The Stacks project

Remark 102.10.7. Let $\mathcal{X}$ be an algebraic stack. Let $\textit{Parasitic}(\mathcal{O}_\mathcal {X}) \subset \textit{Mod}(\mathcal{O}_\mathcal {X})$ denote the full subcategory consiting of parasitic modules. The results of Lemmas 102.10.1 and 102.10.2 imply that

\[ \mathit{QCoh}(\mathcal{O}_\mathcal {X}) = \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X}) / \textit{Parasitic}(\mathcal{O}_\mathcal {X}) \cap \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X}) \]

in words: the category of quasi-coherent modules is the category of locally quasi-coherent modules with the flat base change property divided out by the Serre subcategory consisting of parasitic objects. See Homology, Lemma 12.10.6. The existence of the inclusion functor $i : \mathit{QCoh}(\mathcal{O}_\mathcal {X}) \to \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$ which is left adjoint to the quotient functor is a key feature of the situation. In Derived Categories of Stacks, Section 103.5 and especially Lemma 103.5.4 we prove that a similar result holds on the level of derived categories.


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