The Weak Subintegral Closure of a Monomial Ideal

Les Reid Southwest Missouri University Springfield, MO 65804

Marie A. Vitulli University of Oregon Eugene, OR 97403

March 13, 1998

Abstract

For a monomial ideal I of a polynomial ring A, we give both explicit algebraic and geometric characterizations of the weak subintegral closure of I in A. The geometric characterization enables us to compute examples for monomial ideals in polynomial rings in a small number of indeterminates. In particular, we show that the seminormalization of the Rees ring of an ideal can not, in general, be described in terms of the (2,3)-closure construction of Oda and Yoshida.

1 Introduction

In a recent paper [6] the second author and J.V. Leahy introduced the notion of the weak subintegral closure of an ideal. Their definition was motivated by an element-wise definition of subintegrality that was introduced by L. Roberts and B. Singh in [13] and developed by the first author, L. Roberts, and B. Singh in [10], [9], [11], and [12]. One useful consequence of the notion of the weak subintegral closure of an ideal is that the direct sum of the weak subintegral closures of the powers of an ideal I in a ring A describes the weak normalization of the Rees ring A[It]. In this paper we give algebraic and geometric characterizations of the weak subintegral closure of a monomial ideal in a polynomial ring over a field of characteristic 0; in the characteristic p case an algebraic characterization is given. These characterizations enable one to compute examples in polynomial rings in a small number of indeterminates. We give one such example that shows that the (2,3)-closure of a monomial ideal in a polynomial ring over a field of characteristic 0 can be properly contained in the weak subintegral closure of the ideal. Hence the direct sum of the (2,3)-closures of the powers of the ideal I in the polynomial ring A does not give the seminormalization of the Rees ring A[It].

2 Preliminaries

Conventions. All rings are assumed to be commutative with identity. If a set S is properly contained in a set T we write S ⊂ T. We let N denote the set of nonnegative integers, R_{+} the set of nonnegative real numbers, and e_{1}, …, e_{n} the standard basis vectors in R^{n}. For a ring R we let J(R) denote the Jacobson radical of R. For a prime ideal P of a ring R we let κ(P) denote the quotient field of the integral domain R/P. By a totally ordered abelian monoid we mean a submonoid of a totally ordered abelian group.

Swan [15] defined an extension R ⊆ S of commutative rings to be subintegral if S is integral over R and the induced map Spec(S) → Spec(R) is a bijection that induces isomorphisms on the residue field extensions. Yanagihara [17] defined an extension R ⊆ S of commutative rings to be weakly subintegral if S is integral over R and the induced map Spec(S) → Spec(R) is a bijection that makes the residue field extensions purely inseparable.

Recall that an elementary subintegral extension of rings is a ring extension of the form R ⊆ R[s] where s^{2}, s^{3} ∈ R. An elementary weakly subintegral extension of rings is an extension of the form R ⊆ R[s] where s^{p}, ps ∈ R for some rational prime p.

Consider an arbitrary extension of rings R ⊆ S and let \overline{R} denote the integral closure of R in S. Traverso [16] showed that the ring _{S}^{+}R defined by

_{S}^{+}R = \{s ∈ \overline{R} | ∀ P ∈ Spec(R), \frac{s}{1} ∈ R_{P} + J( \overline{R}_{P})\}

is the largest subintegral extension of R in S and called it the seminormalization of R in S. One says that R is seminormal in S if R = _{S}^{+}R. Hamann [2] showed that R is seminormal in S if and only if R contains each element of S whose square and cube are in R.

Andreotti and Bombieri [1] defined the ring _{S}^{∗}R by

_{S}^{∗}R = \{s ∈ \overline{R} | ∀ P ∈ Spec(R), ∃ n ∈ N such that (\frac{s}{1})^{e^{n}} ∈ R_{P} + J( \overline{R}_{P})\},

where e = 1 if char(κ(P)) = 0 and e = p if char(κ(P)) = p > 0. Andreotti and Bombieri showed that _{S}^{∗}R is the largest weakly subintegral extension of R in S and called it the weak normalization of R in S.

Let R be a reduced ring and assume the total quotient ring K of R is a product of fields. Then _{K}^{+}R (respectively, _{K}^{∗}R) is denoted simply by^{+}R (respectively, ∗R) and is called the seminormalization (respectively, weak normalization) of R.

