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kernel(ComplexMap) -- make the kernel of a map of complexes

Description

The kernel of a chain complex map $f : C \to D$ is the complex $E$ whose $i-th$ term is $kernel(f_i)$, and whose differential is induced from the differential on the source.

In the following example, we first construct a random complex morphism $f : C \to D$. We consider the exact sequence $0 \to D \to cone(f) \to C[-1] \to 0$. For the maps $g : D \to cone(f)$ and $h : cone(f) \to C[-1]$, we compute the kernel.

i1 : S = ZZ/101[a,b,c,d];
 
i2 : C = freeResolution ideal(b^2-a*c, b*c-a*d, c^2-b*d)

      1      3      2
o2 = S  <-- S  <-- S
                    
     0      1      2

o2 : Complex
 
i3 : D = freeResolution ideal(a,b,c)

      1      3      3      1
o3 = S  <-- S  <-- S  <-- S
                           
     0      1      2      3

o3 : Complex
 
i4 : f = randomComplexMap(D, C, Cycle => true, InternalDegree => 0)

          1               1
o4 = 0 : S  <----------- S  : 0
               | -22 |

          3                                                    3
     1 : S  <------------------------------------------------ S  : 1
               {1} | 36b+3c       30b-19c+22d -29b-10c    |
               {1} | -36a-22b+29c -30a-14c    29a+29c+22d |
               {1} | 19a-29b      19a-8b      10a-29b-22c |

          3                                                 2
     2 : S  <--------------------------------------------- S  : 2
               {2} | -29a-30b+31c-22d 29b+6c-36d       |
               {2} | -10a+24b+3c      34b-19c+19d      |
               {2} | 24a-8b+29c       -24a-29b-14c-29d |

o4 : ComplexMap
 
i5 : Cf = cone f

      1      4      6      3
o5 = S  <-- S  <-- S  <-- S
                           
     0      1      2      3

o5 : Complex
 
i6 : g = canonicalMap(Cf, D)

          1             1
o6 = 0 : S  <--------- S  : 0
               | 1 |

          4                     3
     1 : S  <----------------- S  : 1
               {0} | 0 0 0 |
               {1} | 1 0 0 |
               {1} | 0 1 0 |
               {1} | 0 0 1 |

          6                     3
     2 : S  <----------------- S  : 2
               {2} | 0 0 0 |
               {2} | 0 0 0 |
               {2} | 0 0 0 |
               {2} | 1 0 0 |
               {2} | 0 1 0 |
               {2} | 0 0 1 |

          3                 1
     3 : S  <------------- S  : 3
               {3} | 0 |
               {3} | 0 |
               {3} | 1 |

o6 : ComplexMap
 
i7 : h = canonicalMap(C[-1], Cf)

          1                   4
o7 = 1 : S  <--------------- S  : 1
               | 1 0 0 0 |

          3                           6
     2 : S  <----------------------- S  : 2
               {2} | 1 0 0 0 0 0 |
               {2} | 0 1 0 0 0 0 |
               {2} | 0 0 1 0 0 0 |

          2                     3
     3 : S  <----------------- S  : 3
               {3} | 1 0 0 |
               {3} | 0 1 0 |

o7 : ComplexMap
 
i8 : ker g == 0

o8 = true
 
i9 : prune ker h == D

o9 = true

There is a canonical map of complexes from the kernel to the source.

i10 : h1 = canonicalMap(source h, ker h)

           1             1
o10 = 0 : S  <--------- S  : 0
                | 1 |

           4
      1 : S  <----------------- image {0} | 0 0 0 | : 1
                {0} | 0 0 0 |         {1} | 1 0 0 |
                {1} | 1 0 0 |         {1} | 0 1 0 |
                {1} | 0 1 0 |         {1} | 0 0 1 |
                {1} | 0 0 1 |

           6
      2 : S  <----------------- image {2} | 0 0 0 | : 2
                {2} | 0 0 0 |         {2} | 0 0 0 |
                {2} | 0 0 0 |         {2} | 0 0 0 |
                {2} | 0 0 0 |         {2} | 1 0 0 |
                {2} | 1 0 0 |         {2} | 0 1 0 |
                {2} | 0 1 0 |         {2} | 0 0 1 |
                {2} | 0 0 1 |

           3
      3 : S  <------------- image {3} | 0 | : 3
                {3} | 0 |         {3} | 0 |
                {3} | 0 |         {3} | 1 |
                {3} | 1 |

o10 : ComplexMap
 
i11 : ker h == image h1

o11 = true
 
i12 : ker h1 == 0

o12 = true

See also

Ways to use this method:

The source of this document is in Complexes/ChainComplexMapDoc.m2:2406:0.