isCohenMacaulayMA -- Test whether a simplicial monomial algebra is Cohen-Macaulay.

Synopsis

Usage:

isCohenMacaulayMA R

isCohenMacaulayMA B

isCohenMacaulayMA 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 Cohen-Macaulay.

Note that this condition does not depend on K.

i1 : a=3

o1 = 3

i2 : B={{a, 0}, {0, a}, {1, a-1}, {a-1, 1}}

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

o2 : List

i3 : R=QQ[x_0..x_3,Degrees=>B]

o3 = R

o3 : PolynomialRing

i4 : isCohenMacaulayMA R

o4 = true

i5 : decomposeMonomialAlgebra R

o5 = HashTable{| -1 | => {ideal 1, | 2 |}}
               | 1  |              | 1 |
               0 => {ideal 1, 0}
               | 1  | => {ideal 1, | 1 |}
               | -1 |              | 2 |

o5 : HashTable

i6 : a=4

o6 = 4

i7 : B={{a, 0}, {0, a}, {1, a-1}, {a-1, 1}}

o7 = {{4, 0}, {0, 4}, {1, 3}, {3, 1}}

o7 : List

i8 : R=QQ[x_0..x_3,Degrees=>B]

o8 = R

o8 : PolynomialRing

i9 : isCohenMacaulayMA R

o9 = false

i10 : decomposeMonomialAlgebra R

o10 = HashTable{| -1 | => {ideal 1, | 3 |}      }
                | 1  |              | 1 |
                0 => {ideal 1, 0}
                | 1  | => {ideal 1, | 1 |}
                | -1 |              | 3 |
                | 2 | => {ideal (x , x ), | 2 |}
                | 2 |             1   0   | 2 |

o10 : HashTable

i11 : a=4

o11 = 4

i12 : M=monomialAlgebra {{a, 0}, {0, a}, {1, a-1}, {a-1, 1}}

       ZZ
o12 = ---[x ..x ]
      101  0   3

o12 : MonomialAlgebra generated by {{4, 0}, {0, 4}, {1, 3}, {3, 1}}

i13 : isCohenMacaulayMA M

o13 = false

Ways to use isCohenMacaulayMA :

isCohenMacaulayMA(List)
isCohenMacaulayMA(MonomialAlgebra)
isCohenMacaulayMA(PolynomialRing)

For the programmer

The object isCohenMacaulayMA is a method function.