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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 | 27x2+48xy-20y2  23x2-22xy-38y2  |
              | -38x2+32xy+49y2 -24x2-46xy-21y2 |
              | 29x2+37xy-37y2  -24x2+33xy-28y2 |

                            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 | -25x2+39xy-21y2 3x2+4xy+31y2 x3 x2y+28xy2-12y3 3xy2+25y3   y4 0  0  |
              | x2+5xy+33y2     39xy+31y2    0  15xy2+14y3     -43xy2-27y3 0  y4 0  |
              | 16xy+40y2       x2-30xy-22y2 0  -y3            xy2-15y3    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
               | -25x2+39xy-21y2 3x2+4xy+31y2 x3 x2y+28xy2-12y3 3xy2+25y3   y4 0  0  |
               | x2+5xy+33y2     39xy+31y2    0  15xy2+14y3     -43xy2-27y3 0  y4 0  |
               | 16xy+40y2       x2-30xy-22y2 0  -y3            xy2-15y3    0  0  y4 |

          8                                                                            5
     1 : A  <------------------------------------------------------------------------ A  : 2
               {2} | -48xy2-15y3    -48xy2-41y3    48y3       -33y3     -48y3     |
               {2} | 11xy2-44y3     9y3            -11y3      -13y3     -21y3     |
               {3} | -39xy+21y2     17xy-24y2      39y2       -26y2     -23y2     |
               {3} | 39x2+10xy+10y2 -17x2+7xy+12y2 -39xy-31y2 26xy-38y2 23xy-26y2 |
               {3} | -11x2+6xy-30y2 35xy+45y2      11xy+38y2  13xy-45y2 21xy-30y2 |
               {4} | 0              0              x+27y      -25y      -20y      |
               {4} | 0              0              15y        x+y       -29y      |
               {4} | 0              0              -7y        18y       x-28y     |

          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-5y -39y  |
               {2} | 0 -16y x+30y |
               {3} | 1 25   -3    |
               {3} | 0 -15  42    |
               {3} | 0 -36  31    |
               {4} | 0 0    0     |
               {4} | 0 0    0     |
               {4} | 0 0    0     |

          5                                                                            8
     2 : A  <------------------------------------------------------------------------ A  : 1
               {5} | -40 12 0 36y    -46x-26y xy-9y2       -8xy-25y2  -44xy+25y2  |
               {5} | 44  30 0 -6x-9y 43x+32y  -15y2        xy+43y2    43xy-17y2   |
               {5} | 0   0  0 0      0        x2-27xy-11y2 25xy-50y2  20xy+38y2   |
               {5} | 0   0  0 0      0        -15xy+17y2   x2-xy+13y2 29xy-22y2   |
               {5} | 0   0  0 0      0        7xy-26y2     -18xy-8y2  x2+28xy-2y2 |

                   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 :