nullhomotopy -- make a null homotopy

nullhomotopy -- make a null homotopy

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 which 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 | 42x2-50xy+39y2 -39x2+30xy+19y2 |

              | 9x2-15xy-22y2  -38x2+2xy-4y2   |

              | 50x2+45xy-29y2 -36x2-16xy-6y2  |



                            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 | -41x2-26xy+17y2 -50x2-42xy-4y2 x3 x2y-16xy2+50y3 -12xy2-19y3 0  0  y4 |

              | 20xy-5y2        x2+6xy-24y2    0  -12xy2+45y3    18xy2-37y3  0  y4 0  |

              | x2+21xy-5y2     -25xy-6y2      0  16y3           xy2+y3      y4 0  0  |



                            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

               | -25xy-6y2      x2+21xy-5y2     0  16y3           xy2+y3      0  0  y4 |

               | x2+6xy-24y2    20xy-5y2        0  -12xy2+45y3    18xy2-37y3  0  y4 0  |

               | -50x2-42xy-4y2 -41x2-26xy+17y2 x3 x2y-16xy2+50y3 -12xy2-19y3 y4 0  0  |



          8                                                                              5

     1 : A  <-------------------------------------------------------------------------- A  : 2

               {2} | -3xy2+8y3       43xy2+23y3     3y3       16y3       -50y3      |

               {2} | -7xy2-6y3       36y3           7y3       3y3        30y3       |

               {3} | 15xy+24y2       -12xy-49y2     -15y2     20y2       -35y2      |

               {3} | -15x2-11xy+43y2 12x2-33xy+35y2 15xy-13y2 -20xy-39y2 35xy+41y2  |

               {3} | 7x2+8xy+37y2    39xy-40y2      -7xy-2y2  -3xy-47y2  -30xy+13y2 |

               {4} | 0               0              x         -41y       12y        |

               {4} | 0               0              12y       x+11y      10y        |

               {4} | 0               0              -40y      -26y       x-11y      |



          5

     2 : A  <----- 0 : 3

               0



o6 : ChainComplexMap

i7 : s = nullhomotopy (x^3 * id_C)



          8                            3

o7 = 1 : A  <------------------------ A  : 0

               {2} | -20y  x-6y 0 |

               {2} | x-21y 25y  0 |

               {3} | 41    50   1 |

               {3} | -17   -41  0 |

               {3} | 47    38   0 |

               {4} | 0     0    0 |

               {4} | 0     0    0 |

               {4} | 0     0    0 |



          5                                                                               8

     2 : A  <--------------------------------------------------------------------------- A  : 1

               {5} | 9   -50 0 32y      29x+21y xy+50y2    -14xy+35y2   -39xy+44y2   |

               {5} | -19 -19 0 -42x-25y 11x+41y 12y2       xy+43y2      -18xy+42y2   |

               {5} | 0   0   0 0        0       x2+38y2    41xy+45y2    -12xy-37y2   |

               {5} | 0   0   0 0        0       -12xy+35y2 x2-11xy-25y2 -10xy+43y2   |

               {5} | 0   0   0 0        0       40xy+27y2  26xy+24y2    x2+11xy-13y2 |



                   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 :

nullhomotopy ChainComplexMap