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 $\mathcal$ be an Abelian category with enough projectives, and let $A_$ be a chain complex with objects in $\mathcal$. Then a Cartan–Eilenberg resolution of $A_$ is an upper half-plane double complex $P_$ (i.e., $P_ = 0$ for $q < 0$) consisting of projective objects of $\mathcal$ and an "augmentation" chain map $\varepsilon \colon P_ \to A_*$ such that

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

* For any fixed column $P_$,
** The complex of boundaries $B_p(P, d^h) := d^h(P_)$ obtained by applying the horizontal differential to $P_$ (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 $P_$ is a projective resolution of $A_$.

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 $F \colon \mathcal \to \mathcal$, 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.

See also

* Hyperhomology

References

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{{DEFAULTSORT:Cartan-Eilenberg resolution
Homological algebra