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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 | -38x2-48xy+28y2 42x2-49xy-48y2  |
              | 23x2+35xy+32y2  -31x2-36xy+22y2 |
              | -22x2-45xy+43y2 -13x2+31xy-29y2 |

                            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 | 28x2-47xy-42y2 31x2-24xy-36y2 x3 x2y+38xy2   -40xy2-26y3 y4 0  0  |
              | x2-43xy-29y2   -19xy-18y2     0  -36xy2+45y3 22xy2-26y3  0  y4 0  |
              | -19xy+4y2      x2-27xy+16y2   0  32y3        xy2+27y3    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
               | 28x2-47xy-42y2 31x2-24xy-36y2 x3 x2y+38xy2   -40xy2-26y3 y4 0  0  |
               | x2-43xy-29y2   -19xy-18y2     0  -36xy2+45y3 22xy2-26y3  0  y4 0  |
               | -19xy+4y2      x2-27xy+16y2   0  32y3        xy2+27y3    0  0  y4 |

          8                                                                              5
     1 : A  <-------------------------------------------------------------------------- A  : 2
               {2} | -9xy2+y3        -12xy2-46y3    9y3        -43y3     -7y3       |
               {2} | 41xy2-7y3       -47y3          -41y3      -30y3     30y3       |
               {3} | -28xy+49y2      34xy+9y2       28y2       -48y2     -46y2      |
               {3} | 28x2-35xy+7y2   -34x2+3xy-33y2 -28xy-14y2 48xy-5y2  46xy-46y2  |
               {3} | -41x2+43xy-23y2 -2xy-16y2      41xy-36y2  30xy-34y2 -30xy+14y2 |
               {4} | 0               0              x-14y      -33y      39y        |
               {4} | 0               0              25y        x-22y     44y        |
               {4} | 0               0              20y        13y       x+36y      |

          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+43y 19y   |
               {2} | 0 19y   x+27y |
               {3} | 1 -28   -31   |
               {3} | 0 -29   -32   |
               {3} | 0 13    -37   |
               {4} | 0 0     0     |
               {4} | 0 0     0     |
               {4} | 0 0     0     |

          5                                                                                8
     2 : A  <---------------------------------------------------------------------------- A  : 1
               {5} | -12 -28 0 10y    32x+33y  xy+45y2      50xy-13y2    -50xy+47y2   |
               {5} | 16  -24 0 -3x+5y -39x-30y 36y2         xy+31y2      -22xy+26y2   |
               {5} | 0   0   0 0      0        x2+14xy+50y2 33xy-22y2    -39xy+12y2   |
               {5} | 0   0   0 0      0        -25xy-20y2   x2+22xy+29y2 -44xy-25y2   |
               {5} | 0   0   0 0      0        -20xy-43y2   -13xy+27y2   x2-36xy+22y2 |

                   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 :