Editing Spectral sequence (section)

=== Homological spectral sequence ===
Mostly the objects we are talking about are [[chain complex|chain complexes]], that occur with descending (like above) or ascending order. In the latter case, by replacing <math> E_r^{p,q} </math> with <math> E^r_{p,q} </math> and <math> d_r^{p,q} : E_r^{p,q}  \to E_r^{p+r,q-r+1} </math> with <math> d^r_{p,q} : E^r_{p,q}  \to E^r_{p-r,q+r-1} </math> (bidegree <math> (-r,r-1) </math>), one receives the definition of a '''homological spectral sequence''' analogously to the cohomological case.

==== Spectral sequence from a chain complex ====
The most elementary example in the ungraded situation is a [[chain complex]] ''C''<sub>•</sub>. An object ''C''<sub>•</sub> in an abelian category of chain complexes naturally comes with a differential ''d''.  Let ''r''<sub>0</sub> = 0, and let ''E''<sub>0</sub> be ''C''<sub>•</sub>. This forces ''E''<sub>1</sub> to be the complex ''H''(''C''<sub>•</sub>): At the {{prime|''i''}}th location this is the {{prime|''i''}}th homology group of ''C''<sub>•</sub>. The only natural differential on this new complex is the zero map, so we let ''d''<sub>1</sub> = 0. This forces <math>E_2</math> to equal <math>E_1</math>, and again our only natural differential is the zero map. Putting the zero differential on all the rest of our sheets gives a spectral sequence whose terms are:

* ''E''<sub>0</sub> = ''C''<sub>•</sub>
* ''E<sub>r</sub>'' = ''H''(''C''<sub>•</sub>) for all ''r'' ≥ 1.

The terms of this spectral sequence stabilize at the first sheet because its only nontrivial differential was on the zeroth sheet. Consequently, we can get no more information at later steps. Usually, to get useful information from later sheets, we need extra structure on the <math>E_r</math>.