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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 | -31x2+41xy+29y2 -15x2+3xy+42y2  |
              | 19x2+42xy-18y2  -33x2-12xy+38y2 |
              | 10x2-6xy-31y2   33x2+19xy-18y2  |

                            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 | 26x2-4xy-33y2 -2x2+42xy+5y2 x3 x2y-43xy2+47y3 xy2-31y3  y4 0  0  |
              | x2+44xy+26y2  -39xy-31y2    0  -19xy2+12y3    -2xy2+3y3 0  y4 0  |
              | 34xy+14y2     x2-45xy+41y2  0  -31y3          xy2-45y3  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
               | 26x2-4xy-33y2 -2x2+42xy+5y2 x3 x2y-43xy2+47y3 xy2-31y3  y4 0  0  |
               | x2+44xy+26y2  -39xy-31y2    0  -19xy2+12y3    -2xy2+3y3 0  y4 0  |
               | 34xy+14y2     x2-45xy+41y2  0  -31y3          xy2-45y3  0  0  y4 |

          8                                                                             5
     1 : A  <------------------------------------------------------------------------- A  : 2
               {2} | 43xy2+28y3      -26xy2+47y3    -43y3     29y3       0         |
               {2} | 15xy2-28y3      13y3           -15y3     -6y3       3y3       |
               {3} | 44xy+44y2       12xy+28y2      -44y2     31y2       5y2       |
               {3} | -44x2+21xy+22y2 -12x2+31xy-3y2 44xy+36y2 -31xy+16y2 -5xy-4y2  |
               {3} | -15x2+30xy-14y2 -6xy+42y2      15xy-2y2  6xy-28y2   -3xy+46y2 |
               {4} | 0               0              x+33y     -27y       -17y      |
               {4} | 0               0              25y       x-38y      3y        |
               {4} | 0               0              21y       -15y       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-44y 39y   |
               {2} | 0 -34y  x+45y |
               {3} | 1 -26   2     |
               {3} | 0 -31   44    |
               {3} | 0 -48   -49   |
               {4} | 0 0     0     |
               {4} | 0 0     0     |
               {4} | 0 0     0     |

          5                                                                               8
     2 : A  <--------------------------------------------------------------------------- A  : 1
               {5} | 17 -10 0 -37y    -27x-33y xy-10y2      -40xy+31y2   20xy-15y2   |
               {5} | -6 -27 0 42x-42y -2x-24y  19y2         xy           2xy-35y2    |
               {5} | 0  0   0 0       0        x2-33xy-44y2 27xy-14y2    17xy-20y2   |
               {5} | 0  0   0 0       0        -25xy+39y2   x2+38xy+17y2 -3xy-19y2   |
               {5} | 0  0   0 0       0        -21xy+19y2   15xy+29y2    x2-5xy+27y2 |

                   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 :