Sortable simplicial complexes and their associated toric rings

Antonino Ficarra; Somayeh Moradi

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arXiv · 2412.10113

Sortable simplicial complexes and their associated toric rings

Antonino Ficarra·Somayeh Moradi

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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.

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Antonino Ficarra, Somayeh Moradi. 2024-12-13. Sortable simplicial complexes and their associated toric rings. https://arxiv.org/abs/2412.10113

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