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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 | -31x2-38xy+42y2 40x2+18xy-40y2  |
              | 15x2-44xy+11y2  -17x2-37xy-32y2 |
              | -27x2+30xy+8y2  41x2-34xy+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 | -20x2-25xy+8y2 5x2-5xy-42y2 x3 x2y-xy2+16y3 -29xy2-3y3  y4 0  0  |
              | x2+28xy-2y2    -9xy-9y2     0  -31xy2-42y3  -44xy2-34y3 0  y4 0  |
              | 2xy+41y2       x2+15y2      0  49y3         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
               | -20x2-25xy+8y2 5x2-5xy-42y2 x3 x2y-xy2+16y3 -29xy2-3y3  y4 0  0  |
               | x2+28xy-2y2    -9xy-9y2     0  -31xy2-42y3  -44xy2-34y3 0  y4 0  |
               | 2xy+41y2       x2+15y2      0  49y3         xy2-11y3    0  0  y4 |

          8                                                                             5
     1 : A  <------------------------------------------------------------------------- A  : 2
               {2} | 12xy2-41y3      10xy2-4y3       -12y3     -46y3     -49y3     |
               {2} | -29xy2+26y3     24y3            29y3      -14y3     23y3      |
               {3} | 31xy-3y2        29xy-47y2       -31y2     -21y2     -5y2      |
               {3} | -31x2-14xy+15y2 -29x2+16xy-38y2 31xy+17y2 21xy-19y2 5xy-40y2  |
               {3} | 29x2-30xy-23y2  -37xy+23y2      -29xy+4y2 14xy+25y2 -23xy+3y2 |
               {4} | 0               0               x+44y     -43y      -13y      |
               {4} | 0               0               -24y      x+36y     46y       |
               {4} | 0               0               -25y      -31y      x+21y     |

          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-28y 9y  |
               {2} | 0 -2y   x   |
               {3} | 1 20    -5  |
               {3} | 0 -20   -17 |
               {3} | 0 15    -33 |
               {4} | 0 0     0   |
               {4} | 0 0     0   |
               {4} | 0 0     0   |

          5                                                                              8
     2 : A  <-------------------------------------------------------------------------- A  : 1
               {5} | -4 29  0 5y      7x+30y  xy-7y2       3xy-16y2    -41xy-34y2   |
               {5} | 21 -49 0 -7x+50y -4x+32y 31y2         xy-y2       44xy-45y2    |
               {5} | 0  0   0 0       0       x2-44xy-40y2 43xy-7y2    13xy+5y2     |
               {5} | 0  0   0 0       0       24xy-40y2    x2-36xy-7y2 -46xy+5y2    |
               {5} | 0  0   0 0       0       25xy+28y2    31xy+15y2   x2-21xy+47y2 |

                   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 :