Cartan–Eilenberg resolution

In homological algebra, the Cartan–Eilenberg resolution is in a sense, a resolution of a chain complex. It can be used to construct [|hyper-derived functors]. It is named in honor of Henri Cartan and Samuel Eilenberg.

Definition

Let be an Abelian category with enough projectives, and let be a chain complex with objects in. Then a Cartan–Eilenberg resolution of is an upper half-plane double complex consisting of projective objects of and an "augmentation" chain map such that

If then the p-th column is zero, i.e. for all q.
For any fixed column,
* The complex of boundaries obtained by applying the horizontal differential to forms a projective resolution of the boundaries of.
* The complex obtained by taking the homology of each row with respect to the horizontal differential forms a projective resolution of degree p homology of.

It can be shown that for each p, the column is a projective resolution of.
There is an analogous definition using injective resolutions and cochain complexes.
The existence of Cartan–Eilenberg resolutions can be proved via the horseshoe lemma.

Hyper-derived functors

Given a right exact functor, one can define the left hyper-derived functors of on a chain complex by

Constructing a Cartan–Eilenberg resolution,
Applying the functor to, and
Taking the homology of the resulting total complex.

Similarly, one can also define right hyper-derived functors for left exact functors.