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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 | -29x2+21xy+16y2 27x2+21y2      |
              | -46x2+35xy+41y2 -13x2-32xy+4y2 |
              | 12x2+27xy-44y2  -37x2-2xy+32y2 |

                            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 | -4x2-27xy-11y2 -43x2-26xy-24y2 x3 x2y-3xy2-36y3 -35xy2-29y3 y4 0  0  |
              | x2+5xy+24y2    -20xy-14y2      0  -28xy2+27y3   30xy2+40y3  0  y4 0  |
              | -37xy+21y2     x2-47xy-41y2    0  -16y3         xy2+18y3    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
               | -4x2-27xy-11y2 -43x2-26xy-24y2 x3 x2y-3xy2-36y3 -35xy2-29y3 y4 0  0  |
               | x2+5xy+24y2    -20xy-14y2      0  -28xy2+27y3   30xy2+40y3  0  y4 0  |
               | -37xy+21y2     x2-47xy-41y2    0  -16y3         xy2+18y3    0  0  y4 |

          8                                                                            5
     1 : A  <------------------------------------------------------------------------ A  : 2
               {2} | 14xy2+4y3       -11xy2+10y3   -14y3     -43y3      -8y3      |
               {2} | 30xy2-40y3      -45y3         -30y3     43y3       -17y3     |
               {3} | 10xy-43y2       4xy-45y2      -10y2     44y2       47y2      |
               {3} | -10x2+6xy+26y2  -4x2-11xy-8y2 10xy+37y2 -44xy+40y2 -47xy-6y2 |
               {3} | -30x2+25xy-39y2 -22xy+19y2    30xy+15y2 -43xy-5y2  17xy-33y2 |
               {4} | 0               0             x-16y     36y        48y       |
               {4} | 0               0             34y       x+47y      22y       |
               {4} | 0               0             -8y       -38y       x-31y     |

          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-5y 20y   |
               {2} | 0 37y  x+47y |
               {3} | 1 4    43    |
               {3} | 0 -18  6     |
               {3} | 0 18   -40   |
               {4} | 0 0    0     |
               {4} | 0 0    0     |
               {4} | 0 0    0     |

          5                                                                                8
     2 : A  <---------------------------------------------------------------------------- A  : 1
               {5} | 41 -35 0 49y     37x-36y  xy+10y2      -25xy+48y2   -23xy-14y2   |
               {5} | -6 -12 0 25x+36y -42x-42y 28y2         xy+7y2       -30xy+16y2   |
               {5} | 0  0   0 0       0        x2+16xy-15y2 -36xy-y2     -48xy-50y2   |
               {5} | 0  0   0 0       0        -34xy-31y2   x2-47xy-29y2 -22xy-36y2   |
               {5} | 0  0   0 0       0        8xy-7y2      38xy+13y2    x2+31xy+44y2 |

                   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 :