Editing Spectral sequence (section)

=== 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.)