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SimplicialComplexes :: chainComplex(SimplicialComplex)

chainComplex(SimplicialComplex) -- The chain complex of boundary maps.

Synopsis

Description

The ChainComplex of boundary maps from i-faces to (i-1)-faces.

i1 : R = QQ[a..f];
i2 : D = simplicialComplex monomialIdeal(a*b*c,a*b*f,a*c*e,a*d*e,a*d*f,b*c*d,b*d*e,b*e*f,c*d*f,c*e*f);
i3 : R' = ZZ/2[a..f];
i4 : D' = simplicialComplex monomialIdeal(a*b*c,a*b*f,a*c*e,a*d*e,a*d*f,b*c*d,b*d*e,b*e*f,c*d*f,c*e*f);
i5 : c = chainComplex D

       1       6       15       10
o5 = QQ  <-- QQ  <-- QQ   <-- QQ
                               
     -1      0       1        2

o5 : ChainComplex
i6 : c' = chainComplex D'

      ZZ 1      ZZ 6      ZZ 15      ZZ 10
o6 = (--)  <-- (--)  <-- (--)   <-- (--)
       2         2         2          2
                                     
     -1        0         1          2

o6 : ChainComplex
i7 : c.dd_1

o7 = | 1  1  1  1  1  0  0  0  0  0  0  0  0  0  0  |
     | -1 0  0  0  0  1  1  1  1  0  0  0  0  0  0  |
     | 0  -1 0  0  0  -1 0  0  0  1  1  1  0  0  0  |
     | 0  0  -1 0  0  0  -1 0  0  -1 0  0  1  1  0  |
     | 0  0  0  -1 0  0  0  -1 0  0  -1 0  -1 0  1  |
     | 0  0  0  0  -1 0  0  0  -1 0  0  -1 0  -1 -1 |

              6        15
o7 : Matrix QQ  <--- QQ
i8 : c'.dd_1

o8 = | 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 |
     | 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 |
     | 0 1 0 0 0 1 0 0 0 1 1 1 0 0 0 |
     | 0 0 1 0 0 0 1 0 0 1 0 0 1 1 0 |
     | 0 0 0 1 0 0 0 1 0 0 1 0 1 0 1 |
     | 0 0 0 0 1 0 0 0 1 0 0 1 0 1 1 |

             ZZ 6       ZZ 15
o8 : Matrix (--)  <--- (--)
              2          2

See also