Editing Spectral sequence (section)

== Formal definition ==
=== Cohomological spectral sequence ===
Fix an [[abelian category]], such as a category of [[module (mathematics)|module]]s over a [[ring (mathematics)|ring]], and a nonnegative integer <math>r_0</math>. A '''cohomological spectral sequence''' is a sequence <math> \{E_r, d_r\}_{r\geq r_0} </math> of objects <math> E_r </math> and endomorphisms <math> d_r : E_r  \to E_r </math>, such that for every <math> r\geq r_0 </math>
# <math> d_r \circ d_r = 0 </math>,
# <math> E_{r+1}  \cong H_{*}(E_r, d_r) </math>, the [[Homology (mathematics)|homology]] of <math>E_r</math> with respect to <math>d_r</math>.
Usually the isomorphisms are suppressed and we write <math> E_{r+1} = H_{*}(E_r, d_r) </math> instead. An object <math> E_r </math> is called  ''sheet'' (as in a sheet of [[paper]]), or sometimes a ''page'' or a ''term''; an endomorphism <math> d_r </math> is called ''boundary map'' or ''differential''. Sometimes <math>E_{r+1}</math> is called the ''derived object'' of <math>E_r</math>.{{citation needed|date=June 2015}}

=== Bigraded spectral sequence === 
In reality spectral sequences mostly occur in the category of doubly graded [[module (mathematics)|module]]s over a [[ring (mathematics)|ring]] ''R'' (or doubly graded [[sheaf (mathematics)|sheaves]] of modules over a sheaf of rings), i.e. every sheet is a bigraded R-module <math display="inline"> E_r =  \bigoplus_{p,q \in \mathbb{Z}^2} E_r^{p,q}. </math>
So in this case a cohomological spectral sequence is a sequence <math> \{E_r, d_r\}_{r\geq r_0} </math> of bigraded R-modules <math> \{E_r^{p,q}\}_{p,q} </math> and for every module the direct sum of endomorphisms <math> d_r = (d_r^{p,q} : E_r^{p,q}  \to E_r^{p+r,q-r+1})_{p,q \in \mathbb{Z}^2} </math> of bidegree <math> (r,1-r) </math>, such that for every <math> r\geq r_0 </math> it holds that:
# <math> d_r^{p+r,q-r+1} \circ d_r^{p,q} = 0 </math>,
# <math> E_{r+1}  \cong H_{*}(E_r, d_r) </math>.
The notation used here is called  ''complementary degree''. Some authors write  <math> E_r^{d,q} </math> instead, where <math> d = p + q </math> is the ''total degree''. Depending upon the spectral sequence, the boundary map on the first sheet can have a degree which corresponds to ''r'' = 0, ''r'' = 1, or ''r'' = 2. For example, for the spectral sequence of a filtered complex, described below, ''r''<sub>0</sub> = 0, but for the [[Grothendieck spectral sequence]], ''r''<sub>0</sub> = 2. Usually ''r''<sub>0</sub> is zero, one, or two. In the ungraded situation described above, ''r''<sub>0</sub> is irrelevant.

=== 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>.