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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 | -36x2-50xy+35y2 -32xy+25y2    |
              | -37x2+47xy-43y2 3x2+23xy+47y2 |
              | -20x2-49xy-38y2 8x2+25xy-49y2 |

                            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 | -45x2-11xy-17y2 -21x2+38xy-41y2 x3 x2y+36xy2+14y3 50xy2-24y3  y4 0  0  |
              | x2+17xy+30y2    47xy-18y2       0  -47xy2+41y3    -47xy2-29y3 0  y4 0  |
              | -21xy+14y2      x2+11xy-24y2    0  -12y3          xy2+4y3     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
               | -45x2-11xy-17y2 -21x2+38xy-41y2 x3 x2y+36xy2+14y3 50xy2-24y3  y4 0  0  |
               | x2+17xy+30y2    47xy-18y2       0  -47xy2+41y3    -47xy2-29y3 0  y4 0  |
               | -21xy+14y2      x2+11xy-24y2    0  -12y3          xy2+4y3     0  0  y4 |

          8                                                                             5
     1 : A  <------------------------------------------------------------------------- A  : 2
               {2} | 25xy2+5y3      27xy2-45y3     -25y3      -32y3     49y3       |
               {2} | -16xy2+37y3    19y3           16y3       -17y3     17y3       |
               {3} | -49xy-50y2     -50xy-46y2     49y2       -21y2     -3y2       |
               {3} | 49x2-8xy+41y2  50x2-34xy-29y2 -49xy-43y2 21xy+31y2 3xy-29y2   |
               {3} | 16x2-24xy-26y2 37xy+33y2      -16xy-13y2 17xy+2y2  -17xy+36y2 |
               {4} | 0              0              x+16y      -11y      -28y       |
               {4} | 0              0              0          x+47y     -42y       |
               {4} | 0              0              -33y       0         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-17y -47y  |
               {2} | 0 21y   x-11y |
               {3} | 1 45    21    |
               {3} | 0 -10   40    |
               {3} | 0 4     -34   |
               {4} | 0 0     0     |
               {4} | 0 0     0     |
               {4} | 0 0     0     |

          5                                                                               8
     2 : A  <--------------------------------------------------------------------------- A  : 1
               {5} | 25 40  0 -8y     19x-48y xy+34y2      -20xy-22y2   20xy-18y2    |
               {5} | 6  -40 0 -2x-42y 44x-21y 47y2         xy-46y2      47xy+19y2    |
               {5} | 0  0   0 0       0       x2-16xy-32y2 11xy+14y2    28xy-40y2    |
               {5} | 0  0   0 0       0       -28y2        x2-47xy-13y2 42xy-35y2    |
               {5} | 0  0   0 0       0       33xy+36y2    -41y2        x2-38xy+45y2 |

                   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 :