nLab
filtered object

Contents

Contents

Definition

Definition

Given a category 𝒞\mathcal{C}, then a filtered object is an object XX of 𝒞\mathcal{C} equipped with a filtration:

A descending filtration or decreasing filtrations of XX is a sequence of morphisms (often required to be monomorphisms) of the form

X n+1X nX n1X \cdots \longrightarrow X_{n+1} \longrightarrow X_n \longrightarrow X_{n-1} \longrightarrow \cdots \longrightarrow X

An ascending filtration or increasing filtration of XX is of the form

XX n1X nX n+1 X \longrightarrow \cdots \longrightarrow X_{n-1} \longrightarrow X_n \longrightarrow X_{n+1} \longrightarrow \cdots

(In more generality, it is also possible to index using any ordered abelian group.)

Definition

A decreasing filtration {X s} s\{X_s\}_s of XX (def. ) is called

  • exhaustive if lim sX sX\underset{\longrightarrow}{\lim}_s X_s \simeq X (XX is the colimit) of the filter stages)

  • Hausdorff if lim sX s0\underset{\longleftarrow}{\lim}_s X_s \simeq 0 (the limit of the the filter stages is initial (zero in the case of an abelian category))

and for a filtration of abelian groups:

  • complete if lim s 1X n=0\underset{\longleftarrow}{\lim}^1_s X_n = 0 (also the first derived limit (lim^1) vanishes)

(Boardman 99, def. 2.1, see also Rognes 12, section 2.1)

Remark

Beware that for decreasing filtrations (def. ) often exhaustiveness (def. ) is understood by default, and often it is even assumed by default that X s=XX_s = X for s0s \leq 0.

Properties

Proposition

If a decreasing filtration (def. ) of abelian subgroups

A n+1A nA n1A \cdots \hookrightarrow A_{n+1} \hookrightarrow A_n \hookrightarrow A_{n-1} \hookrightarrow \cdots \hookrightarrow A

is exhaustive and complete Hausdorff (def. ) then AA may be reobtained from the subquotients of the filtering as the limit/colimit

A lim s(A/A s) lim s(lim t(A t/A s)). \begin{aligned} A & \simeq \underset{\longleftarrow}{\lim}_s (A/A_s) \\ & \simeq \underset{\longleftarrow}{\lim}_s (\underset{\longrightarrow}{\lim}_t ( A_t / A_s )) \end{aligned} \,.

(Boardman 99, prop. 2.5)

Examples

References

Last revised on September 29, 2018 at 15:36:02. See the history of this page for a list of all contributions to it.