An Introduction to Homological Algebra

Homological algebra is a tool used to prove nonconstructive existence theorems in algebra (and in algebraic topology). It also provides obstructions to carrying out various kinds of constructions; when the obstructions are zero, the construction is possible. Finally, it is detailed enough so that actual calculations may be performed in important cases. The following simple question (taken from Chapter 3) illustrates these points: Given a subgroup A of an abelian group B and an integer n, when is nA the intersection of A and nB? Since the cyclic group ℤ/n is not flat, this is not always the case. The obstruction is the group Tor(B/A, ℤ/n), which explicitly is {x ϵ B/A : nx = 0}.

This book intends to paint a portrait of the landscape of homological algebra in broad brushstrokes. In addition to the “canons” of the subject (Ext, Tor, cohomology of groups, and spectral sequences), the reader will find introductions to several other subjects: sheaves, lim, local cohomology, hypercohomology, profinite groups, the classifying space of a group, Affine Lie algebras, the Dold-Kan correspondence with simplicial modules, triple cohomology, Hochschild and cyclic homology, and the derived category. The historical connections with topology, regular local rings, and semisimple Lie algebras are also described.

After a lengthy gestation period (1890–1940), the birth of homological algebra might be said to have taken place at the beginning of World War II with the crystallization of the notions of homology and cohomology of a topological space.