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Book review

Author(s) Weibel, Ch.
Title An Introduction to Homological Algebra
Publisher Cambridge University Press
Year of publication 1994
Reviewed by Paul Blaga

Homological algebra has the undesirable fame of being one of the most difficult parts of modern mathematics and let me say that this fame is largely based on reality. This is why any attempt to make it more friendly is highly welcome. Of course, it is unreasonable to expect for a book of "homological algebra made easy"; I have serious doubts that such a kind of book could ever be written.

The present book, based on the lectures given by the author at several universities, is, first of all, an attempt describe the classical topics at a level accessible for both graduate students and researchers, not only in mathematics, but also in other fields where the homological methods are important, thus filling a gap, because most of the existent textbooks are a little to technical for the applied mathematician or the theoretical physicist. On the other hand, it contains a great number of new notions and results, some of them for the first time included in a textbook, but also at an accessible level.

Roughly speaking, the contents look as follows: The first four chapters are of a rather introductive nature, dealing, in order, with chain complexes, derived functors, the functors Tor and Ext and the homological dimension of rings. The chapter five introduces the delicate topic of spectral sequences. Chapter six is devoted to the homology and cohomology of groups, with an account of Galois cohomology. Chapter seven concerns the homology and cohomology of Lie algebras, while the following one is dealing with simplicial methods, including a short discussion of André-Quillen homology. The ninth chapter discusses Hochschild and cyclic homology of k-algebras, while the last one is devoted to the derived category of an abelian category.

As for the prerequisites, it is assumed that the reader has taken at least an introductory course in higher algebra. The book is written in a very pedagogical manner. There is a great number of worked examples, applications and exercises, which are a constitutive part of the book. A strength of it is the fact that, unlike most of the classical textbooks in the field, the spectral sequences are introduced rather early, such that they are used intensively throughout the book. It is, also, worth noting that the book includes much new material. For instance, I'd like to mention the discussion of cyclic homology, which is very important and has many applications.

Another thing I would like to point out is the fact that the author is faithful to the prerequisites (and we all know that this is not always the case in the literature). Thus, for example, the chapter of homology of Lie algebras includes an introduction to the Lie algebras themselves. Moreover, the book contains an appendix including the main notions from category theory.

I have to say that the author made no attempt to be exhaustive. Sometimes, he prefers to skip over a long demonstration, providing instead illuminating motivations and examples.

All in all, this is, in my opinion, an excellent textbook and it is very suitable for both self study and courses at graduate level. It will be especially helpful to those who apply homological algebra, preventing an important waste of time. I have no doubt that, in the next future, it will become a serious challenger for the classics.