Lemma 46.8.7. Let $A$ be a ring.

Any $A$-module has a pure injective resolution.

Let $M \to N$ be a map of $A$-modules. Let $M \to M^\bullet $ be a universally exact resolution and let $N \to I^\bullet $ be a pure injective resolution.

There exists a map of complexes $M^\bullet \to I^\bullet $ inducing the given map

\[ M = \mathop{\mathrm{Ker}}(M^0 \to M^1) \to \mathop{\mathrm{Ker}}(I^0 \to I^1) = N \]two maps $\alpha , \beta : M^\bullet \to I^\bullet $ inducing the same map $M \to N$ are homotopic.

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