It can be used to construct hyper-derived functors.
It is named in honor of Henri Cartan and Samuel Eilenberg.
be an Abelian category with enough projectives, and let
be a chain complex with objects in
is an upper half-plane double complex
) consisting of projective objects of
and an "augmentation" chain map
There is an analogous definition using injective resolutions and cochain complexes.
The existence of Cartan–Eilenberg resolutions can be proved via the horseshoe lemma.
, one can define the left hyper-derived functors of