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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 | 48x2-10xy-15y2 -8x2-37xy+32y2  |
              | 31x2-xy+11y2   -43x2+30xy-24y2 |
              | -11x2-43xy+2y2 41x2-46xy+10y2  |

                            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+11xy-4y2 -20x2-14xy+36y2 x3 x2y-41xy2-37y3 -5xy2-y3   y4 0  0  |
              | x2-39xy-49y2  -18xy+17y2      0  -14xy2+34y3    -2xy2+30y3 0  y4 0  |
              | 35xy+y2       x2-26xy+21y2    0  -2y3           xy2        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+11xy-4y2 -20x2-14xy+36y2 x3 x2y-41xy2-37y3 -5xy2-y3   y4 0  0  |
               | x2-39xy-49y2  -18xy+17y2      0  -14xy2+34y3    -2xy2+30y3 0  y4 0  |
               | 35xy+y2       x2-26xy+21y2    0  -2y3           xy2        0  0  y4 |

          8                                                                             5
     1 : A  <------------------------------------------------------------------------- A  : 2
               {2} | 16xy2+40y3     27xy2+29y3     -16y3      21y3       -38y3     |
               {2} | -36xy2+42y3    -47y3          36y3       -7y3       -36y3     |
               {3} | 4xy+20y2       -38xy+19y2     -4y2       50y2       -21y2     |
               {3} | -4x2-39xy-44y2 38x2+50xy+26y2 4xy+19y2   -50xy-25y2 21xy+45y2 |
               {3} | 36x2-31xy-50y2 -14xy-43y2     -36xy-11y2 7xy-7y2    36xy+31y2 |
               {4} | 0              0              x+39y      10y        12y       |
               {4} | 0              0              5y         x-14y      27y       |
               {4} | 0              0              5y         -25y       x-25y     |

          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+39y 18y   |
               {2} | 0 -35y  x+26y |
               {3} | 1 -14   20    |
               {3} | 0 -45   -21   |
               {3} | 0 47    25    |
               {4} | 0 0     0     |
               {4} | 0 0     0     |
               {4} | 0 0     0     |

          5                                                                                8
     2 : A  <---------------------------------------------------------------------------- A  : 1
               {5} | -49 47  0 48y    -14x-46y xy-y2        -3xy+22y2    -xy+15y2     |
               {5} | 4   -45 0 8x-48y -44x+44y 14y2         xy-25y2      2xy+47y2     |
               {5} | 0   0   0 0      0        x2-39xy+15y2 -10xy-50y2   -12xy+34y2   |
               {5} | 0   0   0 0      0        -5xy-43y2    x2+14xy-25y2 -27xy+17y2   |
               {5} | 0   0   0 0      0        -5xy+46y2    25xy+15y2    x2+25xy+10y2 |

                   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 :