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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 | 48x2-47xy+6y2  -31x2+41xy+23y2 |
              | 26x2+43xy+18y2 -17x2-4xy-48y2  |
              | 19x2+17xy-39y2 -30x2+12xy-46y2 |

                            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 | -3x2-10xy-36y2 -4x2-26xy+23y2 x3 x2y-10xy2+5y3 28xy2-15y3  y4 0  0  |
              | x2+21xy+33y2   32xy-7y2       0  35xy2-18y3    -36xy2-31y3 0  y4 0  |
              | -28xy-50y2     x2+2xy-24y2    0  20y3          xy2-12y3    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
               | -3x2-10xy-36y2 -4x2-26xy+23y2 x3 x2y-10xy2+5y3 28xy2-15y3  y4 0  0  |
               | x2+21xy+33y2   32xy-7y2       0  35xy2-18y3    -36xy2-31y3 0  y4 0  |
               | -28xy-50y2     x2+2xy-24y2    0  20y3          xy2-12y3    0  0  y4 |

          8                                                                           5
     1 : A  <----------------------------------------------------------------------- A  : 2
               {2} | 7xy2+16y3      11xy2-18y3     -7y3      37y3       13y3     |
               {2} | 48xy2-y3       -48y3          -48y3     -46y3      -9y3     |
               {3} | -37xy-15y2     -17xy-21y2     37y2      23y2       -6y2     |
               {3} | 37x2+23xy-37y2 17x2+22xy-45y2 -37xy-8y2 -23xy+25y2 6xy+13y2 |
               {3} | -48x2-3xy+42y2 16xy-5y2       48xy+4y2  46xy+19y2  9xy-35y2 |
               {4} | 0              0              x+43y     25y        -16y     |
               {4} | 0              0              -24y      x+y        -30y     |
               {4} | 0              0              19y       -31y       x-44y    |

          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-21y -32y |
               {2} | 0 28y   x-2y |
               {3} | 1 3     4    |
               {3} | 0 -42   23   |
               {3} | 0 12    41   |
               {4} | 0 0     0    |
               {4} | 0 0     0    |
               {4} | 0 0     0    |

          5                                                                              8
     2 : A  <-------------------------------------------------------------------------- A  : 1
               {5} | -25 24  0 2y     -40x-17y xy+48y2      22xy-25y2  -44xy+6y2    |
               {5} | 45  -12 0 6x-24y -8x+48y  -35y2        xy-27y2    36xy-42y2    |
               {5} | 0   0   0 0      0        x2-43xy+36y2 -25xy-20y2 16xy-27y2    |
               {5} | 0   0   0 0      0        24xy-10y2    x2-xy+28y2 30xy-43y2    |
               {5} | 0   0   0 0      0        -19xy+18y2   31xy-10y2  x2+44xy+37y2 |

                   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 :