The Stacks project

Definition 12.19.1. Let $\mathcal{A}$ be an abelian category.

1. A decreasing filtration $F$ on an object $A$ is a family $(F^ nA)_{n \in \mathbf{Z}}$ of subobjects of $A$ such that

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

2. A filtered object of $\mathcal{A}$ is pair $(A, F)$ consisting of an object $A$ of $\mathcal{A}$ and a decreasing filtration $F$ on $A$.

3. A morphism $(A, F) \to (B, F)$ of filtered objects is given by a morphism $\varphi : A \to B$ of $\mathcal{A}$ such that $\varphi (F^ iA) \subset F^ iB$ for all $i \in \mathbf{Z}$.

4. The category of filtered objects is denoted $\text{Fil}(\mathcal{A})$.

5. Given a filtered object $(A, F)$ and a subobject $X \subset A$ the induced filtration on $X$ is the filtration with $F^ nX = X \cap F^ nA$.

6. Given a filtered object $(A, F)$ and a surjection $\pi : A \to Y$ the quotient filtration is the filtration with $F^ nY = \pi (F^ nA)$.

7. A filtration $F$ on an object $A$ is said to be finite if there exist $n, m$ such that $F^ nA = A$ and $F^ mA = 0$.

8. Given a filtered object $(A, F)$ we say $\bigcap F^ iA$ exists if there exists a biggest subobject of $A$ contained in all $F^ iA$. We say $\bigcup F^ iA$ exists if there exists a smallest subobject of $A$ containing all $F^ iA$.

9. The filtration on a filtered object $(A, F)$ is said to be separated if $\bigcap F^ iA = 0$ and exhaustive if $\bigcup F^ iA = A$.