Stable set rings which are Gorenstein on the punctured spectrum (Q2112293)

Given a simple graph, \(G\), with vertex set \(V(G)=\{1,\dots,n\}\), the monomial subalgebra of \(K[x_1,\dots,x_n,t]\) generated by the monomials \[ x_{i_1}\cdots x_{i_k}\cdot t, \] for every stable (or independent) subset \(\{i_1,\dots,i_k\}\subseteq V(G)\), is called the stable set ring of \(G\). In the main result of the present article, the authors show that if \(G\) is perfect, the trace of the canonical module of its the stable set ring is \(\mathfrak{m}\)-primary (where \(\mathfrak{m}\) is the maximal homogeneous ideal) if and only if every connected component of \(G\) is pure. Here \textit{pure} stands for \textit{having pure clix complex}, or, in other words, possessing maximal cliques of equal cardinality.