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

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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