Editing Spectral sequence (section)

== Convergence, degeneration, and abutment ==
=== Interpretation as a filtration of cycles and boundaries ===
Let ''E''<sub>''r''</sub> be a spectral sequence, starting with say ''r'' = 1. Then there is a sequence of subobjects
:<math>0 = B_0 \subset B_1 \subset B_{2} \subset \dots \subset B_r \subset \dots \subset Z_r \subset \dots \subset Z_2 \subset Z_1 \subset Z_0 = E_1</math>
such that <math>E_r \simeq Z_{r-1}/B_{r-1}</math>; indeed, recursively we let <math>Z_0 = E_1, B_0 = 0</math> and let <math>Z_r, B_r</math> be so that  <math>Z_r/B_{r-1}, B_r/B_{r-1}</math> are the kernel and the image of <math>E_r \overset{d_r}\to E_r.</math>

We then  let <math>Z_{\infty} = \cap_r Z_r, B_{\infty} = \cup_r B_r</math> and
:<math>E_{\infty} = Z_{\infty}/B_{\infty}</math>;
it is called the '''limiting term'''. (Of course, such <math>E_{\infty}</math> need not exist in the category, but this is usually a non-issue since for example in the category of modules such limits exist or since in practice a spectral sequence one works with tends to degenerate; there are only finitely many inclusions in the sequence above.)

=== 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.