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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 | -43x2-47xy-15y2 35x2+18xy+48y2  |
              | 20x2-10xy+41y2  -36x2-47xy+28y2 |
              | 31x2+18xy+5y2   -12x2-37xy+5y2  |

                            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 | 5x2+9xy-43y2 41x2+22xy+24y2 x3 x2y+49xy2+50y3 -27xy2+30y3 y4 0  0  |
              | x2-8xy-20y2  -31xy+24y2     0  -48xy2-44y3    -43xy2+50y3 0  y4 0  |
              | 33xy+31y2    x2-37xy+16y2   0  31y3           xy2+47y3    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
               | 5x2+9xy-43y2 41x2+22xy+24y2 x3 x2y+49xy2+50y3 -27xy2+30y3 y4 0  0  |
               | x2-8xy-20y2  -31xy+24y2     0  -48xy2-44y3    -43xy2+50y3 0  y4 0  |
               | 33xy+31y2    x2-37xy+16y2   0  31y3           xy2+47y3    0  0  y4 |

          8                                                                           5
     1 : A  <----------------------------------------------------------------------- A  : 2
               {2} | -4xy2-7y3       49xy2-20y3    4y3      -20y3      -45y3     |
               {2} | 7xy2-32y3       -7y3          -7y3     25y3       -23y3     |
               {3} | 38xy-28y2       -41xy-27y2    -38y2    18y2       -50y2     |
               {3} | -38x2+20xy+30y2 41x2-5xy+43y2 38xy+8y2 -18xy-11y2 50xy+46y2 |
               {3} | -7x2+11xy+43y2  48xy+13y2     7xy+21y2 -25xy-15y2 23xy+22y2 |
               {4} | 0               0             x+17y    45y        0         |
               {4} | 0               0             -46y     x-27y      -30y      |
               {4} | 0               0             -35y     -47y       x+10y     |

          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+8y 31y   |
               {2} | 0 -33y x+37y |
               {3} | 1 -5   -41   |
               {3} | 0 -9   23    |
               {3} | 0 -1   27    |
               {4} | 0 0    0     |
               {4} | 0 0    0     |
               {4} | 0 0    0     |

          5                                                                                 8
     2 : A  <----------------------------------------------------------------------------- A  : 1
               {5} | -29 -25 0 -3y      -29x+14y xy-4y2       -18xy-25y2   -40xy+23y2  |
               {5} | 30  28  0 -32x-19y 15x-8y   48y2         xy-35y2      43xy-43y2   |
               {5} | 0   0   0 0        0        x2-17xy+37y2 -45xy-46y2   -37y2       |
               {5} | 0   0   0 0        0        46xy-5y2     x2+27xy-32y2 30xy+5y2    |
               {5} | 0   0   0 0        0        35xy+5y2     47xy+32y2    x2-10xy-5y2 |

                   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 :