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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+4xy-42y2 40x2-37xy+40y2 |
              | 25x2-46xy+24y2 18x2-49xy-44y2 |
              | 22xy-35y2      30x2+8xy-19y2  |

                            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 | -49x2-16xy-34y2 24x2+5xy+15y2 x3 x2y-13xy2-33y3 38xy2-23y3 y4 0  0  |
              | x2-18xy+5y2     26xy+9y2      0  13xy2+46y3     40xy2-30y3 0  y4 0  |
              | 13xy+39y2       x2-21xy+13y2  0  -14y3          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
               | -49x2-16xy-34y2 24x2+5xy+15y2 x3 x2y-13xy2-33y3 38xy2-23y3 y4 0  0  |
               | x2-18xy+5y2     26xy+9y2      0  13xy2+46y3     40xy2-30y3 0  y4 0  |
               | 13xy+39y2       x2-21xy+13y2  0  -14y3          xy2+26y3   0  0  y4 |

          8                                                                              5
     1 : A  <-------------------------------------------------------------------------- A  : 2
               {2} | -22xy2+25y3     -27xy2+26y3     22y3      10y3      -24y3      |
               {2} | 23xy2-13y3      8y3             -23y3     4y3       7y3        |
               {3} | 13xy-28y2       29xy-7y2        -13y2     4y2       31y2       |
               {3} | -13x2+12xy-8y2  -29x2+24xy-25y2 13xy+16y2 -4xy-11y2 -31xy+34y2 |
               {3} | -23x2-14xy-12y2 38xy+29y2       23xy+27y2 -4xy+2y2  -7xy+4y2   |
               {4} | 0               0               x+20y     -37y      -10y       |
               {4} | 0               0               -31y      x-25y     27y        |
               {4} | 0               0               -27y      -42y      x+5y       |

          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+18y -26y  |
               {2} | 0 -13y  x+21y |
               {3} | 1 49    -24   |
               {3} | 0 -2    35    |
               {3} | 0 -41   -42   |
               {4} | 0 0     0     |
               {4} | 0 0     0     |
               {4} | 0 0     0     |

          5                                                                             8
     2 : A  <------------------------------------------------------------------------- A  : 1
               {5} | 7   -15 0 6y     -22x+10y xy+46y2    13xy-39y2    48xy-35y2   |
               {5} | -45 -6  0 -7x-5y -18x-25y -13y2      xy+2y2       -40xy-49y2  |
               {5} | 0   0   0 0      0        x2-20xy-y2 37xy-y2      10xy-37y2   |
               {5} | 0   0   0 0      0        31xy+32y2  x2+25xy+32y2 -27xy-28y2  |
               {5} | 0   0   0 0      0        27xy+21y2  42xy+21y2    x2-5xy-31y2 |

                   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 :