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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 | -50x2-35xy+23y2 11x2-41xy-22y2  |
              | -26x2-xy+35y2   -48x2+29xy+27y2 |
              | -40x2+29xy-30y2 -28x2+32xy+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 | -9x2-33xy+45y2 -3x2+40xy+3y2 x3 x2y+25xy2-2y3 46xy2+13y3 y4 0  0  |
              | x2+19xy+5y2    -30xy-42y2    0  -48xy2-49y3   -17xy2+6y3 0  y4 0  |
              | -17xy+46y2     x2+28xy-14y2  0  25y3          xy2+34y3   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
               | -9x2-33xy+45y2 -3x2+40xy+3y2 x3 x2y+25xy2-2y3 46xy2+13y3 y4 0  0  |
               | x2+19xy+5y2    -30xy-42y2    0  -48xy2-49y3   -17xy2+6y3 0  y4 0  |
               | -17xy+46y2     x2+28xy-14y2  0  25y3          xy2+34y3   0  0  y4 |

          8                                                                            5
     1 : A  <------------------------------------------------------------------------ A  : 2
               {2} | -4xy2+41y3      -24xy2-34y3    4y3       -17y3     32y3      |
               {2} | 21xy2+21y3      -13y3          -21y3     -12y3     -46y3     |
               {3} | 20xy+38y2       -50xy-41y2     -20y2     -48y2     3y2       |
               {3} | -20x2+41xy+24y2 50x2-11xy-34y2 20xy+22y2 48xy-22y2 -3xy+28y2 |
               {3} | -21x2+32xy-11y2 -29xy+30y2     21xy+48y2 12xy-46y2 46xy+39y2 |
               {4} | 0               0              x+10y     42y       -36y      |
               {4} | 0               0              -11y      x-9y      -45y      |
               {4} | 0               0              -34y      y         x-y       |

          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-19y 30y   |
               {2} | 0 17y   x-28y |
               {3} | 1 9     3     |
               {3} | 0 14    45    |
               {3} | 0 -37   -5    |
               {4} | 0 0     0     |
               {4} | 0 0     0     |
               {4} | 0 0     0     |

          5                                                                            8
     2 : A  <------------------------------------------------------------------------ A  : 1
               {5} | -21 19 0 22y     24x+6y  xy-12y2      15xy-39y2   7xy-24y2   |
               {5} | -41 37 0 -2x-28y 50x+22y 48y2         xy-5y2      17xy+y2    |
               {5} | 0   0  0 0       0       x2-10xy-47y2 -42xy+6y2   36xy+8y2   |
               {5} | 0   0  0 0       0       11xy+4y2     x2+9xy-22y2 45xy+38y2  |
               {5} | 0   0  0 0       0       34xy-14y2    -xy-24y2    x2+xy-32y2 |

                   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 :