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Open AccessArticle

Odd Cycles and Hilbert Functions of Their Toric Rings

1
Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Suita, Osaka 565-0871, Japan
2
Graduate School of Mathematical Sciences, University of Tokyo, Komaba, Meguro-ku, Tokyo 153-8914, Japan
*
Author to whom correspondence should be addressed.
Mathematics 2020, 8(1), 22; https://doi.org/10.3390/math8010022
Received: 3 December 2019 / Revised: 17 December 2019 / Accepted: 18 December 2019 / Published: 20 December 2019
(This article belongs to the Special Issue Current Trends on Monomial and Binomial Ideals)

Abstract

Studying Hilbert functions of concrete examples of normal toric rings, it is demonstrated that for each 1 s 5 , an O-sequence ( h 0 , h 1 , , h 2 s 1 ) Z 0 2 s satisfying the properties that (i) h 0 h 1 h s 1 , (ii) h 2 s 1 = h 0 , h 2 s 2 = h 1 and (iii) h 2 s 1 i = h i + ( 1 ) i , 2 i s 1 , can be the h-vector of a Cohen-Macaulay standard G-domain.
Keywords: O-sequence; h-vector; flawless; toric ring; stable set polytope O-sequence; h-vector; flawless; toric ring; stable set polytope

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 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 [ A 1 ] and (iii) dim K A 1 < . The Hilbert function of A is defined by
H ( A , n ) = dim K A n , n = 0 , 1 , 2 ,
Let dim A = 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 , ) , called the h-vector of A, defined by the formula
( 1 λ ) d n = 0 H ( A , n ) λ n = i = 0 h i λ 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 0 , then we write h ( A ) = ( h 0 , h 1 , , h s ) .
Let Y 1 , , Y r be indeterminates. A non-empty set M of monomials Y 1 a 1 Y r a r in the variables Y 1 , , Y r is said to be an order ideal of monomials if, whenever m M and m divides m, then m M . Equivalently, if Y 1 a 1 Y r a r M and 0 b i a i , then Y 1 b 1 Y r b r M . In particular, since M is non-empty, 1 M . A finite sequence ( h 0 , h 1 , , h s ) of non-negative integers is said to be an O-sequence if there exists an order ideal M of monomials in Y , , Y r with each deg Y i = 1 such that h j = | { m M | deg m = j } | for any 0 j s . In particular, h 0 = 1 . If A is Cohen-Macaulay, then h ( A ) = ( h 0 , h 1 , , h s ) is an O-sequence ([2], p. 60). Furthermore, a finite sequence ( h 0 , h 1 , , h s ) of integers with h 0 = 1 and h s 0 is the h-vector of a Cohen-Macaulay standard G-algebra if and only if ( h 0 , h 1 , , h s ) is an O-sequence ([2], Corollary 3.11).
An O-sequence ( h 0 , h 1 , , h s ) with h s 0 is called flawless ([1], p. 245) if (i) h i h s i for 0 i [ s / 2 ] and (ii) h 0 h 1 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 s 5 , an O-sequence
( h 0 , h 1 , , h s 1 , h s , , h 2 s 2 , h 2 s 1 ) Z 0 2 s
satisfying the properties that
(i)
h 0 h 1 h s 1 ,
(ii)
h 2 s 1 = h 0 , h 2 s 2 = h 1 ,
(iii)
h 2 s 1 i = h i + ( 1 ) i , 2 i 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 2 s + 1 denote a cycle of length 2 s + 1 , where s 1 , on [ 2 s + 1 ] = { 1 , 2 , , 2 s + 1 } with the edges
{ 1 , 2 } , { 2 , 3 } , , { 2 s 1 , 2 s } , { 2 s , 2 s + 1 } , { 2 s + 1 , 1 } .
