components(ZZdFactorizationMap) -- list the components of a direct sum

Function: components
Usage:

components f
Inputs:
- f, a map of ZZ/d-graded factorizations,
Outputs:
- a list, the component maps of a direct sum of maps of ZZ/d-graded factorizations

Description

A map of ZZ/d-graded factorizations stores its component maps.

i1 : S = ZZ/101[a,b,c]

o1 = S

o1 : PolynomialRing

i2 : R = S/(a^2+b^2+c^2);

i3 : m = ideal vars R

o3 = ideal (a, b, c)

o3 : Ideal of R

i4 : C = tailMF m

      4      4      4
o4 = S  <-- S  <-- S
                    
     0      1      0

o4 : ZZdFactorization

i5 : g1 = id_C

          4                       4
o5 = 0 : S  <------------------- S  : 0
               {3} | 1 0 0 0 |
               {3} | 0 1 0 0 |
               {3} | 0 0 1 0 |
               {3} | 0 0 0 1 |

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

o5 : ZZdFactorizationMap

i6 : g2 = randomFactorizationMap(C[1], C[2], Boundary => true)

          4                                                                   4
o6 = 0 : S  <--------------------------------------------------------------- S  : 0
               {4} | 48a-36b-c    -18a-15b-11c -38a-50b-24c -39a-34b-13c |
               {4} | 37a+29b+11c  -28a+10b+32c 11a-23b-22c  -38a-34b-35c |
               {4} | -22a+30b-14c 19a-44b-39c  -25a-34b-8c  16a+19b+9c   |
               {4} | 44a-24b+28c  31a+20b+25c  -15a+10b+29c -21a+23b     |

          4                                                                  4
     1 : S  <-------------------------------------------------------------- S  : 1
               {3} | 11a+20b-9c  -37a-23b+39c -28a-10b+14c -38a+24b-8c  |
               {3} | 25a+15b+35c -22a-34b-32c -19a-50b-11c -16a-36b-22c |
               {3} | 38a-44b     48a-19b+25c  18a+34b+28c  39a-30b-29c  |
               {3} | 15a+10b+13c 44a+34b+11c  -31a+23b+c   21a-29b-24c  |

o6 : ZZdFactorizationMap

i7 : f = g1 ++ g2

          8                                                                           8
o7 = 0 : S  <----------------------------------------------------------------------- S  : 0
               {3} | 1 0 0 0 0            0            0            0            |
               {3} | 0 1 0 0 0            0            0            0            |
               {3} | 0 0 1 0 0            0            0            0            |
               {3} | 0 0 0 1 0            0            0            0            |
               {4} | 0 0 0 0 48a-36b-c    -18a-15b-11c -38a-50b-24c -39a-34b-13c |
               {4} | 0 0 0 0 37a+29b+11c  -28a+10b+32c 11a-23b-22c  -38a-34b-35c |
               {4} | 0 0 0 0 -22a+30b-14c 19a-44b-39c  -25a-34b-8c  16a+19b+9c   |
               {4} | 0 0 0 0 44a-24b+28c  31a+20b+25c  -15a+10b+29c -21a+23b     |

          8                                                                          8
     1 : S  <---------------------------------------------------------------------- S  : 1
               {4} | 1 0 0 0 0           0            0            0            |
               {4} | 0 1 0 0 0           0            0            0            |
               {4} | 0 0 1 0 0           0            0            0            |
               {4} | 0 0 0 1 0           0            0            0            |
               {3} | 0 0 0 0 11a+20b-9c  -37a-23b+39c -28a-10b+14c -38a+24b-8c  |
               {3} | 0 0 0 0 25a+15b+35c -22a-34b-32c -19a-50b-11c -16a-36b-22c |
               {3} | 0 0 0 0 38a-44b     48a-19b+25c  18a+34b+28c  39a-30b-29c  |
               {3} | 0 0 0 0 15a+10b+13c 44a+34b+11c  -31a+23b+c   21a-29b-24c  |

o7 : ZZdFactorizationMap

i8 : assert isWellDefined f

i9 : L = components f

           4                       4           4
o9 = {0 : S  <------------------- S  : 0, 0 : S 
                {3} | 1 0 0 0 |                 
                {3} | 0 1 0 0 |                 
                {3} | 0 0 1 0 |                 
                {3} | 0 0 0 1 |                 

