Cartan–Eilenberg resolution

In homological algebra, the Cartan–Eilenberg resolution is in a sense, a resolution of a complex. It can be used to construct hyper-derived functors.

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., Ppq = 0 for q < 0) consisting of projective objects of \mathcal{A} and a chain map ε : Pp0Ap 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 \colon \mathcal{A} \to \mathcal{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.

See also

References

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