We now present several results on the seminormalization and weak normalization of a graded ring in a graded extension ring. From these we deduce that the weak subintegral closure of a homogeneous ideal in a graded ring is again homogeneous. We begin with a lemma that characterizes the seminormalization _{S}^{+}R of R in S as the unique subintegral extension of R that is seminormal in S, which is similar to Traverso’s characterization of _{S}^{+}R as the largest subintegral extension of R in S [16] and follows from the fact that _{S}^{+}R is the union of all subrings of S that can be obtained from R by a finite sequence of elementary subintegral extensions. The proof of the latter result is due to R. Swan [15, Theorem 2.8]. Subsequently, we state the analogous characterization of the weak normalization _{S}^{∗}R of R in S.

Lemma 2.1 Let R ⊆ S be an arbitrary extension of rings.

(a) _{S}^{+}R is the filtered union of all subrings of S that can be obtained from R by a finite sequence of elementary subintegral extensions.

(b) _{S}^{+}R is the unique subintegral extension of R that is seminormal in S.

Proof. (b) Suppose that T is a subintegral extension of R that is contained in S and that T is seminormal in S. Since R ⊆ T is a subintegral extension, T ⊆ _{S}^{+}R by Traverso’s characterization of the seminormalization. Let R′ be the filtered union of all subrings of S that can be obtained from R by a finite sequence of elementary subintegral extensions. Since T is seminormal in S, if A ⊆ T and A′ is an elementary subintegral extension of A in S, then A′ ⊆ T by Hamann’s criterion for seminormality [2]. Consequently, R′ is contained in T. Since R′ = _{S}^{+}R by part (a), we must have _{S}^{+}R = T, as asserted.

The proof the the corresponding result for weak normalization stated below follows in a similar manner. The statement and proof of part (a) is due to Yanagihara [17, Theorem 2].

Lemma 2.2 Let R ⊆ S be an arbitrary extension of rings.

(a) _{S}^{∗}R is the filtered union of all subrings of S that can be obtained from R by a finite sequence of elementary subintegral or elementary weakly subintegral extensions.

(b) _{S}^{∗}R is the unique weakly subintegral extension of R that is weakly subintegral in S.

The next lemma is a generalization of a result in [5, Proposition 2.4] It is convenient to establish some notation before stating and proving this result.

Notation 2.3 Let H be any totally ordered abelian monoid and let R ⊆ S be an extension of H−graded rings. For an element s ∈ S let l(s) denote the number of nonzero homogeneous components of s. For each element s = s_{h(1)} + ⋯ + s_{h(l)} of S with 0 ≠ s_{h(i)} ∈ S_{h(i)}, l = l(s), and h(1) < ⋯ < h(l), let in(s) = s_{h(1)} and

k(s) = max{i |s_{h(1)}, …, s_{h(i−1)} ∈ R}.

We refer to in(s) as the initial component of s. Notice that k(s) = 1 if and only if the initial component s_{h(1)} of s is not in R.

Lemma 2.4 Let H be any totally ordered abelian monoid and let R ⊆ S be an extension of H−graded rings. Then, R is seminormal in S if and only if R contains each homogeneous element s ∈ S such that s^{2}, s^{3} ∈ R.

Proof. Assume R ⊆ S satisfies the above criterion but R is not seminormal in the overring S.

Among those elements s ∈ S \ R such that s^{2}, s^{3} ∈ R choose one such that l(s) is minimal and if l(t) = l(s), then k(t) ≤ k(s). Say s = s_{h(1)} + ⋯ + s_{h(l)} with l = l(s). Since in(s^{2}) = s_{h(1)}^{2} and in(s^{3}) = s_{h(1)}^{3}, we must have s_{h(1)} ∈ R. Thus k := k(s) ≥ 2.

Now (2s_{h(1)}s)^{2}, (2s_{h(1)}s)^{3} ∈ R and l(2s_{h(1)}s) ≤ l(s). In the case that l(2s_{h(1)}s) < l(s) we must have 2s_{h(1)}s ∈ R by the minimality of l(s). Suppose l(2s_{h(1)}s) = l(s), that is, 2s_{h(1)}s_{h(i)} ≠ 0 for 1 ≤ i ≤ l. Since the (h(1) + h(k))−component of s^{2} is 2s_{h(1)}s_{h(k)} + terms in R and s^{2} ∈ R we may deduce that 2s_{h(1)}s_{h(k)} ∈ R. So under the assumption that l(2s_{h(1)}s) = l(s) we must have k(2s_{h(1)}s) > k(s) and by the maximality of k(s) we may conclude that 2s_{h(1)}s ∈ R. In either case, 2s_{h(1)}s ∈ R.

