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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 | -33x2+41xy-4y2  -3x2-2xy+40y2  |
              | -47x2+48xy-18y2 -8x2-24xy-10y2 |
              | 26x2-29xy+27y2  30x2-43xy+10y2 |

                            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 | 30x2+7xy-15y2 18x2+22xy+31y2 x3 x2y-50xy2+5y3 39xy2+3y3  y4 0  0  |
              | x2-36xy-28y2  30xy-28y2      0  22xy2+33y3    -50xy2-8y3 0  y4 0  |
              | 45xy-46y2     x2-13xy+15y2   0  -16y3         xy2+42y3   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
               | 30x2+7xy-15y2 18x2+22xy+31y2 x3 x2y-50xy2+5y3 39xy2+3y3  y4 0  0  |
               | x2-36xy-28y2  30xy-28y2      0  22xy2+33y3    -50xy2-8y3 0  y4 0  |
               | 45xy-46y2     x2-13xy+15y2   0  -16y3         xy2+42y3   0  0  y4 |

          8                                                                           5
     1 : A  <----------------------------------------------------------------------- A  : 2
               {2} | 30xy2+43y3      -9xy2+10y3  -30y3      35y3      -9y3       |
               {2} | -19xy2-11y3     25y3        19y3       -3y3      18y3       |
               {3} | 50xy-46y2       -5xy+16y2   -50y2      -19y2     -10y2      |
               {3} | -50x2-16xy-30y2 5x2-xy-17y2 50xy-39y2  19xy+39y2 10xy+37y2  |
               {3} | 19x2+48xy+42y2  -45xy+22y2  -19xy-37y2 3xy-22y2  -18xy+39y2 |
               {4} | 0               0           x-26y      -16y      15y        |
               {4} | 0               0           -24y       x+39y     50y        |
               {4} | 0               0           -28y       -29y      x-13y      |

          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+36y -30y  |
               {2} | 0 -45y  x+13y |
               {3} | 1 -30   -18   |
               {3} | 0 26    38    |
               {3} | 0 -38   -11   |
               {4} | 0 0     0     |
               {4} | 0 0     0     |
               {4} | 0 0     0     |

          5                                                                                8
     2 : A  <---------------------------------------------------------------------------- A  : 1
               {5} | 2  -26 0 43y      16x-7y  xy-5y2       -48xy-y2     -15xy        |
               {5} | 35 -5  0 -20x+22y -42x-7y -22y2        xy-39y2      50xy+6y2     |
               {5} | 0  0   0 0        0       x2+26xy+34y2 16xy-37y2    -15xy+29y2   |
               {5} | 0  0   0 0        0       24xy+5y2     x2-39xy-50y2 -50xy+31y2   |
               {5} | 0  0   0 0        0       28xy-30y2    29xy-3y2     x2+13xy+16y2 |

                   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 :