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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 | 15x2-25xy-45y2  -17x2+26xy+4y2  |
              | -30x2+9xy+49y2  -2x2-8xy-37y2   |
              | -22x2-15xy-37y2 -11x2+12xy+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 | 2x2-32xy+37y2 -8x2+31xy+48y2 x3 x2y-45xy2-6y3 -45xy2+6y3 y4 0  0  |
              | x2+34xy+38y2  -33xy+24y2     0  -8xy2+21y3    26xy2+23y3 0  y4 0  |
              | -49xy-48y2    x2+17xy+35y2   0  -29y3         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
               | 2x2-32xy+37y2 -8x2+31xy+48y2 x3 x2y-45xy2-6y3 -45xy2+6y3 y4 0  0  |
               | x2+34xy+38y2  -33xy+24y2     0  -8xy2+21y3    26xy2+23y3 0  y4 0  |
               | -49xy-48y2    x2+17xy+35y2   0  -29y3         xy2+7y3    0  0  y4 |

          8                                                                           5
     1 : A  <----------------------------------------------------------------------- A  : 2
               {2} | 9xy2+34y3      -4xy2+2y3     -9y3       -6y3      -47y3     |
               {2} | 14xy2+36y3     -11y3         -14y3      -38y3     -46y3     |
               {3} | -44xy+37y2     -50xy+27y2    44y2       -47y2     -30y2     |
               {3} | 44x2-7xy-36y2  50x2+9xy+42y2 -44xy-30y2 47xy-2y2  30xy+15y2 |
               {3} | -14x2+26xy+5y2 -48xy-17y2    14xy+39y2  38xy+35y2 46xy+36y2 |
               {4} | 0              0             x-15y      6y        -17y      |
               {4} | 0              0             7y         x-27y     30y       |
               {4} | 0              0             26y        32y       x+42y     |

          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-34y 33y   |
               {2} | 0 49y   x-17y |
               {3} | 1 -2    8     |
               {3} | 0 -13   -31   |
               {3} | 0 -27   -48   |
               {4} | 0 0     0     |
               {4} | 0 0     0     |
               {4} | 0 0     0     |

          5                                                                               8
     2 : A  <--------------------------------------------------------------------------- A  : 1
               {5} | -38 11 0 -22y    36x+41y xy+9y2       46xy+25y2    -40xy+17y2   |
               {5} | -37 -3 0 -2x+48y 37x+33y 8y2          xy-11y2      -26xy+34y2   |
               {5} | 0   0  0 0       0       x2+15xy+27y2 -6xy+12y2    17xy+24y2    |
               {5} | 0   0  0 0       0       -7xy-19y2    x2+27xy+14y2 -30xy+28y2   |
               {5} | 0   0  0 0       0       -26xy+17y2   -32xy+30y2   x2-42xy-41y2 |

                   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 :