# decomposeHomogeneousMA -- Decomposition of one monomial algebra over a subalgebra

## Synopsis

• Usage:
decomposeHomogeneousMA f
decomposeHomogeneousMA R
decomposeHomogeneousMA B
decomposeHomogeneousMA M
• Inputs:
• f, , between multigraded polynomial rings such that the degree monoids are homogeneous and the degrees are minimal generating sets of the degree monoids, or
• R, , 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,
• 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:
• , 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

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 :

• "decomposeHomogeneousMA(List)"
• "decomposeHomogeneousMA(MonomialAlgebra)"
• "decomposeHomogeneousMA(PolynomialRing)"
• "decomposeHomogeneousMA(RingMap)"

## For the programmer

The object decomposeHomogeneousMA is .