Lemma 13.26.5 (05TS)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 13: Derived Categories
Section 13.26: Filtered derived category and injective resolutions
Lemma 13.26.5 (cite)

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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 $u_ n$ . $\square$

Comments (2)

Comment #3766 by Owen B on November 01, 2018 at 16:42

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.

Comment #3896 by Johan on January 22, 2019 at 10:11

Thanks and fixed here.

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