Difference of Hilbert series of homogeneous monoid algebras and their normalizations

Let $Q$ be an affine monoid, $\Bbbk[Q]$ the associated monoid $\Bbbk$-algebra, and $\Bbbk[\overline{Q}]$ its normalization, where we let $\Bbbk$ be a field. In this paper, in the case where $\Bbbk[Q]$ is homogeneous (i.e., standard graded), a difference of the Hilbert series of $\Bbbk[Q]$ and $\Bbbk[\overline{Q}]$ is discussed. More precisely, we prove that if $\Bbbk[Q]$ satisfies Serre's condition $(S_2)$, then the degree of the $h$-polynomial of $\Bbbk[Q]$ is always greater than or equal to that of $\Bbbk[\overline{Q}]$. Moreover, we also show counterexamples of this statement if we drop the assumption $(S_2)$.

1. Introduction 1.1.Backgrounds.Let k be a field throughout this paper.For the fundamental materials on commutative algebra, see [3].
Let R = i∈Z R i be a graded k-algebra with dim k R i < ∞ for each i.Hilbert series of R is one of the most fundamental invariants in the theory of commutative algebra.For the introduction to the theory of Hilbert series, see, e.g., [3,Section 4].Although the Hilbert series just provides the numerical information of R as a k-vector space, it reflects several commutative-algebraic properties.For example, a classical observation claims that if R is Cohen-Macaulay, then the Laurent polynomial appearing in the numerator of the Hilbert series of R has nonnegative coefficients ( [3,Corollary 4.1.10]).Moreover, the Gorensteinness of R is completely characterized in terms of a kind of symmetry of the Hilbert series if R is a domain ([3,Corollary 4.4.6]).Furthermore, by several recent studies, some connections with certain generalized notions of Gorensteinness (e.g.almost Gorenstein, nearly Gorenstein, level) and the Hilbert series have been discovered.(See, e.g., [6,7,10].)Since the Hilbert series of R can be computed from the graded minimal free resolution of R, it also captures other invariants which have been well-discussed in commutative algebra, such as Krull dimension (or codimension), Koszulness, Castelnuovo-Mumford regularity, multiplicity, a-invariant, and so on.
On the other hand, affine monoids and their associated k-algebras have been well studied by many researchers in several contexts.For the introduction to the theory of affine monoids and affine monoid k-algebras, see, e.g., [2] and [3,Section 6.1].Since geometric information of affine monoids is applicable for the analysis of algebraic properties of the associated k-algebras, affine monoid k-algebras have been regarded as useful objects in commutative algebra.For example, the celebrated theorem by Hochster claims that if an affine monoid Q is normal, i.e., Q = Q, then k[Q] is Cohen-Macaulay ([3, Theorem 6.3.5 (a)]).Moreover, Cohen-Macaulayness of the affine monoid k-algebra k[Q] is characterized in terms of geometric information on Q together with the information about reduced homology groups over k of certain simplicial complexes ( [15]).Furthermore, Katthän reveals a strong connection with the structure of holes of affine monoids Q (i.e., Q \ Q) and ring-theoretic properties (e.g., Serre's condition (S 2 ), (R 1 ) and depth).See [9].
By taking these backgrounds into account, in this paper, we study the Hilbert series of In particular, we focus on the difference of them.
1.2.Main Results.To explain the main results of this paper, we introduce the notations used throughout this paper.Given a graded k-algebra R = i∈Z R i with dim k R i < ∞ for each i, let Hilb(R, t) denote the Hilbert series of R, i.e., We say that R is homogeneous (or standard graded ) if it is generated by degree 1 elements.If R is homogeneous and of dimension d, then we see that Hilb(R, t) is of the following form: The following is the first main theorem of this paper: Theorem 1.1.Let Q be a homogeneous affine monoid and assume that k Here, deg(f (t)) denotes the degree of the polynomial f (t).For the definition of Serre's condition, see Subsection 2.2.
The following second main theorem shows the existence of counterexamples of Theorem 1.1 if we drop the assumption (S 2 ).Theorem 1.2.For any positive integer m, there exists a homogeneous affine monoid 1.3.Organization of this paper.In Section 2, we prepare the materials for the proofs of the main results.In Section 3, we give a proof of Theorem 1.1.In Section 4, we give a proof of Theorem 1.2.
Acknowledgements.This paper is partially supported by KAKENHI 20K03513 and 21KK0043.

