Spectral sequence

Remarks

A spectral sequence is first quadrant if $E_a^{p,q} = 0$ unless $p \ge 0$ and $q \ge 0$. Then for fixed $(p, q)$ the differentials into and out of $E_r^{p,q}$ vanish for $r$ large, so $E_r^{p,q}$ is eventually constant; its value is $E_\infty^{p,q}$. The sequence converges to objects $H^n$, written $E_a^{p,q} \Rightarrow H^{p+q}$, if each $H^n$ has a finite filtration $H^n = F^0 H^n \supseteq F^1 H^n \supseteq \cdots \supseteq F^{n+1} H^n = 0$ with $F^p H^n / F^{p+1} H^n \cong E_\infty^{p, n-p}$. Example: if $E_2^{p,q} = 0$ for $q \ne 0$, then every $d_r$ with $r \ge 2$ starts or ends at a zero object, because it changes $q$ by $1 - r \ne 0$; so $E_2 = E_\infty$ and $H^n \cong E_2^{n,0}$. Non-example: let $E_2^{0,1} = E_2^{2,0} = \Z/2\Z$, all other $E_2^{p,q} = 0$, let $d_2^{0,1}$ be the identity of $\Z/2\Z$, and put $E_3 = E_2$. Clause 3 fails, since the homology of $d_2$ is zero at $(0,1)$ and at $(2,0)$, while $E_3^{0,1} = \Z/2\Z$. Homological spectral sequences are defined in the same way with $d^r \colon E^r_{p,q} \to E^r_{p-r, q+r-1}$.