The Stacks project

Proposition 21.43.9. Let $\mathcal{C}$ be a category viewed as a site with the chaotic topology. Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}$. With $\mathit{QC}(\mathcal{O})$ as in Definition 21.43.1 we have

  1. $\mathit{QC}(\mathcal{O})$ is a strictly full, saturated, triangulated subcategory of $D(\mathcal{O})$ preserved by arbitrary direct sums,

  2. any contravariant cohomological functor $H : \mathit{QC}(\mathcal{O}) \to \textit{Ab}$ which transforms direct sums into products is representable,

  3. any exact functor $F : \mathit{QC}(\mathcal{O}) \to \mathcal{D}$ of triangulated categories which transforms direct sums into direct sums has an exact right adjoint, and

  4. the inclusion functor $\mathit{QC}(\mathcal{O}) \to D(\mathcal{O})$ has an exact right adjoint.

Proof. Part (1) is Lemma 21.43.2. Part (2) follows from Lemma 21.43.8 and Derived Categories, Lemma 13.39.1. Part (3) follows from Lemma 21.43.8 and Derived Categories, Proposition 13.39.2. Part (4) is a special case of (3). $\square$

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