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 |
The object decomposeHomogeneousMA is a method function with options.