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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 | -36x2+50xy+47y2 50x2+30xy+29y2 |
              | 29x2+3xy-23y2   32x2-50xy-37y2 |
              | 21x2-22xy+26y2  -7x2+21xy-46y2 |

                            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 | 30x2-11xy+14y2 29x2+32xy+31y2 x3 x2y+44xy2-36y3 -49xy2+9y3  y4 0  0  |
              | x2-44xy-14y2   8xy-y2         0  26xy2+21y3     -29xy2-17y3 0  y4 0  |
              | 48xy+29y2      x2-34y2        0  7y3            xy2+28y3    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
               | 30x2-11xy+14y2 29x2+32xy+31y2 x3 x2y+44xy2-36y3 -49xy2+9y3  y4 0  0  |
               | x2-44xy-14y2   8xy-y2         0  26xy2+21y3     -29xy2-17y3 0  y4 0  |
               | 48xy+29y2      x2-34y2        0  7y3            xy2+28y3    0  0  y4 |

          8                                                                              5
     1 : A  <-------------------------------------------------------------------------- A  : 2
               {2} | -14xy2-44y3    -39xy2+28y3     14y3       -45y3     -13y3      |
               {2} | 15xy2+12y3     -26y3           -15y3      -11y3     -44y3      |
               {3} | -11xy+35y2     49xy-24y2       11y2       -14y2     32y2       |
               {3} | 11x2+44xy+4y2  -49x2+17xy+25y2 -11xy+22y2 14xy-24y2 -32xy-22y2 |
               {3} | -15x2-7xy-13y2 19xy+45y2       15xy-5y2   11xy+37y2 44xy+19y2  |
               {4} | 0              0               x-5y       -24y      -23y       |
               {4} | 0              0               38y        x-13y     -47y       |
               {4} | 0              0               -21y       -38y      x+18y      |

          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+44y -8y |
               {2} | 0 -48y  x   |
               {3} | 1 -30   -29 |
               {3} | 0 -18   6   |
               {3} | 0 -20   14  |
               {4} | 0 0     0   |
               {4} | 0 0     0   |
               {4} | 0 0     0   |

          5                                                                             8
     2 : A  <------------------------------------------------------------------------- A  : 1
               {5} | -35 -26 0 39y      -27x+21y xy+11y2 -6xy-37y2    -24xy        |
               {5} | 33  -49 0 -33x-13y -4x+10y  -26y2   xy+38y2      29xy+39y2    |
               {5} | 0   0   0 0        0        x2+5xy  24xy-7y2     23xy+21y2    |
               {5} | 0   0   0 0        0        -38xy   x2+13xy+33y2 47xy+2y2     |
               {5} | 0   0   0 0        0        21xy    38xy+11y2    x2-18xy-33y2 |

                   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 :