Suppose that $E$ is a spectral sequence with the properties that:
1. $E^2_{p,q} = 0$ for all $p < l$ and all $q \in \mathbb{Z}$;
2. $E^2_{p,q} = 0 $ for all $q < m$ and all $p \in \mathbb{Z}$;
3. $E$ converges to the graded module $\{H_n\}$ for $n \in \mathbb{Z}$.
Then $E$ determines a $5$-term exact sequence $H_{l+m+2} \rightarrow E^2_{l+2,m} \rightarrow E^2_{l,m+1} \rightarrow H_{l+m+1} \rightarrow E^2_{l+1,m} \rightarrow 0$ which we refer to as the edge complex.
Note that the above properties are satisfied if $E$ is the spectral sequence determined by a bounded filtration of a bounded chain complex.
The following is an easy example, of a spectral sequence which arises from a nested chain of simplicial complexes, which illustrates this concept.
i1 : A = QQ[a,b,c,d]; |
i2 : D = simplicialComplex {a*d*c, a*b, a*c, b*c};
|
i3 : F2D = D; |
i4 : F1D = simplicialComplex {a*c, d};
|
i5 : F0D = simplicialComplex {a,d};
|
i6 : K = filteredComplex({F2D, F1D, F0D},ReducedHomology => false);
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i7 : C = K_infinity; |
i8 : prune HH C
o8 = -1 : 0
1
0 : QQ
1
1 : QQ
2 : 0
o8 : GradedModule
|
The second page of the corresponding spectral sequences take the form:
i9 : E = spectralSequence(K); |
i10 : e = prune E; |
i11 : E^2
+------------------------------------------------+------------------------------------------------------------------+---------------------------------------+
o11 = |subquotient (| 1 0 1 0 0 0 0 |, | 1 0 0 0 0 |)|subquotient (| 0 |, | 0 |) |image 0 |
| | 0 0 0 0 0 0 0 | | 0 0 0 0 0 | | | -1 | | -1 | | |
| | 0 0 -1 0 0 0 0 | | -1 0 0 0 0 | | | 1 | | 1 | |{2, 0} |
| | 0 1 0 0 0 0 0 | | 0 0 0 0 0 | | | 0 | | 0 | | |
| | | -1 | | -1 | | |
|{0, 0} | | |
| |{1, 0} | |
+------------------------------------------------+------------------------------------------------------------------+---------------------------------------+
|image 0 |subquotient (| 1 0 0 1 1 1 0 0 1 0 |, | 1 1 1 0 0 1 0 |)|subquotient (| 0 1 0 0 0 |, | 0 0 |)|
| | | 0 0 0 -1 0 0 1 0 0 0 | | -1 0 0 1 0 0 0 | | | 0 -1 1 -1 1 | | -1 1 | |
|{0, -1} | | 0 1 0 0 -1 0 -1 1 0 0 | | 0 -1 0 -1 1 0 0 | | | 1 0 0 1 0 | | 1 0 | |
| | | 0 0 1 0 0 -1 0 -1 0 1 | | 0 0 -1 0 -1 0 1 | | | 0 1 0 0 0 | | 0 0 | |
| | | | 0 0 1 -1 0 | | -1 0 | |
| |{1, -1} | |
| | |{2, -1} |
+------------------------------------------------+------------------------------------------------------------------+---------------------------------------+
o11 : SpectralSequencePage
|
i12 : e^2
+-------+-------+-------+
| 2 | | |
o12 = |QQ |0 |0 |
| | | |
|{0, 0} |{1, 0} |{2, 0} |
+-------+-------+-------+
| | | 2 |
|0 |0 |QQ |
| | | |
|{0, -1}|{1, -1}|{2, -1}|
+-------+-------+-------+
o12 : SpectralSequencePage
|
The acyclic edge complex for this example has the form $H_1(C) \rightarrow E^2_{2,-1} \rightarrow E^2_{0,0} \rightarrow H_0(C) \rightarrow E^2_{1, -1} \rightarrow 0$ and is given by
i13 : edgeComplex E
o13 = subquotient (| 1 0 0 1 1 1 0 0 1 0 |, | 1 1 1 0 0 1 0 |) <-- cokernel | 1 1 1 0 0 | <-- subquotient (| 1 0 1 0 0 0 0 |, | 1 0 0 0 0 |) <-- subquotient (| 0 1 0 0 0 |, | 0 0 |) <-- subquotient (| 1 0 |, | 0 |)
| 0 0 0 -1 0 0 1 0 0 0 | | -1 0 0 1 0 0 0 | | -1 0 0 1 0 | | 0 0 0 0 0 0 0 | | 0 0 0 0 0 | | 0 -1 1 -1 1 | | -1 1 | | -1 1 | | -1 |
| 0 1 0 0 -1 0 -1 1 0 0 | | 0 -1 0 -1 1 0 0 | | 0 -1 0 -1 1 | | 0 0 -1 0 0 0 0 | | -1 0 0 0 0 | | 1 0 0 1 0 | | 1 0 | | 0 -1 | | 1 |
| 0 0 1 0 0 -1 0 -1 0 1 | | 0 0 -1 0 -1 0 1 | | 0 0 -1 0 -1 | | 0 1 0 0 0 0 0 | | 0 0 0 0 0 | | 0 1 0 0 0 | | 0 0 | | 1 0 | | 0 |
| 0 0 1 -1 0 | | -1 0 | | 0 1 | | -1 |
0 1 2
3 4
o13 : ChainComplex
|
i14 : prune edgeComplex E
1 2 2 1
o14 = 0 <-- QQ <-- QQ <-- QQ <-- QQ
0 1 2 3 4
o14 : ChainComplex
|
To see that it is acyclic we can compute
i15 : prune HH edgeComplex E
o15 = 0 : 0
1 : 0
2 : 0
3 : 0
4 : 0
o15 : GradedModule
|
The method currently does not support pruned spectral sequences.