A Sequence of Cohen–Macaulay Standard Graded Domains Whose h-Vectors Have Exponentially Deep Flaws

: Let K be a ﬁeld. In this paper, we construct a sequence of Cohen–Macaulay standard graded K -domains whose h-vectors are non-ﬂawless and have exponentially deep ﬂaws.


Introduction
In 1989, Hibi [1] made several conjectures on the h-vectors of Cohen-Macaulay standard graded algebras over a field. In particular, he conjectured that the h-vector of a standard graded Cohen-Macaulay domain is flawless ( [1], Conjecture 1.4). The hvector (h 0 , h 1 , . . . , h s ), h s = 0, of a Cohen-Macaulay standard graded algebra is flawless if h i ≤ h s−i for 0 ≤ i ≤ s/2 and h i−1 ≤ h i for 1 ≤ i ≤ s/2 . Niesi and Robbiano [2] disproved this conjecture by constructing a Cohen-Macaulay standard graded domain whose h-vector is (1,3,5,4,4,1). Further, Hibi and Tsuchiya [3] showed that the Ehrhart rings of the stable-set polytopes of cycle graphs of length 9 and 11 have non-flawless h-vectors by computation using the software Normaliz [4]. Moreover, the present author showed that the Ehrhart ring of the stable-set polytope of any odd cycle graph whose length is at least 9 has non-flawless h-vectors ( [5], Theorem 5.2) by proving the conjecture of Hibi and Tsuchiya ( [3], Conjecture 1).
However, these examples have the slightest flaws, i.e., there exists i with 0 ≤ i ≤ s/2 and h i = h s−i + 1. In this paper, we construct a sequence of standard graded Cohen-Macaulay domains that have h-vectors with exponentially deep flaws, i.e., we show the following. Theorem 1. Let K be a field and an integer with ≥ 2. Then, there exists a standard graded Cohen-Macaulay domain A over K such that dim A = 8 − 3, a(A ) = −4, and an h-vector (h 0 , h 1 , . . . , h s ), h s = 0, with h s /2 = h s − s /2 + 2 2 −3 . In particular, A 2 , A 3 , . . . is a sequence of Cohen-Macaulay standard graded domains over K who have exponentially deep flaws.
This theorem is proved at the end of this paper.

Preliminaries
In this section, we establish notation and terminology. For unexplained terminology of commutative algebra and graphy theory we consult [6] and [7], respectively.
In this paper, all rings and algebras are assumed to be commutative with an identity element. Further, all graphs are assumed to be finite, simple and without loops. We denote the set of non-negative integers, the set of integers, the set of rational numbers, the set of real numbers and the set of non-negative real numbers by N, Z, Q, R and R ≥0 , respectively.
For a set X, the cardinality of X is denoted by #X. For sets X and Y, we define X \ Y := {x ∈ X | x ∈ Y}. For non-empty sets X and Y, we denote the set of maps from X to Y by Y X . If Y 1 is a subset of Y 2 , then we treat Y X 1 as a subset of Y X 2 . If X is a finite set, we identify R X with the Euclidean space R #X . For f , f 1 , f 2 ∈ R X and a ∈ R, we define maps f 1 ± f 2 and a f by We denote the zero map, i.e., a map which sends all elements of X to 0, by 0. Further, if X 1 is a subset of X, then we treat R X 1 as a coordinate subspace of R X , i.e., we identify R X 1 with { f ∈ R X | f (x) = 0 for any x ∈ X \ X 1 }. For a non-empty subset X of R X , the convex hull (resp. affine span) of X is denoted by convX (resp. affX ). Definition 1. Let X be a finite set and ξ ∈ R X . For B ⊂ X, we set ξ + (B) := ∑ b∈B ξ(b).
For a field K, the polynomial ring with n variables over K is denoted by K [n] . Let R = n∈N R n be an N-graded ring. We say that R is a standard graded K-algebra if R 0 = K and R is generated by R 1 as a K-algebra. Let R = n∈N R n and S = n∈N S n be N-graded rings with R 0 = S 0 = K. We denote the Segre product n∈N R n ⊗ K S n of R and S by R#S.
Let Y be a finite set. Suppose that there is a family {T y } y∈Y of indeterminates indexed , is denoted by T f . A convex polyhedral cone in R Y is a set C of the form C = R ≥0 a 1 + · · · + R ≥0 a r , where a 1 , . . . , a r ∈ R Y . If one can take a 1 , . . . , a r ∈ Q Y , we say that C is rational.
Let C be a rational convex polyhedral cone. For a field K, is a finitely generated K-algebra. In particular, K[Z Y ∩ C] is Noetherian. Further, by the result of Hochster [8], we see that K[Z Y ∩ C] is normal and Cohen-Macaulay.
A subspace W of R Y is rational if there is a basis of W contained in Q Y . Let W 1 and W 2 be rational subspaces of R Y with W 1 ∩ W 2 = {0} and C i be a rational convex polyhedral cone in W i for i = 1, 2. Then, C 1 + C 2 is a rational convex polyhedral cone in R Y that is isomorphic to the Cartesian product Let X be a finite set and let P be a rational convex polytope in R X , i.e., a convex polytope in R X whose vertices are in Q X . In addition, let −∞ be a new element that is not contained in X. We set X − := X ∪ {−∞}. Further, we set C(P) : Then, C(P) is a rational convex polyhedral cone in R X − . We define the Ehrhart ring E K [P] of P over a field K by E K [P] := K[Z X − ∩ C(P)]. We define deg T −∞ = 1 and deg T x = 0 for x ∈ X. Then, E K [P] is an N-graded K-algebra.
Note that if W 1 and W 2 are rational subspaces of R X with W 1 ∩ W 2 = {0} and P i is a rational convex polytope in W i for i = 1, 2, then P 1 + P 2 is a rational convex polytope in R X that is isomorphic to the Cartesian product P 1 × P 2 and It is known that dim E K [P] = dim P + 1. Moreover, by the description of the canonical module of a normal affine semigroup ring by Stanley ([9], p. 82), we have the following.
where relint(C(P)) denotes the interior of C(P) in the topological space aff(C(P)).
The ideal of the above lemma is denoted by ω E K [P] and is called the canonical ideal of E K [P]. Note that the a-invariant (cf. ( [10], Definition 3. A stable set of a graph G = (V, E) is a subset S of V whose no two elements are adjacent. We treat the empty set as a stable set.
We set for any odd cycle C .
Let G = (V, E) be an arbitrary graph and n ∈ Z.
for any odd cycle C without chord and length at least 5 K is a maximal element of K and C is an odd cycle without chords. However, since the values appearing in these inequalities are integers, these inequalities are equivalent to µ(z) ≥ 1, µ + (K) + 1 ≤ µ(−∞) and µ + (C) + 1 ≤ µ(−∞) · #C−1 2 , respectively. Therefore, by Lemma 1, we see that

