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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 | -14x2+47xy-32y2 -17x2+30xy+16y2 |
              | -14x2+40xy+40y2 49x2+41xy-42y2  |
              | -43x2-46xy-35y2 -39x2-24xy-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 | -48x2+36xy-48y2 23x2-41xy-40y2 x3 x2y+2xy2-21y3 -43xy2-38y3 y4 0  0  |
              | x2+15xy+10y2    -34xy+24y2     0  13xy2+49y3    -31xy2-20y3 0  y4 0  |
              | 17xy-39y2       x2-35xy+37y2   0  20y3          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
               | -48x2+36xy-48y2 23x2-41xy-40y2 x3 x2y+2xy2-21y3 -43xy2-38y3 y4 0  0  |
               | x2+15xy+10y2    -34xy+24y2     0  13xy2+49y3    -31xy2-20y3 0  y4 0  |
               | 17xy-39y2       x2-35xy+37y2   0  20y3          xy2+36y3    0  0  y4 |

          8                                                                            5
     1 : A  <------------------------------------------------------------------------ A  : 2
               {2} | 10xy2-37y3     10xy2+29y3    -10y3      7y3       -11y3      |
               {2} | 8xy2+6y3       -7y3          -8y3       -47y3     17y3       |
               {3} | -19xy+27y2     -7xy+40y2     19y2       5y2       -38y2      |
               {3} | 19x2-12xy+2y2  7x2+22xy-17y2 -19xy-15y2 -5xy+14y2 38xy-9y2   |
               {3} | -8x2+12xy+34y2 20y2          8xy-18y2   47xy-23y2 -17xy+27y2 |
               {4} | 0              0             x+19y      -3y       -21y       |
               {4} | 0              0             -40y       x+13y     -24y       |
               {4} | 0              0             46y        35y       x-32y      |

          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-15y 34y   |
               {2} | 0 -17y  x+35y |
               {3} | 1 48    -23   |
               {3} | 0 39    -41   |
               {3} | 0 2     4     |
               {4} | 0 0     0     |
               {4} | 0 0     0     |
               {4} | 0 0     0     |

          5                                                                                 8
     2 : A  <----------------------------------------------------------------------------- A  : 1
               {5} | -25 -50 0 0       -38x+44y xy+10y2      -32xy-12y2   -40xy+50y2   |
               {5} | 19  -46 0 29x-19y 31x+21y  -13y2        xy+46y2      31xy+4y2     |
               {5} | 0   0   0 0       0        x2-19xy+20y2 3xy-23y2     21xy+42y2    |
               {5} | 0   0   0 0       0        40xy+40y2    x2-13xy-46y2 24xy-17y2    |
               {5} | 0   0   0 0       0        -46xy+22y2   -35xy+5y2    x2+32xy+26y2 |

                   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 :