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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 | x2-44xy+4y2     17x2-16xy+30y2  |
              | 46x2-28xy+23y2  39x2-37xy+14y2  |
              | -35x2+44xy+31y2 -37x2+10xy+14y2 |

                            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 | 43x2+28xy+21y2 -7x2-11xy-36y2 x3 x2y-21xy2+19y3 -32xy2-50y3 y4 0  0  |
              | x2-31xy-28y2   12xy-49y2      0  12xy2+37y3     -13xy2+43y3 0  y4 0  |
              | -4xy-49y2      x2+31xy-22y2   0  4y3            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
               | 43x2+28xy+21y2 -7x2-11xy-36y2 x3 x2y-21xy2+19y3 -32xy2-50y3 y4 0  0  |
               | x2-31xy-28y2   12xy-49y2      0  12xy2+37y3     -13xy2+43y3 0  y4 0  |
               | -4xy-49y2      x2+31xy-22y2   0  4y3            xy2+20y3    0  0  y4 |

          8                                                                             5
     1 : A  <------------------------------------------------------------------------- A  : 2
               {2} | -36xy2+19y3     31xy2+36y3     36y3      23y3      37y3       |
               {2} | 29xy2-45y3      -32y3          -29y3     7y3       -29y3      |
               {3} | 20xy+37y2       11xy-40y2      -20y2     -41y2     39y2       |
               {3} | -20x2-38xy+35y2 -11x2-9xy+43y2 20xy+y2   41xy      -39xy+32y2 |
               {3} | -29x2-35xy-15y2 -2xy+4y2       29xy-21y2 -7xy-48y2 29xy+16y2  |
               {4} | 0               0              x-41y     -5y       -13y       |
               {4} | 0               0              49y       x+21y     -35y       |
               {4} | 0               0              27y       19y       x+20y      |

          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+31y -12y  |
               {2} | 0 4y    x-31y |
               {3} | 1 -43   7     |
               {3} | 0 -20   7     |
               {3} | 0 49    26    |
               {4} | 0 0     0     |
               {4} | 0 0     0     |
               {4} | 0 0     0     |

          5                                                                                  8
     2 : A  <------------------------------------------------------------------------------ A  : 1
               {5} | 40  -20 0 38y      -7x+31y  xy+3y2       -49xy+45y2   xy-17y2      |
               {5} | -47 37  0 -46x+42y -24x+15y -12y2        xy-7y2       13xy+6y2     |
               {5} | 0   0   0 0        0        x2+41xy-26y2 5xy-46y2     13xy+44y2    |
               {5} | 0   0   0 0        0        -49xy-6y2    x2-21xy+36y2 35xy+49y2    |
               {5} | 0   0   0 0        0        -27xy-40y2   -19xy+38y2   x2-20xy-10y2 |

                   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 :