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In the present paper, we consider the problem when the toric ring arising from an integral cyclic polytope is Cohen-Macaulay by discussing Serres condition and we give a complete characterization when that is Gorenstein. Moreover, we study the normality of the other semigroup ring arising from an integral cyclic polytope but generated only with its vertices.
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In this paper we study monomial ideals attached to posets, introduce generalized Hibi rings and investigate their algebraic and homological properties. The main tools to study these objects are Groebner basis theory, the concept of sortability due to Sturmfels and the theory of weakly polymatroidal ideals.
We study properties of the Stanley-Reisner rings of simplicial complexes with isolated singularities modulo two generic linear forms. Miller, Novik, and Swartz proved that if a complex has homologically isolated singularities, then its Stanley-Reisner ring modulo one generic linear form is Buchsbaum. Here we examine the case of non-homologically isolated singularities, providing many examples in which the Stanley-Reisner ring modulo two generic linear forms is a quasi-Buchsbaum but not Buchsbaum ring.
Let $G$ be a finite graph allowing loops, having no multiple edge and no isolated vertex. We associate $G$ with the edge polytope ${cal P}_G$ and the toric ideal $I_G$. By classifying graphs whose edge polytope is simple, it is proved that the toric ideals $I_G$ of $G$ possesses a quadratic Grobner basis if the edge polytope ${cal P}_G$ of $G$ is simple. It is also shown that, for a finite graph $G$, the edge polytope is simple but not a simplex if and only if it is smooth but not a simplex. Moreover, the Ehrhart polynomial and the normalized volume of simple edge polytopes are computed.
The aim of this paper is to study integer rounding properties of various systems of linear inequalities to gain insight about the algebraic properties of Rees algebras of monomial ideals and monomial subrings. We study the normality and Gorenstein property--as well as the canonical module and the a-invariant--of Rees algebras and subrings arising from systems with the integer rounding property. We relate the algebraic properties of Rees algebras and monomial subrings with integer rounding properties and present a duality theorem.
The Jacobian algebra $mathsf{A}$ arising from a consistent dimer model is derived equivalent to crepant resolutions of a $3$-dimensional Gorenstein toric singularity $R$, and it is also called a non-commutative crepant resolution of $R$. This algebra $mathsf{A}$ is a maximal Cohen-Macaulay (= MCM) module over $R$, and it is a finite direct sum of rank one MCM $R$-modules. In this paper, we observe a relationship between properties of a dimer model and those of MCM modules appearing in the decomposition of $mathsf{A}$ as an $R$-module. More precisely, we take notice of isoradial dimer models and divisorial ideals which are called conic. Especially, we investigate them for the case of $3$-dimensional Gorenstein toric singularities associated with reflexive polygons.
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