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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 | -7x2+10xy-5y2   x2+31xy-14y2    |
              | 10x2-41xy-44y2  -29x2-34xy-23y2 |
              | -35x2+17xy-33y2 -32x2+10xy-28y2 |

                            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 | 2x2-21xy+15y2 -5x2+37xy-10y2 x3 x2y+6xy2-4y3 24xy2-34y3 y4 0  0  |
              | x2+22xy-32y2  19xy-5y2       0  21xy2-47y3   13xy2+34y3 0  y4 0  |
              | 18xy-31y2     x2-4xy-31y2    0  -36y3        xy2+21y3   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
               | 2x2-21xy+15y2 -5x2+37xy-10y2 x3 x2y+6xy2-4y3 24xy2-34y3 y4 0  0  |
               | x2+22xy-32y2  19xy-5y2       0  21xy2-47y3   13xy2+34y3 0  y4 0  |
               | 18xy-31y2     x2-4xy-31y2    0  -36y3        xy2+21y3   0  0  y4 |

          8                                                                            5
     1 : A  <------------------------------------------------------------------------ A  : 2
               {2} | -42xy2-49y3     12xy2-37y3     42y3     8y3        -y3       |
               {2} | 45xy2-28y3      -44y3          -45y3    -7y3       -41y3     |
               {3} | -xy+15y2        15xy-37y2      y2       48y2       -42y2     |
               {3} | x2-46xy-19y2    -15x2+2xy-48y2 -xy+31y2 -48xy-32y2 42xy-25y2 |
               {3} | -45x2+26xy-26y2 -5xy+29y2      45xy+2y2 7xy-27y2   41xy-y2   |
               {4} | 0               0              x+21y    -24y       -24y      |
               {4} | 0               0              -17y     x+39y      36y       |
               {4} | 0               0              -29y     -49y       x+41y     |

          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-22y -19y |
               {2} | 0 -18y  x+4y |
               {3} | 1 -2    5    |
               {3} | 0 -25   21   |
               {3} | 0 -49   -15  |
               {4} | 0 0     0    |
               {4} | 0 0     0    |
               {4} | 0 0     0    |

          5                                                                                8
     2 : A  <---------------------------------------------------------------------------- A  : 1
               {5} | -37 -34 0 3y       -9x+42y xy-16y2      -38xy+4y2    -35xy-40y2  |
               {5} | 43  -48 0 -27x+37y -41x-2y -21y2        xy-49y2      -13xy+26y2  |
               {5} | 0   0   0 0        0       x2-21xy+30y2 24xy+39y2    24xy-29y2   |
               {5} | 0   0   0 0        0       17xy-44y2    x2-39xy-37y2 -36xy-45y2  |
               {5} | 0   0   0 0        0       29xy+45y2    49xy+8y2     x2-41xy+7y2 |

                   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 :