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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 | 2x2-11xy-49y2  -41x2+26xy-45y2 |
              | 13x2+27xy-10y2 -28x2+38xy-25y2 |
              | 2x2-31xy+45y2  8x2-16xy-21y2   |

                            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 | -36x2+37xy+12y2 33x2-44xy+49y2 x3 x2y+10xy2-45y3 25xy2-12y3 y4 0  0  |
              | x2-37xy+44y2    -49xy+12y2     0  -21xy2+33y3    xy2+25y3   0  y4 0  |
              | 34xy-24y2       x2+16xy+27y2   0  47y3           xy2-17y3   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
               | -36x2+37xy+12y2 33x2-44xy+49y2 x3 x2y+10xy2-45y3 25xy2-12y3 y4 0  0  |
               | x2-37xy+44y2    -49xy+12y2     0  -21xy2+33y3    xy2+25y3   0  y4 0  |
               | 34xy-24y2       x2+16xy+27y2   0  47y3           xy2-17y3   0  0  y4 |

          8                                                                          5
     1 : A  <---------------------------------------------------------------------- A  : 2
               {2} | -10xy2-2y3      -21xy2-9y3  10y3      -37y3      -11y3     |
               {2} | 47xy2-21y3      -36y3       -47y3     10y3       -47y3     |
               {3} | 46xy-49y2       xy-35y2     -46y2     -17y2      -45y2     |
               {3} | -46x2-25xy+9y2  -x2-4xy-8y2 46xy-27y2 17xy+34y2  45xy+35y2 |
               {3} | -47x2-38xy+30y2 -11xy-25y2  47xy-42y2 -10xy+28y2 47xy+11y2 |
               {4} | 0               0           x-41y     2y         y         |
               {4} | 0               0           45y       x-11y      22y       |
               {4} | 0               0           44y       43y        x-49y     |

          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+37y 49y   |
               {2} | 0 -34y  x-16y |
               {3} | 1 36    -33   |
               {3} | 0 -7    13    |
               {3} | 0 17    -23   |
               {4} | 0 0     0     |
               {4} | 0 0     0     |
               {4} | 0 0     0     |

          5                                                                              8
     2 : A  <-------------------------------------------------------------------------- A  : 1
               {5} | 24 21  0 -32y   -43x+36y xy+50y2     -26xy-41y2   xy+33y2      |
               {5} | 0  -44 0 -x-39y -42x-9y  21y2        xy+19y2      -xy-22y2     |
               {5} | 0  0   0 0      0        x2+41xy-3y2 -2xy+40y2    -xy-46y2     |
               {5} | 0  0   0 0      0        -45xy+42y2  x2+11xy+46y2 -22xy+38y2   |
               {5} | 0  0   0 0      0        -44xy-5y2   -43xy+33y2   x2+49xy-43y2 |

                   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 :