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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 | 19x2-37xy-5y2  16x2+10xy+8y2  |
              | 13x2+18xy-13y2 42x2-13xy-23y2 |
              | 37x2-10xy-47y2 -9x2+17xy-10y2 |

                            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 | -10x2-3xy-34y2 -26x2+41xy+20y2 x3 x2y+34xy2+31y3 -10xy2-36y3 y4 0  |
              | x2-26xy+43y2   26xy-10y2       0  -26xy2+31y3    -49xy2-35y3 0  y4 |
              | 30xy+37y2      x2-19xy-17y2    0  -14y3          xy2+44y3    0  0  |

                            3
o4 : A-module, quotient of A
i5 : C = resolution N

      3      7      4
o5 = A  <-- A  <-- A  <-- 0
                           
     0      1      2      3

o5 : ChainComplex
i6 : d = C.dd

          3                                                                              7
o6 = 0 : A  <-------------------------------------------------------------------------- A  : 1
               | -10x2-3xy-34y2 -26x2+41xy+20y2 x3 x2y+34xy2+31y3 -10xy2-36y3 y4 0  |
               | x2-26xy+43y2   26xy-10y2       0  -26xy2+31y3    -49xy2-35y3 0  y4 |
               | 30xy+37y2      x2-19xy-17y2    0  -14y3          xy2+44y3    0  0  |

          7                                                                      4
     1 : A  <------------------------------------------------------------------ A  : 2
               {2} | 38xy2-28y3     25y3       -49xy2-35y3     23y4         |
               {2} | 29y3           -44y3      23xy2+2y3       49y4         |
               {3} | -17xy-7y2      47y2       44xy+17y2       0            |
               {3} | 17x2+11xy-28y2 -47xy+33y2 -44x2+44xy-43y2 3y3          |
               {3} | -22xy-4y2      44xy-39y2  -23x2-22xy+28y2 -49xy2-27y3  |
               {4} | 0              -26x+10y   0               x2-36xy+50y2 |
               {4} | 0              x+36y      0               -22y2        |

          4
     2 : A  <----- 0 : 3
               0

o6 : ChainComplexMap
i7 : s = nullhomotopy (x^3 * id_C)

          7                             3
o7 = 1 : A  <------------------------- A  : 0
               {2} | 0 x+26y -26y  |
               {2} | 0 -30y  x+19y |
               {3} | 1 10    26    |
               {3} | 0 -12   -9    |
               {3} | 0 27    47    |
               {4} | 0 0     0     |
               {4} | 0 0     0     |

          4                                                           7
     2 : A  <------------------------------------------------------- A  : 1
               {5} | 35  27 0 6x+43y 50x-22y  26y2  xy+28y2      |
               {5} | 0   0  0 0      0        22y2  x2-36xy+50y2 |
               {5} | -12 24 0 25y    -22x-18y 33y2  -42xy+45y2   |
               {6} | 0   0  0 0      0        x+36y 26x-10y      |

                   4
     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 |

          7                                    7
     1 : A  <-------------------------------- A  : 1
               {2} | x3 0  0  0  0  0  0  |
               {2} | 0  x3 0  0  0  0  0  |
               {3} | 0  0  x3 0  0  0  0  |
               {3} | 0  0  0  x3 0  0  0  |
               {3} | 0  0  0  0  x3 0  0  |
               {4} | 0  0  0  0  0  x3 0  |
               {4} | 0  0  0  0  0  0  x3 |

          4                           4
     2 : A  <----------------------- A  : 2
               {5} | x3 0  0  0  |
               {5} | 0  x3 0  0  |
               {5} | 0  0  x3 0  |
               {6} | 0  0  0  x3 |

     3 : 0 <----- 0 : 3
              0

o8 : ChainComplexMap
i9 : s^2

          4         3
o9 = 2 : A  <----- A  : 0
               0

                   7
     3 : 0 <----- A  : 1
              0

o9 : ChainComplexMap

Ways to use nullhomotopy :