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The Stacks project

Lemma 13.26.6. Let \mathcal{A} be an abelian category with enough injectives. For any object A of \text{Fil}^ f(\mathcal{A}) there exists a filtered quasi-isomorphism A[0] \to I^\bullet where I^\bullet is a complex of filtered injective objects with I^ n = 0 for n < 0.

Proof. First choose a strict monomorphism u_0 : A \to I^0 of A into a filtered injective object, see Lemma 13.26.5. Next, choose a strict monomorphism u_1 : \mathop{\mathrm{Coker}}(u_0) \to I^1 into a filtered injective object of \mathcal{A}. Denote d^0 the induced map I^0 \to I^1. Next, choose a strict monomorphism u_2 : \mathop{\mathrm{Coker}}(u_1) \to I^2 into a filtered injective object of \mathcal{A}. Denote d^1 the induced map I^1 \to I^2. And so on. This works because each of the sequences

0 \to \mathop{\mathrm{Coker}}(u_ n) \to I^{n + 1} \to \mathop{\mathrm{Coker}}(u_{n + 1}) \to 0

is short exact, i.e., induces a short exact sequence on applying \text{gr}. To see this use Homology, Lemma 12.19.13. \square


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