Editing Spectral sequence (section)

=== Low-degree terms ===
With an obvious notational change, the type of the computations in the previous examples can also be carried out for cohomological spectral sequence. Let <math>E_r^{p, q}</math> be a first-quadrant spectral sequence converging to ''H'' with the decreasing filtration
:<math>0 = F^{n+1} H^n \subset F^n H^n \subset \dots \subset F^0 H^n = H^n</math>
so that <math>E_{\infty}^{p,q} = F^p H^{p+q}/F^{p+1} H^{p+q}.</math>
Since <math>E_2^{p, q}</math> is zero if ''p'' or ''q'' is negative, we have:
:<math>0 \to E^{0, 1}_{\infty} \to E^{0, 1}_2 \overset{d}\to E^{2, 0}_2 \to E^{2, 0}_{\infty} \to 0.</math>
Since <math>E_{\infty}^{1, 0} = E_2^{1, 0}</math> for the same reason and since <math>F^2 H^1 = 0,</math>
:<math>0 \to E_2^{1, 0} \to H^1 \to E^{0, 1}_{\infty} \to 0</math>.
Since <math>F^3 H^2 = 0</math>, <math>E^{2, 0}_{\infty} \subset H^2</math>. Stacking the sequences together, we get the so-called [[five-term exact sequence]]:
:<math>0 \to E^{1, 0}_2 \to H^1 \to E^{0, 1}_2 \overset{d}\to E^{2, 0}_2 \to H^2.</math>