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 | 5x2+30xy+41y2  7x2+38xy+20y2   |
              | 3x2+31xy-38y2  -9x2+15xy-43y2  |
              | -23x2-9xy-25y2 -35x2+18xy-38y2 |

                            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 | 24x2+42xy+41y2 -41x2-9xy-14y2 x3 x2y+31xy2-44y3 -xy2+39y3  y4 0  0  |
              | x2-30xy+7y2    -36xy-15y2     0  33xy2+13y3     -4xy2+26y3 0  y4 0  |
              | 20xy-45y2      x2+3xy+4y2     0  -35y3          xy2-41y3   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
               | 24x2+42xy+41y2 -41x2-9xy-14y2 x3 x2y+31xy2-44y3 -xy2+39y3  y4 0  0  |
               | x2-30xy+7y2    -36xy-15y2     0  33xy2+13y3     -4xy2+26y3 0  y4 0  |
               | 20xy-45y2      x2+3xy+4y2     0  -35y3          xy2-41y3   0  0  y4 |

          8                                                                               5
     1 : A  <--------------------------------------------------------------------------- A  : 2
               {2} | 3xy2-49y3      -47xy2-26y3     -3y3       39y3       -49y3      |
               {2} | -36xy2+46y3    46y3            36y3       -5y3       16y3       |
               {3} | -41xy+50y2     20xy+14y2       41y2       22y2       28y2       |
               {3} | 41x2-5xy+20y2  -20x2+17xy+39y2 -41xy-45y2 -22xy-50y2 -28xy-43y2 |
               {3} | 36x2-16xy-13y2 -8xy+49y2       -36xy-30y2 5xy-17y2   -16xy+2y2  |
               {4} | 0              0               x+19y      26y        -40y       |
               {4} | 0              0               7y         x+37y      -21y       |
               {4} | 0              0               47y        35y        x+45y      |

          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+30y 36y  |
               {2} | 0 -20y  x-3y |
               {3} | 1 -24   41   |
               {3} | 0 34    32   |
               {3} | 0 10    -8   |
               {4} | 0 0     0    |
               {4} | 0 0     0    |
               {4} | 0 0     0    |

          5                                                                              8
     2 : A  <-------------------------------------------------------------------------- A  : 1
               {5} | 39 -11 0 46y    -14x+48y xy-18y2      -31xy+y2    -22xy+39y2   |
               {5} | 16 -26 0 5x+43y 42x-24y  -33y2        xy+35y2     4xy-16y2     |
               {5} | 0  0   0 0      0        x2-19xy-24y2 -26xy-45y2  40xy+25y2    |
               {5} | 0  0   0 0      0        -7xy+11y2    x2-37xy+8y2 21xy+18y2    |
               {5} | 0  0   0 0      0        -47xy+21y2   -35xy-49y2  x2-45xy+16y2 |

                   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 :