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 13.4.16 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 21.19.1. 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 4.24.4.
\square
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