Sortable simplicial complexes and their associated toric rings

TL;DR

This paper studies algebraic properties of toric rings and Rees algebras associated with $d$-flag sortable simplicial complexes, establishing their Koszul, normal, Cohen-Macaulay nature, and characterizing their vertex decomposability.

Contribution

It proves that these algebras are Koszul, normal, Cohen-Macaulay domains and characterizes the Cohen-Macaulay property via vertex decomposability.

Findings

Algebras are Koszul, normal, Cohen-Macaulay domains.

Characterization of Cohen-Macaulay property through vertex decomposability.

Analysis of Gorenstein property, canonical module, and divisor class group.

Abstract

Let $\Gamma$ be a $d$-flag sortable simplicial complex. We consider the toric ring $R_{\Gamma}=K[{\bf x}_Ft:F\in \Gamma]$ and the Rees algebra of the facet ideals $I(\Gamma^{[i]})$ of pure skeletons of $\Gamma$. We show that these algebras are Koszul, normal Cohen-Macaulay domains. Moreover, we study the Gorenstein property, the canonical module, and the $a$-invariant of the normal domain $R_{\Gamma}$ by investigating its divisor class group. Finally, it is shown that any $d$-flag sortable simplicial complex is vertex decomposable, which provides a characterization of the Cohen-Macaulay property of such complexes.