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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 | 6x2-20xy+34y2  -43x2-29xy+39y2 |
              | 27x2-21xy+47y2 48x2-28xy-9y2   |
              | 38x2-35xy-50y2 20x2+42xy-3y2   |

                            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 | -44x2-35xy+42y2 -43x2+27xy-13y2 x3 x2y-32xy2+26y3 -41xy2+40y3 y4 0  0  |
              | x2-37xy-15y2    47xy-20y2       0  34xy2-22y3     -3xy2+41y3  0  y4 0  |
              | -13xy-41y2      x2+3xy-28y2     0  -9y3           xy2-35y3    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
               | -44x2-35xy+42y2 -43x2+27xy-13y2 x3 x2y-32xy2+26y3 -41xy2+40y3 y4 0  0  |
               | x2-37xy-15y2    47xy-20y2       0  34xy2-22y3     -3xy2+41y3  0  y4 0  |
               | -13xy-41y2      x2+3xy-28y2     0  -9y3           xy2-35y3    0  0  y4 |

          8                                                                             5
     1 : A  <------------------------------------------------------------------------- A  : 2
               {2} | -31xy2-y3      -5xy2-33y3     31y3       19y3      15y3       |
               {2} | -50xy2-30y3    12y3           50y3       -10y3     0          |
               {3} | -38xy+30y2     -15xy+12y2     38y2       -33y2     45y2       |
               {3} | 38x2+25xy+46y2 15x2+46xy+26y2 -38xy+46y2 33xy-47y2 -45xy+28y2 |
               {3} | 50x2-50xy+22y2 -43xy-44y2     -50xy-21y2 10xy+15y2 -9y2       |
               {4} | 0              0              x+2y       -3y       12y        |
               {4} | 0              0              5y         x-49y     -2y        |
               {4} | 0              0              27y        -5y       x+47y      |

          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+37y -47y |
               {2} | 0 13y   x-3y |
               {3} | 1 44    43   |
               {3} | 0 0     -2   |
               {3} | 0 -22   32   |
               {4} | 0 0     0    |
               {4} | 0 0     0    |
               {4} | 0 0     0    |

          5                                                                               8
     2 : A  <--------------------------------------------------------------------------- A  : 1
               {5} | -44 -37 0 y       -2x-27y xy-42y2     20xy+38y2    -46y2        |
               {5} | -3  40  0 27x-18y 32x+14y -34y2       xy-6y2       3xy-3y2      |
               {5} | 0   0   0 0       0       x2-2xy+10y2 3xy-20y2     -12xy-12y2   |
               {5} | 0   0   0 0       0       -5xy+14y2   x2+49xy-28y2 2xy-37y2     |
               {5} | 0   0   0 0       0       -27xy-15y2  5xy+30y2     x2-47xy+18y2 |

                   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 :