Editing Spectral sequence (section)

=== 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.