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 $A \to I^\bullet$ 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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