Lemma 13.21.2. Let $\mathcal{A}$ be an abelian category with enough injectives. Let $K^\bullet$ be a bounded below complex. There exists a Cartan-Eilenberg resolution of $K^\bullet$.

Proof. Suppose that $K^ p = 0$ for $p < n$. Decompose $K^\bullet$ into short exact sequences as follows: Set $Z^ p = \mathop{\mathrm{Ker}}(d^ p)$, $B^ p = \mathop{\mathrm{Im}}(d^{p - 1})$, $H^ p = Z^ p/B^ p$, and consider

$\begin{matrix} 0 \to Z^ n \to K^ n \to B^{n + 1} \to 0 \\ 0 \to B^{n + 1} \to Z^{n + 1} \to H^{n + 1} \to 0 \\ 0 \to Z^{n + 1} \to K^{n + 1} \to B^{n + 2} \to 0 \\ 0 \to B^{n + 2} \to Z^{n + 2} \to H^{n + 2} \to 0 \\ \ldots \end{matrix}$

Set $I^{p, q} = 0$ for $p < n$. Inductively we choose injective resolutions as follows:

1. Choose an injective resolution $Z^ n \to J_ Z^{n, \bullet }$.

2. Using Lemma 13.18.9 choose injective resolutions $K^ n \to I^{n, \bullet }$, $B^{n + 1} \to J_ B^{n + 1, \bullet }$, and an exact sequence of complexes $0 \to J_ Z^{n, \bullet } \to I^{n, \bullet } \to J_ B^{n + 1, \bullet } \to 0$ compatible with the short exact sequence $0 \to Z^ n \to K^ n \to B^{n + 1} \to 0$.

3. Using Lemma 13.18.9 choose injective resolutions $Z^{n + 1} \to J_ Z^{n + 1, \bullet }$, $H^{n + 1} \to J_ H^{n + 1, \bullet }$, and an exact sequence of complexes $0 \to J_ B^{n + 1, \bullet } \to J_ Z^{n + 1, \bullet } \to J_ H^{n + 1, \bullet } \to 0$ compatible with the short exact sequence $0 \to B^{n + 1} \to Z^{n + 1} \to H^{n + 1} \to 0$.

4. Etc.

Taking as maps $d_1^\bullet : I^{p, \bullet } \to I^{p + 1, \bullet }$ the compositions $I^{p, \bullet } \to J_ B^{p + 1, \bullet } \to J_ Z^{p + 1, \bullet } \to I^{p + 1, \bullet }$ everything is clear. $\square$

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