Lemma 13.26.2 (05TP)—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 13.26.2 (cite)

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Lemma 13.26.2. Let $\mathcal{A}$ be an abelian category. An object $I$ of $\text{Fil}^ f(\mathcal{A})$ is filtered injective if and only if there exist $a \leq b$ , injective objects $I_ n$ , $a \leq n \leq b$ of $\mathcal{A}$ and an isomorphism $I \cong \bigoplus _{a \leq n \leq b} I_ n$ such that $F^ pI = \bigoplus _{n \geq p} I_ n$ .

Proof. Follows from the fact that any injection $J \to M$ of $\mathcal{A}$ is split if $J$ is an injective object. Details omitted. $\square$

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