Cartan–Eilenberg resolution

Definition

Let A be an Abelian category with enough projectives, and let A_∗ be a chain complex with objects in A . Then a Cartan–Eilenberg resolution of A_∗ is an upper half-plane double complex P_∗∗ (i.e., P_pq = 0 for q < 0) consisting of projective objects of A and a chain map ε : P_p0 → A_p such that

(In fact, it would suffice to require it for the kernels and homology - the case of images follows from these.) In particular, since the kernels, cokernels, and homology will all be projective, they will give a projective resolution of the kernels, cokernels, and homology of the original complex A_•

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 F : A → B , one can define the left hyper-derived functors of F on a chain complex A_∗ by constructing a Cartan–Eilenberg resolution ε : P_∗∗ → A_∗, applying F to P_∗∗, and taking the homology of the resulting total complex.

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