Contemporary Mathematics

Some Normal Monomial Ideals

Marie A. Vitulli

Abstract. In this paper we investigate the question of normality for special monomial ideals in a polynomial ring over a field. We first include some expository sections that give the basics on the integral closure of a ideal, the Rees algebra on an ideal, and some fundamental results on the integral closure of a monomial ideal.

1. Introduction

There has always been considerable interest in monomial ideals in a polynomial ring in n indeterminates. Many properties of the ideal I can be translated into properties about the set Γ(I) of exponents of monomials in the ideal and its convex hull NP(I) (the Newton polyhedron of I) and these sets can largely be understood by the convex geometry of ℝ^{n}. For example, it is well known that the monomial ideal I is integrally closed if and only if Γ(I) = NP(I) ∩ ℕ^{n} (see Theorem 3.10).

Another algebraic notion that has significant geometric consequences is that of a normal ideal. An ideal I of a ring R is said to be normal provided that all its positive powers are integrally closed. If R is a normal integral domain then the ideal I in R is normal if and only if the Rees algebra R[It] is normal. The Rees algebra is the algebraic counterpart to blowing up a scheme along a closed subscheme so various geometric properties of the blow-up can be interpreted as algebraic properties of the Rees algebra. If I is a monomial ideal and R is a finite-dimensional polynomial ring over a field we can again translate the question of normality of I into questions about the exponent set Γ(I) and the Newton polyhedron NP(I) of I. However, no concise answer to the question of when I is normal is known, nor does it appear likely that one can ever be given. The question of normality of monomial ideals is the main focus of this paper. We spend most of our time studying a special class of monomial ideals that are primary to the ideal generated by the indeterminates.

We now give a brief account of the organization of this manuscript. Our aim is to make this paper largely self-contained, thus we include proofs of many results that are well-known as “folklore” but whose proofs are elusive. In Section 2.1 we present several preliminary results about the integral closure of graded rings. We give results for both the relative and absolute notions of integral closure. In Section

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2000 Mathematics Subject Classification. Primary 13C13; Secondary 13A20, 13F20.

This article is dedicated to our friend and colleague, Ruth I. Michler, whose energy and presence enriched our community.

2.2 we present results on the integral closure of a homogeneous ideal in a graded ring. In Section 3 we specialize to the case of monomial ideals in a polynomial ring in n indeterminates over a field K. By introducing an ℕ^{n}-grading on K[x_{1}, …, x_{n}] we can identify monomial ideals and homogeneous ideals of K[x_{1}, …, x_{n}]. We then present some well-known results about monomial ideals. We start with a complete account of the fact that the integral closure of a monomial ideal is again a monomial ideal (see Corollary 3.2) whose exponent set is the set of integral points in the Newton polyhedron of the ideal (see Theorem 3.10). In Section 3.1 we turn our attention to the question of when a monomial ideal is normal. In section 3.2 we are interested in a special class of monomial ideals that arise from vectors λ = (λ_{1}, …, λ_{n}) of positive integers. First define J(λ) = (x_{1}^{λ_{1}}, …, x_{n}^{λ_{n}}) and then define I(λ) to be the integral closure of J(λ). We introduce several recent results of Reid, Roberts, and the current author in [12]. We also discuss a new connection between an arithmetic condition of the additive submonoid ⟨1/λ_{1}, …, 1/λ_{n}⟩ called almost quasinormality and the regularity of the Rees algebra R[It] in codimension one.

2. Preliminaries

2.1. Integral Closure of Graded Rings. In this subsection we present some basic material about the integral closure of graded rings, which we will need when we talk about Rees algebras. The standard technique for deducing results about an ideal I in a ring R is to pass to the Rees algebra R[It] of I, which is the Rsubalgebra of the polynomial algebra R[t] generated by {at | a ∈ I}. We set the stage by proving a standard result about the integral closure of a graded ring.

Conventions. All rings are assumed to be commutative with identity and all ring homomorphisms preserve the identity. We let ℤ_{+} denote the set of positive integers, ℕ the set of nonnegative integers, ℚ_{≥} the set of nonnegative rational numbers, ℚ_{+} the set of positive rational numbers, ℝ_{≥} the set of nonnegative real numbers, and e_{1}, …, e_{n} the standard basis vectors in ℝ^{n}. We write α ≤_{pr} β for vectors α = (a_{1}, …, a_{n}), β = (b_{1}, …, b_{n}) ∈ ℝ^{n} provided that a_{i} ≤ b_{i} for 1 ≤ i ≤ n. Thus α <_{pr} β means that a_{i} ≤ b_{i} for all 1 ≤ i ≤ n and a_{j} < b_{j} for some 1 ≤ j ≤ n. By a semigroup we mean a set with an associative binary operation. By a monoid we mean a semigroup with identity. We say an abelian monoid G is torsion-free provided that whenever f ≠ g are elements of G then nf ≠ ng for all n ∈ ℤ_{+}. By a totally ordered abelian monoid we mean an abelian monoid that is totally ordered and has the property that whenever f < g and h are elements of G then f + h < g + h. For elements g_{1}, …, g_{r} of an abelian monoid G we let ⟨g_{1}, …, g_{r}⟩ denote the submonoid of G generated by g_{1}, …, g_{r}. If G is an additive abelian monoid and K is any ring, we think of the monoid ring K[G] = ⊕_{g∈G}Kx^{g} as a G-graded ring and say that the homogeneous element x^{g} of K[G] has exponent g. If G is a totally ordered abelian monoid then H = grp(G) is naturally a totally ordered abelian group. If R is a G-graded ring we may regard R as an H-graded ring by setting R_{g} = {0} for g ∈ H \ G. By the integral closure, or normalization, \overline{R} of a reduced ring R we mean the integral closure, or normalization, of R in its total quotient ring.

