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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 | -41x2+38xy+13y2 31x2+40xy-2y2 |
              | -27x2+11xy-49y2 30x2+21xy-2y2 |
              | 49x2+3xy+23y2   8x2+34xy+50y2 |

                            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 | -27x2+15xy   42x2-26xy+25y2 x3 x2y-23xy2+30y3 -22xy2-15y3 y4 0  0  |
              | x2+12xy-16y2 46xy-16y2      0  2xy2+16y3      25xy2-26y3  0  y4 0  |
              | 35xy-34y2    x2-26xy-43y2   0  3y3            xy2+36y3    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
               | -27x2+15xy   42x2-26xy+25y2 x3 x2y-23xy2+30y3 -22xy2-15y3 y4 0  0  |
               | x2+12xy-16y2 46xy-16y2      0  2xy2+16y3      25xy2-26y3  0  y4 0  |
               | 35xy-34y2    x2-26xy-43y2   0  3y3            xy2+36y3    0  0  y4 |

          8                                                                            5
     1 : A  <------------------------------------------------------------------------ A  : 2
               {2} | 24xy2-46y3      -9xy2+23y3     -24y3     35y3      11y3      |
               {2} | 42xy2+39y3      27y3           -42y3     -5y3      9y3       |
               {3} | -8xy+45y2       46xy+28y2      8y2       -21y2     -6y2      |
               {3} | 8x2+18xy-27y2   -46x2-16xy+7y2 -8xy+38y2 21xy+31y2 6xy-40y2  |
               {3} | -42x2-16xy+11y2 22xy-42y2      42xy-23y2 5xy+18y2  -9xy-48y2 |
               {4} | 0               0              x-31y     -30y      -48y      |
               {4} | 0               0              -40y      x+48y     15y       |
               {4} | 0               0              11y       32y       x-17y     |

          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-12y -46y  |
               {2} | 0 -35y  x+26y |
               {3} | 1 27    -42   |
               {3} | 0 20    15    |
               {3} | 0 49    6     |
               {4} | 0 0     0     |
               {4} | 0 0     0     |
               {4} | 0 0     0     |

          5                                                                                 8
     2 : A  <----------------------------------------------------------------------------- A  : 1
               {5} | 18  29  0 -25y     12x-43y xy+42y2      41xy-22y2    7xy-8y2      |
               {5} | -14 -10 0 -11x+39y 46x-10y -2y2         xy+y2        -25xy-6y2    |
               {5} | 0   0   0 0        0       x2+31xy+17y2 30xy-26y2    48xy+36y2    |
               {5} | 0   0   0 0        0       40xy-10y2    x2-48xy+45y2 -15xy-39y2   |
               {5} | 0   0   0 0        0       -11xy+10y2   -32xy-45y2   x2+17xy+39y2 |

                   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 :