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Cartan–Eilenberg resolution

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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

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Let be an Abelian category with enough projectives, and let be a chain complex with objects in . Then a Cartan–Eilenberg resolution of is an upper half-plane double complex (i.e., for ) consisting of projective objects of and an "augmentation" chain map such that

  • If then the p-th column is zero, i.e. for all q.
  • For any fixed column ,
    • The complex of boundaries obtained by applying the horizontal differential to (the st column of ) forms a projective resolution of the boundaries of .
    • The complex obtained by taking the homology of each row with respect to the horizontal differential forms a projective resolution of degree p homology of .

It can be shown that for each p, the column is a projective resolution of .

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

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Given a right exact functor , one can define the left hyper-derived functors of on a chain complex by

  • Constructing a Cartan–Eilenberg resolution ,
  • Applying the functor to , and
  • Taking the homology of the resulting total complex.

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

See also

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References

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  • 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