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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 | 39x2+40xy+32y2 49x2+49xy-22y2 |
              | 2x2-11xy+35y2  31x2-38xy-17y2 |
              | -25x2+xy+3y2   -26x2+8xy+44y2 |

                            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 | -31x2-20xy+2y2 17xy+11y2   x3 x2y-44xy2+3y3 13xy2-23y3 y4 0  0  |
              | x2+9xy-14y2    -15xy+50y2  0  -25xy2-48y3   34xy2+12y3 0  y4 0  |
              | -27xy+26y2     x2+18xy+6y2 0  22y3          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
               | -31x2-20xy+2y2 17xy+11y2   x3 x2y-44xy2+3y3 13xy2-23y3 y4 0  0  |
               | x2+9xy-14y2    -15xy+50y2  0  -25xy2-48y3   34xy2+12y3 0  y4 0  |
               | -27xy+26y2     x2+18xy+6y2 0  22y3          xy2-15y3   0  0  y4 |

          8                                                                            5
     1 : A  <------------------------------------------------------------------------ A  : 2
               {2} | -22xy2-44y3    32xy2-3y3      22y3       -36y3     29y3      |
               {2} | -45xy2+36y3    -47y3          45y3       y3        -13y3     |
               {3} | -28xy-22y2     27xy+21y2      28y2       23y2      -35y2     |
               {3} | 28x2-13xy-30y2 -27x2-15xy-3y2 -28xy+35y2 -23xy+6y2 35xy+48y2 |
               {3} | 45x2+37xy+49y2 -10xy+12y2     -45xy+28y2 -xy+6y2   13xy+37y2 |
               {4} | 0              0              x          -21y      -16y      |
               {4} | 0              0              8y         x-35y     -13y      |
               {4} | 0              0              20y        -21y      x+35y     |

          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-9y 15y   |
               {2} | 0 27y  x-18y |
               {3} | 1 31   0     |
               {3} | 0 44   44    |
               {3} | 0 -48  16    |
               {4} | 0 0    0     |
               {4} | 0 0    0     |
               {4} | 0 0    0     |

          5                                                                              8
     2 : A  <-------------------------------------------------------------------------- A  : 1
               {5} | 28 -43 0 -37y     9x-36y  xy-34y2    9xy-32y2     -16xy+2y2    |
               {5} | -3 -28 0 -15x-45y 43x+13y 25y2       xy+50y2      -34xy+21y2   |
               {5} | 0  0   0 0        0       x2+17y2    21xy-40y2    16xy+16y2    |
               {5} | 0  0   0 0        0       -8xy-35y2  x2+35xy+17y2 13xy-27y2    |
               {5} | 0  0   0 0        0       -20xy+27y2 21xy-16y2    x2-35xy-34y2 |

                   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 :