fullCollapse -- Converts a d-fold factorization into a 2-fold factorization

Usage:

fullCollapse(X, n, r)
Inputs:
- X, a ZZ/d graded factorization, The input d-fold factorization
- n, an integer, Starting position for composing maps
- r, an integer, Number of maps to compose
Outputs:
- a ZZ/d graded factorization, The resulting 2-fold factorization

Description

This method converts a d-fold factorization into a 2-fold factorization by composing the maps in the specified positions. It starts composing maps from position n and continues composing r maps. The resulting 2-fold factorization is returned.

i1 : Q = ZZ/101[a,b];

i2 : C = linearMF(a^4+b^4, t)

     / Q[t] \4     / Q[t] \4     / Q[t] \4     / Q[t] \4     / Q[t] \4
o2 = |------|  <-- |------|  <-- |------|  <-- |------|  <-- |------|
     | 2    |      | 2    |      | 2    |      | 2    |      | 2    |
     \t  + 1/      \t  + 1/      \t  + 1/      \t  + 1/      \t  + 1/
                                                              
     0             1             2             3             0

o2 : ZZdFactorization

i3 : C.dd

         / Q[t] \4                             / Q[t] \4
o3 = 3 : |------|  <-------------------------- |------|  : 0
         | 2    |     {0, 2} | b a  0  0   |   | 2    |
         \t  + 1/     {0, 2} | 0 bt a  0   |   \t  + 1/
                      {0, 2} | 0 0  -b a   |
                      {0, 2} | a 0  0  -bt |

         / Q[t] \4                             / Q[t] \4
     0 : |------|  <-------------------------- |------|  : 1
         | 2    |     {0, 2} | b a  0  0   |   | 2    |
         \t  + 1/     {0, 2} | 0 bt a  0   |   \t  + 1/
                      {0, 2} | 0 0  -b a   |
                      {0, 2} | a 0  0  -bt |

         / Q[t] \4                             / Q[t] \4
     1 : |------|  <-------------------------- |------|  : 2
         | 2    |     {0, 2} | b a  0  0   |   | 2    |
         \t  + 1/     {0, 2} | 0 bt a  0   |   \t  + 1/
                      {0, 2} | 0 0  -b a   |
                      {0, 2} | a 0  0  -bt |

         / Q[t] \4                             / Q[t] \4
     2 : |------|  <-------------------------- |------|  : 3
         | 2    |     {0, 2} | b a  0  0   |   | 2    |
         \t  + 1/     {0, 2} | 0 bt a  0   |   \t  + 1/
                      {0, 2} | 0 0  -b a   |
                      {0, 2} | a 0  0  -bt |

o3 : ZZdFactorizationMap

i4 : C.dd^4

         / Q[t] \4                                         / Q[t] \4
o4 = 0 : |------|  <-------------------------------------- |------|  : 0
         | 2    |     {0, 2} | a4+b4 0     0     0     |   | 2    |
         \t  + 1/     {0, 2} | 0     a4+b4 0     0     |   \t  + 1/
                      {0, 2} | 0     0     a4+b4 0     |
                      {0, 2} | 0     0     0     a4+b4 |

         / Q[t] \4                                         / Q[t] \4
     1 : |------|  <-------------------------------------- |------|  : 1
         | 2    |     {0, 2} | a4+b4 0     0     0     |   | 2    |
         \t  + 1/     {0, 2} | 0     a4+b4 0     0     |   \t  + 1/
                      {0, 2} | 0     0     a4+b4 0     |
                      {0, 2} | 0     0     0     a4+b4 |

         / Q[t] \4                                         / Q[t] \4
     2 : |------|  <-------------------------------------- |------|  : 2
         | 2    |     {0, 2} | a4+b4 0     0     0     |   | 2    |
         \t  + 1/     {0, 2} | 0     a4+b4 0     0     |   \t  + 1/
                      {0, 2} | 0     0     a4+b4 0     |
                      {0, 2} | 0     0     0     a4+b4 |

         / Q[t] \4                                         / Q[t] \4
     3 : |------|  <-------------------------------------- |------|  : 3
         | 2    |     {0, 2} | a4+b4 0     0     0     |   | 2    |
         \t  + 1/     {0, 2} | 0     a4+b4 0     0     |   \t  + 1/
                      {0, 2} | 0     0     a4+b4 0     |
                      {0, 2} | 0     0     0     a4+b4 |

o4 : ZZdFactorizationMap

i5 : C' = fullCollapse(C,2,1)

     / Q[t] \4     / Q[t] \4     / Q[t] \4
o5 = |------|  <-- |------|  <-- |------|
     | 2    |      | 2    |      | 2    |
     \t  + 1/      \t  + 1/      \t  + 1/
                                  
     0             1             0

o5 : ZZdFactorization

i6 : isdFactorization C'

             4    4
o6 = (true, a  + b )

o6 : Sequence

i7 : E = Hom(C,C)

     / Q[t] \64     / Q[t] \64     / Q[t] \64     / Q[t] \64     / Q[t] \64
o7 = |------|   <-- |------|   <-- |------|   <-- |------|   <-- |------|
     | 2    |       | 2    |       | 2    |       | 2    |       | 2    |
     \t  + 1/       \t  + 1/       \t  + 1/       \t  + 1/       \t  + 1/
                                                                  
