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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+45xy+23y2 39x2-47xy-39y2  |
              | 19x2+21xy-21y2  -25x2+12xy-44y2 |
              | 4x2+32xy+11y2   3x2-3xy+47y2    |

                            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 | -48x2+22xy-5y2 17x2+33xy-21y2 x3 x2y-48xy2-15y3 -44xy2+34y3 y4 0  0  |
              | x2-34xy-7y2    -10xy+28y2     0  26xy2-12y3     14y3        0  y4 0  |
              | 38xy-19y2      x2-21xy-8y2    0  -38y3          xy2-36y3    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
               | -48x2+22xy-5y2 17x2+33xy-21y2 x3 x2y-48xy2-15y3 -44xy2+34y3 y4 0  0  |
               | x2-34xy-7y2    -10xy+28y2     0  26xy2-12y3     14y3        0  y4 0  |
               | 38xy-19y2      x2-21xy-8y2    0  -38y3          xy2-36y3    0  0  y4 |

          8                                                                           5
     1 : A  <----------------------------------------------------------------------- A  : 2
               {2} | -9xy2+40y3     37xy2-14y3    9y3        23y3     -8y3       |
               {2} | -45xy2+y3      7y3           45y3       -3y3     -40y3      |
               {3} | -12xy-14y2     -18xy+50y2    12y2       -3y2     23y2       |
               {3} | 12x2-26xy+5y2  18x2-36xy+9y2 -12xy+40y2 3xy+17y2 -23xy+43y2 |
               {3} | 45x2-43xy-21y2 -22xy+30y2    -45xy+42y2 3xy-8y2  40xy+29y2  |
               {4} | 0              0             x-40y      -27y     -9y        |
               {4} | 0              0             8y         x-45y    -38y       |
               {4} | 0              0             28y        -37y     x-16y      |

          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 10y   |
               {2} | 0 -38y  x+21y |
               {3} | 1 48    -17   |
               {3} | 0 34    -11   |
               {3} | 0 50    -32   |
               {4} | 0 0     0     |
               {4} | 0 0     0     |
               {4} | 0 0     0     |

          5                                                                               8
     2 : A  <--------------------------------------------------------------------------- A  : 1
               {5} | -46 -15 0 11y     9x-28y  xy+43y2      -27xy-43y2   44xy-9y2    |
               {5} | 40  5   0 -28x+4y -6x-14y -26y2        xy-2y2       48y2        |
               {5} | 0   0   0 0       0       x2+40xy+21y2 27xy+2y2     9xy+15y2    |
               {5} | 0   0   0 0       0       -8xy-27y2    x2+45xy-17y2 38xy+24y2   |
               {5} | 0   0   0 0       0       -28xy-46y2   37xy-14y2    x2+16xy-4y2 |

                   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 :