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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 | 2x2-3xy-24y2   -19x2-23xy+40y2 |
              | 43x2-9xy+5y2   -5x2+38xy-45y2  |
              | 21x2+17xy-37y2 42x2-4xy-22y2   |

                            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 | 28x2-32xy+29y2 -38x2-26xy+2y2 x3 x2y+37xy2+34y3 -40xy2-y3  y4 0  0  |
              | x2+46xy+45y2   16xy-39y2      0  -5xy2+46y3     -6xy2+24y3 0  y4 0  |
              | -24xy-15y2     x2-27xy-48y2   0  -42y3          xy2-2y3    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
               | 28x2-32xy+29y2 -38x2-26xy+2y2 x3 x2y+37xy2+34y3 -40xy2-y3  y4 0  0  |
               | x2+46xy+45y2   16xy-39y2      0  -5xy2+46y3     -6xy2+24y3 0  y4 0  |
               | -24xy-15y2     x2-27xy-48y2   0  -42y3          xy2-2y3    0  0  y4 |

          8                                                                            5
     1 : A  <------------------------------------------------------------------------ A  : 2
               {2} | -39xy2+22y3    -47xy2-44y3   39y3       -28y3      -22y3     |
               {2} | 7xy2+31y3      30y3          -7y3       29y3       -5y3      |
               {3} | -41xy+19y2     -31xy+13y2    41y2       -9y2       -30y2     |
               {3} | 41x2+47xy+24y2 31x2-46xy+4y2 -41xy+35y2 9xy-11y2   30xy+21y2 |
               {3} | -7x2+21xy+y2   43xy+31y2     7xy+49y2   -29xy+27y2 5xy       |
               {4} | 0              0             x-36y      44y        35y       |
               {4} | 0              0             34y        x+27y      31y       |
               {4} | 0              0             -y         -42y       x+9y      |

          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-46y -16y  |
               {2} | 0 24y   x+27y |
               {3} | 1 -28   38    |
               {3} | 0 10    -15   |
               {3} | 0 -37   -11   |
               {4} | 0 0     0     |
               {4} | 0 0     0     |
               {4} | 0 0     0     |

          5                                                                                 8
     2 : A  <----------------------------------------------------------------------------- A  : 1
               {5} | 38  -16 0 50y      -29x-25y xy           -33xy-20y2   44xy+25y2   |
               {5} | -18 -37 0 -13x-17y -4x+32y  5y2          xy-42y2      6xy+33y2    |
               {5} | 0   0   0 0        0        x2+36xy+30y2 -44xy-48y2   -35xy+15y2  |
               {5} | 0   0   0 0        0        -34xy-34y2   x2-27xy+14y2 -31xy-17y2  |
               {5} | 0   0   0 0        0        xy+13y2      42xy-41y2    x2-9xy-44y2 |

                   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 :