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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 | -37x2+2xy+41y2 -34x2-5xy+11y2  |
              | 49x2+24xy-7y2  -29x2-40xy-49y2 |
              | 26x2-46xy-18y2 4x2+40xy+18y2   |

                            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 | -19x2+28xy-35y2 -20x2+25xy-49y2 x3 x2y+14xy2+48y3 2xy2+40y3 y4 0  0  |
              | x2+8xy+23y2     48xy+3y2        0  -25xy2+13y3    6xy2+49y3 0  y4 0  |
              | -20xy+9y2       x2-34xy-6y2     0  27y3           xy2-49y3  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
               | -19x2+28xy-35y2 -20x2+25xy-49y2 x3 x2y+14xy2+48y3 2xy2+40y3 y4 0  0  |
               | x2+8xy+23y2     48xy+3y2        0  -25xy2+13y3    6xy2+49y3 0  y4 0  |
               | -20xy+9y2       x2-34xy-6y2     0  27y3           xy2-49y3  0  0  y4 |

          8                                                                              5
     1 : A  <-------------------------------------------------------------------------- A  : 2
               {2} | -32xy2+32y3    -48xy2+48y3     32y3       -8y3       -16y3     |
               {2} | 8xy2-46y3      -17y3           -8y3       14y3       33y3      |
               {3} | -17xy+20y2     10xy-2y2        17y2       36y2       -48y2     |
               {3} | 17x2+17xy-24y2 -10x2+38xy-22y2 -17xy-37y2 -36xy-47y2 48xy-48y2 |
               {3} | -8x2+39xy-45y2 34xy+40y2       8xy+7y2    -14xy-4y2  -33xy+9y2 |
               {4} | 0              0               x+2y       -6y        -29y      |
               {4} | 0              0               32y        x+40y      48y       |
               {4} | 0              0               -4y        17y        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-8y -48y  |
               {2} | 0 20y  x+34y |
               {3} | 1 19   20    |
               {3} | 0 18   46    |
               {3} | 0 6    0     |
               {4} | 0 0    0     |
               {4} | 0 0    0     |
               {4} | 0 0    0     |

          5                                                                             8
     2 : A  <------------------------------------------------------------------------- A  : 1
               {5} | 26 0  0 -8y     -38x+25y xy+18y2     -27xy-7y2   -42xy+y2     |
               {5} | 46 45 0 10x-16y -4x-7y   25y2        xy+44y2     -6xy+46y2    |
               {5} | 0  0  0 0       0        x2-2xy+29y2 6xy-38y2    29xy-37y2    |
               {5} | 0  0  0 0       0        -32xy+41y2  x2-40xy+2y2 -48xy-14y2   |
               {5} | 0  0  0 0       0        4xy-3y2     -17xy-10y2  x2+42xy-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 :