affineAlgebra -- Define a monomial algebra

Synopsis

Usage:

affineAlgebra P

affineAlgebra B
Inputs:
- P, 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.
Optional inputs:
- CoefficientField => ..., default value ZZ/101, Option to set the coefficient field.
Outputs:
- a ring,

Description

Returns the monomial algebra K[B]=P/binomialIdeal(P) associated to the degree monoid of the polynomial ring P.

i1 : kk=ZZ/101

o1 = kk

o1 : QuotientRing

i2 : B = {{1,2},{3,0},{0,4},{0,5}}

o2 = {{1, 2}, {3, 0}, {0, 4}, {0, 5}}

o2 : List

i3 : S = kk[x_0..x_3, Degrees=> B]

o3 = S

o3 : PolynomialRing

i4 : affineAlgebra S

                            S
o4 = ----------------------------------------------
       3        2   6    2 3     4    3 2   5    4
     (x x  - x x , x  - x x , x x  - x x , x  - x )
       0 2    1 3   0    1 2   1 2    0 3   2    3

o4 : QuotientRing

i5 : affineAlgebra B

                       kk[x ..x ]
                           0   3
o5 = ----------------------------------------------
       3        2   6    2 3     4    3 2   5    4
     (x x  - x x , x  - x x , x x  - x x , x  - x )
       0 2    1 3   0    1 2   1 2    0 3   2    3

o5 : QuotientRing

i6 : M = monomialAlgebra B

o6 = kk[x ..x ]
         0   3

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

i7 : affineAlgebra M

                       kk[x ..x ]
                           0   3
o7 = ----------------------------------------------
       3        2   6    2 3     4    3 2   5    4
     (x x  - x x , x  - x x , x x  - x x , x  - x )
       0 2    1 3   0    1 2   1 2    0 3   2    3

o7 : QuotientRing

Ways to use affineAlgebra:

affineAlgebra(List)
affineAlgebra(MonomialAlgebra)
affineAlgebra(PolynomialRing)

For the programmer

The object affineAlgebra is a method function with options.