Editing Spectral sequence (section)

=== Terms of convergence ===
We say a spectral sequence '''converges weakly''' if there is a graded object <math> H^{\bullet} </math> with a filtration <math> F^{\bullet} H^{n} </math> for every <math> n </math>, and for every <math> p </math> there exists an isomorphism <math> E_{\infty}^{p,q} \cong F^pH^{p+q}/F^{p+1}H^{p+q} </math>.  It '''converges''' to <math> H^{\bullet} </math> if the filtration <math> F^{\bullet} H^{n} </math> is Hausdorff, i.e. <math> \cap_{p}F^pH^{\bullet}=0 </math>. We write
:<math>E_r^{p,q} \Rightarrow_p E_\infty^n</math>

to mean that whenever ''p'' + ''q'' = ''n'', <math>E_r^{p,q}</math> converges to <math>E_\infty^{p,q}</math>.
We say that a spectral sequence <math>E_r^{p,q}</math> '''abuts to''' <math>E_\infty^{p,q}</math> (the '''spectral sequence abutment'''{{Anchor|Abutment}}) if for every <math> p,q </math> there is <math> r(p,q) </math> such that for all <math>r \geq r(p,q)</math>, <math>E_r^{p,q} = E_{r(p,q)}^{p,q}</math>. Then <math>E_{r(p,q)}^{p,q} = E_\infty^{p,q}</math> is the limiting term. The spectral sequence is '''regular''' or '''degenerates at <math> r_0 </math> '''if the differentials <math>d_r^{p,q}</math> are zero for all <math> r \geq r_0 </math>. If in particular there is <math> r_0 \geq 2 </math>, such that the <math> r_0^{th} </math> sheet is concentrated on a single row or a single column, then we say it '''collapses'''. In symbols, we write:

:<math>E_r^{p,q} \Rightarrow_p E_\infty^{p,q}</math>

The ''p'' indicates the filtration index. It is very common to write the <math>E_2^{p,q}</math> term on the left-hand side of the abutment, because this is the most useful term of most spectral sequences. The spectral sequence of an unfiltered chain complex degenerates at the first sheet (see first example): since nothing happens after the zeroth sheet, the limiting sheet <math> E_{\infty} </math> is the same as <math> E_1 </math>.

The [[five-term exact sequence]] of a spectral sequence relates certain low-degree terms and ''E''<sub>∞</sub> terms.