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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 | 4x2+38xy+42y2   40x2-47xy+30y2 |
              | -15x2+35xy+48y2 45x2-38xy-34y2 |
              | 12x2+42xy-28y2  15x2+10xy-49y2 |

                            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 | -45x2-30xy-36y2 3x2+33xy+9y2 x3 x2y+35xy2-34y3 -18xy2+41y3 y4 0  0  |
              | x2-50xy-9y2     33xy+18y2    0  41xy2+19y3     -48xy2-23y3 0  y4 0  |
              | 33xy+18y2       x2-31xy+37y2 0  -42y3          xy2+49y3    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
               | -45x2-30xy-36y2 3x2+33xy+9y2 x3 x2y+35xy2-34y3 -18xy2+41y3 y4 0  0  |
               | x2-50xy-9y2     33xy+18y2    0  41xy2+19y3     -48xy2-23y3 0  y4 0  |
               | 33xy+18y2       x2-31xy+37y2 0  -42y3          xy2+49y3    0  0  y4 |

          8                                                                            5
     1 : A  <------------------------------------------------------------------------ A  : 2
               {2} | -49xy2+19y3    45xy2+24y3     49y3      -49y3     45y3       |
               {2} | 22xy2-45y3     43y3           -22y3     -25y3     20y3       |
               {3} | -5xy+34y2      -26xy+14y2     5y2       -28y2     -42y2      |
               {3} | 5x2-48xy-41y2  26x2-10xy+30y2 -5xy+14y2 28xy-34y2 42xy+34y2  |
               {3} | -22x2-4xy-37y2 -32xy-30y2     22xy+49y2 25xy-15y2 -20xy-41y2 |
               {4} | 0              0              x+25y     41y       35y        |
               {4} | 0              0              -19y      x+7y      -29y       |
               {4} | 0              0              38y       3y        x-32y      |

          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+50y -33y  |
               {2} | 0 -33y  x+31y |
               {3} | 1 45    -3    |
               {3} | 0 -45   5     |
               {3} | 0 36    -7    |
               {4} | 0 0     0     |
               {4} | 0 0     0     |
               {4} | 0 0     0     |

          5                                                                            8
     2 : A  <------------------------------------------------------------------------ A  : 1
               {5} | 20 41 0 -5y    -23x+26y xy+24y2      -31xy-6y2  45xy-10y2    |
               {5} | -6 23 0 35x-5y -15x+40y -41y2        xy-46y2    48xy-38y2    |
               {5} | 0  0  0 0      0        x2-25xy-36y2 -41xy+3y2  -35xy-20y2   |
               {5} | 0  0  0 0      0        19xy+7y2     x2-7xy-9y2 29xy-41y2    |
               {5} | 0  0  0 0      0        -38xy-20y2   -3xy-32y2  x2+32xy+45y2 |

                   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 :