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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 | 28x2-43xy-35y2  44x2+39xy-24y2  |
              | 26x2-40xy-5y2   -4x2-26xy+46y2  |
              | -11x2-28xy-40y2 -11x2+14xy-14y2 |

                            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 | -14x2+4xy+3y2 47x2-5xy+47y2 x3 x2y+4xy2-15y3 -3xy2-38y3  y4 0  0  |
              | x2+13xy-32y2  16xy+35y2     0  -3xy2-35y3    -10xy2-38y3 0  y4 0  |
              | 39xy-48y2     x2-43xy-18y2  0  -43y3         xy2-19y3    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
               | -14x2+4xy+3y2 47x2-5xy+47y2 x3 x2y+4xy2-15y3 -3xy2-38y3  y4 0  0  |
               | x2+13xy-32y2  16xy+35y2     0  -3xy2-35y3    -10xy2-38y3 0  y4 0  |
               | 39xy-48y2     x2-43xy-18y2  0  -43y3         xy2-19y3    0  0  y4 |

          8                                                                           5
     1 : A  <----------------------------------------------------------------------- A  : 2
               {2} | -6xy2-41y3      38xy2-48y3    6y3       46y3      -27y3     |
               {2} | 18xy2+y3        -47y3         -18y3     -50y3     -49y3     |
               {3} | 43xy+44y2       21xy+4y2      -43y2     -17y2     4y2       |
               {3} | -43x2-48xy-45y2 -21x2+6xy-9y2 43xy+4y2  17xy+46y2 -4xy-33y2 |
               {3} | -18x2+28xy+45y2 -15xy+32y2    18xy-29y2 50xy-41y2 49xy+7y2  |
               {4} | 0               0             x-12y     31y       34y       |
               {4} | 0               0             -39y      x+42y     -38y      |
               {4} | 0               0             -11y      -18y      x-30y     |

          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 -16y  |
               {2} | 0 -39y  x+43y |
               {3} | 1 14    -47   |
               {3} | 0 31    -18   |
               {3} | 0 -11   -34   |
               {4} | 0 0     0     |
               {4} | 0 0     0     |
               {4} | 0 0     0     |

          5                                                                               8
     2 : A  <--------------------------------------------------------------------------- A  : 1
               {5} | 5   43  0 -20y    28x+24y xy+38y2      14xy+21y2    42xy+3y2    |
               {5} | -28 -36 0 24x+43y 10x+34y 3y2          xy+43y2      10xy-30y2   |
               {5} | 0   0   0 0       0       x2+12xy-25y2 -31xy+15y2   -34xy+20y2  |
               {5} | 0   0   0 0       0       39xy-45y2    x2-42xy+27y2 38xy+36y2   |
               {5} | 0   0   0 0       0       11xy-48y2    18xy+49y2    x2+30xy-2y2 |

                   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 :