Our first result is about the integral closure of a reduced graded ring in a reduced graded extension ring.

Theorem 2.1. Let G be a totally ordered abelian monoid and R ⊆ S be an extension of reduced G-graded rings. Let U denote the set of homogeneous S-regular elements of R and assume that U^{−1}R is integrally closed in U^{−1}S. Then, the integral closure of R in S is again G-graded.

Proof. First let us assume that R is Noetherian. Suppose that s ∈ S is integral over R. For any element s ∈ S let in(s) denote the initial component of s, that is the (nonzero) homogeneous component of least degree. Then R[s] is a finite R-submodule of U^{−1}R. Hence there exists an element u ∈ U such that us^{i} ∈ R for all i ≥ 0. Since S is reduced and u is not a zero divisor on S, the initial component of us^{i} is uin(s)^{i} and we must have uin(s)^{i} ∈ R for all i ≥ 0. Thus R[in(s)] ⊆ Ru^{−1}. Since R is Noetherian we may deduce that R[in(s)] is a finitelygenerated R-module and hence in(s) is integral over R. Hence s − in(s) is integral over R and by induction on the number of (nonzero) homogeneous components we may assume that each homogeneous component of s − in(s) is integral over R.

Now consider the general case. Suppose s ∈ S satisfies the equation

s^{n} + r_{1}s^{n−1} + ⋯ + r_{n} = 0.

As in the Noetherian case, there exists an element u ∈ U such that us^{i} ∈ R for all i ≥ 0 and hence uin(s)^{i} ∈ R for all i ≥ 0. Let R′ be the ℤ-subalgebra of R generated by u, the elements us_{g}, where s_{g} is a homogeneous component of s, and the homogeneous components of the coefficents r_{i} of the above equation. By the proof of the Noetherian case, in(s) is integral over R′ and by induction on the number of homogeneous components every homogeneous component of s is integral over R′, hence is integral over R. □

Remark 2.2. We point out that the hypotheses of the above theorem are satisfied if G is a finitely-generated totally ordered abelian monoid and R ⊆ S is an extension of G-graded integral domains with S contained in the quotient field of R. In this case we have H := grp(α ∈ G | R_{α} ≠ 0) is a finitely generated, torsion-free abelian group and hence is isomorphic to ℤ^{d} for some positive integer d. The multiplicative subset U above consists of the nonzero homogeneous elements of R. We have U^{−1}R ≅ K[H] ≅ K[x_{1}, …, x_{d}, x_{1}^{−1}, …, x_{d}^{−1}] is a ring of Laurent polynomials over a field by [6, Theorem 1.1.4]. With this we can prove an absolute result about the integral closure of a graded integral domain. The corresponding assertion for ℤ-graded rings appeared in [8, Lemma 2.1].

Corollary 2.3. Let G be a totally ordered abelian monoid and R a G-graded reduced ring with finitely many minimal primes. Then, the integral closure of R is again G-graded.

Proof. First assume that R is an integral domain. Let U denote the set of nonzero homogeneous elements of R and let H = grp(G). We will regard R ⊆ U^{−1}R as an extension of H-graded rings. If we can show that U^{−1}R is integrally closed, then it will follow that the integral closure of R is the integral closure of R in U^{−1}R. Then we can apply Theorem 2.1 to the extension R ⊆ U^{−1}R to deduce the conclusion of this corollary.

Let K denote the quotient field of R and suppose that r/s ∈ K is integral over U^{−1}R. Let H′ be the subgroup of H generated by exponents of the homogeneous components of r, s, and the numerators and denominators of an equation of integral dependence. Let A = ⊕_{g∈H′}R_{g} and V be the set of nonzero homogeneous elements of A. View A ⊆ V^{−1}A as an extension of H′-graded domains. Then r/s is in the integral closure of A and hence is in V^{−1}A by [6, Theorem 1.1.4]. Hence r/s ∈ U^{−1}R as asserted.

In the general case, if P_{1}, … P_{ℓ} are the minimal primes of R, then each P_{i} is homogeneous and \overline{R} = \overline{R/P_{1}}×⋯×\overline{R/P_{ℓ}}. Since each factor \overline{R/P_{i}} is again G-graded so is \overline{R}. □

We will now state a version of Theorem 2.1 that doesn’t require that the rings be integral domains or even reduced.

Theorem 2.4. Let G be a cancellative, torsion-free abelian monoid and R ⊂ S an extension of G-graded rings. Then, the integral closure of R in S is again G-graded.

Proof. This is a generalization due to Y. Yao [7, Theorem 1.7.5] of the corresponding assertion when G is ℤ^{n} [7, Theorem 1.7.3]. Essentially the result for ℤ^{n}-graded rings is proved by a clever use of Vandermonde matrices and then proved in the more general setting by reducing to the ℤ^{n}-case. □

2.2. Integral Closure of Homogeneous Ideals. We now recall some fundamental results on the integral closure of an ideal I in a ring R and the integral closure of a homogeneous ideal in a graded ring.

Definition 2.5. Let I be an ideal in a ring R and x ∈ R. We say that x is integral over I if there is a positive integer n and elements a_{j} ∈ I^{j}(j = 1, …, n) such that

x^{n} + a_{1}x^{n−1} + ⋯ + a_{n} = 0.

The integral closure \overline{I} of I is defined to be the set of all elements of R that are integral over I. The ideal I is said to be integrally closed if I = \overline{I}. The ideal I is said to be normal if every positive power of I is integrally closed.

For an ideal I in a ring R and an element x ∈ R, x is integral over I if and only if xt ∈ R[t] is integral over the Rees algebra R[It], where R[It] is the R-subalgebra of the polynomial algebra R[t] generated by {at | a ∈ I}. We can regard R[t] as a ℕ-graded ring, where R[t]_{n} = Rt^{n} for each n ∈ ℕ. By Theorem 2.1 the integral closure of R[It] in R[t] is again ℕ-graded. This point of view yields the following.

