SpectralSequencePageMap ^ List -- the p,q th map on of a spectral sequence page

Synopsis

Operator: ^
Usage:

d = D ^L
Inputs:
- D, a spectral sequence page map,
- L, a list, A list L = \{p,q\} \ of integers.
Outputs:
- d, a matrix,

Description

Returns the p,q th map on an (upper index) spectral sequence page. The relationship $D^{p,q} = D_{-p,-q}$ holds.

Consider the filtered complex $K$ below, obtained by multiplying the minimal free resolution of the rational quartic space curve by successive powers of the irrelevant ideal.

i1 : B = QQ[a..d];

i2 : J = ideal vars B;

o2 : Ideal of B

i3 : C = complete res monomialCurveIdeal(B,{1,3,4});

i4 : K = filteredComplex(J,C,4);

We compute a map on the third page of the spectral sequence associated to $K$.

i5 : E = spectralSequence K

o5 = E

o5 : SpectralSequence

i6 : E_3 .dd^{1,-2}

o6 = {4} | 0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0 
     {4} | 0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0 
     {4} | 0 0 0 0 -1 0  0  0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0 
     {4} | 0 0 0 0 0  0  0  0 0 0 0 -1 0  0  0 0 0 0 0  0  0  0 0 0 0 0  0 
     {4} | 0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0 
     {4} | 0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0 
     {4} | 0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0 
     {4} | 0 0 0 0 0  1  0  0 0 0 0 0  0  0  0 0 0 0 -1 0  0  0 0 0 0 0  0 
     {4} | 0 0 0 0 0  0  0  0 0 0 0 0  1  0  0 0 0 0 0  0  0  0 0 0 0 -1 0 
     {4} | 0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0 
     {4} | 0 0 0 0 1  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0 
     {4} | 0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0 
     {4} | 0 0 0 0 0  -1 0  0 0 0 0 0  0  0  0 0 0 0 1  0  0  0 0 0 0 0  0 
     {4} | 0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0 
     {4} | 0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0 
     {4} | 0 0 0 0 0  0  -1 0 0 0 0 0  0  0  0 0 0 0 0  -1 0  0 0 0 0 0  0 
     {4} | 0 0 0 0 0  0  1  0 0 0 0 0  0  0  0 0 0 0 0  1  0  0 0 0 0 0  0 
     {4} | 0 0 0 0 0  0  0  0 0 0 0 0  0  1  0 0 0 0 0  0  0  0 0 0 0 0  1 
     {4} | 0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0 
     {4} | 0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0  -1 0 0 0 0 0  0 
     {4} | 0 0 0 0 0  0  0  0 0 0 0 1  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0 
     {4} | 0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0 
     {4} | 0 0 0 0 0  0  0  0 0 0 0 0  -1 0  0 0 0 0 0  0  0  0 0 0 0 1  0 
     {4} | 0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0 
     {4} | 0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0 
     {4} | 0 0 0 0 0  0  0  0 0 0 0 0  0  -1 0 0 0 0 0  0  0  0 0 0 0 0  -1
     {4} | 0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0 
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     {4} | 0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0 
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     {5} | 0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0 
     {4} | 0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0 
     {5} | 0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0 
     {5} | 0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0 
     {5} | 0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0 
     {4} | 0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0 
     {5} | 0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0 
     {5} | 0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0 
     {5} | 0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0 
     {4} | 0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0 
     {5} | 0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0 
     {5} | 0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0 
     {5} | 0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0 
     {4} | 0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0 
     {5} | 0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0 
     {5} | 0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0 
     {5} | 0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0 
     {4} | 0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0 
     {5} | 0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0 
     {5} | 0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0 
     {5} | 0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0 
     {4} | 0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0 
     {5} | 0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0 
     {5} | 0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0 
     {5} | 0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0 
     {4} | 0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0 
     {5} | 0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0 
     {5} | 0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0 
     {5} | 0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0 
     {0} | 0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0  0  0 0 0 0 0  0 
     ------------------------------------------------------------------------
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     ------------------------------------------------------------------------
     0 0 0 0 0 0 0 0 0 |
     0 0 0 0 0 0 0 0 0 |
     0 0 0 0 0 0 0 0 0 |
     0 0 0 0 0 0 0 0 0 |
     0 0 0 0 0 0 0 0 0 |
     0 0 0 0 0 0 0 0 0 |
     0 0 0 0 0 0 0 0 0 |
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     0 0 0 0 0 0 0 0 0 |
     0 0 0 0 0 0 0 0 0 |
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     0 0 0 0 0 0 0 0 0 |
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     0 0 0 0 0 0 0 0 0 |
     0 0 0 0 0 0 0 0 0 |
     0 0 0 0 0 0 0 0 0 |
     0 0 0 0 0 0 0 0 0 |
     0 0 0 0 0 0 0 0 0 |
     0 0 0 0 0 0 0 0 0 |
     0 0 0 0 0 0 0 0 0 |
     0 0 0 0 0 0 0 0 0 |
     0 0 0 0 0 0 0 0 0 |
     0 0 0 0 0 0 0 0 0 |
     0 0 0 0 0 0 0 0 0 |
     0 0 0 0 0 0 0 0 0 |
     0 0 0 0 0 0 0 0 0 |
     0 0 0 0 0 0 0 0 0 |
     0 0 0 0 0 0 0 0 0 |
     0 0 0 0 0 0 0 0 0 |
     0 0 0 0 0 0 0 0 0 |
     0 0 0 0 0 0 0 0 0 |
     0 0 0 0 0 0 0 0 0 |
     0 0 0 0 0 0 0 0 0 |
     0 0 0 0 0 0 0 0 0 |
     0 0 0 0 0 0 0 0 0 |
     0 0 0 0 0 0 0 0 0 |
     0 0 0 0 0 0 0 0 0 |
     0 0 0 0 0 0 0 0 0 |
     0 0 0 0 0 0 0 0 0 |
     0 0 0 0 0 0 0 0 0 |
     0 0 0 0 0 0 0 0 0 |
     0 0 0 0 0 0 0 0 0 |
     0 0 0 0 0 0 0 0 0 |
     0 0 0 0 0 0 0 0 0 |
     0 0 0 0 0 0 0 0 0 |
     0 0 0 0 0 0 0 0 0 |
     0 0 0 0 0 0 0 0 0 |
     0 0 0 0 0 0 0 0 0 |
     0 0 0 0 0 0 0 0 0 |

o6 : Matrix

Ways to use this method: