The Stacks project

Lemma 21.28.2. Let $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D})$ be a morphism of ringed topoi. Consider the full subcategory $D' \subset D(\mathcal{O}_\mathcal {C})$ consisting of objects $K$ such that

\[ Lf^*Rf_*K \longrightarrow K \]

is an isomorphism. Then $D'$ is a saturated triangulated strictly full subcategory of $D(\mathcal{O}_\mathcal {C})$ and the functor $Rf_* : D' \to D(\mathcal{O}_\mathcal {D})$ is fully faithful.

Proof. See Derived Categories, Definition 13.6.1 for the definition of saturated in this setting. See Derived Categories, Lemma 13.4.15 for a discussion of triangulated subcategories. The canonical map of the lemma is the counit of the adjoint pair of functors $(Lf^*, Rf_*)$, see Lemma 21.20.1. Having said this the proof that $D'$ is a saturated triangulated subcategory is omitted; it follows formally from the fact that $Lf^*$ and $Rf_*$ are exact functors of triangulated categories. The final part follows formally from fact that $Lf^*$ and $Rf_*$ are adjoint; compare with Categories, Lemma 4.24.3. $\square$

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