rk1MCM2gen -- Construct every possible rank 1, 2-generated maximal Cohen-Macaulay module over the cubic Fermat hypersurface in 4 variables

Usage:

rk1MCM2gen(L,d)
Inputs:
- L, a list, any permutation of the set {2,3,4}
- d, an integer, any integer specifying the characteristic of the ambient field
Outputs:
- a ZZ/d graded factorization, A matrix factorization corresponding to a rank 1, 2-generated maximal Cohen-Macaulay module over the cubic Fermat hypersurface in 4 variables.

Description

This method constructs every possible rank 1, 2-generated maximal Cohen-Macaulay module (up to isomorphism) over the cubic Fermat hypersurface in 4 variables. It is known that such MCM modules are parametrized by all permutations of $\{ 2, 3 ,4 \}$.

i1 : H = hashTable for i in permutations {2,3,4} list i => rk1MCM2gen(i,0);

i2 : H#{2,3,4}.dd

         /    QQ[x ..x , a, b]    \                                                                     /    QQ[x ..x , a, b]    \
         |        1   4           |2                                                                    |        1   4           |2
o2 = 1 : |------------------------|  <----------------------------------------------------------------- |------------------------|  : 0
         |  2           2         |     {2} | x_1x_4a+x_4^2a+x_1^2-x_4^2 x_2x_3b+x_3^2b+x_2^2-x_3^2 |   |  2           2         |
         \(a  - a + 1, b  - b + 1)/     {3} | x_3b-x_2                   -x_4a+x_1                  |   \(a  - a + 1, b  - b + 1)/

         /    QQ[x ..x , a, b]    \                                                     /    QQ[x ..x , a, b]    \
         |        1   4           |2                                                    |        1   4           |2
     0 : |------------------------|  <------------------------------------------------- |------------------------|  : 1
         |  2           2         |     {3} | -x_4a+x_1 -x_2x_3b-x_3^2b-x_2^2+x_3^2 |   |  2           2         |
         \(a  - a + 1, b  - b + 1)/     {3} | -x_3b+x_2 x_1x_4a+x_4^2a+x_1^2-x_4^2  |   \(a  - a + 1, b  - b + 1)/

o2 : ZZdFactorizationMap

i3 : H#{3,2,4}.dd

         /    QQ[x ..x , a, b]    \                                                                     /    QQ[x ..x , a, b]    \
         |        1   4           |2                                                                    |        1   4           |2
o3 = 1 : |------------------------|  <----------------------------------------------------------------- |------------------------|  : 0
         |  2           2         |     {2} | x_1x_4a+x_4^2a+x_1^2-x_4^2 x_2^2b+x_2x_3b-x_2^2+x_3^2 |   |  2           2         |
         \(a  - a + 1, b  - b + 1)/     {3} | x_2b-x_3                   -x_4a+x_1                  |   \(a  - a + 1, b  - b + 1)/

         /    QQ[x ..x , a, b]    \                                                     /    QQ[x ..x , a, b]    \
         |        1   4           |2                                                    |        1   4           |2
     0 : |------------------------|  <------------------------------------------------- |------------------------|  : 1
         |  2           2         |     {3} | -x_4a+x_1 -x_2^2b-x_2x_3b+x_2^2-x_3^2 |   |  2           2         |
         \(a  - a + 1, b  - b + 1)/     {3} | -x_2b+x_3 x_1x_4a+x_4^2a+x_1^2-x_4^2  |   \(a  - a + 1, b  - b + 1)/

o3 : ZZdFactorizationMap

i4 : H#{2,4,3}.dd

         /    QQ[x ..x , a, b]    \                                                                     /    QQ[x ..x , a, b]    \
         |        1   4           |2                                                                    |        1   4           |2
o4 = 1 : |------------------------|  <----------------------------------------------------------------- |------------------------|  : 0
         |  2           2         |     {2} | x_1x_3a+x_3^2a+x_1^2-x_3^2 x_2x_4b+x_4^2b+x_2^2-x_4^2 |   |  2           2         |
         \(a  - a + 1, b  - b + 1)/     {3} | x_4b-x_2                   -x_3a+x_1                  |   \(a  - a + 1, b  - b + 1)/

         /    QQ[x ..x , a, b]    \                                                     /    QQ[x ..x , a, b]    \
         |        1   4           |2                                                    |        1   4           |2
     0 : |------------------------|  <------------------------------------------------- |------------------------|  : 1
         |  2           2         |     {3} | -x_3a+x_1 -x_2x_4b-x_4^2b-x_2^2+x_4^2 |   |  2           2         |
         \(a  - a + 1, b  - b + 1)/     {3} | -x_4b+x_2 x_1x_3a+x_3^2a+x_1^2-x_3^2  |   \(a  - a + 1, b  - b + 1)/

o4 : ZZdFactorizationMap

i5 : H#{4,3,2}.dd

         /    QQ[x ..x , a, b]    \                                                                     /    QQ[x ..x , a, b]    \
         |        1   4           |2                                                                    |        1   4           |2
o5 = 1 : |------------------------|  <----------------------------------------------------------------- |------------------------|  : 0
         |  2           2         |     {2} | x_1x_2a+x_2^2a+x_1^2-x_2^2 x_3^2b+x_3x_4b-x_3^2+x_4^2 |   |  2           2         |
         \(a  - a + 1, b  - b + 1)/     {3} | x_3b-x_4                   -x_2a+x_1                  |   \(a  - a + 1, b  - b + 1)/

