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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 | -23x2-50xy-25y2 32x2+13xy-16y2  |
              | 42x2-40xy-22y2  x2+28xy-28y2    |
              | 19x2-7xy-27y2   -31x2-16xy+23y2 |

                            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 | 50x2-24xy+4y2 -32x2-24xy+43y2 x3 x2y-40xy2-y3 28xy2+2y3   y4 0  0  |
              | x2-38xy+25y2  50xy+18y2       0  -44xy2+44y3  -18xy2+16y3 0  y4 0  |
              | -2xy-50y2     x2+20xy+40y2    0  31y3         xy2+19y3    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
               | 50x2-24xy+4y2 -32x2-24xy+43y2 x3 x2y-40xy2-y3 28xy2+2y3   y4 0  0  |
               | x2-38xy+25y2  50xy+18y2       0  -44xy2+44y3  -18xy2+16y3 0  y4 0  |
               | -2xy-50y2     x2+20xy+40y2    0  31y3         xy2+19y3    0  0  y4 |

          8                                                                             5
     1 : A  <------------------------------------------------------------------------- A  : 2
               {2} | 32xy2+36y3     -5xy2-21y3     -32y3      -16y3     2y3        |
               {2} | -27xy2+6y3     14y3           27y3       -17y3     49y3       |
               {3} | -31xy-40y2     7xy-42y2       31y2       -34y2     35y2       |
               {3} | 31x2-21xy-43y2 -7x2+12xy-18y2 -31xy-40y2 34xy+29y2 -35xy+26y2 |
               {3} | 27x2+33xy+30y2 -9xy+35y2      -27xy-39y2 17xy+42y2 -49xy+29y2 |
               {4} | 0              0              x+15y      33y       -26y       |
               {4} | 0              0              -29y       x-30y     -15y       |
               {4} | 0              0              8y         y         x+15y      |

          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+38y -50y  |
               {2} | 0 2y    x-20y |
               {3} | 1 -50   32    |
               {3} | 0 6     -35   |
               {3} | 0 -15   -43   |
               {4} | 0 0     0     |
               {4} | 0 0     0     |
               {4} | 0 0     0     |

          5                                                                               8
     2 : A  <--------------------------------------------------------------------------- A  : 1
               {5} | 23 39 0 24y      15x-y    xy+31y2      48xy-50y2    28xy-25y2   |
               {5} | 45 9  0 -29x-30y -49x-18y 44y2         xy-5y2       18xy-12y2   |
               {5} | 0  0  0 0        0        x2-15xy-31y2 -33xy-16y2   26xy+38y2   |
               {5} | 0  0  0 0        0        29xy+12y2    x2+30xy+29y2 15xy-31y2   |
               {5} | 0  0  0 0        0        -8xy+9y2     -xy+47y2     x2-15xy+2y2 |

                   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 :