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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 | -19x2-41xy+20y2 -27x2+43xy-9y2 |
              | -37x2-23xy+27y2 -50x2-14xy+2y2 |
              | -3x2-37xy-18y2  18x2+xy+19y2   |

                            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+13xy-29y2 -44x2-12xy+48y2 x3 x2y+20xy2-47y3 -15xy2-35y3 y4 0  0  |
              | x2-44xy-11y2   -22xy-8y2       0  -23xy2+41y3    21xy2-43y3  0  y4 0  |
              | 45xy+46y2      x2-4xy+11y2     0  -15y3          xy2+18y3    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+13xy-29y2 -44x2-12xy+48y2 x3 x2y+20xy2-47y3 -15xy2-35y3 y4 0  0  |
               | x2-44xy-11y2   -22xy-8y2       0  -23xy2+41y3    21xy2-43y3  0  y4 0  |
               | 45xy+46y2      x2-4xy+11y2     0  -15y3          xy2+18y3    0  0  y4 |

          8                                                                            5
     1 : A  <------------------------------------------------------------------------ A  : 2
               {2} | -13xy2-24y3     -30xy2-35y3  13y3      -22y3      -50y3      |
               {2} | 15xy2-2y3       -12y3        -15y3     -41y3      30y3       |
               {3} | 45xy+8y2        -47xy-49y2   -45y2     25y2       12y2       |
               {3} | -45x2+7xy-44y2  47x2-37xy+y2 45xy-15y2 -25xy+45y2 -12xy+42y2 |
               {3} | -15x2+40xy-19y2 47xy-9y2     15xy-38y2 41xy+16y2  -30xy+2y2  |
               {4} | 0               0            x-29y     -35y       -38y       |
               {4} | 0               0            14y       x-10y      -27y       |
               {4} | 0               0            26y       32y        x+39y      |

          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+44y 22y  |
               {2} | 0 -45y  x+4y |
               {3} | 1 -15   44   |
               {3} | 0 -27   -41  |
               {3} | 0 16    25   |
               {4} | 0 0     0    |
               {4} | 0 0     0    |
               {4} | 0 0     0    |

          5                                                                                8
     2 : A  <---------------------------------------------------------------------------- A  : 1
               {5} | 28 -32 0 27y     -27x+49y xy-11y2      -4xy-35y2    -2xy+38y2    |
               {5} | -7 39  0 43x-37y -28x+16y 23y2         xy+38y2      -21xy+41y2   |
               {5} | 0  0   0 0       0        x2+29xy-31y2 35xy+48y2    38xy-41y2    |
               {5} | 0  0   0 0       0        -14xy-36y2   x2+10xy-42y2 27xy-2y2     |
               {5} | 0  0   0 0       0        -26xy+y2     -32xy+18y2   x2-39xy-28y2 |

                   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 :