Definition 13.11.3. Let $\mathcal{A}$ be an abelian category. Let $\text{Ac}(\mathcal{A})$ and $\text{Qis}(\mathcal{A})$ be as in Lemma 13.11.2. The derived category of $\mathcal{A}$ is the triangulated category

$D(\mathcal{A}) = K(\mathcal{A})/\text{Ac}(\mathcal{A}) = \text{Qis}(\mathcal{A})^{-1} K(\mathcal{A}).$

We denote $H^0 : D(\mathcal{A}) \to \mathcal{A}$ the unique functor whose composition with the quotient functor gives back the functor $H^0$ defined above. Using Lemma 13.6.4 we introduce the strictly full saturated triangulated subcategories $D^{+}(\mathcal{A}), D^{-}(\mathcal{A}), D^ b(\mathcal{A})$ whose sets of objects are

$\begin{matrix} \mathop{\mathrm{Ob}}\nolimits (D^{+}(\mathcal{A})) = \{ X \in \mathop{\mathrm{Ob}}\nolimits (D(\mathcal{A})) \mid H^ n(X) = 0\text{ for all }n \ll 0\} \\ \mathop{\mathrm{Ob}}\nolimits (D^{-}(\mathcal{A})) = \{ X \in \mathop{\mathrm{Ob}}\nolimits (D(\mathcal{A})) \mid H^ n(X) = 0\text{ for all }n \gg 0\} \\ \mathop{\mathrm{Ob}}\nolimits (D^ b(\mathcal{A})) = \{ X \in \mathop{\mathrm{Ob}}\nolimits (D(\mathcal{A})) \mid H^ n(X) = 0\text{ for all }|n| \gg 0\} \end{matrix}$

The category $D^ b(\mathcal{A})$ is called the bounded derived category of $\mathcal{A}$.

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