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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 | 47x2-20xy-5y2   -11x2-13xy-4y2 |
              | -22x2+11xy+7y2  -39x2+43xy+9y2 |
              | -33x2-19xy+18y2 34x2+2xy-3y2   |

                            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 | -15x2+47xy-27y2 30x2+6xy-40y2 x3 x2y+27xy2+6y3 -15xy2-5y3 y4 0  0  |
              | x2-32xy-2y2     21xy-5y2      0  xy2+42y3      19xy2-21y3 0  y4 0  |
              | 5xy-14y2        x2-15xy-31y2  0  -30y3         xy2-11y3   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
               | -15x2+47xy-27y2 30x2+6xy-40y2 x3 x2y+27xy2+6y3 -15xy2-5y3 y4 0  0  |
               | x2-32xy-2y2     21xy-5y2      0  xy2+42y3      19xy2-21y3 0  y4 0  |
               | 5xy-14y2        x2-15xy-31y2  0  -30y3         xy2-11y3   0  0  y4 |

          8                                                                           5
     1 : A  <----------------------------------------------------------------------- A  : 2
               {2} | -8xy2+30y3     -10xy2-18y3   8y3        -37y3     15y3      |
               {2} | -43xy2-47y3    5y3           43y3       -38y3     -47y3     |
               {3} | xy+14y2        -10xy-11y2    -y2        -22y2     -y2       |
               {3} | -x2+10xy+13y2  10x2-5xy+48y2 xy-24y2    22xy-45y2 xy-10y2   |
               {3} | 43x2-14xy-33y2 42xy+34y2     -43xy-40y2 38xy-14y2 47xy-32y2 |
               {4} | 0              0             x-39y      4y        41y       |
               {4} | 0              0             -5y        x+19y     48y       |
               {4} | 0              0             -18y       32y       x+20y     |

          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+32y -21y  |
               {2} | 0 -5y   x+15y |
               {3} | 1 15    -30   |
               {3} | 0 -23   37    |
               {3} | 0 -19   -43   |
               {4} | 0 0     0     |
               {4} | 0 0     0     |
               {4} | 0 0     0     |

          5                                                                                 8
     2 : A  <----------------------------------------------------------------------------- A  : 1
               {5} | -16 1   0 45y      47x+14y xy+42y2      32xy+27y2    13xy-26y2    |
               {5} | 31  -44 0 -10x+50y 35x-18y -y2          xy+11y2      -19xy+27y2   |
               {5} | 0   0   0 0        0       x2+39xy-45y2 -4xy+20y2    -41xy+19y2   |
               {5} | 0   0   0 0        0       5xy+44y2     x2-19xy-42y2 -48xy-50y2   |
               {5} | 0   0   0 0        0       18xy-20y2    -32xy-36y2   x2-20xy-14y2 |

                   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 :