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Complexes :: isNullHomotopyOf(ComplexMap,ComplexMap)

isNullHomotopyOf(ComplexMap,ComplexMap) -- whether the first map of chain complexes is a null homotopy for the second

Synopsis

Description

A map of chain complexes $f \colon C \to D$ is null-homopic if there exists a map of chain complexes $h : C \to D$ of degree $\deg(f)+1$, such that we have the equality \[ f = \operatorname{dd}^D h + (-1)^{\deg(f)} h \operatorname{dd}^C. \]

As a first example, we construct a map of chain complexes in which the null homotopy is given by the identity.

i1 : R = ZZ/101[x,y,z];
i2 : M = cokernel matrix{{x,y,z^2}, {y^2,z,x^2}}

o2 = cokernel | x  y z2 |
              | y2 z x2 |

                            2
o2 : R-module, quotient of R
i3 : C = complex {id_M}

o3 = cokernel | x  y z2 | <-- cokernel | x  y z2 |
              | y2 z x2 |              | y2 z x2 |
                               
     0                        1

o3 : Complex
i4 : h = map(C, C, i -> if i == 0 then id_M, Degree => 1)

o4 = 1 : cokernel | x  y z2 | <----------- cokernel | x  y z2 | : 0
                  | y2 z x2 |    | 1 0 |            | y2 z x2 |
                                 | 0 1 |

o4 : ComplexMap
i5 : isWellDefined h

o5 = true
i6 : assert isNullHomotopyOf(h, id_C)
i7 : assert isNullHomotopic id_C

A random map of chain complexes, arising as a boundary in the associated Hom complex, is automatically null homotopic. We use the method nullHomotopy to construct a witness and verify it is a null homotopy.

i8 : C = (freeResolution M) ** R^1/ideal(x^3, z^3-x)

o8 = cokernel | x3 z3-x 0  0    | <-- cokernel {1} | x3 z3-x 0  0    0  0    | <-- cokernel {5} | x3 z3-x |
              | 0  0    x3 z3-x |              {2} | 0  0    x3 z3-x 0  0    |      
                                               {2} | 0  0    0  0    x3 z3-x |     2
     0                                 
                                      1

o8 : Complex
i9 : f = randomComplexMap(C, C[1], Boundary => true)

o9 = -1 : 0 <----- cokernel | x3 z3-x 0  0    | : -1
               0            | 0  0    x3 z3-x |

     0 : cokernel | x3 z3-x 0  0    | <-------------------------------------------------------------------------- cokernel {1} | x3 z3-x 0  0    0  0    | : 0
                  | 0  0    x3 z3-x |    | -5y+30z 30x2+19xy-10y2-29yz-32z2-22x -29xy+6y2-38yz-16z2+15x       |            {2} | 0  0    x3 z3-x 0  0    |
                                         | 36y+48z 21x2-22y2+19xz-10yz+7z2      -16x2-33y2-29xz-24yz-38z2+36x |            {2} | 0  0    0  0    x3 z3-x |

     1 : cokernel {1} | x3 z3-x 0  0    0  0    | <---------------------------------------------------------------------------------- cokernel {5} | x3 z3-x | : 1
                  {2} | 0  0    x3 z3-x 0  0    |    {1} | 24x2y2+19xy3-10y4+38x2yz-29y3z-19y2z2-19x2z+10xyz+29xz2-29x2-24xy-38xz |
                  {2} | 0  0    0  0    x3 z3-x |    {2} | 16x2y-8y3+8xz-16x                                                      |
                                                     {2} | -39x2y-22y3+22xz+39x                                                   |

o9 : ComplexMap
i10 : assert isNullHomotopic f
i11 : h = nullHomotopy f

o11 = 0 : cokernel | x3 z3-x 0  0    | <---------------------------- cokernel | x3 z3-x 0  0    | : -1
                   | 0  0    x3 z3-x |    | 24        -30        |            | 0  0    x3 z3-x |
                                          | 39x2yz-36 -39x2y2-29 |

      1 : cokernel {1} | x3 z3-x 0  0    0  0    | <--------------------------------------- cokernel {1} | x3 z3-x 0  0    0  0    | : 0
                   {2} | 0  0    x3 z3-x 0  0    |    {1} | 19 19x-10y-29z -29x-24y-38z |            {2} | 0  0    x3 z3-x 0  0    |
                   {2} | 0  0    0  0    x3 z3-x |    {2} | 0  -8          -16          |            {2} | 0  0    0  0    x3 z3-x |
                                                      {2} | 0  -22         -39x2y2+39   |

      2 : cokernel {5} | x3 z3-x | <----- cokernel {5} | x3 z3-x | : 1
                                      0

o11 : ComplexMap
i12 : assert isNullHomotopyOf(h, f)

By assigning debugLevel a positive value, this method provides some information about the nature of the failure to be a null homotopy.

i13 : g1 = randomComplexMap(C, C[1], Degree => 1)

o13 = 0 : cokernel | x3 z3-x 0  0    | <-------------- cokernel | x3 z3-x 0  0    | : -1
                   | 0  0    x3 z3-x |    | 21 19  |            | 0  0    x3 z3-x |
                                          | 34 -47 |

