Cartan–Eilenberg resolution

Definition

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

• If ${\displaystyle A_p=0}$ then the p-th column is zero, i.e. ${\displaystyle P_{p,q}=0}$ for all q.

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

It can be shown that for each p, the column ${\displaystyle P_{p,\ast }}$ is a projective resolution of ${\displaystyle A_p}$.

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

Given a right exact functor ${\displaystyle F:\mathcal{A}\to \mathcal{B}}$, one can define the left hyper-derived functors of ${\displaystyle F}$ on a chain complex ${\displaystyle A_{\ast }}$ by

• Constructing a Cartan–Eilenberg resolution ${\displaystyle \epsilon :P_{\ast ,\ast }\to A_{\ast }}$,
• Applying the functor ${\displaystyle F}$ to ${\displaystyle P_{\ast ,\ast }}$, and
• Taking the homology of the resulting total complex.

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