filtered object




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.)


A decreasing filtration {X s} s\{X_s\}_s of XX (def. 1) 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)


Beware that for decreasing filtrations (def. 1) often exhaustiveness (def. 2) is understood by default, and often it is even assumed by default that X s=XX_s = X for s0s \leq 0.



If a decreasing filtration (def. 1) 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. 2) 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)



Revised on May 6, 2016 16:45:30 by Todd Trimble (