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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 | 16x2-18xy-9y2  -46x2+13xy+16y2 |
              | 3x2+5xy+32y2   -25x2+7xy+3y2   |
              | 31x2+13xy-21y2 -12x2+17xy+30y2 |

                            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 | 7x2-9xy-50y2 -36x2-16xy-15y2 x3 x2y+44xy2+17y3 17xy2-41y3  y4 0  0  |
              | x2+45xy-37y2 -27xy-41y2      0  -18xy2-9y3     -37xy2-12y3 0  y4 0  |
              | 26xy+23y2    x2+37xy-42y2    0  21y3           xy2-31y3    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
               | 7x2-9xy-50y2 -36x2-16xy-15y2 x3 x2y+44xy2+17y3 17xy2-41y3  y4 0  0  |
               | x2+45xy-37y2 -27xy-41y2      0  -18xy2-9y3     -37xy2-12y3 0  y4 0  |
               | 26xy+23y2    x2+37xy-42y2    0  21y3           xy2-31y3    0  0  y4 |

          8                                                                              5
     1 : A  <-------------------------------------------------------------------------- A  : 2
               {2} | -11xy2-21y3     -35xy2-4y3      11y3      -31y3     -21y3      |
               {2} | 48xy2-9y3       -8y3            -48y3     -15y3     -42y3      |
               {3} | 31xy-14y2       30xy+11y2       -31y2     -46y2     37y2       |
               {3} | -31x2-41xy-38y2 -30x2+39xy-19y2 31xy-46y2 46xy+24y2 -37xy-47y2 |
               {3} | -48x2+5xy+23y2  33xy-5y2        48xy+4y2  15xy-49y2 42xy+37y2  |
               {4} | 0               0               x-32y     50y       30y        |
               {4} | 0               0               -8y       x-13y     47y        |
               {4} | 0               0               33y       -21y      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-45y 27y   |
               {2} | 0 -26y  x-37y |
               {3} | 1 -7    36    |
               {3} | 0 -6    10    |
               {3} | 0 -12   2     |
               {4} | 0 0     0     |
               {4} | 0 0     0     |
               {4} | 0 0     0     |

          5                                                                             8
     2 : A  <------------------------------------------------------------------------- A  : 1
               {5} | 25 -21 0 -44y    -40x-y   xy+50y2     -6xy-18y2   -37xy-7y2   |
               {5} | 11 -15 0 37x-31y -26x+32y 18y2        xy-22y2     37xy+13y2   |
               {5} | 0  0   0 0       0        x2+32xy-2y2 -50xy+49y2  -30xy+13y2  |
               {5} | 0  0   0 0       0        8xy-8y2     x2+13xy-6y2 -47xy-49y2  |
               {5} | 0  0   0 0       0        -33xy-9y2   21xy-32y2   x2-45xy+8y2 |

                   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 :