Proposition 2.6. Let I be an ideal of a ring R. Then, the integral closure \overline{I} of I is again an ideal of R. Furthermore, \overline{I} is integrally closed.

The above conclusion is also a consequence of the following well-known theorem (e.g., see [13] or [7, Proposition 5.1.7 ]).

Theorem 2.7. Let R be a ring and t an indeterminate. For an ideal I of R, the integral closure of R[It] in R[t] is the graded ring ⨁_{n≥0} \overline{I^{n}}t^{n}.

If we are working in a G-graded ring R and have a homogeneous ideal I we can prove that \overline{I} is again homogeneous by considering R[t] to be a G × ℕ-graded ring, where R[t]_{(g,n)} = R_{g}t^{n}. We now state the exact result.

Theorem 2.8. Let G be a cancellative, torsion-free abelian monoid and let I be a homogeneous ideal in the G-graded ring R. Then, \overline{I} is again homogeneous.

Proof. Note that G×ℕ is again a torsion-free abelian monoid so by Theorem 2.4 the integral closure of R[It] in R[t] is again G × ℕ-graded. Since an element x ∈ R is in \overline{I} if and only if xt ∈ ∑_{g∈G} \overline{R[It]}_{(g,1)}, we may conclude that \overline{I} is G-homogeneous. □

3. Monomial Ideals

3.1. Basics on Monomial Ideals. In this section we recall some preliminary results about monomial ideals in a polynomial ring over a field of arbitrary characteristic. We give proofs of some well-known results for expository purposes and because we do not know of a complete and correct reference for all of these assertions.

Notation 3.1. Throughout this section R will denote the polynomial ring K[x_{1}, …, x_{n}] over a field K and 𝔪 = (x_{1}, …, x_{n}) will denote the maximal homogeneous ideal of R. In this context, for a vector α = (a_{1}, …, a_{n}) ∈ ℕ^{n} we let x^{α} denote the monomial x_{1}^{a_{1}} ⋯ x_{n}^{a_{n}}.

After some preliminaries on the integral closure of a monomial ideal we turn our attention to the question of normality of a monomial ideal. We will consider the polynomial ring as an ℕ^{n}-graded ring where for a vector α ∈ ℕ^{n} we let R_{α} = Kx^{α}. With this grading homogeneous ideals and monomial ideals coincide.

Corollary 3.2. If I ⊆ K[x_{1}, …, x_{n}] is a monomial ideal in a polynomial ring in n indeterminates over a field k, then \overline{I} is again a monomial ideal. Indeed,

\overline{I} = (x^{α} | x^{mα} ∈ I^{m} ∃m ∈ ℤ_{+}).

Proof. Let R = K[x_{1}, …, x_{n}] and regard R as an ℕ^{n}-graded ring as above. Then \overline{I} is again a monomial ideal by Theorem 2.8. Suppose a monomial x^{α} is integral over I. Consider an equation of integral dependence

x^{nα} + a_{1}x^{(n−1)α} + ⋯ + a_{n} = 0,

where a_{j} ∈ I^{j}. Replacing each a_{j} by its homogeneous component of degree jα we may assume each a_{j} = c_{j}x^{jα} ∈ I^{j} where c_{j} ∈ K. We must have c_{j} ≠ 0 for some j between 1 and n and hence x^{jα} ∈ I^{j} for some positive integer j. □

There is a convex-geometric description of the set of exponents of monomials appearing in the integral closure of a monomial ideal that is well-known but whose proof is difficult to find. To our knowledge the result first appeared for convergent power series rings in [9]. We present the proof for polynomial rings below.

For the reader’s convenience we recall some definitions necessary to our exposition. For the fundamentals on the convex geometry we are using the reader can consult [2], [14] or [16]. Although the concepts of affine combination and affine dependence can be defined for an arbitrary vector space, we restrict our attention to real vector spaces for our convex-geometric definitions.

Definition 3.3. Let V be a real vector space. We say a vector α in V is an affine combination of the vectors α_{1}, …, α_{p} in V if α is a linear combination α = c_{1}α_{1} + ⋯ + c_{p}α_{p} and ∑_{i=1}^{p} c_{i} = 1. If, in addition, each c_{i} ≥ 0 we say α is a convex combination of the vectors α_{1}, …, α_{p}. A nonempty subset of ℝ^{n} that is closed under affine combinations is called an affine subspace. Analogously, a nonempty subset of ℝ^{n} is said to be convex if it is closed under convex combinations.

Notice that if W is an affine subspace of the real vector space V and α ∈ W then the set U := W − α = {β − α | β ∈ W } is a linear subspace of V and W = α + U. The subspace U is uniquely determined by W and by the dimension dim(W) of W we mean the dimension of U.

Definition 3.4. Let S be a subset of a real vector space V. The set of all affine combinations of vectors in S is called the affine hull aff(S) of S. The set of all convex combinations of vectors in S is called the convex hull conv(S) of S. The set of all nonnegative linear combinations (i.e., the coefficients are nonnegative) of vectors in S is called the positive cone pos(S) of S.

Observe that the affine hull of S is the smallest affine subspace containing S. Similarly, the convex hull of S is the smallest convex subset containing S.

Definition 3.5. We say a subset {α_{1}, …, α_{p}} of a real vector space V is affinely dependent if there exist scalars c_{1}, … c_{p}, not all zero, such that ∑_{i=1}^{p} c_{i}α_{i} = 0 and ∑_{i=1}^{p} c_{i} = 0. If {α_{1}, …, α_{p}} is not affinely dependent we say the set is affinely independent.

