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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 | -7x2+31xy-13y2  x2-28xy+14y2    |
              | -46x2-46xy-17y2 -43x2+28xy-32y2 |
              | -6x2+18xy-10y2  22x2+25xy+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 | 13x2+19xy    -48x2-33xy+19y2 x3 x2y+30xy2-37y3 -6xy2-45y3  y4 0  0  |
              | x2+25xy-39y2 18xy-18y2       0  -48xy2+3y3     -45xy2+10y3 0  y4 0  |
              | 8xy-41y2     x2+3xy+13y2     0  15y3           xy2+28y3    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
               | 13x2+19xy    -48x2-33xy+19y2 x3 x2y+30xy2-37y3 -6xy2-45y3  y4 0  0  |
               | x2+25xy-39y2 18xy-18y2       0  -48xy2+3y3     -45xy2+10y3 0  y4 0  |
               | 8xy-41y2     x2+3xy+13y2     0  15y3           xy2+28y3    0  0  y4 |

          8                                                                            5
     1 : A  <------------------------------------------------------------------------ A  : 2
               {2} | 27xy2+23y3      5xy2-47y3      -27y3     7y3       -49y3     |
               {2} | -44xy2+4y3      -6y3           44y3      -42y3     -10y3     |
               {3} | 47xy-40y2       2xy+15y2       -47y2     -26y2     -38y2     |
               {3} | -47x2-42xy-19y2 -2x2-20xy+23y2 47xy-19y2 26xy-9y2  38xy-14y2 |
               {3} | 44x2-9xy+42y2   -4xy+y2        -44xy+5y2 42xy+19y2 10xy-25y2 |
               {4} | 0               0              x-y       7y        -39y      |
               {4} | 0               0              49y       x-40y     19y       |
               {4} | 0               0              -19y      32y       x+41y     |

          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-25y -18y |
               {2} | 0 -8y   x-3y |
               {3} | 1 -13   48   |
               {3} | 0 23    22   |
               {3} | 0 -38   39   |
               {4} | 0 0     0    |
               {4} | 0 0     0    |
               {4} | 0 0     0    |

          5                                                                           8
     2 : A  <----------------------------------------------------------------------- A  : 1
               {5} | 33 6   0 -23y    -39x-5y xy+16y2    22xy+42y2    -14xy+43y2 |
               {5} | 44 -29 0 50x-10y -43x+3y 48y2       xy+y2        45xy+30y2  |
               {5} | 0  0   0 0       0       x2+xy-26y2 -7xy-20y2    39xy-13y2  |
               {5} | 0  0   0 0       0       -49xy-47y2 x2+40xy+26y2 -19xy+27y2 |
               {5} | 0  0   0 0       0       19xy       -32xy        x2-41xy    |

                   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 :