## Tag `0121`

Chapter 12: Homological Algebra > Section 12.16: Filtrations

Definition 12.16.1. Let $\mathcal{A}$ be an abelian category.

- A
decreasing filtration$F$ on an object $A$ is a family $(F^nA)_{n \in \mathbf{Z}}$ of subobjects of $A$ such that $$ A \supset \ldots \supset F^nA \supset F^{n + 1}A \supset \ldots \supset 0 $$- A
filtered object of $\mathcal{A}$is pair $(A, F)$ consisting of an object $A$ of $\mathcal{A}$ and a decreasing filtration $F$ on $A$.- A
morphism $(A, F) \to (B, F)$ of filtered objectsis given by a morphism $\varphi : A \to B$ of $\mathcal{A}$ such that $\varphi(F^iA) \subset F^iB$ for all $i \in \mathbf{Z}$.- The category of filtered objects is denoted $\text{Fil}(\mathcal{A})$.
- Given a filtered object $(A, F)$ and a subobject $X \subset A$ the
induced filtrationon $X$ is the filtration with $F^nX = X \cap F^nA$.- Given a filtered object $(A, F)$ and a surjection $\pi : A \to Y$ the
quotient filtrationis the filtration with $F^nY = \pi(F^nA)$.- A filtration $F$ on an object $A$ is said to be
finiteif there exist $n, m$ such that $F^nA = A$ and $F^mA = 0$.- 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 $F^iA$.
- The filtration on a filtered object $(A, F)$ is said to be
separatedif $\bigcap F^iA = 0$ andexhaustiveif $\bigcup F^iA = A$.

The code snippet corresponding to this tag is a part of the file `homology.tex` and is located in lines 3555–3586 (see updates for more information).

```
\begin{definition}
\label{definition-filtered}
Let $\mathcal{A}$ be an abelian category.
\begin{enumerate}
\item A {\it decreasing filtration} $F$ on an object $A$
is a family $(F^nA)_{n \in \mathbf{Z}}$ of subobjects of $A$ such that
$$
A \supset \ldots \supset F^nA \supset F^{n + 1}A \supset \ldots \supset 0
$$
\item A {\it filtered object of $\mathcal{A}$} is
pair $(A, F)$ consisting of an object $A$ of $\mathcal{A}$
and a decreasing filtration $F$ on $A$.
\item A {\it 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 $i \in \mathbf{Z}$.
\item The category of filtered objects is denoted $\text{Fil}(\mathcal{A})$.
\item Given a filtered object $(A, F)$ and a subobject $X \subset A$ the
{\it induced filtration} on $X$ is the filtration with $F^nX = X \cap F^nA$.
\item Given a filtered object $(A, F)$ and a surjection
$\pi : A \to Y$ the {\it quotient filtration} is the filtration with
$F^nY = \pi(F^nA)$.
\item A filtration $F$ on an object $A$ is said to be {\it finite}
if there exist $n, m$ such that $F^nA = A$ and $F^mA = 0$.
\item 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 $F^iA$.
\item The filtration on a filtered object $(A, F)$ is said to be
{\it separated} if $\bigcap F^iA = 0$ and
{\it exhaustive} if $\bigcup F^iA = A$.
\end{enumerate}
\end{definition}
```

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