Remark 3.6. Suppose that ∑_{i=1}^{p} c_{i}α_{i} = 0 is an equation of affine dependence and that c_{1} ≠ 0. After scaling by a factor of 1/c_{1} we may may and shall assume that c_{1} = 1. Hence α_{1} = −c_{2}α_{2} − ⋯ − c_{p}α_{p} expresses α_{1} as an affine combination of α_{2}, …, α_{p}. Thus the set {α_{1}, …, α_{p}} is affinely dependent if and only if some vector in the set is an affine combination of the remaining vectors. Also notice that {α_{1}, …, α_{p}} is affinely independent if and only if {α_{2} − α_{1}, …, α_{p} − α_{1}} is linearly independent.

Definition 3.7. Let X be any subset of R = K[x_{1}, …, x_{n}]. Then set

Γ(X) = {α ∈ ℕ^{n} | x^{α} ∈ X}.

We refer to Γ(X) as the exponent set of X. If I is a monomial ideal then Γ(I) is an ideal of the monoid ℕ^{n} [5, page 3]. If A is a subalgebra of R generated by monomials then Γ(A) is a submonoid of ℕ^{n}, and A is isomorphic to the monoid ring K[Γ(A)].

Recall that if I is a monomial ideal, then its exponent set Γ(I) is a subset of ℕ^{n}. We also regard it as a subset of ℝ^{n}, when this is convenient.

Definition 3.8. For an arbitrary subset Λ of ℝ^{n} and a positive integer m we let

mΛ = {λ_{1} + ⋯ + λ_{m} | λ_{i} ∈ Λ (i = 1, …, m)}.

If Λ = Γ(I) (respectively, Γ(A)) then conv(Λ) will be denoted NP(I) (respectively, NP(A)), and will be referred to as the Newton polyhedron of I (respectively, of A).

A polyhedron may be defined as the intersection of finitely many half-spaces in ℝ^{n} for some positive integer n. A polyhedron can also be thought of as the sum of the convex hull of a finite point set (a polytope) and the positive cone generated by a finite set of vectors. Indeed these are equivalent notions (e.g., see [16, Theorem 1.2]). An affine hyperplane in ℝ^{n} divides ℝ^{n} into two closed half-spaces. If Σ ⊆ ℝ^{n} is a polyhedron and H is an affine hyperplane in ℝ^{n} such that Σ ∩ H ≠ ∅ and Σ is contained in one of these half-spaces, then H is called a supporting hyperplane of Σ. A face of Σ is the intersection of Σ and a supporting hyperplane. If F is a face and dim(aff(F)) = dim(aff(Σ)) − 1 then F is called a facet of Σ.

The Newton polyhedron of a monomial ideal is an example of an unbounded polyhedron. To prepare for the geometric characterization of the integral closure of a monomial ideal we now state and prove Carathéodory’s Theorem.

Theorem 3.9 (Carathéodory’s Theorem). Let V be a finite-dimensional real vector space and let S ⊆ V be a subset. Then, a vector α ∈ V is in conv(S) if and only if there exist affinely independent vectors {α_{1}, …, α_{p}} in S with α ∈ conv(α_{1}, …, α_{p}).

Proof. Say α = ∑_{i=1}^{p} b_{i}α_{i} is a convex linear combination. Without loss of generality we may and shall assume that α_{i} ≠ 0 and b_{i} > 0 for all i. Suppose that {α_{1}, …, α_{p}} is affinely dependent. We proceed by induction on p, the case p = 1 being vacuously true. Since {α_{1}, …, α_{p}} is affinely dependent there exist scalars c_{1}, …, c_{p}, not all zero, such that ∑_{i=1}^{p} c_{i}α_{i} = 0 and ∑_{i=1}^{p} c_{i} = 0. By multiplying by −1 if necessary, we may and shall assume that c_{i} > 0 for some index i. For each index i such that c_{i} > 0 consider c_{i}/b_{i} and choose an index i^{∗} such that c_{i^{∗}}/b_{i^{∗}} ≥ c_{i}/b_{i} for all indices i such that c_{i} > 0. Notice that c_{i^{∗}}/b_{i^{∗}} = max{c_{1}/b_{1}, …, c_{p}/b_{p}}. Since the coefficient c_{i^{∗}} ≠ 0 we have α_{i^{∗}} = − ∑_{i≠i^{∗}}(c_{i}/c_{i^{∗}})α_{i}, where ∑_{i≠i^{∗}}(c_{i}/c_{i^{∗}}) = −1. Hence

α = ∑_{i≠i^{∗}} b_{i}α_{i} − b_{i^{∗}} ∑_{i≠i^{∗}}(c_{i}/c_{i^{∗}})α_{i} = ∑_{i≠i^{∗}} (b_{i} − b_{i^{∗}}(c_{i}/c_{i^{∗}})) α_{i},

where

∑_{i≠i^{∗}} (b_{i} − b_{i^{∗}}(c_{i}/c_{i^{∗}})) = ∑_{i≠i^{∗}} b_{i} − b_{i^{∗}} ∑_{i≠i^{∗}}(c_{i}/c_{i^{∗}}) = (∑_{i≠i^{∗}} b_{i}) + b_{i^{∗}} = 1,

and b_{i} − b_{i^{∗}}(c_{i}/c_{i^{∗}}) ≥ 0 for i ≠ i^{∗}. By induction, ∑_{i≠i^{∗}}(b_{i} − b_{i^{∗}}(c_{i}/c_{i^{∗}}))α_{i} is a convex combination of affinely independent vectors in S. □

We can now state and prove the geometric characterization of the integral closure of a monomial ideal.

Theorem 3.10. (a) Let I be a monomial ideal in R = K[x_{1}, …, x_{n}]. Then the integral closure \overline{I} of I in R is the monomial ideal defined by Γ(\overline{I}) = NP(I) ∩ ℕ^{n} (so that NP(I) = NP(\overline{I})). Furthermore

Γ(\overline{I}) = \{α ∈ ℕ^{n}|mα ∈ mΓ(I) for some m ∈ ℤ_{+}\}.

