Test whether the simplicial monomial algebra K[B] is Gorenstein.
Note that this condition does not depend on K.
Gorenstein:
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, | 1 |}}
| 0 | | 0 |
| 1 | | 1 |
| 0 | => {ideal 1, | 1 |}
| 1 | | 1 |
| 0 | | 2 |
| 0 | => {ideal 1, | 0 |}
| 1 | | 1 |
| 1 | | 1 |
0 => {ideal 1, 0}
o12 : HashTable
|