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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 | 39x2-7xy+40y2   -11x2-43xy+30y2 |
              | -42x2-47xy-16y2 -25x2+19xy+5y2  |
              | 26x2+35xy+50y2  -43x2-7xy+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 | -34x2-3xy-15y2 -34x2-9xy-46y2 x3 x2y-30xy2+45y3 14xy2+40y3 y4 0  0  |
              | x2+13xy+49y2   -8xy-38y2      0  -24xy2-37y3    29xy2+46y3 0  y4 0  |
              | 6xy+6y2        x2+46xy-35y2   0  23y3           xy2+7y3    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
               | -34x2-3xy-15y2 -34x2-9xy-46y2 x3 x2y-30xy2+45y3 14xy2+40y3 y4 0  0  |
               | x2+13xy+49y2   -8xy-38y2      0  -24xy2-37y3    29xy2+46y3 0  y4 0  |
               | 6xy+6y2        x2+46xy-35y2   0  23y3           xy2+7y3    0  0  y4 |

          8                                                                             5
     1 : A  <------------------------------------------------------------------------- A  : 2
               {2} | -45xy2-26y3     -32xy2-30y3     45y3      14y3     -26y3      |
               {2} | 43xy2-31y3      -42y3           -43y3     -3y3     -18y3      |
               {3} | 37xy-49y2       35xy+22y2       -37y2     0        13y2       |
               {3} | -37x2-22xy+6y2  -35x2-39xy-20y2 37xy-30y2 29y2     -13xy-17y2 |
               {3} | -43x2-20xy-37y2 29xy-32y2       43xy-50y2 3xy+33y2 18xy+45y2  |
               {4} | 0               0               x+27y     -28y     -31y       |
               {4} | 0               0               -23y      x-33y    12y        |
               {4} | 0               0               -28y      24y      x+6y       |

          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-13y 8y    |
               {2} | 0 -6y   x-46y |
               {3} | 1 34    34    |
               {3} | 0 -37   30    |
               {3} | 0 45    -18   |
               {4} | 0 0     0     |
               {4} | 0 0     0     |
               {4} | 0 0     0     |

          5                                                                               8
     2 : A  <--------------------------------------------------------------------------- A  : 1
               {5} | -6  -38 0 13y      -47x-41y xy-20y2      40xy-11y2  38xy+22y2   |
               {5} | -38 -34 0 -26x+21y -34x+4y  24y2         xy-33y2    -29xy+37y2  |
               {5} | 0   0   0 0        0        x2-27xy+19y2 28xy+30y2  31xy-46y2   |
               {5} | 0   0   0 0        0        23xy+4y2     x2+33xy+y2 -12xy-15y2  |
               {5} | 0   0   0 0        0        28xy+39y2    -24xy+35y2 x2-6xy-20y2 |

                   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 :