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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 | -11x2-40xy+28y2 -25x2+22xy+30y2 |
              | 25x2-11xy+14y2  3x2+20xy+9y2    |
              | -38x2+5xy+31y2  15x2-23xy-31y2  |

                            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 | -25x2-40xy-47y2 37x2-4xy-29y2 x3 x2y-19xy2-35y3 11xy2+26y3  y4 0  0  |
              | x2+7xy+16y2     -27xy-43y2    0  22xy2-27y3     -23xy2-12y3 0  y4 0  |
              | 31xy+13y2       x2-xy+29y2    0  -28y3          xy2-50y3    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
               | -25x2-40xy-47y2 37x2-4xy-29y2 x3 x2y-19xy2-35y3 11xy2+26y3  y4 0  0  |
               | x2+7xy+16y2     -27xy-43y2    0  22xy2-27y3     -23xy2-12y3 0  y4 0  |
               | 31xy+13y2       x2-xy+29y2    0  -28y3          xy2-50y3    0  0  y4 |

          8                                                                             5
     1 : A  <------------------------------------------------------------------------- A  : 2
               {2} | -20xy2+5y3      16xy2+49y3     20y3       25y3      -49y3     |
               {2} | -22xy2+7y3      41y3           22y3       35y3      -43y3     |
               {3} | 22xy-20y2       -36xy+13y2     -22y2      y2        -38y2     |
               {3} | -22x2-23xy+19y2 36x2-40xy+10y2 22xy+43y2  -xy-y2    38xy+9y2  |
               {3} | 22x2-36xy+23y2  -34xy+39y2     -22xy+29y2 -35xy+7y2 43xy+43y2 |
               {4} | 0               0              x+6y       -47y      -10y      |
               {4} | 0               0              14y        x-50y     -3y       |
               {4} | 0               0              39y        -8y       x+44y     |

          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-7y 27y |
               {2} | 0 -31y x+y |
               {3} | 1 25   -37 |
               {3} | 0 2    36  |
               {3} | 0 -29  44  |
               {4} | 0 0    0   |
               {4} | 0 0    0   |
               {4} | 0 0    0   |

          5                                                                               8
     2 : A  <--------------------------------------------------------------------------- A  : 1
               {5} | -40 -2 0 -40y     23x+16y  xy-16y2    -3xy+20y2    21xy+31y2    |
               {5} | 26  9  0 -14x-40y -14x-50y -22y2      xy+43y2      23xy-45y2    |
               {5} | 0   0  0 0        0        x2-6xy-2y2 47xy+27y2    10xy+45y2    |
               {5} | 0   0  0 0        0        -14xy-26y2 x2+50xy+48y2 3xy-21y2     |
               {5} | 0   0  0 0        0        -39xy+20y2 8xy+33y2     x2-44xy-46y2 |

                   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 :