isGorensteinMA -- Test whether a simplicial monomial algebra is Gorenstein.

Synopsis

Usage:

isGorensteinMA R

isGorensteinMA B

isGorensteinMA M
Inputs:
- R, a polynomial ring, with B = degrees R and K = coefficientRing R, or
- B, a list, with the generators of an affine semigroup in \mathbb{N}^d.
- M, a MonomialAlgebra,
Outputs:
- a Boolean value,

Description

Test whether the simplicial monomial algebra K[B] is Gorenstein.

Note that this condition does not depend on K.

i1 : R=QQ[x_0..x_2,Degrees=>{{2, 0}, {0, 2}, {1, 1}}]

o1 = R

o1 : PolynomialRing

i2 : isGorensteinMA R

o2 = true

Not Gorenstein:

i3 : B={{3, 0}, {0, 3}, {1, 2}, {2, 1}}

o3 = {{3, 0}, {0, 3}, {1, 2}, {2, 1}}

o3 : List

i4 : R=QQ[x_0..x_3,Degrees=>B]

o4 = R

o4 : PolynomialRing

i5 : isGorensteinMA R

o5 = false

Not even Cohen-Macaulay:

i6 : B={{4, 0}, {0, 4}, {1, 3}, {3, 1}}

o6 = {{4, 0}, {0, 4}, {1, 3}, {3, 1}}

o6 : List

i7 : R=QQ[x_0..x_3,Degrees=>B]

o7 = R

o7 : PolynomialRing

i8 : isGorensteinMA R

o8 = false

i9 : isCohenMacaulayMA R

o9 = false

Gorenstein:

i10 : R=QQ[x_0..x_4,Degrees=>{{1,0,0},{0,2,0},{0,0,2},{1,0,1},{0,1,1}}]

o10 = R

o10 : PolynomialRing

i11 : isGorensteinMA R

o11 = true

i12 : decomposeMonomialAlgebra R

o12 = HashTable{0 => {ideal 1, 0}        }
                | 0 | => {ideal 1, | 1 |}
                | 0 |              | 0 |
                | 1 |              | 1 |
                | 0 | => {ideal 1, | 1 |}
                | 1 |              | 1 |
                | 0 |              | 2 |
                | 0 | => {ideal 1, | 0 |}
                | 1 |              | 1 |
                | 1 |              | 1 |

o12 : HashTable

i13 : R=QQ[x_0..x_4,Degrees=>{{1,0,0},{0,2,0},{0,0,2},{1,0,1},{0,1,1}}]

o13 = R

o13 : PolynomialRing

i14 : M=monomialAlgebra R

o14 = R

o14 : MonomialAlgebra generated by {{1, 0, 0}, {0, 2, 0}, {0, 0, 2}, {1, 0, 1}, {0, 1, 1}}

i15 : isGorensteinMA M

o15 = true

Ways to use isGorensteinMA :

isGorensteinMA(List)
isGorensteinMA(MonomialAlgebra)
isGorensteinMA(PolynomialRing)

For the programmer

The object isGorensteinMA is a method function.