next | previous | forward | backward | up | top | index | toc | Macaulay2 web site
SpectralSequences :: Edge homomorphisms

Edge homomorphisms

Suppose that E is a spectral sequence with the properties that:

1. E2p,q = 0 for all p < l and all q ∈ℤ;

2. E2p,q = 0 for all q < m and all p ∈ℤ;

3. E converges to the graded module {Hn} for n ∈ℤ.

Then E determines a 5-term exact sequence Hl+m+2 →E2l+2,m →E2l,m+1 →Hl+m+1 →E2l+1,m →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 simplical 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);
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 H1(C) →E22,-1 →E20,0 →H0(C) →E21, -1 →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

Caveat

The method currently does not support pruned spectral sequences.

See also