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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