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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 | -42x2-43xy-39y2 13x2-12xy+25y2  |
              | 50x2-22xy-9y2   11x2+50xy-13y2  |
              | -49x2-28xy-5y2  -41x2+49xy+17y2 |

                            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 | 9x2+16xy-y2  -25x2+44xy+41y2 x3 x2y-43xy2+38y3 -38xy2+13y3 y4 0  0  |
              | x2+30xy+34y2 29xy+39y2       0  48xy2-48y3     -18xy2+49y3 0  y4 0  |
              | -47xy+7y2    x2+33xy-y2      0  19y3           xy2-20y3    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
               | 9x2+16xy-y2  -25x2+44xy+41y2 x3 x2y-43xy2+38y3 -38xy2+13y3 y4 0  0  |
               | x2+30xy+34y2 29xy+39y2       0  48xy2-48y3     -18xy2+49y3 0  y4 0  |
               | -47xy+7y2    x2+33xy-y2      0  19y3           xy2-20y3    0  0  y4 |

          8                                                                             5
     1 : A  <------------------------------------------------------------------------- A  : 2
               {2} | -42xy2+6y3     -23xy2-2y3     42y3      9y3        41y3       |
               {2} | 21xy2-31y3     -y3            -21y3     7y3        35y3       |
               {3} | 7xy+39y2       -11xy+48y2     -7y2      47y2       -26y2      |
               {3} | -7x2-33xy-16y2 11x2+26xy-47y2 7xy-6y2   -47xy+29y2 26xy-9y2   |
               {3} | -21x2+6xy+44y2 24xy-15y2      21xy+25y2 -7xy+36y2  -35xy-19y2 |
               {4} | 0              0              x-2y      -30y       3y         |
               {4} | 0              0              -y        x-42y      -38y       |
               {4} | 0              0              -4y       12y        x+44y      |

          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-30y -29y  |
               {2} | 0 47y   x-33y |
               {3} | 1 -9    25    |
               {3} | 0 15    -2    |
               {3} | 0 -39   30    |
               {4} | 0 0     0     |
               {4} | 0 0     0     |
               {4} | 0 0     0     |

          5                                                                               8
     2 : A  <--------------------------------------------------------------------------- A  : 1
               {5} | 27 -13 0 -34y    24x+23y  xy-42y2     -34xy-43y2   32xy+48y2    |
               {5} | 25 47  0 46x-11y -49x+28y -48y2       xy+36y2      18xy-5y2     |
               {5} | 0  0   0 0       0        x2+2xy+22y2 30xy+43y2    -3xy-47y2    |
               {5} | 0  0   0 0       0        xy-6y2      x2+42xy+25y2 38xy+22y2    |
               {5} | 0  0   0 0       0        4xy+22y2    -12xy+43y2   x2-44xy-47y2 |

                   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 :