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

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

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

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