Spectral Sequences - Key Properties

Understanding convergence and edge maps is essential for using spectral sequences effectively.

Definition7.5Convergence

A spectral sequence $E_r^{p,q} \Rightarrow H^n$ converges strongly if:

For each $(p,q)$ , $E_r^{p,q}$ stabilizes: $E_r^{p,q} = E_\infty^{p,q}$ for $r \gg 0$
$H^n$ has a filtration $F^pH^n$ with $E_\infty^{p,q} \cong F^pH^{p+q} / F^{p+1}H^{p+q}$

Definition7.6Edge Homomorphisms

For a spectral sequence $E_2^{p,q} \Rightarrow H^{p+q}$ , the edge homomorphisms are:

Horizontal: $H^n \to E_2^{n,0}$ (when $E_2^{i,n-i} = 0$ for $i > 0$ )
Vertical: $H^n \to E_2^{0,n}$ (when $E_2^{i,n-i} = 0$ for $i < n$ )

These allow us to extract information even when the spectral sequence doesn't fully determine $H^n$ .

Theorem7.7Degeneration

A spectral sequence degenerates at $E_r$ if $d_s = 0$ for all $s \geq r$ . Common cases:

Degeneration at $E_2$ : $d_2 = d_3 = \cdots = 0$
Immediate degeneration: $E_1 = E_\infty$

When degeneration occurs, computations simplify dramatically.

ExampleKünneth Spectral Sequence

For chain complexes $C, D$ over a field: $E_2^{p,q} = \bigoplus_{i+j=q} H_p(C) \otimes H_i(D) \otimes H_j(D) \Rightarrow H_{p+q}(C \otimes D)$

Over a field, this degenerates at $E_2$ , giving the Künneth formula.

Definition7.8Multiplicative Structure

A spectral sequence has multiplicative structure if there are pairings: $E_r^{p,q} \otimes E_r^{p',q'} \to E_r^{p+p',q+q'}$

compatible with differentials and converging to the product on $H^*$ . Many geometric spectral sequences have this structure.

Theorem7.9Comparison Theorem

Given morphisms of spectral sequences $f_r: E_r \to E_r'$ compatible with differentials, if $f_s: E_s^{p,q} \to E_s'^{p,q}$ is an isomorphism for some $s$ , then $f_\infty: E_\infty \to E_\infty'$ is an isomorphism.

Remark

The comparison theorem allows us to deduce isomorphisms on limits from isomorphisms at any finite stage, a powerful tool for inductive arguments.