ABSTRACT
T he cohomology of groups, as already described, had originated in the observation that the fundamental group π1 of an aspherical space would determine all the homology and
cohomology groups of that space-the latter as the cohomology Hn(π1, –) of the group π1. From this result developed the surprising idea that cohomology, originally studied just for spaces, could also apply to algebraic objects such as groups and rings. Given his topological background and enthusiasm, Eilenberg was perhaps the first person to see this clearly. He was in active touch with Gerhard Hochschild, who was then a student of Chevally at Princeton. Eilenberg suggested that there ought to be a cohomology (and a homology) for algebras. This turned out to be the case, and the complex used to describe the cohomology of groups (i.e., the bar resolution) was adapted to define the Hochschild cohomology of algebras. Eilenberg soon saw other possibilities for homology, and he and Henri Cartan wrote the book Homological Algebra, which attracted lively interest among algebraists such as Kaplansky. A leading feature was the general notion of a resolution, say of module M; such a resolution was an exact sequence of free modules Fi
M F0 F1 F2 ... .