     0              1              2              3              0

o7 : ZZdFactorization

i8 : prune HH_1 fullCollapse(E,2,1)

o8 = cokernel | 0         0         -ab2t+ab2  ab2t-ab2 2a2bt-a2b   -3a2bt-a2b 49ab2t+50ab2 -a2b -a3 b3t-b3        ab2t-ab2    50a2bt+48a2b -a2bt+50a2b -a3t+50a3    a2b2t 0 0 0 b3     -ab2 0 a2bt    0   0  0   2a2bt -b3t-2b3     0    -2a2bt   0         b3t+4b3      0    0   -49a2b      -a3t      0        0            b3t+2b3       b3      ab2 -a3b 0 0 0 0 0      0    -2b3t         -2ab2t-ab2 0 0             0   0  0   |
              | 0         0         0          -abt     -a2t-a2     0          0            0    0   -b2t+b2       -abt+ab     a2           0           0            0     0 0 0 b2t-b2 0    0 0       0   0  0   0     2b2          0    0        0         -4b2         0    0   0           0         0        0            -2b2          -b2     -ab a3   0 0 0 0 0      0    b2t+b2        0          0 0             0   0  0   |
              | 0         0         48b2t+50b2 0        -48abt-50ab -2ab       49b2t+49b2   abt  a2t 0             0           -49abt+49ab  -ab         -a2          0     0 0 0 0      0    0 0       0   0  0   0     0            0    0        0         0            0    0   -2abt       -a2t-a2   0        0            0             0       0   0    0 0 0 0 0      0    0             -2b2       0 0             0   0  0   |
              | 0         0         50bt+50b   0        -50at-50a   0          50bt+50b     0    0   0             0           -50at+50a    0           0            0     0 0 0 0      0    0 0       0   0  0   0     0            0    0        0         0            0    0   0           0         0        0            0             0       0   0    0 0 0 0 0      0    0             0          0 0             0   0  0   |
              | 0         0         0          0        0           0          0            0    0   0             0           0            0           0            0     0 0 0 0      0    0 ab2     a2b a3 b3t 0     0            0    0        0         0            0    0   0           0         0        0            0             0       0   0    0 0 0 0 0      0    0             0          0 0             0   0  0   |
              | ab2t+2ab2 a2bt+2a2b 0          0        0           0          0            0    0   -50a2bt-50a2b -50a3t-50a3 0            50b3t-50b3  49ab2t+49ab2 -34a4 0 0 0 0      0    0 b3t+2b3 0   0  0   2b3   50a2bt+50a2b -a3t -2b3t-b3 -a2bt+a2b 49a2bt-50a2b -a2b -a3 -50b3t-50b3 -ab2t-ab2 3ab2t    47ab2t+50ab2 -48a2bt+50a2b a2b     0   ab3  0 0 0 0 b3t+b3 -ab2 -50a2bt+49a2b 33a3       0 -50a2bt+50a2b 0   0  0   |
              | -2ab      -2a2      0          0        0           0          0            0    0   0             0           0            0           2ab          0     0 0 0 0      0    0 -2b2    0   0  0   -2b2  0            0    2b2t+2b2 2a2t      0            0    0   0           2ab       -2abt-ab 2abt+2ab     -2a2t         0       0   0    0 0 0 0 -2b2   0    0             0          0 0             0   0  0   |
              | 0         0         0          0        0           0          0            0    0   -abt+ab       -a2t+a2     0            0           0            33a3  0 0 0 0      0    0 0       0   0  0   0     abt+ab       0    0        0         -3abt-5ab    abt  a2t 0           0         -49b2    -b2          -abt          -abt-ab -a2 -b3t 0 0 0 0 0      0    abt+ab        -33a2      0 abt+ab        0   0  0   |
              | 0         0         0          0        0           0          0            0    0   0             0           0            b           a            0     0 0 0 0      0    0 0       0   0  0   0     0            0    0        0         0            0    0   bt          at+a      0        0            0             0       0   0    0 0 0 0 0      0    0             0          0 0             0   0  0   |
              | 0         0         0          0        0           0          0            0    0   0             0           0            0           0            0     0 0 0 0      0    0 0       0   0  0   0     0            0    0        0         0            0    0   0           0         -bt-50b  bt           -at+a         0       0   0    0 0 0 0 0      0    0             0          0 0             0   0  0   |
              | 0         0         0          0        0           0          0            0    0   -bt+b         -at+a       0            0           0            0     0 0 0 0      0    0 0       0   0  0   0     bt+b         0    0        0         -3bt-3b      0    0   0           0         0        0            -bt-b         -b      -a  0    0 0 0 0 0      0    bt+b          0          0 bt+b          0   0  0   |
              | 0         0         0          0        0           0          0            0    0   0             0           0            0           0            0     0 0 0 0      0    0 0       0   0  0   0     0            0    0        0         0            0    0   0           0         0        0            0             0       0   0    0 0 0 0 0      0    0             0          0 ab2           a2b a3 b3t |

      Q[t]                      / Q[t] \12
o8 : -------module, quotient of |------|
      2                         | 2    |
     t  + 1                     \t  + 1/

Ways to use fullCollapse:

fullCollapse(ZZdFactorization,ZZ,ZZ)

For the programmer

The object fullCollapse is a method function.

The source of this document is in MatrixFactorizations/MatrixFactorizationsDOC.m2:6387:0.

fullCollapse -- Converts a d-fold factorization into a 2-fold factorization

Description

See also

Ways to use fullCollapse:

For the programmer