Let

${C}_{2s+1}$ denote a cycle of length

$2s+1$, where

$s\ge 1$, on

$[2s+1]=\{1,2,\dots ,2s+1\}$ with the edges

A finite set

$W\subset [2s+1]$ is called

stable in

${C}_{2s+1}$ if none of the sets of (1) is a subset of

W. In particular, the empty set ∅ and

$\left\{1\right\},\left\{2\right\},\dots ,\{2s+1\}$ are stable. Let

$S=K[{x}_{1},\dots ,{x}_{2s+1},y]$ denote the polynomial ring in

$2s+2$ variables over

K. The

toric ring of

${C}_{2s+1}$ is the subring

$K\left[{C}_{2s+1}\right]$ of

S which is generated by those squarefree monomials

$({\prod}_{i\in W}{x}_{i})y$ for which

$W\subset [2s+1]$ is stable in

${C}_{2s+1}$. It follows that

$K\left[{C}_{2s+1}\right]$ can be a standard

G-algebra with each

$deg({\prod}_{i\in W}{x}_{i})y=1$. It is shown ([

5], Theorem 8.1) that

$K\left[{C}_{2s+1}\right]$ is normal. In particular,

$K\left[{C}_{2s+1}\right]$ is a Cohen-Macaulay standard

G-domain. Now, we discuss when

$K\left[{C}_{2s+1}\right]$ is Gorenstein. Here a Cohen-Macaulay ring is called Gorenstein if it has finite injective dimension.

Now, we show that

$K\left[{C}_{2s+1}\right]$ is not Gorenstein if

$s\ge 3$. Let

$s\ge 3$. Write

${\mathcal{Q}}_{{C}_{2s+1}}\subset {\mathbb{R}}^{2s+1}$ for the stable set polytope of

${C}_{2s+1}$. Thus

${\mathcal{Q}}_{{C}_{2s+1}}$ is the convex hull of the finite set

where

${\mathbf{e}}_{1},\dots ,{\mathbf{e}}_{2s+1}\in {\mathbb{R}}^{2s+1}$ are the canonical unit coordinate vectors of

${\mathbb{R}}^{2s+1}$ and where

${\sum}_{i\in \varnothing}{\mathbf{e}}_{i}=(0,\dots ,0)\in {\mathbb{R}}^{2s+1}$. One has

$dim{\mathcal{Q}}_{2s+1}=2s+1$. Then ([

6], Theorem 4) says that

${\mathcal{Q}}_{{C}_{2s+1}}$ is defined by the following inequalities:

It then follows that each of

${\mathcal{Q}}_{{C}_{2s+1}}$ and

$2{\mathcal{Q}}_{{C}_{2s+1}}$ has no interior lattice points and that

$(1,\dots ,1)$ is an interior lattice point of

$3{\mathcal{Q}}_{{C}_{2s+1}}$. Furthermore, (Ref. [

7], Theorem 4.2) guarantees that the inequality

defines a facet of

${\mathcal{Q}}_{{C}_{2s+1}}$. Let

${\mathcal{P}}_{s}=3{\mathcal{Q}}_{{C}_{2s+1}}-(1,\dots ,1)$. Thus the origin of

${\mathbb{R}}^{2s+1}$ is an interior lattice point of

${\mathcal{P}}_{s}$ and the inequality

defines a facet of

${\mathcal{P}}_{s}$. This fact together with [

8] implies that

${\mathcal{P}}_{s}$ is not reflexive. In other words, the dual polytope

${\mathcal{P}}_{s}^{\vee}$ of

${\mathcal{P}}_{s}$ defined by

is not a lattice polytope, where

$\langle \mathbf{x},\mathbf{y}\rangle $ is the usual inner product of

${\mathbb{R}}^{2s+1}$. It then follows from ([

9], Theorem (1.1)) (and also from ([

5], Theorem 8.1)) that

$K\left[{C}_{2s+1}\right]$ is not Gorenstein, as desired. □

It is known ([

2], Theorem 4.4) that a Cohen-Macaulay standard

G-domain

A is Gorenstein if and only if the

h-vector

$h(A)=({h}_{0},\dots ,{h}_{s})$ is symmetric, i.e.,

${h}_{i}={h}_{s-i}$ for

$0\le i\le [s/2]$. Hence the

h-vector of the toric ring

$K\left[{C}_{2s+1}\right]$ is not symmetric when

$s\ge 3$.