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

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

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

It can be shown that for each p, the column ${\displaystyle P_{p,*}}${\displaystyle P_{p,*}} is a projective resolution of ${\displaystyle A_{p}}${\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 ${\displaystyle F\colon {\mathcal {A}}\to {\mathcal {B}}}${\displaystyle F\colon {\mathcal {A}}\to {\mathcal {B}}}, one can define the left hyper-derived functors of ${\displaystyle F}${\displaystyle F} on a chain complex ${\displaystyle A_{*}}${\displaystyle A_{*}} by

• Constructing a Cartan–Eilenberg resolution ${\displaystyle \varepsilon :P_{*,*}\to A_{*}}${\displaystyle \varepsilon :P_{*,*}\to A_{*}},
• Applying the functor ${\displaystyle F}${\displaystyle F} to ${\displaystyle P_{*,*}}${\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

• Homological algebra