Toric rings attached to simplicial complexes

Abstract

We consider standard graded toric rings $R_{\Delta }$ whose generators correspond to the faces of a simplicial complex $\Delta$. When $R_{\Delta }$ is normal, it is shown that its divisor class group is free. For a flag complex $\Delta$ which is the clique complex of a perfect graph, a nice description for the class group and the canonical module of $R_{\Delta }$ in terms of the minimal vertex covers of the graph is given. Moreover, for a quasi-forest simplicial complex a quadratic Gröbner basis for the defining ideal of $R_{\Delta }$ is presented. Using this fact we give combinatorial descriptions for the $a$-invariant and the Gorenstein property of $R_{\Delta }$.

15 pages