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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 | 14x2-23xy-32y2 32x2+38xy+50y2 |
              | 42x2+33xy+48y2 28x2+23xy+16y2 |
              | 18x2+19xy-33y2 20x2+45xy-49y2 |

                            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 | 40x2-xy-13y2 -14x2-27xy-50y2 x3 x2y+18xy2+27y3 8xy2-50y3   y4 0  0  |
              | x2+35xy+2y2  38xy-2y2        0  -27xy2+3y3     -39xy2-10y3 0  y4 0  |
              | 39xy-30y2    x2-17xy-16y2    0  13y3           xy2+50y3    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
               | 40x2-xy-13y2 -14x2-27xy-50y2 x3 x2y+18xy2+27y3 8xy2-50y3   y4 0  0  |
               | x2+35xy+2y2  38xy-2y2        0  -27xy2+3y3     -39xy2-10y3 0  y4 0  |
               | 39xy-30y2    x2-17xy-16y2    0  13y3           xy2+50y3    0  0  y4 |

          8                                                                               5
     1 : A  <--------------------------------------------------------------------------- A  : 2
               {2} | -41xy2-15y3     -50xy2+38y3     41y3       3y3       -9y3       |
               {2} | 36xy2-6y3       5y3             -36y3      -8y3      -24y3      |
               {3} | -43xy-2y2       43xy+25y2       43y2       -11y2     35y2       |
               {3} | 43x2+44xy+27y2  -43x2+22xy+39y2 -43xy-42y2 11xy+11y2 -35xy-43y2 |
               {3} | -36x2+24xy-24y2 -21xy-12y2      36xy-18y2  8xy+12y2  24xy+5y2   |
               {4} | 0               0               x-23y      43y       -7y        |
               {4} | 0               0               -6y        x-36y     48y        |
               {4} | 0               0               -21y       27y       x-42y      |

          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-35y -38y  |
               {2} | 0 -39y  x+17y |
               {3} | 1 -40   14    |
               {3} | 0 47    -33   |
               {3} | 0 25    -31   |
               {4} | 0 0     0     |
               {4} | 0 0     0     |
               {4} | 0 0     0     |

          5                                                                               8
     2 : A  <--------------------------------------------------------------------------- A  : 1
               {5} | -47 30 0 19y     14x+18y xy-12y2      -11xy+32y2   -33xy+19y2   |
               {5} | 41  -6 0 -47x+2y 14x+16y 27y2         xy-10y2      39xy-7y2     |
               {5} | 0   0  0 0       0       x2+23xy+14y2 -43xy+y2     7xy-6y2      |
               {5} | 0   0  0 0       0       6xy-48y2     x2+36xy+11y2 -48xy+35y2   |
               {5} | 0   0  0 0       0       21xy-9y2     -27xy+21y2   x2+42xy-25y2 |

                   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 :