Definition 13.13.7. Let $\mathcal{A}$ be an abelian category. The bounded filtered derived category $DF^ b(\mathcal{A})$ is the full subcategory of $DF(\mathcal{A})$ with objects those $X$ such that $\text{gr}(X) \in D^ b(\mathcal{A})$. Similarly for the bounded below filtered derived category $DF^{+}(\mathcal{A})$ and the bounded above filtered derived category $DF^{-}(\mathcal{A})$.
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