Odd Cycles and Hilbert Functions of Their Toric Rings

Abstract

Studying Hilbert functions of concrete examples of normal toric rings, it is demonstrated that for each $1\le s\le 5$, an O-sequence $(h_0,h_1,\dots ,h_{2s-1})\in {\mathbb{Z}}_{\ge 0}^{2s}$ satisfying the properties that (i) $h_0\le h_1\le \cdots \le h_{s-1}$, (ii) $h_{2s-1}=h_0$, $h_{2s-2}=h_1$ and (iii) $h_{2s-1-i}=h_i+{(-1)}^i$, $2\le i\le s-1$, can be the h-vector of a Cohen-Macaulay standard G-domain.

1. Background

In the paper [1] published in 1989, several conjectures on Hilbert functions of Cohen-Macaulay integral domains are studied.

Let $A={⨁}_{n=0}^{\infty }{A}_n$ be a standard G-algebra [2]. Thus A is a Noetherian commutative graded ring for which (i) $A_0=K$ a field, (ii) $A=K\left[A_1\right]$ and (iii) ${dim}_K{A}_1<\infty$. The Hilbert function of A is defined by

$$H(A,n)={dim}_K{A}_n,n=0,1,2,\dots$$

Let $dimA=d$ and $v=H(A,1)={dim}_K{A}_1$. A classical result ([3], Chapter 5, Section 13) says that $H(A,n)$ is a polynomial for n sufficiently large and its degree is $d-1$. It follows that the sequence $h(A)=(h_0,h_1,h_2,\dots )$, called the h-vector of A, defined by the formula

$${(1-\lambda )}^d\sum _{n=0}^{\infty }H(A,n){\lambda }^n=\sum _{i=0}^{\infty }{h}_i{\lambda }^i$$

has finitely many non-zero terms with $h_0=1$ and $h_1=v-d$. If $h_i=0$ for $i>s$ and $h_s\ne 0$, then we write $h(A)=(h_0,h_1,\dots ,h_s)$.

Let $Y_1,\dots ,Y_r$ be indeterminates. A non-empty set M of monomials $Y_1^{a_1}\cdots Y_r^{a_r}$ in the variables $Y_1,\dots ,Y_r$ is said to be an order ideal of monomials if, whenever $m\in M$ and $m^{\prime }$ divides m, then $m^{\prime }\in M$. Equivalently, if $Y_1^{a_1}\cdots Y_r^{a_r}\in M$ and $0\le b_i\le a_i$, then $Y_1^{b_1}\cdots Y_r^{b_r}\in M$. In particular, since M is non-empty, $1\in M$. A finite sequence $(h_0,h_1,\dots ,h_s)$ of non-negative integers is said to be an O-sequence if there exists an order ideal M of monomials in $Y_{,}\dots ,Y_r$ with each $deg{Y}_i=1$ such that $h_j=\left|\{m\in M|degm=j\}\right|$ for any $0\le j\le s$. In particular, $h_0=1$. If A is Cohen-Macaulay, then $h(A)=(h_0,h_1,\dots ,h_s)$ is an O-sequence ([2], p. 60). Furthermore, a finite sequence $(h_0,h_1,\dots ,h_s)$ of integers with $h_0=1$ and $h_s\ne 0$ is the h-vector of a Cohen-Macaulay standard G-algebra if and only if $(h_0,h_1,\dots ,h_s)$ is an O-sequence ([2], Corollary 3.11).

An O-sequence $(h_0,h_1,\dots ,h_s)$ with $h_s\ne 0$ is called flawless ([1], p. 245) if (i) $h_i\le h_{s-i}$ for $0\le i\le [s/2]$ and (ii) $h_0\le h_1\le \cdots \le h_{[s/2]}$. A standard G-domain is a standard G-algebra which is an integral domain. It was conjectured ([1], Conjecture 1.4) that the h-vector of a Cohen-Macaulay standard G-domain is flawless. Niesi and Robbiano ([4], Example 2.4) succeeded in constructing a Cohen-Macaulay standard G-domain with $(1,3,5,4,4,1)$ its h-vector. Thus, in general, the h-vector of a Cohen-Macaulay standard G-domain is not flawless.

