An Introduction to Homological Algebra

According to some sources, homological algebra was supposed to be the lingua franca for expressing many results on mathematics (or at least, of a large part of it). Without passing judgment on these claims, it is certainly true that many parts of algebra, topology, geometry and number theory are formulated (better) in the language of homological algebra. Important conjectures are in the same language, and the categorical language of (derived) functors and natural transformations is an important tool of the trade.

Rotman’s book, whose first incarnation was a set of lecture notes (Van Nostrand, 1970), saw an expanded edition as Introduction to Homological Algebra (Academic, 1979, 400pp, and included in the MAA’s original “basic library list”). It has been one the fine staples of a long list of books on this topic, starting with the classical Homological Algebra of Cartan-Eilenberg (Princeton, 1956), Hilton-Stammbach’s A Course on Homological Algebra (Springer, 1971), Mac Lane’s Homology (Springer, 1966), and more recents additions such as Weibel book with the same title (Cambridge, 1999) or Gelfand and Manin Methods of Homological Algebra (Springer, 1999).

In the Van Nostrand edition, concepts and theorems were introduced in a straightforward fashion working in the category of modules over a given ring, the categorical concepts kept at a minimum, and the applications to group cohomology and to the theory of rings (homological dimension, for example) were designed to give only a taste of the powerful machinery just developed.

The new expanded second edition of the 1979 Academic Press book attempts to cover more ground, basically going from the (concrete) category of modules over a given ring, as in the first edition, to an arbitrary abelian category and to treat the important example of the category of sheaves on a topological space.

Thus, in addition to the derived functors of the Hom and tensor product, now the elements of sheaf cohomology are introduced, and a section of Chapter 6 is devoted to a discussion of Cech cohomology as a tool for computing the sheaf cohomology groups of a paracompact space, quoting the appropriate comparison theorems, for example from Godement’s Topologie Algébrique et Théorie des Faisceaux (Hermann, 1964). There is even an attempt to give a more concrete application discussing the main ideas involved for the Riemann-Roch Theorem for compact Riemann surfaces, with no proofs but giving adequate references.

The Chapter devoted to group cohomology now includes a discussion of Tate cohomology and Brauer groups, and the chapter on applications to ring theory includes Hilbert’s syzygy theorem and a section discussing the theorems of Auslander, Buchsbaum and Serre on regular local rings essentially showing that commutative noetherian local rings have finite global dimension if and only if they are regular local rings, an important result for algebraic geometry.

As the author recalls, the learning of homological algebra is a two-staged affair and once that the language of derived functors is mastered one needs tools to compute these objects. For initial or elementary examples the usual standard resolutions are discussed, but for more advanced situations one needs the language of spectral sequences and the last chapter of the book introduces them. Here the author balances the exposition between the clean straightforward presentation using exact couples and the down-to-earth multiple-indexed bicomplexes that occur naturally in most applications.

This is a beautifully written book that has grown gracefully for almost four decades, and even though the core of the subject comprises just three chapters (6, 7 and 10) it includes an introduction to rings and modules, abelian categories and applications to rings and group cohomology, to properly frame the subject. These additions account, in part, for the size of the book, about 700 pages, but this size is also consequence of the intention of the author to include details, especially at the beginning, to allow the student to see complete proofs avoiding ‘magic tricks’ that could lead to embarrassing mistakes later on, for example in deciding if a certain diagram commutes or if a given morphism is really natural. These details, the exercises at the end of every section, plenty of examples and motivation for the many new concepts set this book apart and make it an ideal textbook for a course on the subject.

Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is fzc@oso.izt.uam.mx.