Lemma 13.31.9 (08BJ)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 13: Derived Categories
Section 13.31: K-injective complexes
Lemma 13.31.9 (cite)

previous lemma

Lemma 13.31.9. 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 $v$ , and
$v$ is exact.

Then $u$ transforms K-injective complexes into K-injective complexes.

Proof. Let $I^\bullet$ be a K-injective complex of $\mathcal{A}$ . Let $M^\bullet$ be a acyclic complex of $\mathcal{B}$ . As $v$ is exact we see that $v(M^\bullet )$ is an acyclic complex. By adjointness we get

$0 = \mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{A})}(v(M^\bullet ), I^\bullet ) = \mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{B})}(M^\bullet , u(I^\bullet ))$

hence the lemma follows. $\square$

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