The Stacks project

Lemma 19.13.6. Let $F : \mathcal{A} \to \mathcal{B}$ be an additive functor of abelian categories. Assume

  1. $\mathcal{A}$ is a Grothendieck abelian category,

  2. $\mathcal{B}$ has exact countable products, and

  3. $F$ commutes with countable products.

Then $RF : D(\mathcal{A}) \to D(\mathcal{B})$ commutes with derived limits.

Proof. Observe that $RF$ exists as $\mathcal{A}$ has enough K-injectives (Theorem 19.12.6 and Derived Categories, Lemma 13.31.6). The statement means that if $K = R\mathop{\mathrm{lim}}\nolimits K_ n$, then $RF(K) = R\mathop{\mathrm{lim}}\nolimits RF(K_ n)$. See Derived Categories, Definition 13.34.1 for notation. Since $RF$ is an exact functor of triangulated categories it suffices to see that $RF$ commutes with countable products of objects of $D(\mathcal{A})$. In the proof of Lemma 19.13.4 we have seen that products in $D(\mathcal{A})$ are computed by taking products of K-injective complexes and moreover that a product of K-injective complexes is K-injective. Moreover, in Derived Categories, Lemma 13.34.2 we have seen that products in $D(\mathcal{B})$ are computed by taking termwise products. Since $RF$ is computed by applying $F$ to a K-injective representative and since we've assumed $F$ commutes with countable products, the lemma follows. $\square$

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