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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 | 9x2-45xy+40y2  x2+21xy+35y2   |
              | -7x2-16xy+28y2 3x2+39xy-16y2  |
              | 28x2-11xy+22y2 38x2+49xy+41y2 |

                            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 | 32x2+36xy+10y2 48x2+11xy+40y2 x3 x2y+41xy2+45y3 -21xy2+36y3 y4 0  0  |
              | x2-24xy+28y2   -21xy+8y2      0  -21xy2-25y3    4xy2-9y3    0  y4 0  |
              | 8xy-17y2       x2-2xy+29y2    0  42y3           xy2-46y3    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
               | 32x2+36xy+10y2 48x2+11xy+40y2 x3 x2y+41xy2+45y3 -21xy2+36y3 y4 0  0  |
               | x2-24xy+28y2   -21xy+8y2      0  -21xy2-25y3    4xy2-9y3    0  y4 0  |
               | 8xy-17y2       x2-2xy+29y2    0  42y3           xy2-46y3    0  0  y4 |

          8                                                                            5
     1 : A  <------------------------------------------------------------------------ A  : 2
               {2} | -45xy2-11y3     -30xy2+26y3    45y3     -38y3      -14y3     |
               {2} | 18xy2-38y3      -49y3          -18y3    7y3        -2y3      |
               {3} | 20xy-50y2       -13xy-y2       -20y2    32y2       -43y2     |
               {3} | -20x2-43xy+14y2 13x2+24xy-23y2 20xy-8y2 -32xy+25y2 43xy+35y2 |
               {3} | -18x2+42xy+39y2 46xy-11y2      18xy-4y2 -7xy+27y2  2xy+9y2   |
               {4} | 0               0              x-34y    23y        38y       |
               {4} | 0               0              -39y     x-43y      -50y      |
               {4} | 0               0              25y      50y        x-24y     |

          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+24y 21y  |
               {2} | 0 -8y   x+2y |
               {3} | 1 -32   -48  |
               {3} | 0 -16   29   |
               {3} | 0 11    9    |
               {4} | 0 0     0    |
               {4} | 0 0     0    |
               {4} | 0 0     0    |

          5                                                                               8
     2 : A  <--------------------------------------------------------------------------- A  : 1
               {5} | -5  -50 0 33y     28x-5y  xy+35y2     -6xy-44y2    45xy-10y2    |
               {5} | -22 -17 0 -31x+7y 12x-46y 21y2        xy-10y2      -4xy+15y2    |
               {5} | 0   0   0 0       0       x2+34xy-3y2 -23xy+28y2   -38xy-21y2   |
               {5} | 0   0   0 0       0       39xy+36y2   x2+43xy-33y2 50xy+50y2    |
               {5} | 0   0   0 0       0       -25xy+34y2  -50xy-48y2   x2+24xy+36y2 |

                   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 :