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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 | -6x2-32xy+22y2 -41x2+2xy+42y2  |
              | 20x2-30xy+18y2 -28x2+32xy-17y2 |
              | 26x2+11xy-29y2 50x2-4xy+34y2   |

                            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 | -13x2-40xy-36y2 2x2+14xy+13y2 x3 x2y-26xy2-36y3 -5xy2-45y3 y4 0  0  |
              | x2-10xy+21y2    -9xy-31y2     0  -42xy2-13y3    49xy2-22y3 0  y4 0  |
              | -14xy-y2        x2-16xy-12y2  0  14y3           xy2+49y3   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
               | -13x2-40xy-36y2 2x2+14xy+13y2 x3 x2y-26xy2-36y3 -5xy2-45y3 y4 0  0  |
               | x2-10xy+21y2    -9xy-31y2     0  -42xy2-13y3    49xy2-22y3 0  y4 0  |
               | -14xy-y2        x2-16xy-12y2  0  14y3           xy2+49y3   0  0  y4 |

          8                                                                               5
     1 : A  <--------------------------------------------------------------------------- A  : 2
               {2} | 37xy2-7y3       38xy2-45y3     -37y3      -11y3      -50y3      |
               {2} | 46xy2+39y3      -20y3          -46y3      9y3        36y3       |
               {3} | -41xy+12y2      -49xy+21y2     41y2       30y2       48y2       |
               {3} | 41x2+xy+30y2    49x2+30xy+21y2 -41xy-13y2 -30xy+24y2 -48xy-29y2 |
               {3} | -46x2-34xy+40y2 -33xy-18y2     46xy-5y2   -9xy+43y2  -36xy-12y2 |
               {4} | 0               0              x-13y      -37y       -14y       |
               {4} | 0               0              -19y       x-50y      10y        |
               {4} | 0               0              40y        -23y       x-38y      |

          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+10y 9y    |
               {2} | 0 14y   x+16y |
               {3} | 1 13    -2    |
               {3} | 0 41    -30   |
               {3} | 0 -39   -10   |
               {4} | 0 0     0     |
               {4} | 0 0     0     |
               {4} | 0 0     0     |

          5                                                                               8
     2 : A  <--------------------------------------------------------------------------- A  : 1
               {5} | -25 -9 0 40y     -11x+35y xy+9y2      2xy-14y2     8xy+13y2     |
               {5} | -50 37 0 33x-31y 36x-14y  42y2        xy-29y2      -49xy+50y2   |
               {5} | 0   0  0 0       0        x2+13xy+9y2 37xy+27y2    14xy+41y2    |
               {5} | 0   0  0 0       0        19xy-19y2   x2+50xy+44y2 -10xy-8y2    |
               {5} | 0   0  0 0       0        -40xy+13y2  23xy+39y2    x2+38xy+48y2 |

                   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 :