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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 | -9x2+31xy+6y2  x2-6xy-12y2    |
              | 28x2+33xy+34y2 17x2+36xy-16y2 |
              | -19x2-35xy+9y2 13x2-45xy-27y2 |

                            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 | 15x2+31xy-43y2 -4x2-41xy+32y2 x3 x2y+38xy2+38y3 26xy2-35y3 y4 0  0  |
              | x2+10xy-17y2   13xy+21y2      0  -4xy2-37y3     34xy2+11y3 0  y4 0  |
              | 15xy+40y2      x2+8xy         0  -45y3          xy2-46y3   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
               | 15x2+31xy-43y2 -4x2-41xy+32y2 x3 x2y+38xy2+38y3 26xy2-35y3 y4 0  0  |
               | x2+10xy-17y2   13xy+21y2      0  -4xy2-37y3     34xy2+11y3 0  y4 0  |
               | 15xy+40y2      x2+8xy         0  -45y3          xy2-46y3   0  0  y4 |

          8                                                                            5
     1 : A  <------------------------------------------------------------------------ A  : 2
               {2} | 46xy2-13y3     43xy2-27y3    -46y3     43y3       37y3       |
               {2} | -16xy2-15y3    11y3          16y3      14y3       17y3       |
               {3} | 4xy+2y2        -36xy-y2      -4y2      -18y2      9y2        |
               {3} | -4x2-10xy+22y2 36x2+8xy-47y2 4xy+8y2   18xy       -9xy-6y2   |
               {3} | 16x2+9xy-46y2  -46xy-28y2    -16xy+6y2 -14xy+15y2 -17xy+40y2 |
               {4} | 0              0             x+42y     7y         49y        |
               {4} | 0              0             21y       x-31y      14y        |
               {4} | 0              0             -49y      -20y       x-11y      |

          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-10y -13y |
               {2} | 0 -15y  x-8y |
               {3} | 1 -15   4    |
               {3} | 0 -42   2    |
               {3} | 0 28    -44  |
               {4} | 0 0     0    |
               {4} | 0 0     0    |
               {4} | 0 0     0    |

          5                                                                                 8
     2 : A  <----------------------------------------------------------------------------- A  : 1
               {5} | -27 28  0 -15y     19x-46y xy+16y2      -37xy+y2     20xy+22y2    |
               {5} | 15  -25 0 -14x-21y 47x+29y 4y2          xy+36y2      -34xy-28y2   |
               {5} | 0   0   0 0        0       x2-42xy+15y2 -7xy+6y2     -49xy+y2     |
               {5} | 0   0   0 0        0       -21xy+50y2   x2+31xy+20y2 -14xy+37y2   |
               {5} | 0   0   0 0        0       49xy-20y2    20xy-8y2     x2+11xy-35y2 |

                   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 :