The Stacks project

Definition 64.7.1. Let $\mathcal{A}$ be an abelian category.

1. Let $\text{Fil}(\mathcal{A})$ be the category of filtered objects $(A, F)$ of $\mathcal{A}$, where $F$ is a filtration of the form

   \[ A \supset \ldots \supset F^ n A \supset F^{n+1}A \supset \ldots \supset 0. \]

   This is an additive category.

2. We denote $\text{Fil}^ f(\mathcal{A})$ the full subcategory of $\text{Fil}(\mathcal{A})$ whose objects $(A, F)$ have finite filtration. This is also an additive category.

3. An object $I \in \text{Fil}^ f(\mathcal{A})$ is called filtered injective (respectively projective) provided that $\text{gr}^ p(I) = \text{gr}_ F^ p(I) = F^ pI/F^{p+1}I$ is injective (resp. projective) in $\mathcal{A}$ for all $p$.

4. The category of complexes $\text{Comp}(\text{Fil}^ f(\mathcal{A})) \supset \text{Comp}^+(\text{Fil}^ f(\mathcal{A}))$ and its homotopy category $K(\text{Fil}^ f(\mathcal{A})) \supset K^+(\text{Fil}^ f(\mathcal A))$ are defined as before.

5. A morphism $\alpha : K^\bullet \to L^\bullet $ of complexes in $\text{Comp}(\text{Fil}^ f(\mathcal{A}))$ is called a filtered quasi-isomorphism provided that

   \[ \text{gr}^ p(\alpha ): \text{gr}^ p(K^\bullet ) \to \text{gr}^ p(L^\bullet ) \]

   is a quasi-isomorphism for all $p \in \mathbf{Z}$.

6. We define $DF(\mathcal{A})$ (resp. $DF^+(\mathcal{A})$) by inverting the filtered quasi-isomorphisms in $K(\text{Fil}^ f(\mathcal{A}))$ (resp. $K^+(\text{Fil}^ f(\mathcal{A}))$).