(b) Let A be a subalgebra of R generated by a finite number of monomials. Then the integral closure \overline{A} of A in R is the semigroup ring defined by Γ(\overline{A}) = NP(A)∩ℕ^{n}. Furthermore NP(A) is the positive cone spanned by Γ(A) (or by the exponents of a (finite) set of algebra generators of A) and

Γ(\overline{A}) = \{α ∈ ℕ^{n}|mα ∈ Γ(A) for some m ∈ ℤ_{+}\}.

Proof. (a) Suppose first that x^{α} ∈ \overline{I}. Thus x^{mα} ∈ I^{m} for some positive integer m by Corollary 3.2. Thus there exist monomials x^{α^{1}}, …, x^{α^{m}} ∈ I such that mα = α_{1} + ⋯ + α_{m}. Hence

α = \frac{1}{m}α_{1} + ⋯ + \frac{1}{m}α_{m} ∈ NP(I) ∩ ℕ^{n}.

Now assume that α ∈ NP(I) ∩ ℕ^{n}. By Carathéodory’s Theorem there exist affinely independent vectors α_{0}, …, α_{m} ∈ Γ(I) such that α ∈ conv(α_{0}, …, α_{m}).

We may assume α = c_{0}α_{0} + ⋯ + c_{m}α_{m}, where the coefficients are positive real numbers whose sum is 1. Then

α − α_{0} = c_{1}(α_{1} − α_{0}) + ⋯ + c_{m}(α_{m} − α_{0}),

has a solution in ℝ^{m} and hence must have a solution in ℚ^{m}. Since the vectors α_{1} − α_{0}, …, α_{m} − α_{0} are linearly independent the solution is unique. Hence the coefficients c_{j} are positive rational numbers. Letting d be a common denominator for the coefficients and writing each c_{j} = a_{j}/d with a_{j} ∈ ℤ_{+} we have dα = a_{0}α_{0} + ⋯ + a_{m}α_{m} and hence x^{dα} ∈ I^{d} since ∑ a_{j} = d.

(b) This is [2, Proposition 6.1.2]. See also [10, 3.1] for a form closer to what we want here. □

With this geometric characterization and the algebraic characterization of Corollary 3.2 in hand, we have a pretty good understanding of the integral closure of a monomial ideal. A vector α ∈ ℕ^{n} is in Γ(\overline{I}) if and only if mα ∈ mΓ(I) for some positive integer m. Thus the monomial ideal I is integrally closed if and only if whenever α ∈ ℕ^{n} and mα ∈ mΓ(I) for some positive integer m, the vector α ∈ Γ(I).

3.2. Normal 𝔪-primary monomial ideals. We now turn our attention to the question of normality for monomial ideals. Recall that an ideal I is normal if all its positive powers are integrally closed. Thus a monomial ideal I in a polynomial ring R = K[x_{1}, …, x_{n}] is normal if and only if the Rees algebra R[It] is a normal domain by Theorem 2.7. Although we have a good characterization of an integrally closed monomial ideal no such characterization exists for a normal monomial ideal, nor does it appear that a general characterization can ever be given. Thus we will focus on a special class of monomial ideals, which we will now introduce. One can check whether a particular monomial ideal is normal using [3].

Notation 3.11. Let λ = (λ_{1}, …, λ_{n}) denote a vector of positive integers. Set J(λ) = (x_{1}^{λ_{1}}, …, x_{n}^{λ_{n}}) and I(λ) = \overline{J(λ)}.

After a general result on the normality of a monomial ideal we turn our attention to the question of normality of the 𝔪-primary ideal I(λ).

We begin by recalling some recent results of Reid, Roberts, and the current author in [12].

Proposition 3.12. [12, Proposition 3.1] Let I ⊆ R = K[x_{1}, …, x_{n}] be a monomial ideal. If I^{m} is integrally closed for m = 1, …, n − 1, then I is normal.

Thus an integrally closed monomial ideal in K[x_{1}, x_{2}] is automatically normal (this is also a consequence of Zariski’s work on complete ideals, which is described in Appendix 5 of [15]). Thus the first case of interest is an integrally closed monomial ideal in K[x_{1}, x_{2}, x_{3}]. An integrally closed monomial ideal I in K[x_{1}, x_{2}, x_{3}] is normal provided that I^{2} is also integrally closed.

Notation 3.13. Let L = lcm(λ_{1}, …, λ_{n}), ω_{i} = L/λ_{i}, 1/λ = (1/λ_{1}, …, 1/λ_{n}) and ω = (ω_{1}, …, ω_{n}), so that L/λ = ω. We will denote Γ(I(λ)) (Definition 3.7) simply by Γ.

Observe that NP(I(λ)) = NP(J(λ)) has one bounded facet with vertices λ_{1}e_{1}, …, λ_{n}e_{n}. For a general vector α = (a_{1}, …, a_{n}) of ℝ^{n} the hyperplane defined by the equation (1/λ) · α = 1 passes through these vertices, and upon multiplication by L, the equation of this hyperplane becomes ω · α = L. We point out that NP(I(λ)^{p}) = {pα | α ∈ NP(I(λ))} also has one bounded facet with vertices pλ_{1}e_{1}, …, pλ_{n}e_{n} and supporting hyperplane ω · α = pL by [12, Lemma 2.5]. For a precise description of the faces of the Newton polyhedron of a general monomial ideal I the reader can consult [11]. These remarks lead to the two lemmas that follow.

Lemma 3.14. [12, Lemma 4.1] The exponent set Γ of the monomial ideal I(λ) can be described by Γ = {α ∈ ℕ^{n} | (1/λ) · α ≥ 1} = {α ∈ ℕ^{n} | ω · α ≥ L}.

The next lemma gives necessary and sufficient conditions for I(λ) to be normal in terms of the exponent set Γ.

