Torsionfreeness for divisor class groups of toric rings of integral polytopes

TL;DR

This paper establishes conditions under which the divisor class group of toric rings from integral polytopes is torsionfree, focusing on compressed and (0,1)-polytopes with few facets, and characterizes cases with free class groups.

Contribution

It provides new sufficient conditions for torsionfreeness of divisor class groups in toric rings derived from specific classes of integral polytopes.

Findings

Cl({K}[P]) is torsionfree if P is compressed.

Cl({K}[P]) is torsionfree if P is a (0,1)-polytope with at most dim P + 2 facets.

Characterization of toric rings of (0,1)-polytopes with class group isomorphic to .

Abstract

In the present paper, we give some sufficient conditions for $\operatorname{Cl}(\Bbbk[P])$ to be torsionfree, where $\operatorname{Cl}(\Bbbk[P])$ denote the divisor class group of the toric ring $\Bbbk[P]$ of an integral polytope $P$. We prove that $\operatorname{Cl}(\Bbbk[P])$ is torsionfree if $P$ is compressed, and $\operatorname{Cl}(\Bbbk[P])$ is torsionfree if $P$ is a $(0,1)$-polytope which has at most $\dim P+2$ facets. Moreover, we characterize the toric rings of $(0,1)$-polytopes in the case $\operatorname{Cl}(\Bbbk[P])\cong \mathbb{Z}$.