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# isBuchsbaumMA -- Test whether a simplicial monomial algebra is Buchsbaum.

## Synopsis

• Usage:
isBuchsbaumMA R
isBuchsbaumMA B
isBuchsbaumMA M
• Inputs:
• R, , with B = degrees R and K = coefficientRing R, or
• B, a list, with the generators of an affine semigroup in \mathbb{N}^d.
• M, ,
• Outputs:

## Description

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

Note that this condition does not depend on K.

For the definition of Buchsbaum see:

J. Stueckrad, W. Vogel: Castelnuovo Bounds for Certain Subvarieties in \mathbb{P}^n, Math. Ann. 276 (1987), 341-352.

 i1 : R=QQ[x_0..x_3,Degrees=>{{6,0},{0,6},{4,2},{1,5}}] o1 = R o1 : PolynomialRing i2 : isBuchsbaumMA R o2 = false i3 : decomposeMonomialAlgebra R o3 = HashTable{| -1 | => {ideal 1, | 5 |} } | 1 | | 7 | | -2 | => {ideal 1, | 4 |} | 2 | | 2 | 0 => {ideal 1, 0} | 1 | => {ideal 1, | 1 |} | -1 | | 5 | | 2 | => {ideal (x , x ), | 2 |} | -2 | 0 1 | 4 | | 3 | => {ideal (x , x ), | 3 |} | 3 | 0 1 | 9 | o3 : HashTable

 i4 : R=QQ[x_0..x_3,Degrees=>{{4,0},{0,4},{3,1},{1,3}}] o4 = R o4 : PolynomialRing i5 : isBuchsbaumMA R o5 = true i6 : decomposeMonomialAlgebra R o6 = HashTable{| -1 | => {ideal 1, | 3 |} } | 1 | | 1 | 0 => {ideal 1, 0} | 1 | => {ideal 1, | 1 |} | -1 | | 3 | | 2 | => {ideal (x , x ), | 2 |} | 2 | 0 1 | 2 | o6 : HashTable

 i7 : R=QQ[x_0..x_3,Degrees=>{{5,0},{0,5},{4,1},{1,4}}] o7 = R o7 : PolynomialRing i8 : isBuchsbaumMA R o8 = false i9 : decomposeMonomialAlgebra R o9 = HashTable{| -1 | => {ideal 1, | 4 |} } | 1 | | 1 | 2 | -2 | => {ideal (x , x ), | 3 |} | 2 | 0 1 | 2 | 0 => {ideal 1, 0} | 1 | => {ideal 1, | 1 |} | -1 | | 4 | 2 | 2 | => {ideal (x , x ), | 2 |} | -2 | 0 1 | 3 | o9 : HashTable

 i10 : R=QQ[x_0..x_3,Degrees=>{{5,0},{0,5},{4,1},{1,4}}] o10 = R o10 : PolynomialRing i11 : M=monomialAlgebra R o11 = R o11 : MonomialAlgebra generated by {{5, 0}, {0, 5}, {4, 1}, {1, 4}} i12 : isBuchsbaumMA M o12 = false

## Ways to use isBuchsbaumMA :

• isBuchsbaumMA(List)
• isBuchsbaumMA(MonomialAlgebra)
• isBuchsbaumMA(PolynomialRing)

## For the programmer

The object isBuchsbaumMA is .