Macaulay2 » Documentation
Packages » SimplicialModules :: kernel(SimplicialModuleMap)
next | previous | forward | backward | up | index | toc

kernel(SimplicialModuleMap) -- make the kernel of a map of simplicial modules

Description

The kernel of a simplicial module map $f : C \to D$ is the simplicial module $E$ whose $i$-th term is $kernel(f_i)$, and whose face/degeneracy map is induced from the face/degeneracy map on the source.

In the following example, we first construct a random simplicial morphism $f : C \to D$.

i1 : S = ZZ/101[a,b,c,d];
 
i2 : C = simplicialModule(freeResolution ideal(b^2-a*c, b*c-a*d, c^2-b*d), 3, Degeneracy => true)

      1      4      9      16
o2 = S  <-- S  <-- S  <-- S  <-- ...
                           
     0      1      2      3

o2 : SimplicialModule
 
i3 : D = simplicialModule(freeResolution ideal(a,b,c), Degeneracy => true)

      1      4      10      20
o3 = S  <-- S  <-- S   <-- S  <-- ...
                            
     0      1      2       3

o3 : SimplicialModule
 
i4 : f = randomSimplicialMap(D, C, Boundary => true, InternalDegree => 0)

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

          4                                           4
     1 : S  <--------------------------------------- S  : 1
               {0} | 0 0        0        0       |
               {1} | 0 -24b+36c 29b-19c  10b+29c |
               {1} | 0 24a+30c  -29a-19c -10a+8c |
               {1} | 0 -36a-30b 19a+19b  -29a-8b |

          10                                                                                                  9
     2 : S   <---------------------------------------------------------------------------------------------- S  : 2
                {0} | 0 0        0        0       0        0        0       0            0               |
                {1} | 0 -24b+36c 29b-19c  10b+29c 0        0        0       0            0               |
                {1} | 0 24a+30c  -29a-19c -10a+8c 0        0        0       0            0               |
                {1} | 0 -36a-30b 19a+19b  -29a-8b 0        0        0       0            0               |
                {1} | 0 0        0        0       -24b+36c 29b-19c  10b+29c 0            0               |
                {1} | 0 0        0        0       24a+30c  -29a-19c -10a+8c 0            0               |
                {1} | 0 0        0        0       -36a-30b 19a+19b  -29a-8b 0            0               |
                {2} | 0 0        0        0       0        0        0       10a-29b-46c  -10b+24d        |
                {2} | 0 0        0        0       0        0        0       29a+41b+36c  -19c-36d        |
                {2} | 0 0        0        0       0        0        0       -14a+19b+30c -29a-8b-19c-30d |

          20                                                                                                                                                                                      16
     3 : S   <---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- S   : 3
                {0} | 0 0        0        0       0        0        0       0        0        0       0            0               0            0               0            0               |
                {1} | 0 -24b+36c 29b-19c  10b+29c 0        0        0       0        0        0       0            0               0            0               0            0               |
                {1} | 0 24a+30c  -29a-19c -10a+8c 0        0        0       0        0        0       0            0               0            0               0            0               |
                {1} | 0 -36a-30b 19a+19b  -29a-8b 0        0        0       0        0        0       0            0               0            0               0            0               |
                {1} | 0 0        0        0       -24b+36c 29b-19c  10b+29c 0        0        0       0            0               0            0               0            0               |
                {1} | 0 0        0        0       24a+30c  -29a-19c -10a+8c 0        0        0       0            0               0            0               0            0               |
                {1} | 0 0        0        0       -36a-30b 19a+19b  -29a-8b 0        0        0       0            0               0            0               0            0               |
                {1} | 0 0        0        0       0        0        0       -24b+36c 29b-19c  10b+29c 0            0               0            0               0            0               |
                {1} | 0 0        0        0       0        0        0       24a+30c  -29a-19c -10a+8c 0            0               0            0               0            0               |
                {1} | 0 0        0        0       0        0        0       -36a-30b 19a+19b  -29a-8b 0            0               0            0               0            0               |
                {2} | 0 0        0        0       0        0        0       0        0        0       10a-29b-46c  -10b+24d        0            0               0            0               |
                {2} | 0 0        0        0       0        0        0       0        0        0       29a+41b+36c  -19c-36d        0            0               0            0               |
                {2} | 0 0        0        0       0        0        0       0        0        0       -14a+19b+30c -29a-8b-19c-30d 0            0               0            0               |
                {2} | 0 0        0        0       0        0        0       0        0        0       0            0               10a-29b-46c  -10b+24d        0            0               |
                {2} | 0 0        0        0       0        0        0       0        0        0       0            0               29a+41b+36c  -19c-36d        0            0               |
                {2} | 0 0        0        0       0        0        0       0        0        0       0            0               -14a+19b+30c -29a-8b-19c-30d 0            0               |
                {2} | 0 0        0        0       0        0        0       0        0        0       0            0               0            0               10a-29b-46c  -10b+24d        |
                {2} | 0 0        0        0       0        0        0       0        0        0       0            0               0            0               29a+41b+36c  -19c-36d        |
                {2} | 0 0        0        0       0        0        0       0        0        0       0            0               0            0               -14a+19b+30c -29a-8b-19c-30d |
                {3} | 0 0        0        0       0        0        0       0        0        0       0            0               0            0               0            0               |

