Book Review: Weibel's "An Introduction to Homological Algebra"

Let us recall some classic words:

Our subject starts with homology, homomorphisms, and tensors.

Saunders Mac Lane, in "Homology"

And while Mac Lane's "Homology" and its friend by Cartan and Eilenberg are certainly fairly comprehensive sources of homological algebra, viewpoint shifts in the subject have made more recent approaches desirable. Weibel's 'An Introduction to Homological Algebra' (author website, Amazon), or IHA, is just that: a modern textbook on homological algebra. Aside from a few busy semesters, during the last two years I have been slowly reading it, as I was determined to read this book cover-to-cover. Now that I have finished this, it is my pleasure to write a short review of this book.

According to Charles Weibel, IHA was a practice book for his K-Theory text:

I needed a warm-up exercise, a practice book if you will. The result, An introduction to homological algebra, took over five years to write.

Charles Weibel

IHA is very readable, and I feel definitely suitable for a first introduction to homological algebra as long as the reader is comfortable with basic category theory basic module theory including projective and injective modules. The use of abelian categories is just enough to set the subject in its proper context, without being technical; reasoning in the framework of abstract abelian categories is not necessary as explained at the end of Chapter 1, which contains a rough sketch of the Freyd-Mitchell embedding lemma.

Weibel's book covers the standard homological algebra very well. The main question is, what does this book offer that other introductory texts do not? One is the use and emphasis of universal $ \delta$-functors; this viewpoint is certainly convenient and virtually necessary if one wants to make serious use of derived functors. A few other books do mention $ \delta$-functors (e.g. Hilton-Stammbach, Cartan-Eilenberg). Spectral sequences are covered in most books, although I found IHA certainly one of the clearest, and an exceptional feature of this book is that spectral sequences are used several times in later chapters, often to prove theorems.

There are, however, four features of IHA that stand out amongst all other introductory books: dimension theory, simplicial methods, Hochschild homology, and derived categories. What is dimension theory?

Fix a ring $ R$. The projective dimension $ \mathrm{pd}_R(A)$ of a right $ R$-module $ A$ is minimum over the length of all projective resolutions of $ A$, and the (right)-global dimension of $ R$ is the supremum over the projective dimension ranging over all right $ R$-modules. These notions can be used to show for instance that the localisation of a regular local ring is again a regular local ring. Homological dimension can be found in a few other books such as Kaplansky's book "Commutative Rings" and Osofsky's book "Homological Dimension of Modules".

In my opinion, the part of this text that distinguishes it from all others is Chapter 8 of IHA, on simplicial methods. Much of this material is also available in Gelfand and Manin's "Methods of Homological Algebra", but certainly not so clearly. The crowning result of this chapter is certainly the Dold-Kan correspondence: the category of bounded below chain complexes in an abelian category $\mathcal{A}$ is equivalent to the category of simplicial objects in $\mathcal{A}$; this together with the fact that additive adjoints generate simplicial objects truly completes the picture of standard resolutions and derived functors. Moreover, these results truly explain that all this algebraic stuff is really geometric as well, and that homotopy is really fundamental (after all, it seems to be everywhere!)

Hochschild homology is a very useful example coming from a cotriple. If $ R$ is a $ k$-algebra where $ k$ is a commutative ring, then we can form a simplicial object with $ M\otimes R^{\otimes n}$ in degree $ n$ for any $ R$-module $ M$, which corresponds by Dold-Kan to a chain complex. The homology of this chain complex is the Hochschild homology (taking Hom-sets gives a cohomology theory). One observes that $H_1(R,R) \cong \Omega_{R/k}$, and from this observations many connections between separable algebras and Hochschild homology can be made. Readers interested in this will surely want to absorb more detail in J.-L. Loday's book, "Cyclic Homology".

The final chapter on derived categories explains the idea of localisation in a category. Just as in a ring in which we can invert a set of elements that form a multiplicative set, one can also invert morphisms in a category. The main example is the set of quasiisomorphisms in the homotopy category of the category of chain complexes. The main question in this localisation business is: when does a functor between categories extend to a functor of on derived categories? This question is answered via the formalism of derived functors, which resemble the classical derived functors. The theory of derived categories is important in algebraic topology, and in algebraic geometry (e.g. Hartshorne's "Residues and Duality", and perverse cohomology).

The latest edition of this book, which is the second, has a few typographical errors and a missing hypothesis here and there, but almost all of these are corrected and listed on Weibel's website. I feel that if a third edition is published then there will be very few errors left in this book.

One thing I would have liked to see in this text is more applications of derived categories and more motivation for external products. However, given that there are plenty of examples already easily accessible, this should not be a detriment to the student. I feel also that a mention of the Eilenberg-Watts theorem on the characterisation of the tensor and $ \mathrm{Hom}$ functor would make the big picture a bit clearer. There are certainly other topics that were left out, such as explicit descriptions for $ \mathrm{Tor}$, but the addition of them would have probably destroyed the broad nature of this book, and would have hindered the main outline of homological algebra. I feel that the balance between brevity and detail, one which must be difficult to achieve, is very nearly perfect in IHA.

I enjoyed the exercises in this book. There are easy ones to get used to definitions and it has enough harder ones that kept me interested after I felt comfortable with the basics. For those needing serious amounts of homological algebra, this book will definitely give the best preparation, and yet it has enough detail so that it would be suitable for introductory classes as well.

Aside from these specifics, perhaps the most important aspect of this book is its style. Weibel takes a general, categorical approach when necessary, and yet never ventures into abstraction for the sake of abstraction. Yes, computations can be made, and yes, there is an explicit formula for the connecting morphism. This balance shows the power of generality, and at the same time avoids the laborious element-theoretic style that was dominant in the days of Mac Lane.

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