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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 | 50x2-40xy-12y2 -35x2-37xy-15y2 |
              | 16x2-32xy+50y2 -43x2+48xy+5y2  |
              | -29x2-48xy-8y2 -6x2+xy         |

                            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 | 27x2-16xy+2y2 48x2+3xy+5y2 x3 x2y+7xy2-18y3 27xy2+49y3 y4 0  0  |
              | x2+28xy+22y2  27xy-7y2     0  22xy2+22y3    -9xy2+18y3 0  y4 0  |
              | 13xy-42y2     x2-26xy-2y2  0  6y3           xy2-25y3   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
               | 27x2-16xy+2y2 48x2+3xy+5y2 x3 x2y+7xy2-18y3 27xy2+49y3 y4 0  0  |
               | x2+28xy+22y2  27xy-7y2     0  22xy2+22y3    -9xy2+18y3 0  y4 0  |
               | 13xy-42y2     x2-26xy-2y2  0  6y3           xy2-25y3   0  0  y4 |

          8                                                                             5
     1 : A  <------------------------------------------------------------------------- A  : 2
               {2} | 48xy2+11y3     -41xy2+40y3    -48y3      -13y3     -33y3      |
               {2} | -46xy2-50y3    -42y3          46y3       -24y3     -44y3      |
               {3} | -35xy+3y2      -34xy+26y2     35y2       -15y2     31y2       |
               {3} | 35x2+28xy-35y2 34x2+35xy+23y2 -35xy-31y2 15xy+43y2 -31xy+12y2 |
               {3} | 46x2-22xy-35y2 -33xy-26y2     -46xy-29y2 24xy-46y2 44xy-36y2  |
               {4} | 0              0              x+22y      43y       44y        |
               {4} | 0              0              -44y       x         -6y        |
               {4} | 0              0              -39y       18y       x-22y      |

          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-28y -27y  |
               {2} | 0 -13y  x+26y |
               {3} | 1 -27   -48   |
               {3} | 0 -18   -17   |
               {3} | 0 -33   19    |
               {4} | 0 0     0     |
               {4} | 0 0     0     |
               {4} | 0 0     0     |

          5                                                                             8
     2 : A  <------------------------------------------------------------------------- A  : 1
               {5} | -41 -7  0 -22y   11x+22y  xy+44y2     39xy-12y2  21xy+4y2     |
               {5} | 16  -22 0 3x-25y -44x-28y -22y2       xy+45y2    9xy-50y2     |
               {5} | 0   0   0 0      0        x2-22xy+7y2 -43xy+21y2 -44xy+45y2   |
               {5} | 0   0   0 0      0        44xy-27y2   x2+20y2    6xy+14y2     |
               {5} | 0   0   0 0      0        39xy+16y2   -18xy+48y2 x2+22xy-27y2 |

                   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 :