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Macaulay2Doc :: nullhomotopy

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 | 32x2-31xy+28y2 25x2+46xy+9y2 |
              | 11x2+13xy+32y2 35x2-9xy-4y2  |
              | -17x2-8xy+8y2  24x2+xy+41y2  |

                            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 | -38x2-25xy+34y2 27x2-50xy-45y2 x3 x2y-2xy2+y3 -44xy2+41y3 y4 0  0  |
              | x2+15xy-14y2    3xy-5y2        0  -28xy2+14y3 -5xy2+32y3  0  y4 0  |
              | -47xy-40y2      x2-24xy+39y2   0  -4y3        xy2-17y3    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
               | -38x2-25xy+34y2 27x2-50xy-45y2 x3 x2y-2xy2+y3 -44xy2+41y3 y4 0  0  |
               | x2+15xy-14y2    3xy-5y2        0  -28xy2+14y3 -5xy2+32y3  0  y4 0  |
               | -47xy-40y2      x2-24xy+39y2   0  -4y3        xy2-17y3    0  0  y4 |

          8                                                                             5
     1 : A  <------------------------------------------------------------------------- A  : 2
               {2} | -48xy2+36y3     3xy2-47y3      48y3      -34y3      23y3      |
               {2} | 14xy2+5y3       45y3           -14y3     -12y3      31y3      |
               {3} | -8xy-13y2       -47xy-10y2     8y2       25y2       -28y2     |
               {3} | 8x2+39xy+28y2   47x2+16xy-40y2 -8xy-26y2 -25xy+15y2 28xy+42y2 |
               {3} | -14x2-10xy+39y2 -19xy+45y2     14xy+5y2  12xy+36y2  -31xy-5y2 |
               {4} | 0               0              x-17y     34y        -32y      |
               {4} | 0               0              -2y       x+21y      49y       |
               {4} | 0               0              23y       -18y       x-4y      |

          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-15y -3y   |
               {2} | 0 47y   x+24y |
               {3} | 1 38    -27   |
               {3} | 0 4     -5    |
               {3} | 0 -42   -8    |
               {4} | 0 0     0     |
               {4} | 0 0     0     |
               {4} | 0 0     0     |

          5                                                                              8
     2 : A  <-------------------------------------------------------------------------- A  : 1
               {5} | -3  -15 0 21y     36x+44y xy+26y2      -28xy-49y2  5xy-44y2    |
               {5} | -49 22  0 43x-44y 39x-27y 28y2         xy-29y2     5xy-33y2    |
               {5} | 0   0   0 0       0       x2+17xy-10y2 -34xy+5y2   32xy+15y2   |
               {5} | 0   0   0 0       0       2xy+8y2      x2-21xy-4y2 -49xy-12y2  |
               {5} | 0   0   0 0       0       -23xy-43y2   18xy-29y2   x2+4xy+14y2 |

                   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 :