Definition 103.5.1. Let $\mathcal{X}$ be an algebraic stack. With notation as above we define the *derived category of $\mathcal{O}_\mathcal {X}$-modules with quasi-coherent cohomology sheaves* as the Verdier quotient^{1}

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

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