Preliminaries
In this section, we recall several materials used in this paper.
2.1.Affine monoids and hole modules.An affine monoid is a finitely generated submonoid of Z d for some d.Given an affine monoid Q ⊂ Z d ≥0 , we can associate the k-algebra k where for α = (α 1 , . . ., α d ) ∈ Z d ≥0 , we let We call this k-algebra k[Q] the monoid algebra of Q. Affine monoids and monoid algebras have been called as affine semigroups and affine semigroup rings (e.g., in [3]), but it is becoming to be called them as affine monoids and monoid algebras, respectively.Those are the same notions, but we employ the terminology "monoid" in this paper.
We recall some fundamental notions on affine monoids and their monoid k-algebras.Let Q ⊂ Z d ≥0 be an affine monoid.The minimal generating system of Q is the minimal In this case, we use • Let ZQ denote the free abelian group generated by Q, i.e., ZQ = s i=1 The dimension of a face F is defined to be the rank of the free abelian group ZF .
• We say that an affine monoid Q is homogeneous if the minimal generating set of Q lies on the same hyperplane not containing the origin.Throughout this paper, affine monoids are always assumed to be positive.We recall the following statement, which will play a crucial role in the proof of Theorem 1.1.
Theorem 2.1 ([9, Theorem 3.1 and Proposition 5.5]).Let Q be an affine monoid.Then there exists a (not-necessarily disjoint) decomposition: Note that Theorem 3.1 and Proposition 5.5 in [9] claim the similar statement for nonnecessarily positive affine monoids, but we convert the statement into the case of positive affine monoids.

2.2.
Serre's condition.Let R be a Noetherian ring and let M be a finitely generated R-module.For a nonnegative integer m, we say that M satisfies Serre's condition where Spec R denotes the set of all prime ideals of R. Namely, M satisfies (S m ) if and only if M p is Cohen-Macaulay for any p ∈ Spec R with depth M p < m.
Regarding (S 2 ) for monoid algebras k[Q], we konw the following: ≥0 , then we may translate P by a certain lattice point γ ∈ Z N ≥0 to make P + γ ⊂ R N ≥0 .Since most properties of lattice polytopes are preserved by the traslation by a lattice point, we may assume that P ⊂ R N ≥0 without loss of generality.
Given a lattice polytope P ⊂ R N ≥0 , we can associate a homogeneous affine monoid Q P as follows: . Hence, we can associate the k-algebra k[Q P ], known as the toric ring of P .
On the other hand, we can also define the k-algebra associated to P as follows: , where nP = {nα : α ∈ P }.This k-algebra Ehr k (P ) is called the Ehrhart ring of P .We see by definition that Hilb(Ehr k (P ), t) coincides with the Ehrhart series of P , which is the generating function n≥0 |nP ∩ Z N |t n .For the introduction to Ehrhart theory, see [1].
Given a homogeneous affine monoid Q ⊂ Z d ≥0 , a cross section polytope of Q, denoted by P Q ⊂ R d ≥0 , is the lattice polytope obtained by the intersection of Q and a hyperplane including all the generators of Q.
Example 2.4.The Ehrhart ring of P does not necessarily coincide with the normalization of k[Q P ].In fact, let P be the convex hull of {(0, 0, 0), (1, 1, 0), (0, 1, 1), (1, 0, 1)}.Then we see that In particular, the normalization k[Q P ] is its own.On the other hand, the following holds: where we regard deg We say that a lattice polytope P ⊂ R d ≥0 is spanning if ZQ P = Z d+1 .If P is spanning, then we see that k[Q P ] = Ehr k (P ).The notion of spanning polytopes was introduced in [8] and the Ehrhart theory for spanning polytopes was developed there.
We recall a well-known notion for lattice polytopes P ⊂ R d .The codegree of P , denoted by codeg(P ), is defined as follows: where P • denotes the relative interior of P .
Assume that P ⊂ R d ≥0 and P is spanning.Then it is known that deg(h and , h s is the leading coefficient of h Q P (t) and ℓ = codeg(P ).Those are implicitly explained in [1, Section 4] in terms of lattice polytopes or their Ehrhart series.
2.4.Edge rings.Throughout this paper, all graphs are finite and simple.Let G be a connected graph on the vertex set V (G) with the edge set E(G).Then we can associate a homogeneous affine monoid Q G by setting where e u denotes the unit vector of R V (G) and e u,v = e u + e v .We call the associated k-algebra k[Q G ] the edge ring of G.
