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+31xy+10y2 -37x2-8xy-44y2  |
              | 34x2-7xy-20y2 -8x2+22xy+48y2  |
              | -20x2-35xy    -39x2-49xy+32y2 |

                            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 | 39x2+12xy-2y2 -20x2+34xy-14y2 x3 x2y+44xy2+34y3 -41xy2+22y3 y4 0  0  |
              | x2+42xy-41y2  -4xy+2y2        0  -26xy2+5y3     -30xy2+19y3 0  y4 0  |
              | -32xy+36y2    x2+13xy+41y2    0  -18y3          xy2-10y3    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
               | 39x2+12xy-2y2 -20x2+34xy-14y2 x3 x2y+44xy2+34y3 -41xy2+22y3 y4 0  0  |
               | x2+42xy-41y2  -4xy+2y2        0  -26xy2+5y3     -30xy2+19y3 0  y4 0  |
               | -32xy+36y2    x2+13xy+41y2    0  -18y3          xy2-10y3    0  0  y4 |

          8                                                                            5
     1 : A  <------------------------------------------------------------------------ A  : 2
               {2} | 40xy2+17y3     -xy2+49y3       -40y3     -48y3      -50y3    |
               {2} | 2xy2-25y3      -3y3            -2y3      43y3       -4y3     |
               {3} | 35xy+y2        35xy-14y2       -35y2     -40y2      -9y2     |
               {3} | -35x2+38xy+6y2 -35x2-23xy-11y2 35xy-39y2 40xy+17y2  9xy+22y2 |
               {3} | -2x2+23xy-42y2 48xy-8y2        2xy+2y2   -43xy+13y2 4xy-34y2 |
               {4} | 0              0               x-38y     46y        46y      |
               {4} | 0              0               36y       x+38y      9y       |
               {4} | 0              0               32y       -3y        x        |

          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-42y 4y    |
               {2} | 0 32y   x-13y |
               {3} | 1 -39   20    |
               {3} | 0 44    -46   |
               {3} | 0 22    -47   |
               {4} | 0 0     0     |
               {4} | 0 0     0     |
               {4} | 0 0     0     |

          5                                                                               8
     2 : A  <--------------------------------------------------------------------------- A  : 1
               {5} | 11 27  0 -18y     50x-19y  xy+3y2       29xy+30y2    2xy+15y2   |
               {5} | 22 -11 0 -26x+12y -50x+34y 26y2         xy+5y2       30xy+33y2  |
               {5} | 0  0   0 0        0        x2+38xy+27y2 -46xy-37y2   -46xy-21y2 |
               {5} | 0  0   0 0        0        -36xy-15y2   x2-38xy+43y2 -9xy-22y2  |
               {5} | 0  0   0 0        0        -32xy-11y2   3xy+45y2     x2+31y2    |

                   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 :