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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 | -40x2+18xy-34y2 14x2+5xy+25y2  |
              | 33x2+14xy+y2    -37x2+35xy+3y2 |
              | -9x2+30y2       16x2+13xy-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 | -42x2+39xy-41y2 30x2+40xy+33y2 x3 x2y-44xy2+30y3 33xy2+6y3 y4 0  0  |
              | x2+24xy-35y2    -37xy-5y2      0  28xy2-23y3     6xy2+47y3 0  y4 0  |
              | 41xy+39y2       x2-18xy+5y2    0  34y3           xy2+14y3  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
               | -42x2+39xy-41y2 30x2+40xy+33y2 x3 x2y-44xy2+30y3 33xy2+6y3 y4 0  0  |
               | x2+24xy-35y2    -37xy-5y2      0  28xy2-23y3     6xy2+47y3 0  y4 0  |
               | 41xy+39y2       x2-18xy+5y2    0  34y3           xy2+14y3  0  0  y4 |

          8                                                                          5
     1 : A  <---------------------------------------------------------------------- A  : 2
               {2} | -49xy2+35y3     -9xy2-5y3     49y3     -8y3      47y3      |
               {2} | 33xy2-41y3      -3y3          -33y3    30y3      0         |
               {3} | 2xy-8y2         -40xy-50y2    -2y2     -50y2     -38y2     |
               {3} | -2x2-19xy-19y2  40x2+18xy+6y2 2xy+27y2 50xy+35y2 38xy+10y2 |
               {3} | -33x2+43xy+21y2 22xy-14y2     33xy-2y2 -30xy-8y2 12y2      |
               {4} | 0               0             x-23y    3y        40y       |
               {4} | 0               0             43y      x+41y     -2y       |
               {4} | 0               0             -10y     -7y       x-18y     |

          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-24y 37y   |
               {2} | 0 -41y  x+18y |
               {3} | 1 42    -30   |
               {3} | 0 -19   -36   |
               {3} | 0 5     14    |
               {4} | 0 0     0     |
               {4} | 0 0     0     |
               {4} | 0 0     0     |

          5                                                                                8
     2 : A  <---------------------------------------------------------------------------- A  : 1
               {5} | 43  -21 0 32y     -49x-26y xy-4y2       45xy+26y2   28y2         |
               {5} | -10 15  0 48x-20y 43x-y    -28y2        xy-45y2     -6xy-39y2    |
               {5} | 0   0   0 0       0        x2+23xy-45y2 -3xy-24y2   -40xy-30y2   |
               {5} | 0   0   0 0       0        -43xy-14y2   x2-41xy+6y2 2xy-43y2     |
               {5} | 0   0   0 0       0        10xy+8y2     7xy+11y2    x2+18xy+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 :