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 \mathcal{A} be an Abelian category with enough projectives, and let A_{} be a chain complex with objects in \mathcal{A}. Then a Cartan–Eilenberg resolution of A_{} is an upper half-plane double complex P_{,} (i.e., P_{p,q} = 0 for q < 0) consisting of projective objects of \mathcal{A} and an "augmentation" chain map such that It can be shown that for each p, the column P_{p, *} is a projective resolution of A_{p}. 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 F on a chain complex A_{*} by Similarly, one can also define right hyper-derived functors for left exact functors.