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 $$\mathcal{A}$$ 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 $$\varepsilon \colon P_{p,*} \to A_p$$ such that


 * If $$A_{p} = 0 $$ then the p-th column is zero, i.e. $$P_{p, q} = 0$$ for all q.


 * For any fixed column $$P_{p, *}$$,
 * The complex of boundaries $$B_p(P, d^h) := d^h(P_{p+1. *})$$ obtained by applying the horizontal differential to $$P_{p+1, *}$$ (the $$p+1$$st column of $$P_{*,*}$$) forms a projective resolution $$B_p(\varepsilon): B_p(P, d^h) \to B_p(A)$$ of the boundaries of $$A_p$$.
 * The complex $$H_p(P, d^h)$$ obtained by taking the homology of each row with respect to the horizontal differential forms a projective resolution $$H_p(\varepsilon): H_p(P, d^h) \to H_p(A)$$ of degree p homology of $$A$$.

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 $$F \colon \mathcal{A} \to \mathcal{B}$$, one can define the left hyper-derived functors of $$F$$ on a chain complex $$A_{*}$$ by


 * Constructing a Cartan–Eilenberg resolution $$\varepsilon: P_{*, *} \to A_{*}$$,
 * Applying the functor $$F$$ to $$P_{*, *}$$, and
 * Taking the homology of the resulting total complex.

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