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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 | -50x2+44xy+15y2 43x2-30xy-28y2 |
              | -24x2-19xy+14y2 36x2-37xy-4y2  |
              | 26x2+49xy+5y2   35x2-45xy+11y2 |

                            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+36xy+13y2 -25x2-35xy+32y2 x3 x2y-28xy2+4y3 -36xy2+45y3 y4 0  0  |
              | x2-29xy+43y2    23xy+48y2       0  -32xy2-48y3   40xy2+33y3  0  y4 0  |
              | 23xy+19y2       x2+27xy-25y2    0  11y3          xy2-10y3    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+36xy+13y2 -25x2-35xy+32y2 x3 x2y-28xy2+4y3 -36xy2+45y3 y4 0  0  |
               | x2-29xy+43y2    23xy+48y2       0  -32xy2-48y3   40xy2+33y3  0  y4 0  |
               | 23xy+19y2       x2+27xy-25y2    0  11y3          xy2-10y3    0  0  y4 |

          8                                                                             5
     1 : A  <------------------------------------------------------------------------- A  : 2
               {2} | -17xy2+13y3     2xy2-45y3      17y3      -19y3     -42y3      |
               {2} | 3xy2+46y3       43y3           -3y3      3y3       -26y3      |
               {3} | 39xy+35y2       -19xy-8y2      -39y2     -43y2     13y2       |
               {3} | -39x2+31xy+50y2 19x2-16xy+39y2 39xy+35y2 43xy-11y2 -13xy-17y2 |
               {3} | -3x2-44xy-5y2   5xy-39y2       3xy-2y2   -3xy-46y2 26xy+50y2  |
               {4} | 0               0              x+27y     43y       4y         |
               {4} | 0               0              48y       x+47y     -18y       |
               {4} | 0               0              5y        -4y       x+27y      |

          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+29y -23y  |
               {2} | 0 -23y  x-27y |
               {3} | 1 25    25    |
               {3} | 0 13    -3    |
               {3} | 0 36    -30   |
               {4} | 0 0     0     |
               {4} | 0 0     0     |
               {4} | 0 0     0     |

          5                                                                               8
     2 : A  <--------------------------------------------------------------------------- A  : 1
               {5} | 12 -10 0 -7y     -34x-50y xy-4y2       -xy+6y2     -25xy+27y2   |
               {5} | 8  -31 0 16x+31y -6x-15y  32y2         xy+32y2     -40xy+39y2   |
               {5} | 0  0   0 0       0        x2-27xy-15y2 -43xy+35y2  -4xy+48y2    |
               {5} | 0  0   0 0       0        -48xy+28y2   x2-47xy+2y2 18xy-29y2    |
               {5} | 0  0   0 0       0        -5xy-23y2    4xy+20y2    x2-27xy+13y2 |

                   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 :