Spectral Sequences - Main Theorem

The convergence theorem establishes when spectral sequences provide complete information about their targets.

Theorem7.11Convergence Theorem for First Quadrant

Let $E_r^{p,q}$ be a first quadrant spectral sequence (i.e., $E_r^{p,q} = 0$ for $p < 0$ or $q < 0$ ). Then:

For each $(p,q)$ , $E_r^{p,q}$ stabilizes at some $r_0 = r_0(p,q)$
The spectral sequence converges to a graded module $H^*$ with filtration
$E_\infty^{p,q} \cong \text{gr}^p H^{p+q} = F^p H^{p+q} / F^{p+1} H^{p+q}$

Moreover, if the filtration is finite at each degree, then $H^n \cong \bigoplus_{p+q=n} E_\infty^{p,q}$ .

Proof

For first quadrant spectral sequences, $E_r^{p,q} = 0$ unless both $p, q \geq 0$ . The differential $d_r: E_r^{p,q} \to E_r^{p+r, q-r+1}$ requires $q - r + 1 \geq 0$ , so $d_r = 0$ for $r > q + 1$ .

Thus $E_r^{p,q}$ stabilizes for $r > q + 1$ , giving convergence. The filtration and associated graded follow from the construction via filtered complexes.

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Theorem7.12Zeeman Comparison Theorem

Let $f: E_r \to E_r'$ be a morphism of spectral sequences. If $f_s: E_s^{p,q} \to E_s'^{p,q}$ is an isomorphism for some $s$ and all $p, q$ , then: $f_\infty: E_\infty^{p,q} \to E_\infty'^{p,q}$ is an isomorphism for all $p, q$ .

Remark

The Zeeman Comparison Theorem is powerful because it allows us to conclude isomorphisms at the $E_\infty$ page from isomorphisms at any finite stage, often $E_2$ .

Theorem7.13Spectral Sequence of Exact Couple

An exact couple consists of bigraded modules $D, E$ with maps: $D \xrightarrow{i} D \xrightarrow{j} E \xrightarrow{k} D$

where the triangle is exact at each term. Every exact couple gives rise to a spectral sequence with:

$E_1 = E$ with differential $d_1 = j \circ k$
$D_1 = i(D)$ derived from $D$

This provides a general framework for constructing spectral sequences.

ExampleBockstein Spectral Sequence

For a space $X$ and coefficient sequence $0 \to \mathbb{Z} \xrightarrow{\times p} \mathbb{Z} \to \mathbb{Z}/p\mathbb{Z} \to 0$ : $E_1^{n} = H^n(X; \mathbb{Z}/p\mathbb{Z}) \Rightarrow H^n(X; \mathbb{Z})_{(p)}$

where $(p)$ denotes $p$ -localization. The differential $d_1$ is the Bockstein homomorphism.