         /    QQ[x ..x , a, b]    \                                                     /    QQ[x ..x , a, b]    \
         |        1   4           |2                                                    |        1   4           |2
     0 : |------------------------|  <------------------------------------------------- |------------------------|  : 1
         |  2           2         |     {3} | -x_2a+x_1 -x_3^2b-x_3x_4b+x_3^2-x_4^2 |   |  2           2         |
         \(a  - a + 1, b  - b + 1)/     {3} | -x_3b+x_4 x_1x_2a+x_2^2a+x_1^2-x_2^2  |   \(a  - a + 1, b  - b + 1)/

o5 : ZZdFactorizationMap

i6 : H#{4,2,3}.dd

         /    QQ[x ..x , a, b]    \                                                                     /    QQ[x ..x , a, b]    \
         |        1   4           |2                                                                    |        1   4           |2
o6 = 1 : |------------------------|  <----------------------------------------------------------------- |------------------------|  : 0
         |  2           2         |     {2} | x_1x_3a+x_3^2a+x_1^2-x_3^2 x_2^2b+x_2x_4b-x_2^2+x_4^2 |   |  2           2         |
         \(a  - a + 1, b  - b + 1)/     {3} | x_2b-x_4                   -x_3a+x_1                  |   \(a  - a + 1, b  - b + 1)/

         /    QQ[x ..x , a, b]    \                                                     /    QQ[x ..x , a, b]    \
         |        1   4           |2                                                    |        1   4           |2
     0 : |------------------------|  <------------------------------------------------- |------------------------|  : 1
         |  2           2         |     {3} | -x_3a+x_1 -x_2^2b-x_2x_4b+x_2^2-x_4^2 |   |  2           2         |
         \(a  - a + 1, b  - b + 1)/     {3} | -x_2b+x_4 x_1x_3a+x_3^2a+x_1^2-x_3^2  |   \(a  - a + 1, b  - b + 1)/

o6 : ZZdFactorizationMap

i7 : H#{3,4,2}.dd

         /    QQ[x ..x , a, b]    \                                                                     /    QQ[x ..x , a, b]    \
         |        1   4           |2                                                                    |        1   4           |2
o7 = 1 : |------------------------|  <----------------------------------------------------------------- |------------------------|  : 0
         |  2           2         |     {2} | x_1x_2a+x_2^2a+x_1^2-x_2^2 x_3x_4b+x_4^2b+x_3^2-x_4^2 |   |  2           2         |
         \(a  - a + 1, b  - b + 1)/     {3} | x_4b-x_3                   -x_2a+x_1                  |   \(a  - a + 1, b  - b + 1)/

         /    QQ[x ..x , a, b]    \                                                     /    QQ[x ..x , a, b]    \
         |        1   4           |2                                                    |        1   4           |2
     0 : |------------------------|  <------------------------------------------------- |------------------------|  : 1
         |  2           2         |     {3} | -x_2a+x_1 -x_3x_4b-x_4^2b-x_3^2+x_4^2 |   |  2           2         |
         \(a  - a + 1, b  - b + 1)/     {3} | -x_4b+x_3 x_1x_2a+x_2^2a+x_1^2-x_2^2  |   \(a  - a + 1, b  - b + 1)/

o7 : ZZdFactorizationMap

i8 : for i in permutations {2,3,4} list potential H#i

       3    3    3    3   3    3    3    3   3    3    3    3   3    3    3  
o8 = {x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  +
       1    2    3    4   1    2    3    4   1    2    3    4   1    2    3  
     ------------------------------------------------------------------------
      3   3    3    3    3   3    3    3    3
     x , x  + x  + x  + x , x  + x  + x  + x }
      4   1    2    3    4   1    2    3    4

o8 : List

Ways to use rk1MCM2gen:

rk1MCM2gen(List)
rk1MCM2gen(List,ZZ)

For the programmer

The object rk1MCM2gen is a method function.

The source of this document is in MatrixFactorizations/MatrixFactorizationsDOC.m2:6280:0.

rk1MCM2gen -- Construct every possible rank 1, 2-generated maximal Cohen-Macaulay module over the cubic Fermat hypersurface in 4 variables

Description

See also

Ways to use rk1MCM2gen:

For the programmer