Let P G be the convex hull of {e u,v : {u, v} ∈ E(G)}.This polytope is known as the edge polytope of G.
We know by [12 Remark 2.5.For a non-bipartite connected graph G with d vertices, the edge polytope P G is always assumed to be spanning in the following sense.Fix a vertex v of G. Let P ′ be the image of P G by the projection π : R V (G) → R V (G)\{v} ignoring the entry corresponding to v. Then Q G is isomorphic to Q P ′ as affine monoids.Moreover, let S be a (d−1)-simplex in P G described in [12,Lemma 1.4].Then we can check that {(α, 1) : α ∈ π(S) ∩ Z V (G)\{v} } forms a Z-basis for Z V (G) if we choose a vertex v properly.Therefore, we conclude that P ′ is spanning.Since Q G is isomorphic to Q P ′ , we can claim like "P G is spanning".In particular, k[Q G ] = Ehr k (P G ) holds.Even if G is not connected, we may apply the same procedure for each connected component G i and obtain a Z-basis for Z V (G i ) .By combining these Z-bases for Z V (G i ) for each i, we obtain a Z-basis for Z V (G) .Namely, P G is spanning for any non-bipartite (non-necessarily connected) graph G.
A similar discussion can be applied for bipartite graphs and we can also claim the spanning property for edge polytopes of bipartite graphs, but we omit the detail since we do not use it in this paper.
An exceptional pair in G is a pair (C, C ′ ) of two odd cycles C and C ′ in G such that C and C ′ have no common vertex and there is no bridge between C and C ′ (i.e., no edge Edge rings have been intensively studied by several people since the following theorem was established: Theorem 2.6 ( [12,14]).Let G be a connected graph.Then Q G is described as follows: where e C = v∈V (C) e v .In particlar, Q G is normal if and only if there is no exceptional pair in G.
We prepare the following proposition for the proof later.
Then we can write α like α = {u,v}∈E(G) a u,v e u,v , where a u,v > 0 for each {u, v} ∈ E(G) and a u,v = ℓ.Thus, in particular, This means that codeg(P G ) ≥ d/2.Moreover, by this discussion, if codeg(P G ) = d/2, then α must be v∈V (G) e u .This implies that h d/2 = 1.□

Proof of Theorem 1.1
This section is devoted to giving a proof of Theorem 1.1.
Proof of Theorem 1.1.Let Q ⊂ Z d ≥0 be a homogeneous affine monoid of dimension d.Then we know the following short exact sequence of graded k[Q]-modules: By the way, Theorem 2.1 describes the structure of k Since we assume (S 2 ) for k[Q], we know that each family of holes is of dimension d − 1 (see Theorem 2.3).Let us consider the decomposition of Q \ Q in (2.1), where q i ∈ Q and F i is a face of Q of dimension d − 1 for each i.Note that this decomposition is not necessarily disjoint.Hence, we have to apply an "inclusion-exclusion type" formula to get the Hilbert series of k Here, we recall some materials on hyperplanes arrangements.(See, e.g., [13] for the introduction to the theory of hyperplane arrangements.)Let H i = q i + RF i be an affine hyperplane, and let A = {H 1 , . . ., H ℓ } be a hyperplane arrangement.Let L(A) denote the intersection lattice of A, i.e., equipped with a partial order defined by reverse inclusion.We regard i∈∅ H i as R d , i.e., R d ∈ L(A) is the minimal element.We define a map µ : L(A) → Z (known as the Möbius function) as follows: Then we see from (2.1) that where a i = deg x q i and we let X = X ∩ Q.Note that X is a normal homogeneous submonoid of Q.Hence, it follows from (3.1) that Since X is normal, we know that k[X] is Cohen-Macaulay, so h X (t) has positive coefficients.Thus, we can conclude the desired conclusion deg(h Actually, this is known to be true.See, e.g., [13,Corollary 3.4].□ Regarding another relationship between h Q (t) and h Q (t), we know the following: Proof.Since the structure of k[Q]/k[Q] can be captured by the decomposition (2.1) and each face F i is of dimension at most d − 1, we see that where g(t) is some polynomial in t.Then it follows from (3.1) that In particular, h In this section, we give a proof of Theorem 1.2.For the proof of Theorem 1.2, we use the following example.Let Q = Q G .By using Macaulay2 (for k[Q]) together with Normaliz (for k[Q]), we see the following: 10  .