      1 : cokernel {1} | x3 z3-x 0  0    0  0    | <---------------------------------------- cokernel {1} | x3 z3-x 0  0    0  0    | : 0
                   {2} | 0  0    x3 z3-x 0  0    |    {1} | -39 -18x-13y-43z -47x+38y+2z |            {2} | 0  0    x3 z3-x 0  0    |
                   {2} | 0  0    0  0    x3 z3-x |    {2} | 0   -15          16          |            {2} | 0  0    0  0    x3 z3-x |
                                                      {2} | 0   -28          22          |

      2 : cokernel {5} | x3 z3-x | <-------------- cokernel {5} | x3 z3-x | : 1
                                      {5} | 45 |

o13 : ComplexMap
i14 : g2 = randomComplexMap(C, C[1], Degree => -1)

o14 = -2 : 0 <----- cokernel | x3 z3-x 0  0    | : -1
                0            | 0  0    x3 z3-x |

      -1 : 0 <----- cokernel {1} | x3 z3-x 0  0    0  0    | : 0
                0            {2} | 0  0    x3 z3-x 0  0    |
                             {2} | 0  0    0  0    x3 z3-x |

      0 : cokernel | x3 z3-x 0  0    | <--------------------------------------------------------------------- cokernel {5} | x3 z3-x | : 1
                   | 0  0    x3 z3-x |    | -34x2y3+47xy4+7y5-48x2y2z+19xy3z+15y4z-47x2yz2-16xy2z2-23y3z2 |
                                          | 39x2y3-11xy4+35y5+43x2y2z+48xy3z+11y4z-17x2yz2+36xy2z2-38y3z2 |

o14 : ComplexMap
i15 : debugLevel = 1

o15 = 1
i16 : assert not isNullHomotopyOf(g1, f)
 -- 1 : (ContainmentHooks) with Strategy => Inhomogeneous from -*Function[/usr/share/Macaulay2/Core/modules2.m2:332:61-332:99]*-
 -- 1 : (ContainmentHooks) with Strategy => Inhomogeneous from -*Function[/usr/share/Macaulay2/Core/modules2.m2:332:61-332:99]*-
 -- 1 : (ContainmentHooks) with Strategy => Inhomogeneous from -*Function[/usr/share/Macaulay2/Core/modules2.m2:332:61-332:99]*-
 -- 1 : (ContainmentHooks) with Strategy => Inhomogeneous from -*Function[/usr/share/Macaulay2/Core/modules2.m2:332:61-332:99]*-
 -- 1 : (ContainmentHooks) with Strategy => Inhomogeneous from -*Function[/usr/share/Macaulay2/Core/modules2.m2:332:61-332:99]*-
 -- 1 : (ContainmentHooks) with Strategy => Inhomogeneous from -*Function[/usr/share/Macaulay2/Core/modules2.m2:332:61-332:99]*-
 -- 1 : (ReduceHooks) with Strategy => Default from -*Function[/usr/share/Macaulay2/Core/matrix.m2:76:55-76:85]*-
 -- 1 : (ReduceHooks) with Strategy => Default from -*Function[/usr/share/Macaulay2/Core/matrix.m2:76:55-76:85]*-
 -- 1 : (ReduceHooks) with Strategy => Default from -*Function[/usr/share/Macaulay2/Core/matrix.m2:76:55-76:85]*-
 -- 1 : (ReduceHooks) with Strategy => Default from -*Function[/usr/share/Macaulay2/Core/matrix.m2:76:55-76:85]*-
 -- 1 : (ReduceHooks) with Strategy => Default from -*Function[/usr/share/Macaulay2/Core/matrix.m2:76:55-76:85]*-
 -- 1 : (ReduceHooks) with Strategy => Default from -*Function[/usr/share/Macaulay2/Core/matrix.m2:76:55-76:85]*-
fails to be a null homotopy at location 0
fails to be a null homotopy at location 1
i17 : assert not isNullHomotopyOf(g2, f)
 -- 1 : (ContainmentHooks) with Strategy => Inhomogeneous from -*Function[/usr/share/Macaulay2/Core/modules2.m2:332:61-332:99]*-
 -- 1 : (ContainmentHooks) with Strategy => Inhomogeneous from -*Function[/usr/share/Macaulay2/Core/modules2.m2:332:61-332:99]*-
 -- 1 : (ContainmentHooks) with Strategy => Inhomogeneous from -*Function[/usr/share/Macaulay2/Core/modules2.m2:332:61-332:99]*-
 -- 1 : (ContainmentHooks) with Strategy => Inhomogeneous from -*Function[/usr/share/Macaulay2/Core/modules2.m2:332:61-332:99]*-
 -- 1 : (ContainmentHooks) with Strategy => Inhomogeneous from -*Function[/usr/share/Macaulay2/Core/modules2.m2:332:61-332:99]*-
 -- 1 : (ContainmentHooks) with Strategy => Inhomogeneous from -*Function[/usr/share/Macaulay2/Core/modules2.m2:332:61-332:99]*-
expected degree of first map to be one more than degree of the second

See also