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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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