Cartan–Eilenberg resolution explained

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

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

• If

A _{p} = 0 then the p-th column is zero, i.e. P _{p,} = 0 for all q.

• For any fixed column

P _{p,} ,

• • The complex of boundaries

B _{p(P,} d ^{h)} : = d ^{h(P} _{p +1 . *} ) obtained by applying the horizontal differential to P _{p+1,} (the p +1 st column of P _{*,*} ) forms a projective resolution B _{p(\varepsilon):} B _{p(P,} d ^{h)} \to B _{p(A)} of the boundaries of A _{p} .

• • The complex

H _{p(P,} d ^{h)} obtained by taking the homology of each row with respect to the horizontal differential forms a projective resolution H _{p(\varepsilon):} H _{p(P,} d ^{h)} \to H _{p(A)} of degree p homology of A .

It can be shown that for each p, the column

P _{p,} is a projective resolution of 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.

Hyper-derived functors

F \colon l{A} \to l{B} , one can define the left hyper-derived functors of F on a chain complex A _{*} by

• Constructing a Cartan–Eilenberg resolution

\varepsilon: P _{*,} \to A _{*} ,

• Applying the functor

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