In the present paper, it is shown that, for each $1\le s\le 5$, an O-sequence

$$(h_0,h_1,\dots ,h_{s-1},h_s,\dots ,h_{2s-2},h_{2s-1})\in {\mathbb{Z}}_{\ge 0}^{2s}$$

satisfying the properties that

(i)

$h_0\le h_1\le \cdots \le h_{s-1}$,

(ii)

$h_{2s-1}=h_0$, $h_{2s-2}=h_1$,

(iii)

$h_{2s-1-i}=h_i+{(-1)}^i$, $2\le i\le s-1$

can be the h-vector of a normal toric ring arising from a cycle of odd length. In particular, the above O-sequence, which is non-flawless for each of $s=4$ and $s=5$, can be the h-vector of a Cohen-Macaulay standard G-domain.

2. Toric Rings Arising from Odd Cycles

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

$$\begin{array}{c}\hfill \{1,2\},\{2,3\},\dots ,\{2s-1,2s\},\{2s,2s+1\},\{2s+1,1\}.\end{array}$$

(1)

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.

Theorem 1.

The toric ring $K\left[C_{2s+1}\right]$ is Gorenstein if and only if either $s=1$ or $s=2$.

Proof. 

Since the h-vector of $K\left[C_3\right]$ is $(1,1)$ and since the h-vector of $K\left[C_5\right]$ is $(1,6,6,1)$, it follows from ([2], Theorem 4.4) that each of $K\left[C_3\right]$ and $K\left[C_5\right]$ is Gorenstein.

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

$$\left\{\sum _{i\in W}{\mathbf{e}}_i:W\mathrm{is}\mathrm{a}\mathrm{stable}\mathrm{set}\mathrm{of}G\right\}\subset {\mathbb{R}}^{2s+1},$$

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:

• $0\le x_i\le 1$ for all $1\le i\le 2s+1$;
• $x_i+x_{i+1}\le 1$ for all $1\le i\le 2s$;
• $x_1+x_{2s+1}\le 1$;
• $x_1+\cdots +x_{2s+1}\le s$.

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

$$x_1+\cdots +x_{2s+1}\le s$$

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

$$x_1+\cdots +x_{2s+1}\le s-1$$

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

$${\mathcal{P}}_s^{\vee }=\{\mathbf{y}\in {\mathbb{R}}^{2s+1}:\langle \mathbf{x},\mathbf{y}\rangle \le 1\mathrm{for}\mathrm{all}\mathbf{x}\in {\mathcal{P}}_s\}$$

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 $s\ge 3$.

Example 1.

By using Normaliz [10], the h-vector of the toric ring $K\left[C_7\right]$ is $(1,21,84,85,21,1)$.

3. Non-Flawless $\mathbf{O}$-Sequences of Normal Toric Rings

We now come to concrete examples of non-flawless O-sequences which can be the h-vectors of normal toric rings.

Example 2.

The h-vector of the toric ring $K\left[C_9\right]$ is

$$(1,66,744,2305,2304,745,66,1).$$

Furthermore,

$$(1,187,5049,37247,96448,96449,37246,5050,187,1)$$

is the h-vector of the toric ring $K\left[C_{11}\right]$.

We conclude the present paper with the following

Conjecture 1.

The h-vector of the toric ring $K\left[C_{2s+1}\right]$ of $C_{2s+1}$ is of the form

$$(1,h_1,h_2,h_3,\dots ,h_i,\dots ,h_{s-1},h_{s-1}+{(-1)}^{s-1},\dots ,h_i+{(-1)}^i,\dots ,h_3-1,h_2+1,h_1,1).$$