Toric rings attached to simplicial complexes

TL;DR

This paper studies the algebraic properties of toric rings associated with simplicial complexes, providing new descriptions of their class groups, canonical modules, and Gorenstein conditions, especially for flag complexes and quasi-forest complexes.

Contribution

It offers novel descriptions of class groups and canonical modules for toric rings from simplicial complexes, and introduces quadratic Gr"obner bases for quasi-forest complexes.

Findings

Class group of normal toric rings is free.

Explicit description of class group and canonical module for flag complexes of perfect graphs.

Quadratic Gr"obner basis for the defining ideal of quasi-forest 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 Gr\"obner 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 $R_\Delta$.