspects features new research results that borrow methods from neighboring fields such as combinatorics, homological algebra, polyhedral geometry, symbolic computation, and topology. This book consists of articles presented during two conferences held in Spain and Portugal in June, 2003. It encompasses a variety of topics, including blowup algebras, Castelnuovo-Mumford regularity, integral closure and normality, Koszul homology, liaison theory, multiplicities, polarization, and reductions of ideals. This comprehensive volume will stimulate further research in the field. ... Read more

Product Description
An account of a seminar held at the Mathematical Institute in Tbilisi, USSR from 1987 to 1988 which dealt with such subjects as algebraic K-theory of monoid algebras and coefficients. ... Read more

Product Description
This book presents an account of several conjectures arising in commutative algebra from the pioneering work of Serre and Auslander-Buchsbaum. The approach is via Hochster's 'Big Cohen-Macaulay modules', though the complementary view point of Peskine-Szpiro and Roberts, who study the homology of certain complexes, is not neglected. Various refinements of Hochster's construction, obtained in collaboration with Bartijn, are included. A special feature is a long chapter written by Van den Dries which explains how a certain type of result can be 'lifted' from prime characteristic to characteristic zero. Though this is primarily a research monograph, it does provide introductions to most of the topics treated. Non-experts may therefore find it an appealing guide into an active area of algebra. ... Read more

Product Description
This book exposes methods of non-abelian homological algebra,such as the theory of satellites in abstract categories with respectto presheaves of categories and the theory of non-abelian derivedfunctors of group valued functors. Applications to K-theory, bivariantK-theory and non-abelian homology of groups are given. The cohomologyof algebraic theories and monoids are also investigated. The work isbased on the recent work of the researchers at the A. RazmadzeMathematical Institute in Tbilisi, Georgia.
Audience: This volume will be of interest to graduate studentsand researchers whose work involves category theory, homologicalalgebra, algebraic K-theory, associative rings and algebras; algebraictopology, and algebraic geometry. ... Read more

Product Description
This volume constitutes a record of the course in homological algebra given at the Virginia Polytechnic Institute in July 1970 under the auspices of the National Science Foundation's Regional Conference project. The nature of the audience required that the course begin with an introduction to the notion of modules over a unitary ring, but permitted rapid development of the theory from that starting point. The first three chapters may be regarded as containing material essential to any introductory course in homological algebra, while the later chapters reflect the choices actually made by the audience among many possible special topics accessible to those who had mastered the early material. Thus it may be claimed that the course achieved depth of penetration on a narrow front, while it is admitted that breadth of coverage of the entire domain of homological algebra was neither attempted nor achieved. ... Read more

Product Description
Homological algebra, because of its fundamental nature, is relevant to many branches of pure mathematics, including number theory, geometry, group theory and ring theory. Professor Northcott's aim is to introduce homological ideas and methods and to show some of the results which can be achieved. The early chapters provide the results needed to establish the theory of derived functors and to introduce torsion and extension functors. The new concepts are then applied to the theory of global dimensions, in an elucidation of the structure of commutative Noetherian rings of finite global dimension and in an account of the homology and cohomology theories of monoids and groups. A final section is devoted to comments on the various chapters, supplementary notes and suggestions for further reading. This book is designed with the needs and problems of the beginner in mind, providing a helpful and lucid account for those about to begin research, but will also be a useful work of reference for specialists. It can also be used as a textbook for an advanced course. ... Read more

Customer Reviews (2)

still one of the best
This book was first published in 1960. It is one of the first few to appear that treated a very abstract subject matter but whose applications are numerous (number thepory, algebraic geometry, topology, group theory, ring theory, etc.). It took a couple of decades to see the full potential of homological algebra.

Author is a very good writer; theorems are clearly stated and most of them proved. The author has another text called A First Course of Homological Algebra, which I think is a bit better and a bit more modern.
Nonetheless, this is a valuable text, even today, after more than 40 years of its first appearance.

Avoid this book by this author like the plague!
This book may be one of the worst written texts I've seen
on a very difficult abstraction of algebra and topology.
I don't have a real desire to read it further
but I'm getting an idea of the subject
( not a lot of thanks to the way it was written).
I went online and read Mathworld definitions for some of the terms and that helped.
I don't know if there is any application to this besides maybe some number theory ones that are almost as abstract.
If you took a course where this was the text, I can only say,
I'm very sorry.
How Mathematics managed to get this far removed from reality is hard to say?
It is ironic that the son of Eli Cartan who is the creator of some of the most useful algebra/ group theory
should be the sponsor of this homological algebra.
With no problems and no examples there is no way to get context for a student , much less understanding.
Avoidbooks by this author like the plague!
He's still writing and may have improved some, but anyone who published this has to be really numb.
There really should be a zero stars for books like this one. ... Read more

Product Description
Categories, homological algebra, sheaves and their cohomology furnish useful methods for attacking problems in a variety of mathematical fields. This textbook provides an introduction to these methods, describing their elements and illustrating them by examples. ... Read more

Product Description
Based on a series of lectures given at Sheffield during 1971-72, this text is designed to introduce the student to homological algebra avoiding the elaborate machinery usually associated with the subject. This book presents a number of important topics and develops the necessary tools to handle them on an ad hoc basis. The final chapter contains some previously unpublished material and will provide additional interest both for the keen student and his tutor. Some easily proven results and demonstrations are left as exercises for the reader and additional exercises are included to expand the main themes. Solutions are provided to all of these. A short bibliography provides references to other publications in which the reader may follow up the subjects treated in the book. Graduate students will find this an invaluable course text as will those undergraduates who come to this subject in their final year. ... Read more

Customer Reviews (1)

Every algebraist's shelf needs one
I really enjoyed the content of this book and the author's delivery.I would recommend it to anyone familiar with rings and modules who wants to get their feet wet with homological algebra.The other great feature is the supply of numerous exercises with solutions! ... Read more