Cartan–Eilenberg resolution

Summary

A Cartan–Eilenberg resolution of a chain complex $C$ is a double complex of projective objects. It resolves not only each term of $C$ but also its boundaries, cycles and homology, all at once. It is the tool for defining derived functors of complexes, called hyperderived functors. It is also the main step in the proof of the Grothendieck spectral sequence.