Simplicial affine semigroups with monomial minimal reduction ideals (Q2117373)

Let \(K\) be a field and \(K[x_1, x_2, \ldots, x_d]\) be the polynomial ring of dimension \(d\). If \(S\) is a sub-semigroup of \(\mathbb N^d\), then the subring \(R=K[S]\), called the semigroup ring of \(S,\) is generated over \(K\) by the monomials \(x^a\) where \(a\in S.\) Let \(\dim K[S]=d\) and \(S\) be simplicial. This means when the cone over \(S\), denoted as \(\text{cone}(S)\), has \(d\) extremal rays. Let \(\mathbf{a}_1, \mathbf{a}_2, \ldots, \mathbf{a}_d \) be the extremal rays of \(S\) and \(S=(\mathbf{a}_1, \mathbf{a}_2, \ldots, \mathbf{a}_d , \mathbf{a}_{d+1}, \ldots, \mathbf{a}_{d+s}).\) The grading on \(K[S]\) is the one induced from that of the standard grading on \(R.\) It is proved that if \(K\) is infinite then \((x^{a_1}, x^{a_2}, \ldots, x^{a_d})\) is the unique monomial reduction of the maximal homogeneous ideal \(\mathfrak m\) of \(K[S].\) The Apery set of \(S\) with respect to \(b\in S\) is defined to be The set \(Ap(S, b)=\{ a\in S\mid a-b\not\in S\}.\) Put \(A(S, E)=\cap_{j=1}^d Ap(S, a_j).\) Consider the conditions: (1) \(e(R)=|Ap(S, E)|,\) (2) \(\{x^{a_1}, \ldots, x^{a_d}\}\) is a minimal reduction of the maximal homogeneous ideal \(\mathfrak m\) of \(K[S]\) (3) \(R\) is Cohen-Macaulay. It is proved that any two of \((1), (2), (3)\) imply the remaining condition. Several concrete conditions are provided for the Cohen-Macaulay and Gorenstein property of \(R\) and its associated graded ring with respect to its maximal homogeneous ideal using the Apery sets.