Editing Spectral sequence (section)

=== The spectral sequence of a double complex ===

Another common spectral sequence is the spectral sequence of a double complex. A '''double complex''' is a collection of objects ''C<sub>i,j</sub>'' for all integers ''i'' and ''j'' together with two differentials, {{mvar|d}}{{i sup|{{rn|I}}}} and {{mvar|d}}{{i sup|{{rn|II}}}}.  {{mvar|d}}{{i sup|{{rn|I}}}} is assumed to decrease ''i'', and {{mvar|d}}{{i sup|{{rn|II}}}} is assumed to decrease ''j''. Furthermore, we assume that the differentials ''anticommute'', so that ''d''{{i sup|{{rn|I}}}} ''d''{{i sup|{{rn|II}}}} + ''d''{{i sup|{{rn|II}}}} ''d''{{i sup|{{rn|I}}}} = 0. Our goal is to compare the iterated homologies <math>H^\textrm{I}_i(H^\textrm{II}_j(C_{\bullet,\bullet}))</math> and <math>H^\textrm{II}_j(H^\textrm{I}_i(C_{\bullet,\bullet}))</math>. We will do this by filtering our double complex in two different ways.  Here are our filtrations:

:<math>(C_{i,j}^\textrm{I})_p = \begin{cases}
0 & \text{if } i < p \\
C_{i,j} & \text{if } i \ge p \end{cases}</math>
:<math>(C_{i,j}^\textrm{II})_p = \begin{cases}
0 & \text{if } j < p \\
C_{i,j} & \text{if } j \ge p \end{cases}</math>

To get a spectral sequence, we will reduce to the previous example. We define the ''total complex'' ''T''(''C''<sub>•,•</sub>) to be the complex whose {{prime|''n''}}th term is <math>\bigoplus_{i+j=n} C_{i,j}</math> and whose differential is {{mvar|d}}{{i sup|{{rn|I}}}} + {{mvar|d}}{{i sup|{{rn|II}}}}.  This is a complex because {{mvar|d}}{{i sup|{{rn|I}}}} and {{mvar|d}}{{i sup|{{rn|II}}}} are anticommuting differentials. The two filtrations on ''C<sub>i,j</sub>'' give two filtrations on the total complex:

:<math>T_n(C_{\bullet,\bullet})^\textrm{I}_p = \bigoplus_{i+j=n \atop i > p-1} C_{i,j}</math>
:<math>T_n(C_{\bullet,\bullet})^\textrm{II}_p = \bigoplus_{i+j=n \atop j > p-1} C_{i,j}</math>

To show that these spectral sequences give information about the iterated homologies, we will work out the ''E''{{i sup|0}}, ''E''{{i sup|1}}, and ''E''{{i sup|2}} terms of the {{rn|I}} filtration on ''T''(''C''<sub>•,•</sub>).  The ''E''{{i sup|0}} term is clear:

:<math>{}^\textrm{I}E^0_{p,q} =
T_n(C_{\bullet,\bullet})^\textrm{I}_p / T_n(C_{\bullet,\bullet})^\textrm{I}_{p+1} =
\bigoplus_{i+j=n \atop i > p-1} C_{i,j} \Big/
\bigoplus_{i+j=n \atop i > p} C_{i,j} =
C_{p,q},</math>
where {{nowrap|''n'' {{=}} ''p'' + ''q''}}.

To find the ''E''{{i sup|1}} term, we need to determine {{mvar|d}}{{i sup|{{rn|I}}}} + {{mvar|d}}{{i sup|{{rn|II}}}} on ''E''{{i sup|0}}. Notice that the differential must have degree &minus;1 with respect to ''n'', so we get a map

:<math>d^\textrm{I}_{p,q} + d^\textrm{II}_{p,q} :
T_n(C_{\bullet,\bullet})^\textrm{I}_p / T_n(C_{\bullet,\bullet})^\textrm{I}_{p+1} =
C_{p,q} \rightarrow
T_{n-1}(C_{\bullet,\bullet})^\textrm{I}_p / T_{n-1}(C_{\bullet,\bullet})^\textrm{I}_{p+1} =
C_{p,q-1}</math>

Consequently, the differential on ''E''{{i sup|0}} is the map ''C''<sub>''p'',''q''</sub> → ''C''<sub>''p'',''q''&minus;1</sub> induced by {{mvar|d}}{{i sup|{{rn|I}}}} + {{mvar|d}}{{i sup|{{rn|II}}}}.  But  {{mvar|d}}{{i sup|{{rn|I}}}} has the wrong degree to induce such a map, so {{mvar|d}}{{i sup|{{rn|I}}}} must be zero on ''E''{{i sup|0}}. That means the differential is exactly {{mvar|d}}{{i sup|{{rn|II}}}}, so we get

:<math>{}^\textrm{I}E^1_{p,q} = H^\textrm{II}_q(C_{p,\bullet}).</math>

To find ''E''{{i sup|2}}, we need to determine

:<math>d^\textrm{I}_{p,q} + d^\textrm{II}_{p,q} :
H^\textrm{II}_q(C_{p,\bullet}) \rightarrow
H^\textrm{II}_q(C_{p+1,\bullet})</math>

Because ''E''{{i sup|1}} was exactly the homology with respect to {{mvar|d}}{{i sup|{{rn|II}}}}, {{mvar|d}}{{i sup|{{rn|II}}}} is zero on ''E''{{i sup|1}}. Consequently, we get

:<math>{}^\textrm{I}E^2_{p,q} = H^\textrm{I}_p(H^\textrm{II}_q(C_{\bullet,\bullet})).</math>

Using the other filtration gives us a different spectral sequence with a similar ''E''{{i sup|2}} term:

:<math>{}^\textrm{II}E^2_{p,q} = H^\textrm{II}_q(H^{I}_p(C_{\bullet,\bullet})).</math>

What remains is to find a relationship between these two spectral sequences. It will turn out that as ''r'' increases, the two sequences will become similar enough to allow useful comparisons.