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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 | -31x2-6xy+38y2  -20x2-4xy+42y2 |
              | 25x2+42xy-5y2   12x2-38xy-12y2 |
              | -42x2-21xy+30y2 24x2+17xy-45y2 |

                            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 | -19x2-42xy-36y2 -25x2+4xy+45y2 x3 x2y-2xy2-9y3 12xy2-36y3  y4 0  0  |
              | x2-17xy+17y2    49xy-9y2       0  -17xy2-41y3  -45xy2-34y3 0  y4 0  |
              | -30xy-6y2       x2-39xy+39y2   0  12y3         xy2+36y3    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
               | -19x2-42xy-36y2 -25x2+4xy+45y2 x3 x2y-2xy2-9y3 12xy2-36y3  y4 0  0  |
               | x2-17xy+17y2    49xy-9y2       0  -17xy2-41y3  -45xy2-34y3 0  y4 0  |
               | -30xy-6y2       x2-39xy+39y2   0  12y3         xy2+36y3    0  0  y4 |

          8                                                                           5
     1 : A  <----------------------------------------------------------------------- A  : 2
               {2} | 46xy2+6y3     -46xy2+34y3    -46y3     -32y3      21y3      |
               {2} | -4xy2         34y3           4y3       -2y3       -37y3     |
               {3} | -4xy-36y2     -27xy+3y2      4y2       25y2       -34y2     |
               {3} | 4x2-38xy-17y2 27x2-15xy+29y2 -4xy-27y2 -25xy-25y2 34xy+7y2  |
               {3} | 4x2+22xy-27y2 -21xy-24y2     -4xy-22y2 2xy-2y2    37xy-29y2 |
               {4} | 0             0              x-43y     -46y       26y       |
               {4} | 0             0              -27y      x+39y      25y       |
               {4} | 0             0              -23y      -45y       x+4y      |

          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+17y -49y  |
               {2} | 0 30y   x+39y |
               {3} | 1 19    25    |
               {3} | 0 4     40    |
               {3} | 0 -31   12    |
               {4} | 0 0     0     |
               {4} | 0 0     0     |
               {4} | 0 0     0     |

          5                                                                                8
     2 : A  <---------------------------------------------------------------------------- A  : 1
               {5} | 33  -3  0 3y      -25x-17y xy+44y2      50xy-14y2    16xy-23y2   |
               {5} | -20 -16 0 15x-37y -15x-41y 17y2         xy-27y2      45xy+10y2   |
               {5} | 0   0   0 0       0        x2+43xy-32y2 46xy+24y2    -26xy-43y2  |
               {5} | 0   0   0 0       0        27xy+38y2    x2-39xy+22y2 -25xy-31y2  |
               {5} | 0   0   0 0       0        23xy-9y2     45xy+32y2    x2-4xy+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 :