next | previous | forward | backward | up | top | index | toc | Macaulay2 web site
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 | -9x2-19xy-37y2 35x2+37xy+49y2 |
              | -9x2+28y2      35x2+18xy+25y2 |
              | -29x2+3xy-35y2 9x2-43xy-16y2  |

                            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 | x2+28xy-9y2  -15xy+18y2   x3 x2y+43xy2+15y3 20xy2+46y3  y4 0  0  |
              | x2+33xy+31y2 42xy-28y2    0  2xy2+17y3      -29xy2-44y3 0  y4 0  |
              | 32xy-15y2    x2-17xy-22y2 0  19y3           xy2+33y3    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
               | x2+28xy-9y2  -15xy+18y2   x3 x2y+43xy2+15y3 20xy2+46y3  y4 0  0  |
               | x2+33xy+31y2 42xy-28y2    0  2xy2+17y3      -29xy2-44y3 0  y4 0  |
               | 32xy-15y2    x2-17xy-22y2 0  19y3           xy2+33y3    0  0  y4 |

          8                                                                           5
     1 : A  <----------------------------------------------------------------------- A  : 2
               {2} | 44xy2-18y3    46xy2+7y3      -44y3      45y3      -37y3     |
               {2} | -30xy2-43y3   -33y3          30y3       16y3      -35y3     |
               {3} | -9xy+24y2     23xy-20y2      9y2        2y2       -21y2     |
               {3} | 9x2-45xy-13y2 -23x2-47xy+2y2 -9xy+21y2  -2xy-43y2 21xy+50y2 |
               {3} | 30x2-6xy-11y2 8xy+29y2       -30xy+49y2 -16xy+4y2 35xy+44y2 |
               {4} | 0             0              x+30y      -28y      48y       |
               {4} | 0             0              -37y       x-40y     41y       |
               {4} | 0             0              4y         -5y       x+10y     |

          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-33y -42y  |
               {2} | 0 -32y  x+17y |
               {3} | 1 -1    0     |
               {3} | 0 5     -44   |
               {3} | 0 22    39    |
               {4} | 0 0     0     |
               {4} | 0 0     0     |
               {4} | 0 0     0     |

          5                                                                              8
     2 : A  <-------------------------------------------------------------------------- A  : 1
               {5} | 6   29 0 -48y     -37x+7y xy+30y2     14xy-45y2   -18xy+30y2   |
               {5} | -15 27 0 -22x+13y 47x+23y -2y2        xy-50y2     29xy+47y2    |
               {5} | 0   0  0 0        0       x2-30xy+7y2 28xy+40y2   -48xy-36y2   |
               {5} | 0   0  0 0        0       37xy+29y2   x2+40xy+7y2 -41xy+24y2   |
               {5} | 0   0  0 0        0       -4xy+42y2   5xy+38y2    x2-10xy-14y2 |

                   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 :