nLab
conditional convergence

Contents

Context

Homological algebra

homological algebra

(also nonabelian homological algebra)

Introduction

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Homology theories

Theorems

Stable Homotopy theory

Contents

Idea

In the context of algebraic topology, conditional convergence refers to a more general kind of convergence of spectral sequences than actual (“strong”) convergence (def.). It is the right concept of convergence for spectral sequences that are not concentrated in the first or third quadrant.

Notably an EE-Adams spectral sequence is in general only conditionally convergent (to the EE-nilpotent completion), unless finiteness conditions are imposed

Definition

Given a spectral sequence induced from an unrolled exact couple of the form

𝒟 3, i 2 𝒟 2, i 1 𝒟 1, i 0 𝒟 0, k 2 j 2 k 1 j 1 k 0 j 0 3, 2, 1, 0, \array{ \cdots &\stackrel{}{\longrightarrow}& \mathcal{D}^{3,\bullet} &\stackrel{i_2}{\longrightarrow}& \mathcal{D}^{2,\bullet} &\stackrel{i_1}{\longrightarrow} & \mathcal{D}^{1,\bullet} &\stackrel{i_0}{\longrightarrow}& \mathcal{D}^{0,\bullet} \\ && \downarrow^{\mathrlap{}} &{}_{\mathllap{k_2}}\nwarrow & {}^{\mathllap{j_2}}\downarrow &{}_{\mathllap{k_1}}\nwarrow & {}^{\mathllap{j_1}}\downarrow &{}_{\mathllap{k_0}}\nwarrow & {}_{\mathllap{j_0}}\downarrow \\ && \mathcal{E}^{3,\bullet} && \mathcal{E}^{2,\bullet} && \mathcal{E}^{1,\bullet} && \mathcal{E}^{0,\bullet} }

then it is said to converge conditionally to 𝒟 0,\mathcal{D}^{0,\bullet} if, when regarding {𝒟 s,}\{\mathcal{D}^{s,\bullet}\} as a filtering for 𝒟 0,\mathcal{D}^{0,\bullet} it is exhaustive and complete, in that:

  1. lim s𝒟 s,=0\underset{\longleftarrow}{\lim}_s \mathcal{D}^{s,\bullet} = 0 (the limit over the filtering vanishes);

  2. lim s 1𝒟 s,=0\underset{\longleftarrow}{\lim}^1_s \mathcal{D}^{s,\bullet} = 0 (the lim^1 over the filtering vanishes).

(Boardman 99, def. 5.10, see also Rognes 12, def. 2.20)

The point is that: Given a spectral sequence {E r ,,d r} r\{E_r^{\bullet,\bullet}, d_r\}_r as above, that converges conditionally, then sufficient condition that it converges strongly (def.) is that also the lim^1 over its pages vanishes:

lim r 1E r=0 \underset{\longleftarrow}{\lim}^1_r E_r = 0

(Boardman 99, theorem 7.3, see also Rognes 12, theorem 2.24)

References

Due to

Lecture notes include

Last revised on May 6, 2016 at 12:37:18. See the history of this page for a list of all contributions to it.