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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 | -37x2-18xy-36y2 34x2+39xy+32y2  |
              | -32x2-48xy+47y2 -8x2+39xy-48y2  |
              | 43x2+13xy-13y2  -17x2+43xy-40y2 |

                            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 | 6x2+22xy+46y2 13x2+23xy+24y2 x3 x2y-26xy2-46y3 -27xy2-35y3 y4 0  0  |
              | x2+41xy-41y2  20xy+34y2      0  25xy2-31y3     -30xy2-27y3 0  y4 0  |
              | 12xy+23y2     x2-26xy+45y2   0  -28y3          xy2-27y3    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
               | 6x2+22xy+46y2 13x2+23xy+24y2 x3 x2y-26xy2-46y3 -27xy2-35y3 y4 0  0  |
               | x2+41xy-41y2  20xy+34y2      0  25xy2-31y3     -30xy2-27y3 0  y4 0  |
               | 12xy+23y2     x2-26xy+45y2   0  -28y3          xy2-27y3    0  0  y4 |

          8                                                                             5
     1 : A  <------------------------------------------------------------------------- A  : 2
               {2} | 27xy2+37y3     -15xy2+6y3     -27y3      -29y3     17y3       |
               {2} | -48xy2+20y3    27y3           48y3       -34y3     -27y3      |
               {3} | -4xy-23y2      -41xy-36y2     4y2        -46y2     41y2       |
               {3} | 4x2-34xy-21y2  41x2-20xy-25y2 -4xy-44y2  46xy+3y2  -41xy+13y2 |
               {3} | 48x2+18xy+38y2 -12xy-10y2     -48xy-38y2 34xy-47y2 27xy-13y2  |
               {4} | 0              0              x-32y      37y       9y         |
               {4} | 0              0              22y        x+3y      -50y       |
               {4} | 0              0              41y        2y        x+29y      |

          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-41y -20y  |
               {2} | 0 -12y  x+26y |
               {3} | 1 -6    -13   |
               {3} | 0 -24   -39   |
               {3} | 0 -45   -38   |
               {4} | 0 0     0     |
               {4} | 0 0     0     |
               {4} | 0 0     0     |

          5                                                                              8
     2 : A  <-------------------------------------------------------------------------- A  : 1
               {5} | -18 5  0 -8y      40x-48y  xy-11y2      -47xy-17y2  31xy+48y2  |
               {5} | -42 -2 0 -32x-37y -31x-29y -25y2        xy-49y2     30xy-26y2  |
               {5} | 0   0  0 0        0        x2+32xy-15y2 -37xy-45y2  -9xy+42y2  |
               {5} | 0   0  0 0        0        -22xy+39y2   x2-3xy+16y2 50xy+12y2  |
               {5} | 0   0  0 0        0        -41xy+22y2   -2xy-35y2   x2-29xy-y2 |

                   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 :