Lemma 3.15. [12, Lemma 4.3] For the ideal I(λ) defined above the following are equivalent.

(a) I(λ) is normal.

(b) Whenever ω · α ≥ pL for α ∈ ℕ^{n} and p ∈ ℕ, there exist vectors β_{j} ∈ Γ (j = 1, …, p) such that α = ∑ β_{j}.

(c) For all integers p with 1 ≤ p < n and vectors α = (a_{1}, …, a_{n}) ∈ ℕ^{n} with λ_{i} > a_{i} (i = 1, …, n) satisfying ω · α ≥ pL, there exist vectors β_{j} ∈ Γ (j = 1, …, p) such that α = ∑ β_{j}.

Assigning deg(x_{i}) = ω_{i} (i = 1, …, n) we now have I(λ) = R_{≥ L} := ⨁_{n≥L} R_{n}. Rings of this type have recently been studied by K. Smith and her collaborators (e.g., see [4]). One fruitful consequence of this point of view is the following result.

Corollary 3.16. [12, Corollary 4.4] Let λ = (λ_{1}, …, λ_{n}) ∈ ℤ_{+}^{n}, n ≥ 3, and suppose that gcd(λ_{1}, …, λ_{n}) > n − 2. Then the monomial ideal I(λ) ⊆ K[x_{1}, …, x_{n}] is normal. In particular, if n = 3 and the integers λ_{1}, λ_{2}, λ_{3} are not relatively prime, then the ideal I(λ) is normal.

In [1] Bruns-Gubeladze define a submonoid S of ℚ_{≥} to be 1-normal if whenever x ∈ S and x ≤ p for some p ∈ ℕ, there exist rational numbers y_{1}, …, y_{p} in S with y_{i} ≤ 1 for all i such that x = y_{1} + ⋯ + y_{p}. Then they relate the normality of K[S(λ)] to the 1-normality of the submonoid Λ of ℚ_{≥}. In [12] their program was modified as follows.

Definition 3.17. A submonoid S of ℚ_{≥} is quasinormal provided that whenever x ∈ S and x ≥ p for some p ∈ ℕ, there exist rational numbers y_{1}, …, y_{p} in S with y_{i} ≥ 1 for all i such that x = y_{1} + ⋯ + y_{p}.

We have the following necessary condition for I(λ) to be normal.

Lemma 3.18. [12, Lemma 4.6, Proposition 4.7] Let Λ = ⟨1/λ_{1}, …, 1/λ_{n}⟩, the additive submonoid of ℚ_{≥} generated by 1/λ_{1}, …, 1/λ_{n}. If I(λ) is normal then Λ is quasinormal. Furthermore, if integers λ_{1}, …, λ_{n} are pairwise relatively prime the converse is true.

Thus in the special case where the integers λ_{1}, …, λ_{n} are pairwise relatively prime, the normality condition on the n-dimensional monoid Γ is reduced to the quasinormality condition on the 1-dimensional monoid Λ. Notice that in K[x_{1}, x_{2}, x_{3}] the only case where we do not have a good answer to the question of when I(λ) is normal is when gcd(λ_{1}, λ_{2}, λ_{3}) = 1 and the integers λ_{1}, λ_{2}, λ_{3} are not pairwise relatively prime. This remaining case is being investigated by the author and H. Coughlin.

The next result enables one to quickly conclude that certain monomial ideals I(λ) are not normal and leads to another definition.

Proposition 3.19. [12, Propositions 4.8 & 4.9] Let λ ∈ ℤ_{+}^{n}. If Λ = ⟨1/λ_{1}, …, 1/λ_{n}⟩ is quasinormal, then 1 + 1/L ∈ Λ (and hence, L + 1 ∈ ⟨ω_{1}, …, ω_{n}⟩ ).

This condition leads us to our next definition.

Definition 3.20. We say that the submonoid Λ = ⟨1/λ_{1}, …, 1/λ_{n}⟩ of ℚ_{≥} is almost quasinormal provided that 1+1/L ∈ Λ, equivalently, if L+1 ∈ ⟨ω_{1}, …, ω_{n}⟩.

For the discussion below we need to recall yet another definition.

Definition 3.21. An affine semigroup is a finitely generated monoid that is isomorphic to a submonoid of ℤ^{n}, for some positive integer n.

At first almost quasinormality was viewed primarily as a way to produce examples of integrally closed but not normal monomial ideals. However, it turns out that the condition is closely related to whether the Rees algebra of the ideal satisfies condition R_{1} of Serre. In order to discuss this connection we observe that the Rees algebra R[It] of a monomial ideal I of the polynomial ring R = K[x_{1}, …, x_{n}] can always be identified with an affine semigroup ring over K. Namely, if I = (x^{β^{1}}, …, x^{β^{r}}) and we let S(I) = ⟨(e_{1}, 0), …, (e_{n}, 0), (β_{1}, 1), …, (β_{r}, 1)⟩ ⊆ ℕ^{n+1}, then R[It] ≅ K[S(I)]. If I = I(λ) then S(I) may be described as the submonoid of ℕ^{n+1} generated by

{(a_{1}, …, a_{n}, d) ∈ ℕ^{n+1} | a_{1}/λ_{1} + ⋯ + a_{n}/λ_{n} ≥ d for d ≤ 1}.

Throughout the rest of this section S will denote an affine semigroup, K a field, R = K[S] the affine semigroup ring over K, and C = pos(S) the positive cone generated by S. Each supporting hyperplane H of C is a linear subspance and is the zero set of some integral linear form σ. If the coefficients of σ are relatively prime integers we refer to σ as the primitive linear form defining H; this form is unique up to a factor of −1. Let F be a facet of C and σ_{F} the corresponding primitive linear form. Let H_{F} = {α ∈ ℝS | σ_{F} (α) = 0}. For a facet F of C let P_{F} denote the height one monomial prime generated by the set of monomials {x^{β} | β ∈ S \ F } = {x^{β} | β ∈ S, σ_{F} (β) > 0}. For definitions and details the reader should consult [2, Chapter 6].

