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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 | -17x2-14xy+32y2 31x2+5xy-44y2 |
              | -30x2-40xy+3y2  45x2-15xy+5y2 |
              | -2x2-40xy-20y2  -3x2-9xy+29y2 |

                            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 | 27x2+16xy-10y2 -43x2-31xy+33y2 x3 x2y+2xy2+10y3 -31xy2-45y3 y4 0  0  |
              | x2-50xy-23y2   -38xy-10y2      0  4xy2-34y3     -32xy2-9y3  0  y4 0  |
              | -23xy+13y2     x2-39xy+17y2    0  -3y3          xy2+18y3    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
               | 27x2+16xy-10y2 -43x2-31xy+33y2 x3 x2y+2xy2+10y3 -31xy2-45y3 y4 0  0  |
               | x2-50xy-23y2   -38xy-10y2      0  4xy2-34y3     -32xy2-9y3  0  y4 0  |
               | -23xy+13y2     x2-39xy+17y2    0  -3y3          xy2+18y3    0  0  y4 |

          8                                                                            5
     1 : A  <------------------------------------------------------------------------ A  : 2
               {2} | 19xy2+8y3       -xy2-20y3      -19y3     45y3      43y3      |
               {2} | 20xy2+17y3      0              -20y3     41y3      -15y3     |
               {3} | -12xy+37y2      25xy-30y2      12y2      11y2      17y2      |
               {3} | 12x2-31xy-40y2  -25x2+6xy-22y2 -12xy-6y2 -11xy+6y2 -17xy+6y2 |
               {3} | -20x2-20xy+22y2 3xy+22y2       20xy+3y2  -41xy+6y2 15xy-19y2 |
               {4} | 0               0              x-42y     14y       10y       |
               {4} | 0               0              -6y       x-14y     -37y      |
               {4} | 0               0              10y       42y       x-45y     |

          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+50y 38y   |
               {2} | 0 23y   x+39y |
               {3} | 1 -27   43    |
               {3} | 0 27    -25   |
               {3} | 0 14    -46   |
               {4} | 0 0     0     |
               {4} | 0 0     0     |
               {4} | 0 0     0     |

          5                                                                               8
     2 : A  <--------------------------------------------------------------------------- A  : 1
               {5} | 31 28  0 41y    5x-41y   xy-14y2      3xy-y2       26xy-49y2    |
               {5} | 20 -32 0 4x-44y -38x-35y -4y2         xy-15y2      32xy+3y2     |
               {5} | 0  0   0 0      0        x2+42xy-38y2 -14xy+40y2   -10xy+26y2   |
               {5} | 0  0   0 0      0        6xy-34y2     x2+14xy-28y2 37xy+2y2     |
               {5} | 0  0   0 0      0        -10xy-11y2   -42xy-15y2   x2+45xy-35y2 |

                   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 :