Lemma 12.29.3 (0160)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 12: Homological Algebra
Section 12.29: Injectives and adjoint functors
Lemma 12.29.3 (cite)

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Lemma 12.29.3. Let $\mathcal{A}$ and $\mathcal{B}$ be abelian categories. Let $u : \mathcal{A} \to \mathcal{B}$ and $v : \mathcal{B} \to \mathcal{A}$ be additive functors. Assume

$u$ is right adjoint to $v$ ,
$v$ transforms injective maps into injective maps,
$\mathcal{A}$ has enough injectives, and
$vB = 0$ implies $B = 0$ for any $B \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{B})$ .

Then $\mathcal{B}$ has enough injectives.

Proof. Pick $B \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{B})$ . Pick an injection $vB \to I$ for $I$ an injective object of $\mathcal{A}$ . According to Lemma 12.29.1 and the assumptions the corresponding map $B \to uI$ is the injection of $B$ into an injective object. $\square$

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