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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 | 42x2-40xy-15y2  39x2-36xy-42y2  |
              | -15x2+17xy-18y2 -23x2+7xy+33y2  |
              | -37x2-17xy+38y2 -49x2-11xy+16y2 |

                            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 | 41x2+12xy-6y2 15x2-42xy-19y2 x3 x2y+29xy2+2y3 45xy2-25y3  y4 0  0  |
              | x2+45xy-19y2  -46xy          0  -44xy2-40y3   -34xy2+43y3 0  y4 0  |
              | 12xy-50y2     x2+42xy-49y2   0  22y3          xy2+45y3    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
               | 41x2+12xy-6y2 15x2-42xy-19y2 x3 x2y+29xy2+2y3 45xy2-25y3  y4 0  0  |
               | x2+45xy-19y2  -46xy          0  -44xy2-40y3   -34xy2+43y3 0  y4 0  |
               | 12xy-50y2     x2+42xy-49y2   0  22y3          xy2+45y3    0  0  y4 |

          8                                                                            5
     1 : A  <------------------------------------------------------------------------ A  : 2
               {2} | 28xy2+25y3     -4xy2-24y3     -28y3     32y3       25y3      |
               {2} | -40xy2+32y3    -3y3           40y3      -40y3      -29y3     |
               {3} | -34xy-23y2     46xy-27y2      34y2      21y2       -8y2      |
               {3} | 34x2+22xy-26y2 -46x2+10xy+9y2 -34xy+y2  -21xy+16y2 8xy-6y2   |
               {3} | 40x2-24xy+8y2  -48xy+33y2     -40xy-8y2 40xy-42y2  29xy+42y2 |
               {4} | 0              0              x-14y     -34y       -46y      |
               {4} | 0              0              -47y      x+24y      45y       |
               {4} | 0              0              -11y      -34y       x-10y     |

          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-45y 46y   |
               {2} | 0 -12y  x-42y |
               {3} | 1 -41   -15   |
               {3} | 0 -7    -2    |
               {3} | 0 -17   49    |
               {4} | 0 0     0     |
               {4} | 0 0     0     |
               {4} | 0 0     0     |

          5                                                                                 8
     2 : A  <----------------------------------------------------------------------------- A  : 1
               {5} | 8   -29 0 7y      48x+38y  xy+28y2      -xy+35y2     22xy-35y2    |
               {5} | -14 40  0 -11x-6y -26x+41y 44y2         xy-35y2      34xy-22y2    |
               {5} | 0   0   0 0       0        x2+14xy-23y2 34xy+12y2    46xy-22y2    |
               {5} | 0   0   0 0       0        47xy+45y2    x2-24xy+38y2 -45xy-36y2   |
               {5} | 0   0   0 0       0        11xy+44y2    34xy-y2      x2+10xy-15y2 |

                   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 :