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.

See also

• Hyperhomology

References

• Weibel, Charles A. (1994), An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, ISBN 978-0-521-55987-4, MR 1269324