nLab conditional convergence of spectral sequences

This entry is about conditional convergence of spectral sequences in homological algebra and stable homotopy theory. For conditional convergence of series in real analysis and functional analysis, see conditional convergence.

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

Definition

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

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

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

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

2. $\underset{\longleftarrow}{\lim}^1_s \mathcal{D}^{s,\bullet} = 0$ (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 $\{E_r^{\bullet,\bullet}, d_r\}_r$ as above, that converges conditionally, then sufficient condition that it converges strongly (def.) is that also the lim^1 over its pages vanishes:

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

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

References

Due to

• Michael Boardman, Conditionally convergent spectral sequences, 1999 (pdf)

Lecture notes include

• John Rognes, p 44 of The Adams spectral sequence, 2012 (pdf)