The Stacks project

Definition 104.5.1. 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 quotient1

\[ D_\mathit{QCoh}(\mathcal{O}_\mathcal {X}) = D_{\textit{LQCoh}^{fbc}}(\mathcal{O}_\mathcal {X})/ D_{\textit{Parasitic} \cap \textit{LQCoh}^{fbc}}(\mathcal{O}_\mathcal {X}) \]
[1] This definition is different from the one in the literature, see [6.3, olsson_sheaves], but it agrees with that definition by Lemma 104.5.3.

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View Definition 104.5.1 as pdf