decomposeHomogeneousMA -- Decomposition of one monomial algebra over a subalgebra

Synopsis

• Usage:

  decomposeHomogeneousMA f

  decomposeHomogeneousMA R

  decomposeHomogeneousMA B

  decomposeHomogeneousMA M

• Inputs:
  • f, a ring map, between multigraded polynomial rings such that the degree monoids are homogeneous and the degrees are minimal generating sets of the degree monoids, or
  • R, a polynomial ring, multigraded, with homogeneous degree monoid, such that B = degrees R are minimal generators of the degree monoid, and set K = coefficientRing R, or
  • B, a list, with the generators of an affine semigroup in \mathbb{N}^d. In the case B is specified, K is set via the Option CoefficientField. If a list of positive integers is given, the function uses adjoinPurePowers and homogenizeSemigroup to convert it into a list of elements of \mathbb{N}^2.
  • M, a MonomialAlgebra,
• Optional inputs:
  • CoefficientField => ..., default value ZZ/101, Option to set the coefficient field.
  • ReturnMingens => ..., default value false, Option to return the minimal generating set B_A of K[B] as K[A]-module.
  • Verbose => ..., default value 0, Option to print intermediate results.
• Outputs:
  • a hash table, with the following data: Let B be the degree monoid of the target of f and analogously A for the source. The keys are representatives of congruence classes in G(B) / G(A). The value associated to a key k is a tuple whose first component is a monomial ideal of K[A] isomorphic to the K[A]-submodule of K[B] consisting of elements in the class k, and whose second component is an element of ZZ that is the twist between the ideal and the submodule of K[B] with respect to the standard grading (which gives the minimal generators degree 1).

Description

i1 : B = {{4,0},{3,1},{1,3},{0,4}}

o1 = {{4, 0}, {3, 1}, {1, 3}, {0, 4}}

o1 : List

i2 : S = ZZ/101[x_0..x_(#B-1), Degrees=>B];

i3 : decomposeHomogeneousMA S

o3 = HashTable{| -1 | => {ideal 1, 1}      }
               | 1  |
               0 => {ideal 1, 0}
               | 1  | => {ideal 1, 1}
               | -1 |
               | 2 | => {ideal (x , x ), 1}
               | 2 |             0   3

o3 : HashTable

i4 : decomposeHomogeneousMA B

o4 = HashTable{| -1 | => {ideal 1, 1}      }
               | 1  |
               0 => {ideal 1, 0}
               | 1  | => {ideal 1, 1}
               | -1 |
               | 2 | => {ideal (x , x ), 1}
               | 2 |             0   3

o4 : HashTable

i5 : decomposeHomogeneousMA {{2,0,1},{0,2,1},{1,1,1},{2,2,1},{2,1,1},{1,4,1}}

                                 2
o5 = HashTable{0 => {ideal (x , x x ), -1}    }
                             3   0 5
                                       2
               | 0 | => {ideal (x x , x ), -1}
               | 1 |             0 5   3
               | 0 |

o5 : HashTable

i6 : M = monomialAlgebra {{2,0,1},{0,2,1},{1,1,1},{2,2,1},{2,1,1},{1,4,1}}

      ZZ
o6 = ---[x ..x ]
     101  0   5

o6 : MonomialAlgebra generated by {{2, 0, 1}, {0, 2, 1}, {1, 1, 1}, {2, 2, 1}, {2, 1, 1}, {1, 4, 1}}

i7 : decomposeHomogeneousMA M

                                 2
o7 = HashTable{0 => {ideal (x , x x ), -1}    }
                             3   0 5
                                       2
               | 0 | => {ideal (x x , x ), -1}
               | 1 |             0 5   3
               | 0 |

o7 : HashTable