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Tag 0121

Chapter 12: Homological Algebra > Section 12.16: Filtrations

Definition 12.16.1. 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 $\bigcup F^iA = A$.

    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}

    Comments (2)

    Comment #2610 by Ko Aoki on June 24, 2017 a 7:14 am UTC

    Typo in the definition of a separated filtration: "$\bigcap_i F^iA = 0 $" should be replaced by "$\bigcap F^iA = 0$" for consistency of notation.

    Comment #2633 by Johan (site) on July 7, 2017 a 12:52 pm UTC

    Thanks, fixed here.

    There are also 2 comments on Section 12.16: Homological Algebra.

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