Book 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 (4)

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

Book Description
Homological algebra first arose as a language for describing topological prospects of geometrical objects. As with every successful language it quickly expanded its coverage and semantics, and its contemporary applications are many and diverse. 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. Relations with standard cohomology theory (sheaf cohomology, spectral sequences, etc.) are described. In most cases complete proofs are given. Basic concepts and results of homotopical algebra are also presented. The book addresses people who want to learn a modern approach to homological algebra and to use it in their work. For the second edition the authors have made numerous corrections. ... Read more

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

Book 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

Book 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 (1)

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

Customer Reviews (1)

New Edition
I have just begun writing a second edition of Homological Algebra. The new bookwill be published by Springer Verlag, New York, andit should be in print by the end of 2005. ... Read more

Book Description
This volume of the Encyclopaedia presents a modern approach to homologicalalgebra, which is based on the systematic use of the terminology and ideasof derived categories and derived functors. The book contains applicationsof homological algebra to the theory of sheaves on topological spaces,to Hodge theory, and to the theory of sheaves on topological spaces, to Hodge theory, and to the theory of modules over rings of algebraicdifferential 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 famousfor his work in algebraic geometry and mathematical physics. The book isan excellent reference for graduate students and researchers in mathematics and also for physicists who use methods from algebraic geomtry and algebraictopology. ... Read more

Customer Reviews (1)

Homology Category
This is an excellent book for whom are familiar with the previous homological algebra techniques (as Cartan and Eilenberg's Homological algebra as first generation, Hilton and Stammbach's A course in Homological algebra as second generation, this book serves as third generation of derived category. This is not written for graduate students who are trying to learn this material as the book is in part of encyclopaedia.

The book explains well why, how and what the derived functors and derived category are introduced. Many concrete exmaples in algebra and geometry are listed in Chapter 3, in particular, Chapter 6 (Mixed Hodge structure), chapter 7(perverse sheaf) and chapter 8(D-modules) are served very good samples on the Derived categories and their equivalence which interprete many deep and fundamental theorems in these areas from category point of view.

Chapter 4(Derived categories and derived functors) and Chapter 5 (Triangulated categories) are the core of the homology category theory. The authors used the localization of homotopy category to construct the universal derived category which is familiar construction for category people under the name of fraction category. In fact many results can be obtained from calculus of fractions (see chapter 19 of the book by H. Schubert "Categories").

It would be better to include the rational homotopy model of Quillen and Sullivan's results and many other more explicit examples from infinitesimally topological calculations. There are few typos and undefined notations. Overall this is a very good reference book indeed. ... Read more

Book 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.

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

Book Description
Featuring contributions from leading international experts, Abelian Groups, Rings, and Modules, and Homological Algebra presents an introduction to this newly developing field. It reviews basic results from homological and commutative algebra, considering Gorenstein rings from a homological point of view and providing examples of Iwanaga-Gorenstein rings. It develops relative homologica algebra and shows how Tate cohomology arises from the connection between Gorenstein homological algebra and usual variety. The final three chapters look at torsion-free Abelian groups, modular representations and the properties of universal enveloping algebras, and indicate directions for future research. ... Read more

Book Description
This book provides a self-contained systematic treatment of the subject of relative homological algebra. It is designed for graduate students as well as researchers and specialists. It contains twelve chapters with abundant supply of important results with complete proofs covering material that is essential to understanding topics in algebra, algebraic geometry, and algebraic topology. The text also contains results that are in book form for the first time and thus provides essential reading for researchers and specialists. At the end of each section of each chapter, there are exercises that provide practice problems for students as well as additional important results for specialists.

The book can be used as a text for graduate students and as a handbook for researchers and specialists. The material in the first three chapters constitute notes from lectures of the authors at their respective universities and is suitable for an introductory course in module and ring theory. The following chapters are suitable for a course in relative homological algebra and its applications to commutative and non-commutative algebra. The last three chapters give applications to ring theory. These chapters together with Chapter 7 contain recent achievements including a solution to the flat cover conjecture. ... 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

Book 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

Book 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