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 ${\displaystyle {\mathcal {A}}}$ be an Abelian category with enough projectives, and let ${\displaystyle A_{*}}$ be a chain complex with objects in ${\displaystyle {\mathcal {A}}}$. Then a Cartan–Eilenberg resolution of ${\displaystyle A_{*}}$ is an upper half-plane double complex ${\displaystyle P_{**}}$ (i.e., ${\displaystyle P_{pq}=0}$ for ${\displaystyle q<0}$) consisting of projective objects of ${\displaystyle {\mathcal {A}}}$ and a chain map ${\displaystyle \varepsilon \colon P_{p0}\to A_{p}}$ such that

• A_p = 0 implies that the pth column is zero (P_pq = 0 for all q).
• For each p, the column P_p* is a projective resolution of A_p.
• For any fixed column,
  • the kernels of each of the horizontal maps starting at that column (which themselves form a complex) are in fact exact,
  • the same is true for the images of those maps, and
  • the same is true for the homology of those maps.

(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 ${\displaystyle F\colon {\mathcal {A}}\to {\mathcal {B}}}$, 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

• Hyperhomology

References

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