o4 : SimplicialModuleMap
 
i5 : prune ker f

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

o5 : SimplicialModule
 
i6 : h1 = inducedMap(source f, ker f)

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

          4
     1 : S  <------------------------- image {0} | 1 0           | : 1
               {0} | 1 0           |         {2} | 0 18a+b+39c   |
               {2} | 0 18a+b+39c   |         {2} | 0 25a-40b-35c |
               {2} | 0 25a-40b-35c |         {2} | 0 a-23b-40c   |
               {2} | 0 a-23b-40c   |

          9
     2 : S  <------------------------------------- image {0} | 1 0           0           | : 2
               {0} | 1 0           0           |         {2} | 0 18a+b+39c   0           |
               {2} | 0 18a+b+39c   0           |         {2} | 0 25a-40b-35c 0           |
               {2} | 0 25a-40b-35c 0           |         {2} | 0 a-23b-40c   0           |
               {2} | 0 a-23b-40c   0           |         {2} | 0 0           18a+b+39c   |
               {2} | 0 0           18a+b+39c   |         {2} | 0 0           25a-40b-35c |
               {2} | 0 0           25a-40b-35c |         {2} | 0 0           a-23b-40c   |
               {2} | 0 0           a-23b-40c   |         {3} | 0 0           0           |
               {3} | 0 0           0           |         {3} | 0 0           0           |
               {3} | 0 0           0           |

          16
     3 : S   <------------------------------------------------- image {0} | 1 0           0           0           | : 3
                {0} | 1 0           0           0           |         {2} | 0 18a+b+39c   0           0           |
                {2} | 0 18a+b+39c   0           0           |         {2} | 0 25a-40b-35c 0           0           |
                {2} | 0 25a-40b-35c 0           0           |         {2} | 0 a-23b-40c   0           0           |
                {2} | 0 a-23b-40c   0           0           |         {2} | 0 0           18a+b+39c   0           |
                {2} | 0 0           18a+b+39c   0           |         {2} | 0 0           25a-40b-35c 0           |
                {2} | 0 0           25a-40b-35c 0           |         {2} | 0 0           a-23b-40c   0           |
                {2} | 0 0           a-23b-40c   0           |         {2} | 0 0           0           18a+b+39c   |
                {2} | 0 0           0           18a+b+39c   |         {2} | 0 0           0           25a-40b-35c |
                {2} | 0 0           0           25a-40b-35c |         {2} | 0 0           0           a-23b-40c   |
                {2} | 0 0           0           a-23b-40c   |         {3} | 0 0           0           0           |
                {3} | 0 0           0           0           |         {3} | 0 0           0           0           |
                {3} | 0 0           0           0           |         {3} | 0 0           0           0           |
                {3} | 0 0           0           0           |         {3} | 0 0           0           0           |
                {3} | 0 0           0           0           |         {3} | 0 0           0           0           |
                {3} | 0 0           0           0           |         {3} | 0 0           0           0           |
                {3} | 0 0           0           0           |

o6 : SimplicialModuleMap
 
i7 : ker f == image h1

o7 = true
 
i8 : ker h1 == 0

o8 = true

See also

Ways to use this method:

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/SimplicialModules/SimplicialModuleDOC.m2:2854:0.