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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 | -50x2+40xy+17y2 33x2+43xy-44y2  |
              | -13x2+34xy+36y2 11x2+11xy-17y2  |
              | 24x2-47xy+25y2  -23x2+30xy-21y2 |

                            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 | 16x2-50xy+19y2 15x2-17xy+11y2 x3 x2y+49xy2+34y3 -8xy2+34y3 y4 0  0  |
              | x2-27xy-12y2   -9xy-5y2       0  13xy2+19y3     19xy2-23y3 0  y4 0  |
              | -7xy-28y2      x2-31xy+49y2   0  -46y3          xy2+38y3   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
               | 16x2-50xy+19y2 15x2-17xy+11y2 x3 x2y+49xy2+34y3 -8xy2+34y3 y4 0  0  |
               | x2-27xy-12y2   -9xy-5y2       0  13xy2+19y3     19xy2-23y3 0  y4 0  |
               | -7xy-28y2      x2-31xy+49y2   0  -46y3          xy2+38y3   0  0  y4 |

          8                                                                             5
     1 : A  <------------------------------------------------------------------------- A  : 2
               {2} | 8xy2+12y3     39xy2-35y3     -8y3       21y3       42y3       |
               {2} | 6xy2-24y3     -32y3          -6y3       45y3       6y3        |
               {3} | -47xy+36y2    3xy-18y2       47y2       -2y2       10y2       |
               {3} | 47x2+21xy+2y2 -3x2+46xy-48y2 -47xy+44y2 2xy+46y2   -10xy-17y2 |
               {3} | -6x2+30xy+5y2 -35xy-32y2     6xy-6y2    -45xy+11y2 -6xy+45y2  |
               {4} | 0             0              x+37y      -4y        2y         |
               {4} | 0             0              11y        x-43y      -27y       |
               {4} | 0             0              -y         -20y       x+6y       |

          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+27y 9y    |
               {2} | 0 7y    x+31y |
               {3} | 1 -16   -15   |
               {3} | 0 18    14    |
               {3} | 0 30    -35   |
               {4} | 0 0     0     |
               {4} | 0 0     0     |
               {4} | 0 0     0     |

          5                                                                                8
     2 : A  <---------------------------------------------------------------------------- A  : 1
               {5} | 5  11  0 30y      -17x+49y xy-2y2       43xy+24y2    -xy-y2      |
               {5} | 17 -25 0 -34x+16y 3x-46y   -13y2        xy+y2        -19xy+19y2  |
               {5} | 0  0   0 0        0        x2-37xy+10y2 4xy-16y2     -2xy-8y2    |
               {5} | 0  0   0 0        0        -11xy-39y2   x2+43xy+22y2 27xy+11y2   |
               {5} | 0  0   0 0        0        xy+40y2      20xy+37y2    x2-6xy-32y2 |

                   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 :