next | previous | forward | backward | up | top | index | toc | Macaulay2 web site
MonomialAlgebras :: monomialAlgebra

monomialAlgebra -- Create a monomial algebra

Synopsis

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 ℕ2. The field K is selected via the option CoefficientField.

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

Specifing 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 , x , x ]
     101  0   1   2   3

o2 : MonomialAlgebra generated by {{1, 2}, {3, 0}, {0, 4}, {0, 5}}
i3 : monomialAlgebra(B, CoefficientField=>QQ)

o3 = QQ[x , x , x , x ]
         0   1   2   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 , x , x ]
         0   1   2   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 , x , x , x , x ]
         0   1   2   3   4   5

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

Ways to use monomialAlgebra :