Lemma 13.13.9. Let \mathcal{A} be an abelian category. The subcategories \text{FAc}^{+}(\mathcal{A}), \text{FAc}^{-}(\mathcal{A}), resp. \text{FAc}^ b(\mathcal{A}) are strictly full saturated triangulated subcategories of K^{+}(\text{Fil}^ f\mathcal{A}), K^{-}(\text{Fil}^ f\mathcal{A}), resp. K^ b(\text{Fil}^ f\mathcal{A}). The corresponding saturated multiplicative systems (see Lemma 13.6.10) are the sets \text{FQis}^{+}(\mathcal{A}), \text{FQis}^{-}(\mathcal{A}), resp. \text{FQis}^ b(\mathcal{A}).
The kernel of the functor K^{+}(\text{Fil}^ f\mathcal{A}) \to DF^{+}(\mathcal{A}) is \text{FAc}^{+}(\mathcal{A}) and this induces an equivalence of triangulated categories
K^{+}(\text{Fil}^ f\mathcal{A})/\text{FAc}^{+}(\mathcal{A}) = \text{FQis}^{+}(\mathcal{A})^{-1}K^{+}(\text{Fil}^ f\mathcal{A}) \longrightarrow DF^{+}(\mathcal{A})The kernel of the functor K^{-}(\text{Fil}^ f\mathcal{A}) \to DF^{-}(\mathcal{A}) is \text{FAc}^{-}(\mathcal{A}) and this induces an equivalence of triangulated categories
K^{-}(\text{Fil}^ f\mathcal{A})/\text{FAc}^{-}(\mathcal{A}) = \text{FQis}^{-}(\mathcal{A})^{-1}K^{-}(\text{Fil}^ f\mathcal{A}) \longrightarrow DF^{-}(\mathcal{A})The kernel of the functor K^ b(\text{Fil}^ f\mathcal{A}) \to DF^ b(\mathcal{A}) is \text{FAc}^ b(\mathcal{A}) and this induces an equivalence of triangulated categories
K^ b(\text{Fil}^ f\mathcal{A})/\text{FAc}^ b(\mathcal{A}) = \text{FQis}^ b(\mathcal{A})^{-1}K^ b(\text{Fil}^ f\mathcal{A}) \longrightarrow DF^ b(\mathcal{A})
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