Editing Spectral sequence (section)

==== Construction ====
<math> C^{\bullet} </math> has only a single grading and a filtration, so we first construct a doubly graded object for the first page of the spectral sequence. To get the second grading, we will take the associated graded object with respect to the filtration.  We will write it in an unusual way which will be justified at the <math> E_1 </math> step:

:<math>Z_{-1}^{p,q} = Z_0^{p,q} = F^p C^{p+q}</math>
:<math>B_0^{p,q} = 0</math>
:<math>E_0^{p,q} = \frac{Z_0^{p,q}}{B_0^{p,q} + Z_{-1}^{p+1,q-1}} = \frac{F^p C^{p+q}}{F^{p+1} C^{p+q}}</math>
:<math>E_0 = \bigoplus_{p,q\in\mathbf{Z}} E_0^{p,q}</math>

Since we assumed that the boundary map was compatible with the filtration, <math> E_0 </math> is a doubly graded object and there is a natural doubly graded boundary map <math> d_0 </math> on <math> E_0 </math>.  To get <math> E_1 </math>, we take the homology of <math> E_0 </math>.

:<math>\bar{Z}_1^{p,q} = \ker d_0^{p,q} : E_0^{p,q} \rightarrow E_0^{p,q+1} = \ker d_0^{p,q} : F^p C^{p+q}/F^{p+1} C^{p+q} \rightarrow F^p C^{p+q+1}/F^{p+1} C^{p+q+1}</math>
:<math>\bar{B}_1^{p,q} = \mbox{im } d_0^{p,q-1} : E_0^{p,q-1} \rightarrow E_0^{p,q} = \mbox{im } d_0^{p,q-1} : F^p C^{p+q-1}/F^{p+1} C^{p+q-1} \rightarrow F^p C^{p+q}/F^{p+1} C^{p+q}</math>
:<math>E_1^{p,q} = \frac{\bar{Z}_1^{p,q}}{\bar{B}_1^{p,q}} = \frac{\ker d_0^{p,q} : E_0^{p,q} \rightarrow E_0^{p,q+1}}{\mbox{im } d_0^{p,q-1} : E_0^{p,q-1} \rightarrow E_0^{p,q}}</math>
:<math>E_1 = \bigoplus_{p,q\in\mathbf{Z}} E_1^{p,q} = \bigoplus_{p,q\in\mathbf{Z}} \frac{\bar{Z}_1^{p,q}}{\bar{B}_1^{p,q}}</math>

Notice that <math>\bar{Z}_1^{p,q}</math> and <math>\bar{B}_1^{p,q}</math> can be written as the images in <math>E_0^{p,q}</math> of
 
:<math>Z_1^{p,q} = \ker d_0^{p,q} : F^p C^{p+q} \rightarrow C^{p+q+1}/F^{p+1} C^{p+q+1}</math>
:<math>B_1^{p,q} = (\mbox{im } d_0^{p,q-1} : F^p C^{p+q-1} \rightarrow C^{p+q}) \cap F^p C^{p+q}</math>

and that we then have

:<math>E_1^{p,q} = \frac{Z_1^{p,q}}{B_1^{p,q} + Z_0^{p+1,q-1}}.</math>

<math>Z_1^{p,q}</math> are exactly the elements which the differential pushes up one level in the filtration, and <math>B_1^{p,q}</math> are exactly the image of the elements which the differential pushes up zero levels in the filtration.  This suggests that we should choose <math>Z_r^{p,q}</math> to be the elements which the differential pushes up ''r'' levels in the filtration and <math>B_r^{p,q}</math> to be image of the elements which the differential pushes up ''r-1'' levels in the filtration. In other words, the spectral sequence should satisfy

:<math>Z_r^{p,q} = \ker d_0^{p,q} : F^p C^{p+q} \rightarrow C^{p+q+1}/F^{p+r} C^{p+q+1}</math>
:<math>B_r^{p,q} = (\mbox{im } d_0^{p-r+1,q+r-2} : F^{p-r+1} C^{p+q-1} \rightarrow C^{p+q}) \cap F^p C^{p+q}</math>
:<math>E_r^{p,q} = \frac{Z_r^{p,q}}{B_r^{p,q} + Z_{r-1}^{p+1,q-1}}</math>

and we should have the relationship

:<math>B_r^{p,q} = d_0^{p,q}(Z_{r-1}^{p-r+1,q+r-2}).</math>

For this to make sense, we must find a differential <math> d_r </math> on each <math> E_r </math> and verify that it leads to homology isomorphic to <math> E_{r+1} </math>.  The differential 
 
:<math>d_r^{p,q} : E_r^{p,q} \rightarrow E_r^{p+r,q-r+1}</math>

is defined by restricting the original differential <math> d </math> defined on <math>C^{p+q}</math> to the subobject <math>Z_r^{p,q}</math>. It is straightforward to check that the homology of <math> E_r </math> with respect to this differential is <math> E_{r+1} </math>, so this gives a spectral sequence. Unfortunately, the differential is not very explicit. Determining differentials or finding ways to work around them is one of the main challenges to successfully applying a spectral sequence.