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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 | -5x2-27xy-28y2  -4x2+5xy+12y2  |
              | -3x2+24xy+29y2  25x2-38xy+43y2 |
              | -21x2-45xy-37y2 -8x2-2xy-47y2  |

                            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 | -x2+5xy+20y2 10x2+15xy-40y2 x3 x2y-15xy2+15y3 -37xy2+34y3 y4 0  0  |
              | x2+28xy-10y2 -34xy-24y2     0  9xy2-7y3       -45xy2+14y3 0  y4 0  |
              | 21xy-18y2    x2+28xy+38y2   0  -25y3          xy2+24y3    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
               | -x2+5xy+20y2 10x2+15xy-40y2 x3 x2y-15xy2+15y3 -37xy2+34y3 y4 0  0  |
               | x2+28xy-10y2 -34xy-24y2     0  9xy2-7y3       -45xy2+14y3 0  y4 0  |
               | 21xy-18y2    x2+28xy+38y2   0  -25y3          xy2+24y3    0  0  y4 |

          8                                                                               5
     1 : A  <--------------------------------------------------------------------------- A  : 2
               {2} | -25xy2+40y3     -29xy2-40y3    25y3       46y3       -35y3      |
               {2} | -31xy2-3y3      -20y3          31y3       13y3       15y3       |
               {3} | 33xy-33y2       8xy-37y2       -33y2      14y2       15y2       |
               {3} | -33x2-40xy-32y2 -8x2-11xy-27y2 33xy-28y2  -14xy+33y2 -15xy+45y2 |
               {3} | 31x2+29xy-7y2   25xy+3y2       -31xy-26y2 -13xy+46y2 -15xy-4y2  |
               {4} | 0               0              x+24y      -35y       -47y       |
               {4} | 0               0              -50y       x-45y      -23y       |
               {4} | 0               0              4y         -46y       x+21y      |

          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-28y 34y   |
               {2} | 0 -21y  x-28y |
               {3} | 1 1     -10   |
               {3} | 0 -25   -4    |
               {3} | 0 -18   32    |
               {4} | 0 0     0     |
               {4} | 0 0     0     |
               {4} | 0 0     0     |

          5                                                                                8
     2 : A  <---------------------------------------------------------------------------- A  : 1
               {5} | -32 12  0 -28y    -13x+41y xy+21y2      33xy-45y2    7xy-6y2     |
               {5} | -10 -50 0 -38x-9y 41x-9y   -9y2         xy+28y2      45xy-25y2   |
               {5} | 0   0   0 0       0        x2-24xy+17y2 35xy-32y2    47xy+3y2    |
               {5} | 0   0   0 0       0        50xy+49y2    x2+45xy-15y2 23xy-27y2   |
               {5} | 0   0   0 0       0        -4xy-45y2    46xy-46y2    x2-21xy-2y2 |

                   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 :