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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 | -37x2+48xy+8y2  -34x2+37xy-25y2 |
              | -45x2+37xy+10y2 x2-30xy-41y2    |
              | -10x2+3xy-19y2  29x2-10xy+2y2   |

                            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 | -28x2-48xy-y2 -42x2-11xy+34y2 x3 x2y+26xy2-25y3 11xy2-5y3   y4 0  0  |
              | x2-30xy+27y2  29y2            0  29xy2-19y3     -41xy2+17y3 0  y4 0  |
              | -40xy+21y2    x2+8xy-32y2     0  46y3           xy2-30y3    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
               | -28x2-48xy-y2 -42x2-11xy+34y2 x3 x2y+26xy2-25y3 11xy2-5y3   y4 0  0  |
               | x2-30xy+27y2  29y2            0  29xy2-19y3     -41xy2+17y3 0  y4 0  |
               | -40xy+21y2    x2+8xy-32y2     0  46y3           xy2-30y3    0  0  y4 |

          8                                                                         5
     1 : A  <--------------------------------------------------------------------- A  : 2
               {2} | -15xy2-15y3   -30xy2-38y3  15y3       46y3      4y3       |
               {2} | -12xy2+8y3    -37y3        12y3       33y3      9y3       |
               {3} | -14xy-31y2    -8xy+30y2    14y2       -4y2      -15y2     |
               {3} | 14x2+25xy+5y2 8x2+33xy+5y2 -14xy+6y2  4xy+4y2   15xy-3y2  |
               {3} | 12x2+12xy-8y2 -16xy+34y2   -12xy-20y2 -33xy-2y2 -9xy+36y2 |
               {4} | 0             0            x-40y      24y       5y        |
               {4} | 0             0            4y         x+32y     -28y      |
               {4} | 0             0            y          48y       x+8y      |

          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+30y 0    |
               {2} | 0 40y   x-8y |
               {3} | 1 28    42   |
               {3} | 0 43    -22  |
               {3} | 0 -50   -5   |
               {4} | 0 0     0    |
               {4} | 0 0     0    |
               {4} | 0 0     0    |

          5                                                                              8
     2 : A  <-------------------------------------------------------------------------- A  : 1
               {5} | 21 -8 0 17y     -42x+39y xy-30y2      28xy-23y2    26xy-43y2   |
               {5} | 21 42 0 38x-35y 23x+6y   -29y2        xy+50y2      41xy-31y2   |
               {5} | 0  0  0 0       0        x2+40xy-16y2 -24xy+48y2   -5xy-24y2   |
               {5} | 0  0  0 0       0        -4xy+41y2    x2-32xy-22y2 28xy+11y2   |
               {5} | 0  0  0 0       0        -xy-42y2     -48xy+25y2   x2-8xy+38y2 |

                   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 :