could be represented as a long exact sequence of modules (with n intermediate terms):

All these various examples of the construction of new functors as “derived” functors of given ones were at hand for Eilenberg. He saw how they could be used to determine a homological “dimension” for algebraic objects, and he established the connection with the Hilbert notion of a syzygy in a 1956 paper [3]. This provided the background for the influential Cartan-Eilenberg book [1] on homological algebra. This text emphasized how the derived functors for a module M could be calculated from any “resolution” of M by free modules, a long exact sequence

with all X_j free. One simply applies the functor to the resolution with the M term dropped and then takes the homology or cohomology of the resulting complex. This effectively generalized the computation from specific “bar resolutions” used to define the cohomology of a group. The ideas of homological algebra were presented in two pioneering books by Cartan-Eilenberg [1] and Mac Lane [4]. The Cartan-Eilenberg treatise had a widespread and decisive influence in algebra. This again illustrates the genius of Eilenberg: If essentially the same idea crops up in different places, follow it out and find out where it lives.