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.
Specifing B:
i1 : B = {{1,2},{3,0},{0,4},{0,5}}
o1 = {{1, 2}, {3, 0}, {0, 4}, {0, 5}}
o1 : List
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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}}
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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
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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}}
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The object monomialAlgebra is a method function with options.