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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 | -45x2+29xy+50y2 -10x2-11xy-7y2 |
              | 48x2+26xy-18y2  -20x2-2xy+10y2 |
              | 31x2+5xy-7y2    23x2+47xy-24y2 |

                            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 | -17x2+43xy-50y2 -24x2-7xy-35y2 x3 x2y+20xy2+12y3 15xy2+28y3  y4 0  0  |
              | x2-50xy+12y2    -26xy+46y2     0  8xy2+34y3      -37xy2+34y3 0  y4 0  |
              | -33xy+40y2      x2+48xy+3y2    0  34y3           xy2-8y3     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
               | -17x2+43xy-50y2 -24x2-7xy-35y2 x3 x2y+20xy2+12y3 15xy2+28y3  y4 0  0  |
               | x2-50xy+12y2    -26xy+46y2     0  8xy2+34y3      -37xy2+34y3 0  y4 0  |
               | -33xy+40y2      x2+48xy+3y2    0  34y3           xy2-8y3     0  0  y4 |

          8                                                                           5
     1 : A  <----------------------------------------------------------------------- A  : 2
               {2} | -10xy2+3y3    36xy2+3y3     10y3       4y3        -2y3      |
               {2} | -8xy2-15y3    19y3          8y3        32y3       -47y3     |
               {3} | -13xy+32y2    -46xy         13y2       -3y2       -3y2      |
               {3} | 13x2+34xy+4y2 46x2-5xy-12y2 -13xy+35y2 3xy+44y2   3xy-8y2   |
               {3} | 8x2-6xy+39y2  9xy+9y2       -8xy+21y2  -32xy-45y2 47xy+39y2 |
               {4} | 0             0             x-26y      32y        -14y      |
               {4} | 0             0             32y        x+29y      21y       |
               {4} | 0             0             -32y       9y         x-3y      |

          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+50y 26y   |
               {2} | 0 33y   x-48y |
               {3} | 1 17    24    |
               {3} | 0 -17   4     |
               {3} | 0 26    28    |
               {4} | 0 0     0     |
               {4} | 0 0     0     |
               {4} | 0 0     0     |

          5                                                                              8
     2 : A  <-------------------------------------------------------------------------- A  : 1
               {5} | 22  47 0 -25y    38x+6y  xy+49y2      4xy+26y2     32xy-47y2   |
               {5} | -30 2  0 11x+39y 20x-32y -8y2         xy+24y2      37xy-23y2   |
               {5} | 0   0  0 0       0       x2+26xy+27y2 -32xy-30y2   14xy-33y2   |
               {5} | 0   0  0 0       0       -32xy+30y2   x2-29xy+34y2 -21xy-3y2   |
               {5} | 0   0  0 0       0       32xy+4y2     -9xy+18y2    x2+3xy+40y2 |

                   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 :