The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

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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