           4                       4           4
      1 : S  <------------------- S  : 1  1 : S 
                {4} | 1 0 0 0 |                 
                {4} | 0 1 0 0 |                 
                {4} | 0 0 1 0 |                 
                {4} | 0 0 0 1 |                 
     ------------------------------------------------------------------------
                                                                       4
     <--------------------------------------------------------------- S  : 0}
        {4} | 48a-36b-c    -18a-15b-11c -38a-50b-24c -39a-34b-13c |
        {4} | 37a+29b+11c  -28a+10b+32c 11a-23b-22c  -38a-34b-35c |
        {4} | -22a+30b-14c 19a-44b-39c  -25a-34b-8c  16a+19b+9c   |
        {4} | 44a-24b+28c  31a+20b+25c  -15a+10b+29c -21a+23b     |

                                                                      4
     <-------------------------------------------------------------- S  : 1
        {3} | 11a+20b-9c  -37a-23b+39c -28a-10b+14c -38a+24b-8c  |
        {3} | 25a+15b+35c -22a-34b-32c -19a-50b-11c -16a-36b-22c |
        {3} | 38a-44b     48a-19b+25c  18a+34b+28c  39a-30b-29c  |
        {3} | 15a+10b+13c 44a+34b+11c  -31a+23b+c   21a-29b-24c  |

o9 : List

i10 : L_0 === g1

o10 = true

i11 : L_1 === g2

o11 = true

i12 : indices f

o12 = {0, 1}

o12 : List

i13 : f' = (chicken => g1) ++ (nuggets => g2)

           8                                                                           8
o13 = 0 : S  <----------------------------------------------------------------------- S  : 0
                {3} | 1 0 0 0 0            0            0            0            |
                {3} | 0 1 0 0 0            0            0            0            |
                {3} | 0 0 1 0 0            0            0            0            |
                {3} | 0 0 0 1 0            0            0            0            |
                {4} | 0 0 0 0 48a-36b-c    -18a-15b-11c -38a-50b-24c -39a-34b-13c |
                {4} | 0 0 0 0 37a+29b+11c  -28a+10b+32c 11a-23b-22c  -38a-34b-35c |
                {4} | 0 0 0 0 -22a+30b-14c 19a-44b-39c  -25a-34b-8c  16a+19b+9c   |
                {4} | 0 0 0 0 44a-24b+28c  31a+20b+25c  -15a+10b+29c -21a+23b     |

           8                                                                          8
      1 : S  <---------------------------------------------------------------------- S  : 1
                {4} | 1 0 0 0 0           0            0            0            |
                {4} | 0 1 0 0 0           0            0            0            |
                {4} | 0 0 1 0 0           0            0            0            |
                {4} | 0 0 0 1 0           0            0            0            |
                {3} | 0 0 0 0 11a+20b-9c  -37a-23b+39c -28a-10b+14c -38a+24b-8c  |
                {3} | 0 0 0 0 25a+15b+35c -22a-34b-32c -19a-50b-11c -16a-36b-22c |
                {3} | 0 0 0 0 38a-44b     48a-19b+25c  18a+34b+28c  39a-30b-29c  |
                {3} | 0 0 0 0 15a+10b+13c 44a+34b+11c  -31a+23b+c   21a-29b-24c  |

o13 : ZZdFactorizationMap

i14 : components f'

            4                       4           4
o14 = {0 : S  <------------------- S  : 0, 0 : S 
                 {3} | 1 0 0 0 |                 
                 {3} | 0 1 0 0 |                 
                 {3} | 0 0 1 0 |                 
                 {3} | 0 0 0 1 |                 

            4                       4           4
       1 : S  <------------------- S  : 1  1 : S 
                 {4} | 1 0 0 0 |                 
                 {4} | 0 1 0 0 |                 
                 {4} | 0 0 1 0 |                 
                 {4} | 0 0 0 1 |                 
      -----------------------------------------------------------------------
                                                                        4
      <--------------------------------------------------------------- S  : 0
         {4} | 48a-36b-c    -18a-15b-11c -38a-50b-24c -39a-34b-13c |
         {4} | 37a+29b+11c  -28a+10b+32c 11a-23b-22c  -38a-34b-35c |
         {4} | -22a+30b-14c 19a-44b-39c  -25a-34b-8c  16a+19b+9c   |
         {4} | 44a-24b+28c  31a+20b+25c  -15a+10b+29c -21a+23b     |

