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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 | -30x2-16xy-32y2 2x2+31xy-41y2   |
              | 6x2-29xy-3y2    -50x2+47xy+39y2 |
              | -33x2+28xy+11y2 17x2+18xy-38y2  |

                            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 | 41x2-35xy+33y2 -10x2-12xy-42y2 x3 x2y+23xy2+40y3 -15xy2+24y3 y4 0  0  |
              | x2-29xy+36y2   -35xy+25y2      0  -36xy2-22y3    -9xy2+44y3  0  y4 0  |
              | -11xy-35y2     x2+40xy-22y2    0  25y3           xy2+26y3    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
               | 41x2-35xy+33y2 -10x2-12xy-42y2 x3 x2y+23xy2+40y3 -15xy2+24y3 y4 0  0  |
               | x2-29xy+36y2   -35xy+25y2      0  -36xy2-22y3    -9xy2+44y3  0  y4 0  |
               | -11xy-35y2     x2+40xy-22y2    0  25y3           xy2+26y3    0  0  y4 |

          8                                                                            5
     1 : A  <------------------------------------------------------------------------ A  : 2
               {2} | 22xy2+36y3      -39xy2+10y3    -22y3     13y3       -11y3    |
               {2} | -32xy2+11y3     -49y3          32y3      47y3       -6y3     |
               {3} | 13xy-29y2       -41xy+40y2     -13y2     44y2       -y2      |
               {3} | -13x2-10xy+23y2 41x2+10xy+40y2 13xy+39y2 -44xy+42y2 xy-47y2  |
               {3} | 32x2-6xy+24y2   9xy+19y2       -32xy-5y2 -47xy-21y2 6xy+38y2 |
               {4} | 0               0              x+24y     -35y       -32y     |
               {4} | 0               0              -41y      x+3y       -39y     |
               {4} | 0               0              -2y       -25y       x-27y    |

          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+29y 35y   |
               {2} | 0 11y   x-40y |
               {3} | 1 -41   10    |
               {3} | 0 -34   -5    |
               {3} | 0 15    -13   |
               {4} | 0 0     0     |
               {4} | 0 0     0     |
               {4} | 0 0     0     |

          5                                                                               8
     2 : A  <--------------------------------------------------------------------------- A  : 1
               {5} | 9  -28 0 9y       -41x+37y xy-8y2       -8xy+9y2   44xy+13y2    |
               {5} | 16 -26 0 -32x-46y -13x+32y 36y2         xy-35y2    9xy+35y2     |
               {5} | 0  0   0 0        0        x2-24xy-46y2 35xy-44y2  32xy+47y2    |
               {5} | 0  0   0 0        0        41xy-19y2    x2-3xy-5y2 39xy+26y2    |
               {5} | 0  0   0 0        0        2xy+21y2     25xy-37y2  x2+27xy-50y2 |

                   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 :