Editing Spectral sequence (section)

== Visualization ==
[[Image:SpectralSequence.png|frame|The E<sub>2</sub> sheet of a cohomological spectral sequence]]
A doubly graded spectral sequence has a tremendous amount of data to keep track of, but there is a common visualization technique which makes the structure of the spectral sequence clearer. We have three indices, ''r'', ''p'', and ''q''. An object <math>E_r</math> can be seen as the {{mvar|r}}{{sup|th}} checkered page of a book. On these sheets, we will take ''p'' to be the horizontal direction and ''q'' to be the vertical direction. At each lattice point we have the object <math>E_r^{p,q}</math>. Now turning to the next page means taking homology, that is the <math>(r+1)</math>{{sup|th}} page is a subquotient of the {{mvar|r}}{{sup|th}} page. The total degree ''n'' = ''p'' + ''q'' runs diagonally, northwest to southeast, across each sheet. In the homological case, the differentials have bidegree {{math|(&minus;''r'', ''r'' &minus; 1)}}, so they decrease ''n'' by one. In the cohomological case, ''n'' is increased by one. The differentials change their direction with each turn with respect to r.
[[File:Spectral Sequence Visualization.jpg|thumb|center|upright=3.0|Four pages of a cohomological spectral sequence]]
The red arrows demonstrate the case of a first quadrant sequence (see example [[spectral sequence#First-quadrant sheet|below]]), where only the objects of the first quadrant are non-zero. While turning pages, either the domain or the codomain of all the differentials become zero.