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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 | 42x2+37xy  -39x2-19xy+16y2 |
              | -37xy-27y2 39x2+20xy+10y2  |
              | 29xy+30y2  10x2-41xy-32y2  |

                            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 | x2-40xy    34xy+42y2     0  -35xy2-15y3    -48xy2-3y3 y4 0  0  |
              | 40xy+21y2  14x2-44xy-8y2 x3 x2y-37xy2+33y3 -49xy2-7y3 0  y4 0  |
              | -45xy+44y2 x2-18xy+27y2  0  25y3           xy2-36y3   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
               | x2-40xy    34xy+42y2     0  -35xy2-15y3    -48xy2-3y3 y4 0  0  |
               | 40xy+21y2  14x2-44xy-8y2 x3 x2y-37xy2+33y3 -49xy2-7y3 0  y4 0  |
               | -45xy+44y2 x2-18xy+27y2  0  25y3           xy2-36y3   0  0  y4 |

          8                                                                           5
     1 : A  <----------------------------------------------------------------------- A  : 2
               {2} | -22xy2-25y3   -xy2-28y3      22y3       -17y3     21y3      |
               {2} | -15xy2+11y3   -19y3          15y3       -49y3     36y3      |
               {3} | xy+26y2       26xy+41y2      -y2        -16y2     -24y2     |
               {3} | -x2-27xy-39y2 -26x2+7xy-18y2 xy+y2      16xy+43y2 24xy+34y2 |
               {3} | 15x2+xy+46y2  18xy+26y2      -15xy-12y2 49xy+20y2 -36xy-y2  |
               {4} | 0             0              x-45y      36y       5y        |
               {4} | 0             0              46y        x-y       31y       |
               {4} | 0             0              -12y       -y        x+46y     |

          5
     2 : A  <----- 0 : 3
               0

o6 : ChainComplexMap
i7 : s = nullhomotopy (x^3 * id_C)

          8                             3
o7 = 1 : A  <------------------------- A  : 0
               {2} | x+40y 0 -34y  |
               {2} | 45y   0 x+18y |
               {3} | 0     1 -14   |
               {3} | 37    0 -6    |
               {3} | 41    0 -21   |
               {4} | 0     0 0     |
               {4} | 0     0 0     |
               {4} | 0     0 0     |

          5                                                                              8
     2 : A  <-------------------------------------------------------------------------- A  : 1
               {5} | 4  -39 0 42y      27x+y    xy+43y2      -10xy-32y2 -38xy-3y2   |
               {5} | 44 44  0 -35x-46y -36x-30y 44y2         xy+37y2    49xy+4y2    |
               {5} | 0  0   0 0        0        x2+45xy-15y2 -36xy-45y2 -5xy+10y2   |
               {5} | 0  0   0 0        0        -46xy+37y2   x2+xy+10y2 -31xy+9y2   |
               {5} | 0  0   0 0        0        12xy+43y2    xy+28y2    x2-46xy+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 :