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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 | -27x2-42xy-41y2 50x2-43xy+33y2  |
              | -49x2+19xy+4y2  -24x2+28xy+28y2 |
              | 49x2-15xy-43y2  13x2+15xy-15y2  |

                            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 | -20x2+37xy+14y2 -2x2-36xy-26y2 x3 x2y-38xy2-28y3 49xy2-2y3 y4 0  0  |
              | x2-42xy-21y2    -21xy+10y2     0  -30xy2-7y3     24xy2-4y3 0  y4 0  |
              | 38xy+23y2       x2+48xy+18y2   0  -27y3          xy2-15y3  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
               | -20x2+37xy+14y2 -2x2-36xy-26y2 x3 x2y-38xy2-28y3 49xy2-2y3 y4 0  0  |
               | x2-42xy-21y2    -21xy+10y2     0  -30xy2-7y3     24xy2-4y3 0  y4 0  |
               | 38xy+23y2       x2+48xy+18y2   0  -27y3          xy2-15y3  0  0  y4 |

          8                                                                            5
     1 : A  <------------------------------------------------------------------------ A  : 2
               {2} | 41xy2+4y3      -12xy2+40y3   -41y3      -y3        13y3      |
               {2} | -13xy2-14y3    48y3          13y3       46y3       -24y3     |
               {3} | 32xy-45y2      -40xy         -32y2      20y2       -23y2     |
               {3} | -32x2-4xy-36y2 40x2-33xy+4y2 32xy+49y2  -20xy-50y2 23xy+11y2 |
               {3} | 13x2+27xy+7y2  -27xy+4y2     -13xy-13y2 -46xy+34y2 24xy+22y2 |
               {4} | 0              0             x+36y      -21y       -50y      |
               {4} | 0              0             7y         x+12y      -29y      |
               {4} | 0              0             19y        -29y       x-48y     |

          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+42y 21y   |
               {2} | 0 -38y  x-48y |
               {3} | 1 20    2     |
               {3} | 0 20    -44   |
               {3} | 0 3     -27   |
               {4} | 0 0     0     |
               {4} | 0 0     0     |
               {4} | 0 0     0     |

          5                                                                                8
     2 : A  <---------------------------------------------------------------------------- A  : 1
               {5} | -8  -29 0 22y     -31x+2y  xy-19y2     -12xy-4y2    37xy-10y2    |
               {5} | -17 -12 0 48x+37y -45x-26y 30y2        xy+39y2      -24xy+21y2   |
               {5} | 0   0   0 0       0        x2-36xy-3y2 21xy+38y2    50xy-3y2     |
               {5} | 0   0   0 0       0        -7xy-13y2   x2-12xy+30y2 29xy-13y2    |
               {5} | 0   0   0 0       0        -19xy-27y2  29xy+39y2    x2+48xy-27y2 |

                   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 :