Definition 13.13.5. Let $\mathcal{A}$ be an abelian category. Let $\text{FAc}(\mathcal{A})$ and $\text{FQis}(\mathcal{A})$ be as in Lemma 13.13.4. The *filtered derived category of $\mathcal{A}$* is the triangulated category

\[ DF(\mathcal{A}) = K(\text{Fil}^ f(\mathcal{A}))/\text{FAc}(\mathcal{A}) = \text{FQis}(\mathcal{A})^{-1} K(\text{Fil}^ f(\mathcal{A})). \]

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