Lemma 13.26.5. Let $\mathcal{A}$ be an abelian category with enough injectives. For any object $A$ of $\text{Fil}^ f(\mathcal{A})$ there exists a strict monomorphism $A \to I$ where $I$ is a filtered injective object.

Proof. Pick $a \leq b$ such that $\text{gr}^ p(A) = 0$ unless $p \in \{ a, a + 1, \ldots , b\}$. For each $n \in \{ a, a + 1, \ldots , b\}$ choose an injection $u_ n : A/F^{n + 1}A \to I_ n$ with $I_ n$ an injective object. Set $I = \bigoplus _{a \leq n \leq b} I_ n$ with filtration $F^ pI = \bigoplus _{n \geq p} I_ n$ and set $u : A \to I$ equal to the direct sum of the maps $u_ n$. $\square$

Comment #3766 by Owen B on

typos: $I_n$ an injective object, not and $I=\bigoplus I_n$, not $I_p$ additionally, it appears that the indexing is off. I think it is fixed if you let $u_n:A/F^{n+1}\rightarrow I_n$ rather than what is written.

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