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

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

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

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

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

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 [math]\displaystyle{ F \colon \mathcal{A} \to \mathcal{B} }[/math], one can define the left hyper-derived functors of [math]\displaystyle{ F }[/math] on a chain complex [math]\displaystyle{ A_{*} }[/math] by

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