monomialAlgebra -- Create a monomial algebra

Synopsis

Usage:

monomialAlgebra R

monomialAlgebra B
Inputs:
- R, a polynomial ring, multigraded, with B = degrees R and K = coefficientRing R, or
- B, a list, 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.
Optional inputs:
- CoefficientField => ..., default value ZZ/101, Option to set the coefficient field.
Outputs:
- a MonomialAlgebra,

Description

Create a monomial algebra K[B] by either specifying

- the semigroup B as a list of generators. The field K is selected via the option CoefficientField.

- a list of positive integers which is converted by adjoinPurePowers and homogenizeSemigroup into a list B of elements of \mathbb{N}^2. The field K is selected via the option CoefficientField.

- a multigraded polynomial ring R with Degrees R = B.

Specifying B:

i1 : B = {{1,2},{3,0},{0,4},{0,5}}

o1 = {{1, 2}, {3, 0}, {0, 4}, {0, 5}}

o1 : List

i2 : monomialAlgebra B

      ZZ
o2 = ---[x ..x ]
     101  0   3

o2 : MonomialAlgebra generated by {{1, 2}, {3, 0}, {0, 4}, {0, 5}}

i3 : monomialAlgebra(B, CoefficientField=>QQ)

o3 = QQ[x ..x ]
         0   3

o3 : MonomialAlgebra generated by {{1, 2}, {3, 0}, {0, 4}, {0, 5}}

Specifying R:

i4 : kk=ZZ/101

o4 = kk

o4 : QuotientRing

i5 : B = {{1,2},{3,0},{0,4},{0,5}}

o5 = {{1, 2}, {3, 0}, {0, 4}, {0, 5}}

o5 : List

i6 : monomialAlgebra(kk[x_0..x_3, Degrees=> B])

o6 = kk[x ..x ]
         0   3

o6 : MonomialAlgebra generated by {{1, 2}, {3, 0}, {0, 4}, {0, 5}}

Specifying a list of integers to define a monomial curve:

i7 : M = monomialAlgebra {1,4,8,9,11}

o7 = kk[x ..x ]
         0   5

o7 : MonomialAlgebra generated by {{11, 0}, {0, 11}, {1, 10}, {4, 7}, {8, 3}, {9, 2}}

Ways to use monomialAlgebra :

monomialAlgebra(List)
monomialAlgebra(PolynomialRing)

For the programmer

The object monomialAlgebra is a method function with options.