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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 | -27x2+11xy+y2   -6x2-27xy+48y2 |
              | -44x2-22xy-15y2 47x2+12xy-31y2 |
              | -6x2+8xy+20y2   17x2-23xy+25y2 |

                            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 | -36x2+11xy+13y2 16x2-32xy-45y2 x3 x2y-5xy2-15y3 26xy2-5y3  y4 0  0  |
              | x2-44xy+12y2    -44xy-35y2     0  -34xy2+20y3   41xy2+33y3 0  y4 0  |
              | -13xy-47y2      x2-7xy-29y2    0  -35y3         xy2-26y3   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
               | -36x2+11xy+13y2 16x2-32xy-45y2 x3 x2y-5xy2-15y3 26xy2-5y3  y4 0  0  |
               | x2-44xy+12y2    -44xy-35y2     0  -34xy2+20y3   41xy2+33y3 0  y4 0  |
               | -13xy-47y2      x2-7xy-29y2    0  -35y3         xy2-26y3   0  0  y4 |

          8                                                                           5
     1 : A  <----------------------------------------------------------------------- A  : 2
               {2} | -40xy2-9y3     -46xy2-8y3    40y3       -22y3     -37y3     |
               {2} | -48xy2+33y3    30y3          48y3       -21y3     17y3      |
               {3} | -27xy+46y2     37xy-26y2     27y2       8y2       -20y2     |
               {3} | 27x2-13xy-23y2 -37x2+3xy+9y2 -27xy-33y2 -8xy-32y2 20xy+49y2 |
               {3} | 48x2-9xy-9y2   -4xy-35y2     -48xy-24y2 21xy+35y2 -17xy-4y2 |
               {4} | 0              0             x+15y      46y       42y       |
               {4} | 0              0             26y        x+24y     -11y      |
               {4} | 0              0             -22y       -35y      x-39y     |

          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+44y 44y  |
               {2} | 0 13y   x+7y |
               {3} | 1 36    -16  |
               {3} | 0 -49   -11  |
               {3} | 0 3     44   |
               {4} | 0 0     0    |
               {4} | 0 0     0    |
               {4} | 0 0     0    |

          5                                                                               8
     2 : A  <--------------------------------------------------------------------------- A  : 1
               {5} | -19 -43 0 -48y    40x-16y xy+28y2     -32xy+46y2   -27xy+7y2    |
               {5} | -7  12  0 30x+22y 21x+6y  34y2        xy+28y2      -41xy+21y2   |
               {5} | 0   0   0 0       0       x2-15xy-8y2 -46xy+21y2   -42xy+y2     |
               {5} | 0   0   0 0       0       -26xy+44y2  x2-24xy+36y2 11xy+45y2    |
               {5} | 0   0   0 0       0       22xy+22y2   35xy+18y2    x2+39xy-28y2 |

                   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 :