The following two results were communicated to the author by W. Bruns. Lemma 3.22. Let S be an affine semigroup and F be a facet of the positive

cone C of S such that R_{P} is regular, where P = P_{F}. Then we have:

i. there exists an element β ∈ S \ F such that σ_{F} (β) = 1; and ii. grp(S ∩ F) = grp(S) ∩ H_{F}.

Proof. Let P = P_{F}. By our assumption R_{P} is a discrete rank one valuation ring and P R_{P} is generated by a single monomial x^{β}. Let v be the associated valuation. Then, 1 = v(x^{β}) = σ_{F} (β). Now let α, α′ ∈ S be such that α − α′ ∈ F. Then we must have α − α′ ∈ grp(S) ∩ H_{F}. Conversely assume that α, α′ ∈ S and suppose that α − α′ ∈ H_{F}. Then let m = σ_{F} (α) = σ_{F} (α′). If m = 0 then we have α, α′ ∈ S ∩ F and hence α − α′ ∈ grp(S ∩ F). Suppose that m > 0. Then x^{α}/x^{mβ} and x^{mβ}/x^{α′} are invertible monomials in R_{P} since P R_{P} = x^{β}R_{P}. Hence α − α′ = (α − mβ) + (mβ − α′) ∈ grp(S ∩ F) as asserted. □

Proposition 3.23. The monoid ring K[S] of an affine semigroup S satisfies condition R_{1} of Serre iff the following two conditions hold for every facet F of the positive cone C of S:

i. there exists an element β ∈ S such that σ_{F} (β) = 1; and

ii. grp(S ∩ F) = grp(S) ∩ H_{F}.

Proof. First assume that R = K[S] satisfies condition R_{1} of Serre. Let P = P_{F} be a height one prime corresponding to the facet F of C. By Lemma 3.22 conditions (i)-(ii) are satisfied.

To prove the converse, assume that conditions (i) - (ii) hold. Let us first notice that R satisfies condition R_{1} if and only R_{P} is regular for the the prime ideals P = P_{F} corresponding to facets F of C. In fact, if Q is a height 1 prime ideal different from all the P_{F}, it cannot contain a monomial (the minimal primes of a monomial ideal are generated by monomials and therefore among the P_{F}). Suppose S = ⟨β_{1}, …, β_{r}⟩ and let β = β_{1}+⋯+β_{r} and f = x^{β}. Then R[f^{−1}] = K[S, f^{−1}] = K[grp(S)] as follows. If α = m_{1}β_{1} +⋯+m_{r}β_{r} ∈ S then m(β_{1} +⋯+β_{r})−α ∈ S, where m = max{m_{i}}. Thus −α ∈ ⟨S, −β⟩. So R_{Q} is a localization of K[grp(S)] which is Laurent polynomial ring, and therefore regular.

Now suppose that P is a height one prime of R containing a monomial x^{α}. Then P is a minimal overprime of x^{α} and hence is a height one monomial prime. Thus P = P_{F} = (x^{α} | α ∈ S \ F) for some facet F of C. We wish to see that R_{P} is regular. By condition (i) there is a vector γ ∈ S \ F with σ_{F} (γ) = 1. We claim that P R_{P} = (x^{γ}). For suppose that x^{α} ∈ P R_{P}. Then α ∈ S \ F and hence σ_{F} (α) = m > 0. Thus α − mγ ∈ grp(S) ∩ H_{F} = grp(S ∩ F) by condition (ii). So we have x^{α} = (x^{α}/x^{mγ}) x^{mγ} ∈ x^{γ}R_{P} as desired. □

We now wish to see what this means for the Rees algebra of an ideal I = I(λ). As we alluded to in the preceding paragraphs the counterpart of the Newton polyhedron of a monomial ideal in the convex-geometric description of the affine semigroup ring K[S(I)] is the positive cone pos(S) that is associated with the monoid S = S(I). The facets of NP(I) are cut out by the supporting hyperplanes H_{ω,L}, H_{1}, …, H_{n} where H_{ω,L} = {α ∈ ℝ^{n} | ω · α = L} and H_{i} is the coordinate hyperplane H_{i} = {α ∈ ℝ^{n} | e_{i} · α = 0}. The facets of C = pos(S) are cut out by the supporting hyperplanes H_{σ}, H_{1}, …, H_{n+1} where σ(α, a_{n+1}) = ω · α − La_{n+1} and H_{1}, …, H_{n+1} are the coordinate hyperplanes in ℝ^{n+1}. Since the height one monomial primes in R[It] and K[S(I)] can be identified we have the following description of the height one monomial primes of R[It].

Lemma 3.24. For a monomial ideal I = I(λ) the height one monomial primes of R[It] are as follows:

P_{i} = (x_{i}) + (x^{β^{j}}t | e_{i} ≤_{pr} β_{j}) for (i = 1, …, n);
P_{n+1} = (x^{β^{1}}t, …, x^{β^{r}}t); and
P_{σ} = (x_{1}, …, x_{n}) + (x^{β^{j}}t | σ(β_{j}, 1) > 0).

We can now characterize which vectors λ lead to Rees algebras that satisfy condition R_{1} of Serre. First we show that condition (ii) of 3.23 is always satisfied by an affine semigroup of the form S(I(λ)).

Lemma 3.25. Let λ ∈ ℤ_{+}^{n}. I = I(λ), and let S = S(I) ⊆ ℕ^{n+1}. Then,

grp(S ∩ H_{σ}) = grp(S) ∩ H_{σ}.

