Editing Homological algebra (section)

===Spectral sequence===
{{Main|Spectral sequence}}
Fix an [[abelian category]], such as a category of modules over a ring.  A '''spectral sequence''' is a choice of a nonnegative integer ''r''<sub>0</sub> and a collection of three sequences:
# For all integers ''r'' ≥ ''r''<sub>0</sub>, an object ''E<sub>r</sub>'', called a ''sheet'' (as in a sheet of [[paper]]), or sometimes a ''page'' or a ''term'',
# Endomorphisms ''d<sub>r</sub>'' : ''E<sub>r</sub>'' → ''E<sub>r</sub>'' satisfying ''d<sub>r</sub>'' <small>o</small> ''d<sub>r</sub>'' = 0, called ''boundary maps'' or ''differentials'',
# Isomorphisms of ''E<sub>r+1</sub>'' with ''H''(''E<sub>r</sub>''), the homology of ''E<sub>r</sub>'' with respect to ''d<sub>r</sub>''.

[[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''.  For each ''r'', imagine that we have a sheet of graph paper.  On this sheet, 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>.

It is very common for ''n'' = ''p'' + ''q'' to be another natural index in the spectral sequence. ''n'' runs diagonally, northwest to southeast, across each sheet.  In the homological case, the differentials have bidegree (&minus;''r'',&nbsp;''r''&nbsp;&minus;&nbsp;1), so they decrease ''n'' by one.  In the cohomological case, ''n'' is increased by one.  When ''r'' is zero, the differential moves objects one space down or up.  This is similar to the differential on a chain complex.  When ''r'' is one, the differential moves objects one space to the left or right.  When ''r'' is two, the differential moves objects just like a [[knight (chess)|knight]]'s move in [[chess]].  For higher ''r'', the differential acts like a generalized knight's move.