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.

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., ${\displaystyle P_{p,q}=0}$ for ${\displaystyle q<0}$) consisting of projective objects of ${\displaystyle {\mathcal {A}}}$ and an "augmentation" chain map ${\displaystyle \varepsilon \colon P_{p,*}\to A_{p}}$ such that

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

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

It can be shown that for each p, the column ${\displaystyle P_{p,*}}$ is a projective resolution of ${\displaystyle A_{p}}$.

There is an analogous definition using injective resolutions and cochain complexes.

The existence of Cartan–Eilenberg resolutions can be proved via the horseshoe lemma.

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

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

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

• Hyperhomology

• 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