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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 | 25x2+47xy+31y2 3x2-38xy+36y2  |
              | 10x2-39y2      12x2+43xy+28y2 |
              | -9x2-37xy-24y2 -6x2-5xy-7y2   |

                            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 | 29x2-13xy-35y2 7x2+14xy+25y2 x3 x2y-49xy2+29y3 -40xy2-47y3 y4 0  0  |
              | x2+27xy+48y2   30xy+24y2     0  -6y3           29xy2+35y3  0  y4 0  |
              | 10xy+8y2       x2+4xy+34y2   0  -46y3          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
               | 29x2-13xy-35y2 7x2+14xy+25y2 x3 x2y-49xy2+29y3 -40xy2-47y3 y4 0  0  |
               | x2+27xy+48y2   30xy+24y2     0  -6y3           29xy2+35y3  0  y4 0  |
               | 10xy+8y2       x2+4xy+34y2   0  -46y3          xy2+8y3     0  0  y4 |

          8                                                                              5
     1 : A  <-------------------------------------------------------------------------- A  : 2
               {2} | 40xy2-43y3      -13y3          -40y3      -21y3     19y3       |
               {2} | -23xy2+2y3      17y3           23y3       -46y3     32y3       |
               {3} | 24xy+20y2       -27xy-37y2     -24y2      -26y2     -4y2       |
               {3} | -24x2+38xy-30y2 27x2+47xy-25y2 24xy+43y2  26xy+5y2  4xy-41y2   |
               {3} | 23x2+18xy-37y2  13xy-y2        -23xy-20y2 46xy+10y2 -32xy+20y2 |
               {4} | 0               0              x-21y      33y       -26y       |
               {4} | 0               0              3y         x-26y     0          |
               {4} | 0               0              -41y       -37y      x+47y      |

          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-27y -30y |
               {2} | 0 -10y  x-4y |
               {3} | 1 -29   -7   |
               {3} | 0 -43   -25  |
               {3} | 0 -1    -21  |
               {4} | 0 0     0    |
               {4} | 0 0     0    |
               {4} | 0 0     0    |

          5                                                                                8
     2 : A  <---------------------------------------------------------------------------- A  : 1
               {5} | 22  -43 0 -48y    22x-19y xy+49y2      -2xy-11y2    -3xy+18y2    |
               {5} | -20 45  0 15x+21y 42x-5y  0            xy-40y2      -29xy-15y2   |
               {5} | 0   0   0 0       0       x2+21xy-10y2 -33xy+17y2   26xy+31y2    |
               {5} | 0   0   0 0       0       -3xy-40y2    x2+26xy-33y2 23y2         |
               {5} | 0   0   0 0       0       41xy+35y2    37xy-9y2     x2-47xy+43y2 |

                   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 :