Construction
Let K be a field. In this section, for each integer ≥ 2, we construct a standard graded Cohen-Macaulay K-algebra, A , which has a non-flawless h-vector. The flaw of the h-vector is computed in the next section.
Let be an integer with ≥ 2. We define a graph G = (V , E ) by the following way. Set The cases where = 3 and 4 are as follows.
In addition, set where G(C ) is the induced subgraph of G by C . In the following, up to the end of the proof of Lemma 5, we fix and write G , V , E , C , B , A , R and I ,i as just G, V, E, C, B, A, R and I i , respectively. Further, we consider the subscripts of c i , I i and the first subscript of b i,k modulo 2 + 1. For example, c 2 +1 = c 0 , We set e i := {c i , c i+1 } and K i,k := {c i , c i+1 , b i,k } for 0 ≤ i ≤ 2 and k ∈ I i . We also consider the subscript of e i and the first subscript of K i,k modulo 2 + 1.

Proposition 1.
The ring A is a standard graded K-algebra.
. Next, suppose that µ(c i ) = 0 for some i. Take i 0 with µ(c i 0 ) = 0. We define µ ∈ Z V − by the following way.

Remark 1.
The functions µ J j+2 and µ in the proof of Proposition 1 are the characteristic function of some stable set of G. Therefore, the above proof shows that G is a t-perfect graph.

Structure of the Canonical Module
In this section, we study the generators and the structure of the canonical module of A. First, we set Then, W is a codimension 1 vector subspace of R V − with W ⊃ R B . Further, we set Then, A (0) is a K-subalgebra of A (we denote this ring by (A ) (0) when it is necessary to express ). Further, since We denote this ring by R (0) . Note that µ J i ∈ tU Further, we see the following.

Lemma 2. It holds that
Proof. It is clear that In order to prove the inclusion , it is enough to show that for any µ ∈ tU 0 . In fact, (µ − µ J i )(z) ≥ 0 for any z ∈ V by the choice of i and the definition of J. If j ≡ i − 2 (mod 2 + 1) or j ≡ i − 2 (mod 2 + 1) and k ∈ J, then (µ J i ) + (K j,k ) = 1. Thus, . If j ≡ i − 2 (mod 2 + 1) and k ∈ J, then µ(b i−2,k ) = µ J i (b i−2,k ) = 0 by the definition of J. Therefore, by the choice of i, we see that

Lemma 4. It holds that
Further, Imϕ is a rank-1-free A-module with basis T η 1 .
Proof. This lemma is proved almost identically to Lemma 4.2 in [5].
Then, the following holds.
By the second proof of Theorem 4.1 in [9], we see that where H(M, λ) denotes the Hilbert series of a graded module M.
Funding: This research was funded by JSPS KAKENHI JP20K03556.
Institutional Review Board Statement: Not applicable.