Proof. Since (e_{1}, 0), …, (e_{n}, 0), (λ_{1}e_{1}, 1) are in S the vectors (e_{1}, 0), …, (e_{n}, 0), (0, 1) are in grp(S) and hence grp(S) = ℤ^{n+1}.

It is clear that grp(S ∩ H_{σ}) ⊆ grp(S) ∩ H_{σ}. To prove the opposite containment suppose that (α, a_{n+1}) ∈ ℤ^{n+1} is on the hyperplane H_{σ}, i.e., σ(α, a_{n+1}) = 0, where α = (a_{1}, …, a_{n}). Notice that S ∩ H_{σ} is generated by those generators (β_{i}, 1) of S for which σ(β_{i}, 1) = 0. Also observe that {(λ_{1}e_{1}, 1), …, (λ_{n}e_{n}, 1)} ⊆ grp(S ∩ H_{σ}). Write a_{i} = q_{i}λ_{i} + r_{i} in ℤ with 0 ≤ r_{i} < λ_{i} for i = 1, …, n. Then (α, a_{n+1}) − ∑_{i=1}^{n} q_{i}(λ_{i}e_{i}, 1) = (r_{1}, …, r_{n}, d) where d = a_{n+1} − ∑_{i=1}^{n} q_{i}. Thus it suffices to show that (r_{1}, …, r_{n}, d) ∈ grp(S ∩H_{σ}). But (r_{1}, …, r_{n}, d) is necessarily in S and σ(r_{1}, …, r_{n}, d) = 0 implies (r_{1}, …, r_{n}, d) = (β_{i(1)}, 1) + ⋯ + (β_{i(d)}, 1) where (β_{i(j)}, 1) ∈ S ∩ H_{σ} (j = 1, …, d); no generator of the form (e_{i}, 0) can occur since one such summand would force the entire sum to have positive value under σ. Thus (r_{1}, …, r_{n}, d) ∈ grp(S ∩ H_{σ}) as desired. □

With this lemma in hand we can prove the connection between the conditions of quasinormality and regularity in codimension one.

Proposition 3.26. For a monomial ideal I = I(λ) of K[x_{1}, …, x_{n}] the Rees algebra R := R[It] satisfies condition R_{1} of Serre if and only if the additive semigroup Λ = ⟨1/λ_{1}, …, 1/λ_{n}⟩ is almost quasinormal.

Proof. We first show that R_{P_{i}} is a 1-dimensional regular local ring for i = 1, …, n + 1 by showing that the unique maximal ideal is principal.

Let S = S(λ). Suppose that 1 ≤ i ≤ n. We claim that x_{i} generates the unique maximal ideal of R_{P_{i}}. Consider β_{j}, where e_{i} ≤_{pr} β_{j}. Write β_{j} = e_{i} + α, where α ∈ ℕ^{n} and choose m ∈ {1, …, î, …, n}. Then

x^{β^{j}}t = (x^{α}t)x_{i} = \frac{x^{α}}{x_{m}^{λ_{m}}} (x_{m}^{λ_{m}}t)x_{i} ∈ x_{i}R_{P_{i}}.

Hence P_{i}R_{P_{i}} = x_{i}R_{P_{i}}. Notice that for the linear form σ_{i}(a_{1}, …, a_{n+1}) = a_{i} cutting out the facet F_{i} = S ∩ H_{i} we have σ_{i}(e_{i}, 0) = 1.

Now consider P := P_{n+1} = (x^{β^{1}}t, …, x^{β^{r}}t). We claim that P R_{P} = x^{β^{1}}tR_{P}. This follows since

x^{β^{j}}t = \frac{x^{β^{j}}}{x^{β^{1}}} x^{β^{1}}t ∈ x^{β^{1}}tR_{P} .

Notice that for the linear form σ_{n+1}(a_{1}, …, a_{n+1}) = a_{n+1} cutting out the facet F_{n+1} = pos(S) ∩ H_{n+1} we have σ_{n+1}(β_{1}, 1) = 1. Notice that P R_{P} = x^{β^{i}}tR_{P} for any i between 1 and r by the same argument.

Thus by Proposition 3.23 we know that R satisfies condition R_{1} of Serre if and only R_{P_{σ}} is regular. By Lemma 3.25 this is the case if and only if there exists a generator of S with σ-value equal to 1; the generator is either of the form (e_{i}, 0) where ω·e_{i} = ω_{i} = 1 or (β_{j}, 1) where ω·β_{j} −L = 1. To finish the proof observe that Λ is almost quasinormal if and only if there exists α ∈ ℕ^{n} with ω · α = L + 1. This happens if and only if for some β_{j} and some γ ∈ ℕ^{n} we have σ(β_{j}, 1)+σ(γ, 0) = 1; notice that γ is either 0 or e_{i} for some i. Thus Λ is almost quasinormal if and only if S has a generator with σ-value equal to 1. □

In closing we mention one last remarkable result from [12]. Our notation continues as usual: R = K[x_{1}, …, x_{n}] for a field K, λ = (λ_{1}, …, λ_{n}) for arbitrary positive integers λ_{j}, and L = lcm(λ_{1}, …, λ_{n}). Define λ′ = (λ_{1}, …, λ_{i−1}, λ_{i} + ℓ, λ_{i+1} …, λ_{n}), where ℓ = lcm(λ_{1}, …, λ̂_{i}, …, λ_{n}). There is no loss of generality in taking i = n, so that ℓ = lcm(λ_{1}, …, λ_{n−1}) and λ′ = (λ_{1}, …, λ_{n−1}, λ_{n} + ℓ). We now state the result.

Theorem 3.27. [12, Theorem 5.1] If I(λ′) is normal then I(λ) is normal. If λ_{n} ≥ ℓ and I(λ) is normal so is I(λ′).

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Department of Mathematics, 1222 University of Oregon, Eugene, OR 97403-1222

E-mail address: vitulli@math.uoregon.edu