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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 | 13x2-28xy+y2    -27x2+20xy+2y2  |
              | -12x2-45xy-32y2 -25x2-16xy+47y2 |
              | -33x2+45xy+15y2 34x2+18xy-32y2  |

                            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 | 31x2-39xy-24y2 22x2-37xy+24y2 x3 x2y-26xy2+20y3 -30xy2+13y3 y4 0  0  |
              | x2-3xy+2y2     30xy-12y2      0  30xy2-5y3      -18y3       0  y4 0  |
              | -29xy+26y2     x2+36y2        0  -y3            xy2+10y3    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
               | 31x2-39xy-24y2 22x2-37xy+24y2 x3 x2y-26xy2+20y3 -30xy2+13y3 y4 0  0  |
               | x2-3xy+2y2     30xy-12y2      0  30xy2-5y3      -18y3       0  y4 0  |
               | -29xy+26y2     x2+36y2        0  -y3            xy2+10y3    0  0  y4 |

          8                                                                             5
     1 : A  <------------------------------------------------------------------------- A  : 2
               {2} | 5xy2-38y3       -49xy2+22y3   -5y3       -47y3      -49y3     |
               {2} | -46xy2+34y3     -6y3          46y3       -46y3      -34y3     |
               {3} | 17xy-8y2        -5xy-43y2     -17y2      22y2       -5y2      |
               {3} | -17x2-15xy+21y2 5x2-25xy+40y2 17xy+23y2  -22xy+45y2 5xy-17y2  |
               {3} | 46x2+38xy+29y2  4xy+29y2      -46xy+29y2 46xy-27y2  34xy-40y2 |
               {4} | 0               0             x-41y      33y        -5y       |
               {4} | 0               0             -13y       x-12y      -4y       |
               {4} | 0               0             25y        -39y       x-48y     |

          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+3y -30y |
               {2} | 0 29y  x    |
               {3} | 1 -31  -22  |
               {3} | 0 15   -43  |
               {3} | 0 -40  3    |
               {4} | 0 0    0    |
               {4} | 0 0    0    |
               {4} | 0 0    0    |

          5                                                                                 8
     2 : A  <----------------------------------------------------------------------------- A  : 1
               {5} | 36  -33 0 -29y     11x-22y xy-45y2      -xy-22y2     30xy+47y2    |
               {5} | -22 -43 0 -20x-37y -3x+24y -30y2        xy+24y2      24y2         |
               {5} | 0   0   0 0        0       x2+41xy+16y2 -33xy-39y2   5xy+10y2     |
               {5} | 0   0   0 0        0       13xy-17y2    x2+12xy-28y2 4xy+2y2      |
               {5} | 0   0   0 0        0       -25xy-y2     39xy+34y2    x2+48xy+12y2 |

                   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 :