The Stacks project

Remark 103.5.5. The result of Lemma 103.5.4 tells us that

\[ D_{\textit{Parasitic} \cap \textit{LQCoh}^{fbc}}(\mathcal{O}_\mathcal {X}) \subset D_{\textit{LQCoh}^{fbc}}(\mathcal{O}_\mathcal {X}) \]

is a left admissible subcategory, see Derived Categories, Section 13.40. In particular, if $\mathcal{A} \subset D_{\textit{LQCoh}^{fbc}}(\mathcal{O}_\mathcal {X})$ denotes its left orthogonal, then Derived Categories, Proposition 13.40.10 implies that $\mathcal{A}$ is right admissible in $D_{\textit{LQCoh}^{fbc}}(\mathcal{O}_\mathcal {X})$ and that the composition

\[ \mathcal{A} \longrightarrow D_{\textit{LQCoh}^{fbc}}(\mathcal{O}_\mathcal {X}) \longrightarrow D_\mathit{QCoh}(\mathcal{O}_\mathcal {X}) \]

is an equivalence. This means that we can view $D_\mathit{QCoh}(\mathcal{O}_\mathcal {X})$ as a strictly full saturated triangulated subcategory of $D_{\textit{LQCoh}^{fbc}}(\mathcal{O}_\mathcal {X})$ and also of $D(\mathcal{X}_{fppf}, \mathcal{O}_\mathcal {X})$.


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