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MonomialAlgebras :: decomposeHomogeneousMA

decomposeHomogeneousMA -- Decomposition of one monomial algebra over a subalgebra

Synopsis

Description

Let K[B] be the monomial algebra of the degree monoid of the target of f and let analogously K[A] for source of f. Assume that K[B] is finite as a K[A]-module.

The monomial algebra K[B] is decomposed as a direct sum of monomial ideals in K[A] with twists in ZZ.

If B or R with degrees B is specified then A is computed via findGeneratorsOfSubalgebra.

Note that the shift chosen by the function depends on the monomial ordering of K[A] (in the non-simplicial case).

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

Ways to use decomposeHomogeneousMA :

For the programmer

The object decomposeHomogeneousMA is a method function with options.