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Macaulay2Doc :: nullhomotopy

nullhomotopy -- make a null homotopy

Description

nullhomotopy f -- produce a nullhomotopy for a map f of chain complexes.

Whether f is null homotopic is not checked.

Here is part of an example provided by Luchezar Avramov. We construct a random module over a complete intersection, resolve it over the polynomial ring, and produce a null homotopy for the map that is multiplication by one of the defining equations for the complete intersection.

i1 : A = ZZ/101[x,y];
i2 : M = cokernel random(A^3, A^{-2,-2})

o2 = cokernel | 23x2+5xy+18y2 46x2-25xy-5y2   |
              | 8x2+2xy+25y2  11x2-20xy-22y2  |
              | 41x2-26xy-2y2 -49x2-10xy+37y2 |

                            3
o2 : A-module, quotient of A
i3 : R = cokernel matrix {{x^3,y^4}}

o3 = cokernel | x3 y4 |

                            1
o3 : A-module, quotient of A
i4 : N = prune (M**R)

o4 = cokernel | -38x2+21xy+28y2 40x2-36xy+27y2 x3 x2y-15xy2-23y3 -46xy2+49y3 y4 0  0  |
              | x2+14xy+15y2    -15xy+10y2     0  -21xy2-46y3    33xy2+13y3  0  y4 0  |
              | 50xy+29y2       x2-3xy+14y2    0  9y3            xy2+19y3    0  0  y4 |

                            3
o4 : A-module, quotient of A
i5 : C = resolution N

      3      8      5
o5 = A  <-- A  <-- A  <-- 0
                           
     0      1      2      3

o5 : ChainComplex
i6 : d = C.dd

          3                                                                                 8
o6 = 0 : A  <----------------------------------------------------------------------------- A  : 1
               | -38x2+21xy+28y2 40x2-36xy+27y2 x3 x2y-15xy2-23y3 -46xy2+49y3 y4 0  0  |
               | x2+14xy+15y2    -15xy+10y2     0  -21xy2-46y3    33xy2+13y3  0  y4 0  |
               | 50xy+29y2       x2-3xy+14y2    0  9y3            xy2+19y3    0  0  y4 |

          8                                                                              5
     1 : A  <-------------------------------------------------------------------------- A  : 2
               {2} | -21xy2-47y3   35xy2+5y3       21y3       -30y3      37y3       |
               {2} | -20xy2-46y3   -11y3           20y3       24y3       27y3       |
               {3} | -16xy-46y2    32xy-32y2       16y2       -33y2      7y2        |
               {3} | 16x2-4xy+31y2 -32x2-27xy-19y2 -16xy+50y2 33xy+18y2  -7xy-11y2  |
               {3} | 20x2+7xy-26y2 -37xy-49y2      -20xy+39y2 -24xy+14y2 -27xy+18y2 |
               {4} | 0             0               x+30y      21y        29y        |
               {4} | 0             0               -35y       x+48y      -50y       |
               {4} | 0             0               41y        5y         x+23y      |

          5
     2 : A  <----- 0 : 3
               0

o6 : ChainComplexMap
i7 : s = nullhomotopy (x^3 * id_C)

          8                            3
o7 = 1 : A  <------------------------ A  : 0
               {2} | 0 x-14y 15y  |
               {2} | 0 -50y  x+3y |
               {3} | 1 38    -40  |
               {3} | 0 33    -19  |
               {3} | 0 16    -48  |
               {4} | 0 0     0    |
               {4} | 0 0     0    |
               {4} | 0 0     0    |

          5                                                                               8
     2 : A  <--------------------------------------------------------------------------- A  : 1
               {5} | -21 -38 0 -10y    -5x+35y xy+20y2      -19xy-22y2  -34xy-40y2   |
               {5} | 0   -48 0 41x+33y 48x-35y 21y2         xy+10y2     -33xy-15y2   |
               {5} | 0   0   0 0       0       x2-30xy+41y2 -21xy-35y2  -29xy-18y2   |
               {5} | 0   0   0 0       0       35xy-33y2    x2-48xy+6y2 50xy-20y2    |
               {5} | 0   0   0 0       0       -41xy-22y2   -5xy+4y2    x2-23xy-47y2 |

                   5
     3 : 0 <----- A  : 2
              0

o7 : ChainComplexMap
i8 : s*d + d*s

          3                    3
o8 = 0 : A  <---------------- A  : 0
               | x3 0  0  |
               | 0  x3 0  |
               | 0  0  x3 |

          8                                       8
     1 : A  <----------------------------------- A  : 1
               {2} | x3 0  0  0  0  0  0  0  |
               {2} | 0  x3 0  0  0  0  0  0  |
               {3} | 0  0  x3 0  0  0  0  0  |
               {3} | 0  0  0  x3 0  0  0  0  |
               {3} | 0  0  0  0  x3 0  0  0  |
               {4} | 0  0  0  0  0  x3 0  0  |
               {4} | 0  0  0  0  0  0  x3 0  |
               {4} | 0  0  0  0  0  0  0  x3 |

          5                              5
     2 : A  <-------------------------- A  : 2
               {5} | x3 0  0  0  0  |
               {5} | 0  x3 0  0  0  |
               {5} | 0  0  x3 0  0  |
               {5} | 0  0  0  x3 0  |
               {5} | 0  0  0  0  x3 |

     3 : 0 <----- 0 : 3
              0

o8 : ChainComplexMap
i9 : s^2

          5         3
o9 = 2 : A  <----- A  : 0
               0

                   8
     3 : 0 <----- A  : 1
              0

o9 : ChainComplexMap

Ways to use nullhomotopy :