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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 | -4x2-33xy+22y2 36x2+34xy+50y2 |
              | -5x2+48xy-44y2 5x2+33xy-3y2   |
              | 39x2+33xy-5y2  4x2+45xy+42y2  |

                            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 | 16x2+50xy+50y2 -11x2+47xy-50y2 x3 x2y+xy2-35y3 16xy2-26y3 y4 0  0  |
              | x2-9xy-11y2    -31xy+13y2      0  -15xy2+41y3  43xy2-8y3  0  y4 0  |
              | -15xy+11y2     x2+30xy+22y2    0  -39y3        xy2-20y3   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
               | 16x2+50xy+50y2 -11x2+47xy-50y2 x3 x2y+xy2-35y3 16xy2-26y3 y4 0  0  |
               | x2-9xy-11y2    -31xy+13y2      0  -15xy2+41y3  43xy2-8y3  0  y4 0  |
               | -15xy+11y2     x2+30xy+22y2    0  -39y3        xy2-20y3   0  0  y4 |

          8                                                                            5
     1 : A  <------------------------------------------------------------------------ A  : 2
               {2} | 24xy2+32y3     20xy2+31y3     -24y3     28y3       -12y3     |
               {2} | 31xy2+7y3      27y3           -31y3     -39y3      -49y3     |
               {3} | -7xy-9y2       -35xy-19y2     7y2       21y2       -5y2      |
               {3} | 7x2-50xy-30y2  35x2-33xy+10y2 -7xy-42y2 -21xy+35y2 5xy-25y2  |
               {3} | -31x2-15xy+3y2 22xy+2y2       31xy+8y2  39xy+36y2  49xy+40y2 |
               {4} | 0              0              x+4y      23y        32y       |
               {4} | 0              0              6y        x-29y      32y       |
               {4} | 0              0              -27y      9y         x+25y     |

          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+9y 31y   |
               {2} | 0 15y  x-30y |
               {3} | 1 -16  11    |
               {3} | 0 -29  36    |
               {3} | 0 -23  30    |
               {4} | 0 0    0     |
               {4} | 0 0    0     |
               {4} | 0 0    0     |

          5                                                                               8
     2 : A  <--------------------------------------------------------------------------- A  : 1
               {5} | -4  14  0 38y     13x-14y  xy-31y2    -2xy-37y2    -31xy-25y2   |
               {5} | -33 -50 0 26x+40y -43x+27y 15y2       xy+33y2      -43xy-29y2   |
               {5} | 0   0   0 0       0        x2-4xy-3y2 -23xy+16y2   -32xy+48y2   |
               {5} | 0   0   0 0       0        -6xy-4y2   x2+29xy-46y2 -32xy-37y2   |
               {5} | 0   0   0 0       0        27xy-22y2  -9xy+50y2    x2-25xy+49y2 |

                   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 :