On the other hand, we see that 3 where we let A = k[x 1 , . . ., x 14 ] and regard k[Q] as A/I by taking the defining ideal , the toric ideal of Q).This implies that k[Q] does not satisfy (S 2 ) (see Proposition 2.2).Note that we can also check non-(S 2 )-ness by using the structure of the hole module k This example shows that Theorem 1.1 does not hold if we drop the assumption (S 2 ).
Actually, we can generalize this example as follows: Proposition 4.2.Given a positive integer k, let G k be the graph as depicted in Figure 2: We also have depth k where e u,v = e u + e v .In the left-hand side of this equality, all edges of G k appear and each of the coefficients is positive.Moreover the sum of the coefficients is equal to k + 3.This implies that (k + 3)P • ∩ Z d ̸ = ∅, i.e., codeg P ≤ k + 3, which implies that deg(h On the other hand, Proposition 2.7 implies that deg(h Q (t)) ≤ (2k + 6)/2 = k + 3. Therefore, deg(h Q (t)) = k + 3.At the same time, we can also see that h k+3 = 1, where h k+3 denotes the leading coefficient of h Q (t).
The second step: Next, we compute the family of holes of Q.Let G ′ be the (nonconnected) graph obtained from G k by removing the vertices v 1 , . . ., v k together with the incident 3k edges.Let q = 6 i=1 e u i and let Note that the pair of 3-cycles (u 1 , u 2 , u 3 ) and (u 4 , u 5 , u 6 ) is a unique exceptional pair in G k ."(⊂)" By (2.3), we have Hence, Q \ Q ⊂ q + Q holds.Moreover, for each i = 1, . . ., k, we see the following: q + e v i ,u 3 = e u 1 ,u 3 + e u 2 ,u 3 + e v i ,u 6 + e u 4 ,u 5 , q + e v i ,u 6 = e u 1 ,u 2 + e v i ,u 3 + e u 4 ,u 6 + e u 5 ,u 6 , and and look at the entries α ′′ u 4 , α ′′ u 5 and α ′′ u 6 .Then α ′′ u 4 + α ′′ u 5 + α ′′ u 6 is always odd (with at least 3).On the other hand, α ′′ v i = 0 for each i by definition of Q ′ .This implies that q + α ′ cannot be decomposed into e u,v 's for some edges {u, v} in G k , i.e., q + α ′ ̸ ∈ Q.
Moreover, since {e u,v : {u, v} ∈ G ′ k } is linearly independent and consists of k +6 vectors, we see that The third step: By the second step together with (3.1), we see the following: .
(Note that q corresponds to a monomial of degree 3 in k This implies the desired conclusion (4.1).The fourth step: Lastly, we discuss the depth of k , where each x v i = 0 becomes a supporting hyperplane of Q.)This implies that (4.2) gives the decomposition (2.1).Hence, k[Q] does not satisfy (S 2 ) by Theorem 2.3.Moreover, we can apply the latter statement of Theorem 2.1 and obtain that the depth of k □ For the proof of Theorem 1.2, we recall the notion of join for lattice polytopes and apply the same idea to homogeneous affine monoids.Given two lattice polytopes P ⊂ R d and P ⊂ R d ′ , let P ⋆ P ′ be te convex hull of We call P ⋆ P ′ the join of P and P ′ .
Similarly, given two homogeneous affine monoids Then it is straightforward to see that Q ⋆ Q ′ is also a homogeneous affine monoid.Let us call Q ⋆ Q ′ the join of homogeneous affine monoids Q and Q ′ . and Proof.Let P (resp.P ′ ) be the cross section polytope of ) is the hyperplane containing the generators of Q (resp.Q ′ ).Hence, the latter equality of the statement is directly obtained from [5,Lemma 1.3].
For the former, similarly to the proof of [5,Lemma 1.3], it suffices to show that but this directly follows from the description of Q ⋆ Q ′ as follows: for each k ∈ Z >0 , □ Now, we are in the position to give a proof of Theorem 1.2.
Proof of Theorem 1.2.Given a positive integer m, let , where Q is the same homogeneous affine monoid as in Example 4.
Proposition 2.7.For a connected graph with d vertices, we have deg(h Q G (t)) ≤ d/2.Moreover, if the equality holds, then h d/2 = 1, where h d/2 is the leading coefficient of h Q G (t).