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 \( {\displaystyle A_{*}} \) be a chain complex with objects in \( {\mathcal {A}} \). 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 \( {\mathcal {A}} \) and a chain map \( {\displaystyle \varepsilon \colon P_{p0}\to A_{p}} \) such that
Ap = 0 implies that the pth column is zero (Ppq = 0 for all q).
For each p, the column Pp* is a projective resolution of Ap.
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 \( F\colon {\mathcal {A}}\to {\mathcal {B}} \), 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
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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