Editing Spectral sequence (section)

=== 2 non-zero adjacent columns ===
Let <math>E^r_{p, q}</math> be a homological spectral sequence such that <math>E^2_{p, q} = 0</math> for all ''p'' other than 0, 1. Visually, this is the spectral sequence with <math>E^2</math>-page
:<math>\begin{matrix}
& \vdots & \vdots & \vdots & \vdots & \\
\cdots & 0 & E^2_{0,2} & E^2_{1,2} & 0 & \cdots \\
\cdots & 0 & E^2_{0,1} & E^2_{1,1} & 0 & \cdots \\
\cdots & 0 & E^2_{0,0} & E^2_{1,0} & 0 & \cdots \\
\cdots & 0 & E^2_{0,-1} & E^2_{1,-1} & 0 & \cdots \\
& \vdots & \vdots & \vdots & \vdots & 
\end{matrix}</math>
The differentials on the second page have degree (-2, 1), so they are of the form
:<math>d^2_{p,q}:E^2_{p,q} \to E^2_{p-2,q+1}</math>
These maps are all zero since they are
:<math>d^2_{0,q}:E^2_{0,q} \to 0</math>, <math>d^2_{1,q}:E^2_{1,q} \to 0</math>
hence the spectral sequence degenerates: <math>E^{\infty} = E^2</math>. Say, it converges to <math>H_*</math> with a filtration
:<math>0 = F_{-1} H_n \subset F_0 H_n \subset \dots \subset F_n H_n = H_n</math>
such that <math>E^{\infty}_{p, q} = F_p H_{p+q}/F_{p-1} H_{p+q}</math>. Then <math>F_0 H_n = E^2_{0, n}</math>, <math>F_1 H_n / F_0 H_n = E^2_{1, n -1}</math>, <math>F_2 H_n / F_1 H_n = 0</math>, <math>F_3 H_n / F_2 H_n = 0</math>, etc. Thus, there is the exact sequence:<ref>{{harvnb|Weibel|1994|loc=Exercise 5.2.1.}}; there are typos in the exact sequence, at least in the 1994 edition.</ref>
:<math>0 \to E^2_{0, n} \to H_n \to E^2_{1, n - 1} \to 0</math>.
Next, let <math>E^r_{p, q}</math> be a spectral sequence whose second page consists only of two lines ''q'' = 0, 1. This need not degenerate at the second page but it still degenerates at the third page as the differentials there have degree (-3, 2). Note <math>E^3_{p, 0} = \operatorname{ker} (d: E^2_{p, 0} \to E^2_{p - 2, 1})</math>, as the denominator is zero. Similarly, <math>E^3_{p, 1} = \operatorname{coker}(d: E^2_{p+2, 0} \to E^2_{p, 1})</math>. Thus,
:<math>0 \to E^{\infty}_{p, 0} \to E^2_{p, 0} \overset{d}\to E^2_{p-2, 1} \to E^{\infty}_{p-2, 1} \to 0</math>.
Now, say, the spectral sequence converges to ''H'' with a filtration ''F'' as in the previous example. Since <math>F_{p-2} H_{p} / F_{p-3} H_{p} = E^{\infty}_{p-2, 2} = 0</math>, <math>F_{p-3} H_p / F_{p-4} H_p = 0</math>, etc., we have: <math>0 \to E^{\infty}_{p - 1, 1} \to H_p \to E^{\infty}_{p, 0} \to 0</math>. Putting everything together, one gets:<ref>{{harvnb|Weibel|1994|loc=Exercise 5.2.2.}}</ref>
:<math>\cdots \to H_{p+1} \to E^2_{p + 1, 0} \overset{d}\to E^2_{p - 1, 1} \to H_p \to E^2_{p, 0} \overset{d}\to E^2_{p - 2, 1} \to H_{p-1} \to \dots.</math>