A finite set W [ 2 s + 1 ] is called stable in C 2 s + 1 if none of the sets of (1) is a subset of W. In particular, the empty set ∅ and { 1 } , { 2 } , , { 2 s + 1 } are stable. Let S = K [ x 1 , , x 2 s + 1 , y ] denote the polynomial ring in 2 s + 2 variables over K. The toric ring of C 2 s + 1 is the subring K [ C 2 s + 1 ] of S which is generated by those squarefree monomials ( i W x i ) y for which W [ 2 s + 1 ] is stable in C 2 s + 1 . It follows that K [ C 2 s + 1 ] can be a standard G-algebra with each deg ( i W x i ) y = 1 . It is shown ([5], Theorem 8.1) that K [ C 2 s + 1 ] is normal. In particular, K [ C 2 s + 1 ] is a Cohen-Macaulay standard G-domain. Now, we discuss when K [ C 2 s + 1 ] is Gorenstein. Here a Cohen-Macaulay ring is called Gorenstein if it has finite injective dimension.
Theorem 1.
The toric ring K [ C 2 s + 1 ] is Gorenstein if and only if either s = 1 or s = 2 .
Proof. 
Since the h-vector of K [ C 3 ] is ( 1 , 1 ) and since the h-vector of K [ C 5 ] is ( 1 , 6 , 6 , 1 ) , it follows from ([2], Theorem 4.4) that each of K [ C 3 ] and K [ C 5 ] is Gorenstein.
Now, we show that K [ C 2 s + 1 ] is not Gorenstein if s 3 . Let s 3 . Write Q C 2 s + 1 R 2 s + 1 for the stable set polytope of C 2 s + 1 . Thus Q C 2 s + 1 is the convex hull of the finite set
i W e i : W is   a   stable   set   of   G R 2 s + 1 ,
where e 1 , , e 2 s + 1 R 2 s + 1 are the canonical unit coordinate vectors of R 2 s + 1 and where i e i = ( 0 , , 0 ) R 2 s + 1 . One has dim Q 2 s + 1 = 2 s + 1 . Then ([6], Theorem 4) says that Q C 2 s + 1 is defined by the following inequalities:
  • 0 x i 1 for all 1 i 2 s + 1 ;
  • x i + x i + 1 1 for all 1 i 2 s ;
  • x 1 + x 2 s + 1 1 ;
  • x 1 + + x 2 s + 1 s .
It then follows that each of Q C 2 s + 1 and 2 Q C 2 s + 1 has no interior lattice points and that ( 1 , , 1 ) is an interior lattice point of 3 Q C 2 s + 1 . Furthermore, (Ref. [7], Theorem 4.2) guarantees that the inequality
x 1 + + x 2 s + 1 s
defines a facet of Q C 2 s + 1 . Let P s = 3 Q C 2 s + 1 ( 1 , , 1 ) . Thus the origin of R 2 s + 1 is an interior lattice point of P s and the inequality
x 1 + + x 2 s + 1 s 1
defines a facet of P s . This fact together with [8] implies that P s is not reflexive. In other words, the dual polytope P s of P s defined by
P s = { y R 2 s + 1 : x , y 1 for all x P s }
is not a lattice polytope, where x , y is the usual inner product of R 2 s + 1 . It then follows from ([9], Theorem (1.1)) (and also from ([5], Theorem 8.1)) that K [ C 2 s + 1 ] 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 , , h s ) is symmetric, i.e., h i = h s i for 0 i [ s / 2 ] . Hence the h-vector of the toric ring K [ C 2 s + 1 ] is not symmetric when s 3 .
Example 1.
By using Normaliz [10], the h-vector of the toric ring K [ C 7 ] is ( 1 , 21 , 84 , 85 , 21 , 1 ) .

3. Non-Flawless 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 [ C 9 ] 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 [ C 11 ] .
We conclude the present paper with the following
Conjecture 1.
The h-vector of the toric ring K [ C 2 s + 1 ] of C 2 s + 1 is of the form
( 1 , h 1 , h 2 , h 3 , , h i , , h s 1 , h s 1 + ( 1 ) s 1 , , h i + ( 1 ) i , , h 3 1 , h 2 + 1 , h 1 , 1 ) .

Author Contributions

All authors made equal and significant contributions to writing this article, and approved the final manuscript. All authors have read and agreed to the published version of the manuscript.

Funding

Takayuki Hibi was partially supported by JSPS KAKENHI 19H00637. Akiyoshi Tsuchiya was partially supported by JSPS KAKENHI 19K14505 and 19J00312.

Conflicts of Interest

The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

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