Studying Hilbert functions of concrete examples of normal toric rings, it is demonstrated that for each , an O-sequence satisfying the properties that (i) , (ii) , and (iii) , , can be the h-vector of a Cohen-Macaulay standard G-domain.
In the paper  published in 1989, several conjectures on Hilbert functions of Cohen-Macaulay integral domains are studied.
Let be a standard G-algebra . Thus A is a Noetherian commutative graded ring for which (i) a field, (ii) and (iii) . The Hilbert function of A is defined by
Let and . A classical result (, Chapter 5, Section 13) says that is a polynomial for n sufficiently large and its degree is . It follows that the sequence , called the h-vector of A, defined by the formula
has finitely many non-zero terms with and . If for and , then we write .
Let be indeterminates. A non-empty set M of monomials in the variables is said to be an order ideal of monomials if, whenever and divides m, then . Equivalently, if and , then . In particular, since M is non-empty, . A finite sequence of non-negative integers is said to be an O-sequence if there exists an order ideal M of monomials in with each such that for any . In particular, . If A is Cohen-Macaulay, then is an O-sequence (, p. 60). Furthermore, a finite sequence of integers with and is the h-vector of a Cohen-Macaulay standard G-algebra if and only if is an O-sequence (, Corollary 3.11).
An O-sequence with is called flawless (, p. 245) if (i) for and (ii) . A standard G-domain is a standard G-algebra which is an integral domain. It was conjectured (, Conjecture 1.4) that the h-vector of a Cohen-Macaulay standard G-domain is flawless. Niesi and Robbiano (, Example 2.4) succeeded in constructing a Cohen-Macaulay standard G-domain with 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 , an O-sequence
satisfying the properties that
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 and , can be the h-vector of a Cohen-Macaulay standard G-domain.
2. Toric Rings Arising from Odd Cycles
Let denote a cycle of length , where , on with the edges
A finite set is called stable in if none of the sets of (1) is a subset of W. In particular, the empty set ∅ and are stable. Let denote the polynomial ring in variables over K. The toric ring of is the subring of S which is generated by those squarefree monomials for which is stable in . It follows that can be a standard G-algebra with each . It is shown (, Theorem 8.1) that is normal. In particular, is a Cohen-Macaulay standard G-domain. Now, we discuss when is Gorenstein. Here a Cohen-Macaulay ring is called Gorenstein if it has finite injective dimension.
The toric ring is Gorenstein if and only if either or .
Since the h-vector of is and since the h-vector of is , it follows from (, Theorem 4.4) that each of and is Gorenstein.
Now, we show that is not Gorenstein if . Let . Write for the stable set polytope of . Thus is the convex hull of the finite set
where are the canonical unit coordinate vectors of and where . One has . Then (, Theorem 4) says that is defined by the following inequalities:
for all ;
for all ;
It then follows that each of and has no interior lattice points and that is an interior lattice point of . Furthermore, (Ref. , Theorem 4.2) guarantees that the inequality
defines a facet of . Let . Thus the origin of is an interior lattice point of and the inequality
defines a facet of . This fact together with  implies that is not reflexive. In other words, the dual polytope of defined by
is not a lattice polytope, where is the usual inner product of . It then follows from (, Theorem (1.1)) (and also from (, Theorem 8.1)) that is not Gorenstein, as desired. □
It is known (, Theorem 4.4) that a Cohen-Macaulay standard G-domain A is Gorenstein if and only if the h-vector is symmetric, i.e., for . Hence the h-vector of the toric ring is not symmetric when .
By using Normaliz , the h-vector of the toric ring is .
3. Non-Flawless -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.
The h-vector of the toric ring is
is the h-vector of the toric ring .
We conclude the present paper with the following
The h-vector of the toric ring of is of the form
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.
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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