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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 | -17x2-50xy+21y2 8x2+15xy-20y2   |
              | -37x2+26xy-32y2 34x2+39xy-8y2   |
              | -7x2+15xy-y2    -37x2-20xy-21y2 |

                            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 | -20x2-4xy-44y2 36x2-15xy+42y2 x3 x2y-13xy2+20y3 -12xy2-47y3 y4 0  0  |
              | x2-2xy+43y2    -22xy+37y2     0  41xy2+8y3      -32xy2-39y3 0  y4 0  |
              | -35xy+y2       x2-48xy-34y2   0  31y3           xy2+41y3    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
               | -20x2-4xy-44y2 36x2-15xy+42y2 x3 x2y-13xy2+20y3 -12xy2-47y3 y4 0  0  |
               | x2-2xy+43y2    -22xy+37y2     0  41xy2+8y3      -32xy2-39y3 0  y4 0  |
               | -35xy+y2       x2-48xy-34y2   0  31y3           xy2+41y3    0  0  y4 |

          8                                                                               5
     1 : A  <--------------------------------------------------------------------------- A  : 2
               {2} | 28xy2-30y3      -7xy2+13y3      -28y3     36y3       8y3        |
               {2} | -2xy2-18y3      -11y3           2y3       34y3       y3         |
               {3} | 41xy            22xy+9y2        -41y2     -12y2      33y2       |
               {3} | -41x2+22xy-40y2 -22x2-31xy-19y2 41xy-22y2 12xy-49y2  -33xy-14y2 |
               {3} | 2x2-30xy-48y2   44xy            -2xy+48y2 -34xy-25y2 -xy-23y2   |
               {4} | 0               0               x-34y     -39y       14y        |
               {4} | 0               0               47y       x+45y      46y        |
               {4} | 0               0               22y       28y        x-11y      |

          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+2y 22y   |
               {2} | 0 35y  x+48y |
               {3} | 1 20   -36   |
               {3} | 0 -4   40    |
               {3} | 0 32   -23   |
               {4} | 0 0    0     |
               {4} | 0 0    0     |
               {4} | 0 0    0     |

          5                                                                               8
     2 : A  <--------------------------------------------------------------------------- A  : 1
               {5} | -16 -39 0 y       -50x+28y xy+23y2      17xy-40y2    -50xy-11y2 |
               {5} | 48  24  0 -23x+3y -17x+9y  -41y2        xy-y2        32xy+4y2   |
               {5} | 0   0   0 0       0        x2+34xy+35y2 39xy-37y2    -14xy      |
               {5} | 0   0   0 0       0        -47xy+14y2   x2-45xy-35y2 -46xy      |
               {5} | 0   0   0 0       0        -22xy+23y2   -28xy-7y2    x2+11xy    |

                   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 :