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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 | -24x2-47xy-32y2 -36x2-34xy-33y2 |
              | -32x2+2xy+24y2  25x2+40xy+37y2  |
              | -36x2+19xy+21y2 17x2+50xy-42y2  |

                            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 | -15x2+19xy-6y2 14x2-24xy-5y2 x3 x2y-3xy2+26y3 -6xy2-42y3  y4 0  0  |
              | x2+24xy-11y2   -27xy-47y2    0  50xy2+31y3    -28xy2+30y3 0  y4 0  |
              | 21xy+43y2      x2-22xy-36y2  0  -36y3         xy2-44y3    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
               | -15x2+19xy-6y2 14x2-24xy-5y2 x3 x2y-3xy2+26y3 -6xy2-42y3  y4 0  0  |
               | x2+24xy-11y2   -27xy-47y2    0  50xy2+31y3    -28xy2+30y3 0  y4 0  |
               | 21xy+43y2      x2-22xy-36y2  0  -36y3         xy2-44y3    0  0  y4 |

          8                                                                              5
     1 : A  <-------------------------------------------------------------------------- A  : 2
               {2} | -15xy2+17y3     46xy2-33y3     15y3       39y3      -30y3      |
               {2} | -34xy2-31y3     16y3           34y3       -43y3     10y3       |
               {3} | 15xy-15y2       9xy-24y2       -15y2      7y2       5y2        |
               {3} | -15x2+21xy+10y2 -9x2-20xy+37y2 15xy-6y2   -7xy+10y2 -5xy+42y2  |
               {3} | 34x2+49xy-19y2  7xy-32y2       -34xy-18y2 43xy-24y2 -10xy+27y2 |
               {4} | 0               0              x-37y      -37y      13y        |
               {4} | 0               0              -36y       x+30y     48y        |
               {4} | 0               0              -25y       18y       x+7y       |

          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-24y 27y   |
               {2} | 0 -21y  x+22y |
               {3} | 1 15    -14   |
               {3} | 0 16    20    |
               {3} | 0 -1    -47   |
               {4} | 0 0     0     |
               {4} | 0 0     0     |
               {4} | 0 0     0     |

          5                                                                              8
     2 : A  <-------------------------------------------------------------------------- A  : 1
               {5} | 3 40  0 36y      3x-33y  xy-11y2      -28xy-41y2   30xy-19y2   |
               {5} | 8 -42 0 -45x+40y -5x+17y -50y2        xy-49y2      28xy+30y2   |
               {5} | 0 0   0 0        0       x2+37xy-48y2 37xy-12y2    -13xy-45y2  |
               {5} | 0 0   0 0        0       36xy-39y2    x2-30xy-35y2 -48xy-5y2   |
               {5} | 0 0   0 0        0       25xy+y2      -18xy-25y2   x2-7xy-18y2 |

                   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 :