The Stacks project

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

  1. $u$ is right adjoint to $v$, and

  2. $v$ 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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