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.

References

{{Citation

| last=Weibel

| first=Charles A.

| authorlink = Charles Weibel

| title=An introduction to homological algebra

| publisher=Cambridge University Press

| isbn=978-0-521-55987-4

| mr=1269324

| year=1994

| series=Cambridge Studies in Advanced Mathematics

| volume=38

}}

Category:Homological algebra

Cartan–Eilenberg resolution

Definition

= Hyper-derived functors =

See also

References