Definition 13.21.1. Let $\mathcal{A}$ be an abelian category. Let $K^\bullet $ be a bounded below complex. A *Cartan-Eilenberg resolution* of $K^\bullet $ is given by a double complex $I^{\bullet , \bullet }$ and a morphism of complexes $\epsilon : K^\bullet \to I^{\bullet , 0}$ with the following properties:

There exists a $i \ll 0$ such that $I^{p, q} = 0$ for all $p < i$ and all $q$.

We have $I^{p, q} = 0$ if $q < 0$.

The complex $I^{p, \bullet }$ is an injective resolution of $K^ p$.

The complex $\mathop{\mathrm{Ker}}(d_1^{p, \bullet })$ is an injective resolution of $\mathop{\mathrm{Ker}}(d_ K^ p)$.

The complex $\mathop{\mathrm{Im}}(d_1^{p, \bullet })$ is an injective resolution of $\mathop{\mathrm{Im}}(d_ K^ p)$.

The complex $H^ p_ I(I^{\bullet , \bullet })$ is an injective resolution of $H^ p(K^\bullet )$.

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