Sortable simplicial complexes and their associated toric rings

Abstract

Let $\Gamma$ be a $d$-flag sortable simplicial complex. We consider the toric ring $R_{\Gamma }=K[{\mathbf{x}}_{F}t:F\in \Gamma ]$ 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.