                                                                       4
      <-------------------------------------------------------------- S  : 1
         {3} | 11a+20b-9c  -37a-23b+39c -28a-10b+14c -38a+24b-8c  |
         {3} | 25a+15b+35c -22a-34b-32c -19a-50b-11c -16a-36b-22c |
         {3} | 38a-44b     48a-19b+25c  18a+34b+28c  39a-30b-29c  |
         {3} | 15a+10b+13c 44a+34b+11c  -31a+23b+c   21a-29b-24c  |
      -----------------------------------------------------------------------
      }

o14 : List

i15 : indices f'

o15 = {chicken, nuggets}

o15 : List

The names of the components are called indices, and are used to access the relevant inclusion and projection maps.

i16 : f'_[chicken]

           8                       4
o16 = 0 : S  <------------------- S  : 0
                {3} | 1 0 0 0 |
                {3} | 0 1 0 0 |
                {3} | 0 0 1 0 |
                {3} | 0 0 0 1 |
                {4} | 0 0 0 0 |
                {4} | 0 0 0 0 |
                {4} | 0 0 0 0 |
                {4} | 0 0 0 0 |

           8                       4
      1 : S  <------------------- S  : 1
                {4} | 1 0 0 0 |
                {4} | 0 1 0 0 |
                {4} | 0 0 1 0 |
                {4} | 0 0 0 1 |
                {3} | 0 0 0 0 |
                {3} | 0 0 0 0 |
                {3} | 0 0 0 0 |
                {3} | 0 0 0 0 |

o16 : ZZdFactorizationMap

i17 : f'^[nuggets]

           4                                                                           8
o17 = 0 : S  <----------------------------------------------------------------------- S  : 0
                {4} | 0 0 0 0 48a-36b-c    -18a-15b-11c -38a-50b-24c -39a-34b-13c |
                {4} | 0 0 0 0 37a+29b+11c  -28a+10b+32c 11a-23b-22c  -38a-34b-35c |
                {4} | 0 0 0 0 -22a+30b-14c 19a-44b-39c  -25a-34b-8c  16a+19b+9c   |
                {4} | 0 0 0 0 44a-24b+28c  31a+20b+25c  -15a+10b+29c -21a+23b     |

           4                                                                          8
      1 : S  <---------------------------------------------------------------------- S  : 1
                {3} | 0 0 0 0 11a+20b-9c  -37a-23b+39c -28a-10b+14c -38a+24b-8c  |
                {3} | 0 0 0 0 25a+15b+35c -22a-34b-32c -19a-50b-11c -16a-36b-22c |
                {3} | 0 0 0 0 38a-44b     48a-19b+25c  18a+34b+28c  39a-30b-29c  |
                {3} | 0 0 0 0 15a+10b+13c 44a+34b+11c  -31a+23b+c   21a-29b-24c  |

o17 : ZZdFactorizationMap

i18 : f^[0]

           4                               8
o18 = 0 : S  <--------------------------- S  : 0
                {3} | 1 0 0 0 0 0 0 0 |
                {3} | 0 1 0 0 0 0 0 0 |
                {3} | 0 0 1 0 0 0 0 0 |
                {3} | 0 0 0 1 0 0 0 0 |

           4                               8
      1 : S  <--------------------------- S  : 1
                {4} | 1 0 0 0 0 0 0 0 |
                {4} | 0 1 0 0 0 0 0 0 |
                {4} | 0 0 1 0 0 0 0 0 |
                {4} | 0 0 0 1 0 0 0 0 |

o18 : ZZdFactorizationMap

i19 : f_[0]

           8                       4
o19 = 0 : S  <------------------- S  : 0
                {3} | 1 0 0 0 |
                {3} | 0 1 0 0 |
                {3} | 0 0 1 0 |
                {3} | 0 0 0 1 |
                {4} | 0 0 0 0 |
                {4} | 0 0 0 0 |
                {4} | 0 0 0 0 |
                {4} | 0 0 0 0 |

           8                       4
      1 : S  <------------------- S  : 1
                {4} | 1 0 0 0 |
                {4} | 0 1 0 0 |
                {4} | 0 0 1 0 |
                {4} | 0 0 0 1 |
                {3} | 0 0 0 0 |
                {3} | 0 0 0 0 |
                {3} | 0 0 0 0 |
                {3} | 0 0 0 0 |

o19 : ZZdFactorizationMap

Ways to use this method:

components(ZZdFactorizationMap) -- list the components of a direct sum

The source of this document is in MatrixFactorizations/MatrixFactorizationsDOC.m2:3049:0.

components(ZZdFactorizationMap) -- list the components of a direct sum

Description

See also

Ways to use this method: