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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 | 49x2+36xy-40y2  39x2-33xy+9y2   |
              | 6x2-5xy+24y2    -46x2-10xy-33y2 |
              | -43x2-25xy+41y2 -25x2+39xy-12y2 |

                            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 | 22x2-15xy-25y2 -38x2+37xy-19y2 x3 x2y+24xy2+6y3 35xy2+11y3 y4 0  0  |
              | x2-43xy-29y2   27xy-14y2       0  50xy2+16y3    6xy2+33y3  0  y4 0  |
              | 3xy+47y2       x2+xy+15y2      0  -19y3         xy2-4y3    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
               | 22x2-15xy-25y2 -38x2+37xy-19y2 x3 x2y+24xy2+6y3 35xy2+11y3 y4 0  0  |
               | x2-43xy-29y2   27xy-14y2       0  50xy2+16y3    6xy2+33y3  0  y4 0  |
               | 3xy+47y2       x2+xy+15y2      0  -19y3         xy2-4y3    0  0  y4 |

          8                                                                             5
     1 : A  <------------------------------------------------------------------------- A  : 2
               {2} | -26xy2+18y3     -41xy2+46y3    26y3      47y3      20y3       |
               {2} | 31xy2+3y3       9y3            -31y3     -46y3     48y3       |
               {3} | 20xy            -19xy+14y2     -20y2     -40y2     31y2       |
               {3} | -20x2-18xy+24y2 19x2+28xy-11y2 20xy+18y2 40xy+y2   -31xy-30y2 |
               {3} | -31x2+45xy+33y2 -30xy-30y2     31xy-48y2 46xy+41y2 -48xy+19y2 |
               {4} | 0               0              x-24y     46y       -31y       |
               {4} | 0               0              0         x-44y     -6y        |
               {4} | 0               0              -y        -23y      x-33y      |

          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+43y -27y |
               {2} | 0 -3y   x-y  |
               {3} | 1 -22   38   |
               {3} | 0 -35   14   |
               {3} | 0 29    -34  |
               {4} | 0 0     0    |
               {4} | 0 0     0    |
               {4} | 0 0     0    |

          5                                                                           8
     2 : A  <----------------------------------------------------------------------- A  : 1
               {5} | 27 38 0 -22y    13x+7y xy-20y2    8xy+38y2     18xy-21y2    |
               {5} | 9  18 0 16x+33y 19x-3y -50y2      xy+50y2      -6xy-2y2     |
               {5} | 0  0  0 0       0      x2+24xy+y2 -46xy+9y2    31xy-24y2    |
               {5} | 0  0  0 0       0      6y2        x2+44xy-47y2 6xy-43y2     |
               {5} | 0  0  0 0       0      xy-44y2    23xy+8y2     x2+33xy+46y2 |

                   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 :