Definition 104.5.1 (07B6)—The Stacks project

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Table of contents
Part 7: Algebraic Stacks
Chapter 104: Derived Categories of Stacks
Section 104.5: Derived categories of quasi-coherent modules
Definition 104.5.1 (cite)

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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 quotient ¹

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

[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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