We have that (3s_{h(1)}^{2}s)^{2}, (3s_{h(1)}^{2}s)^{3} ∈ R and l(3s_{h(1)}^{2}s) ≤ l(s). In the case that l(3s_{h(1)}^{2}s) < l(s) we must have 3s_{h(1)}^{2}s ∈ R by the minimality of l(s). Suppose l(3s_{h(1)}^{2}s) = l(s), that is, 3s_{h(1)}^{2}s_{h(i)} ≠ 0 for 1 ≤ i ≤ l. Since the (2h(1) + h(k))−component of s^{3} is 3s_{h(1)}^{2}s_{h(k)} + terms in R and s^{3} ∈ R, we may conclude that 3s_{h(1)}^{2}s_{h(k)} ∈ R. Since l(3s_{h(1)}^{2}s) = l(s), we must have k(3s_{h(1)}^{2}s) > k(s). By the maximality of k(s) we may deduce that 3s_{h(1)}^{2}s ∈ R. In either case, 3s_{h(1)}^{2}s ∈ R.

Consider s − s_{h(1)} ∈ S \ R. Now

(s − s_{h(1)})^{2} = s^{2} − 2s_{h(1)}s + s_{h(1)}^{2} ∈ R

and

(s − s_{h(1)})^{3} = s^{3} − 3s^{2}s_{h(1)} + 3ss_{h(1)}^{2} − s_{h(1)}^{3} ∈ R.

Since l(s − s_{h(1)}) < l(s), this is a contradiction.

Corollary 2.5 Let H be a totally ordered abelian monoid and let R ⊆ S be an extension of H−graded rings. Then, the seminormalization of R in S is again H−graded.

Proof. Let R′ be the filtered union of all subrings of S that can be obtained from R by a finite sequence of extensions of the form T ⊂ T [s] where T is an H−graded subring of S, s ∈ S is homogeneous, and s^{2}, s^{3} ∈ T. Then, R′ is an H−graded subring of S containing R and R′ ⊆ _{S}^{+}R by Lemma 2.1. In particular, R ⊆ R′ is a subintegral extension.

By the preceding lemma, R′ is seminormal in S. Since _{S}^{+}R is the unique subintegral extension of R that is seminormal in S, R′ = _{S}^{+}R.

We now present the corresponding statements for the weak normalization.

Lemma 2.6 Let H be a totally ordered abelian monoid and R ⊆ S an extension of H−graded rings. Then, R is weakly normal in S if and only if R contains each homogeneous element s of S such that either s^{2}, s^{3} ∈ R or s^{p}, ps ∈ R for some rational prime p.

Proof. The proof of [6, Proposition 3.1] holds for H−graded rings as well as for N-graded rings.

Corollary 2.7 Let H be a totally ordered abelian monoid and R ⊆ S an extension of H−graded rings. The weak normalization of R in S is again an H−graded subring of S.

Proof. Let R′ denote the filtered union of all subrings of S that can be obtained from R by a finite sequence of extensions of the form T ⊂ T [s] where T is an H−graded subring of S, s ∈ S is homogeneous, and either s^{2}, s^{3} ∈ T or s^{p}, ps ∈ T for some rational prime p. By the preceding lemma, R′ is weakly normal in S. Then R′ ⊆ _{S}^{∗}R and hence R ⊆ R′ is a weakly subintegral extension. Since R′ is weakly normal in S we must have R′ = _{S}^{∗}R by Lemma 2.2.

We now recall the definition of the weak subintegral closure of an ideal introduced by the second author and J.V. Leahy in [6]. One should consult their original paper for several equivalent characterizations of the weak subintegral closure of an ideal.

Definition 2.8 Let A ⊆ B be an extension of rings. For an ideal I of A we say b ∈ B is weakly subintegral over I if there exists an integer q ∈ N and elements a_{i} ∈ I^{i} (1 ≤ i ≤ 2q + 1) such that

b^{n} + ∑_{i=1}^{n} \binom{n}{i}a_{i}b^{n−i} = 0 (q + 1 ≤ n ≤ 2q + 1).

We let _{B}^{∗}I denote the set of all elements of B that are weakly subintegral over I. We call _{B}^{∗}I the weak subintegral closure of I in B. We define the weak subintegral closure of I to be the weak subintegral closure of I in A and denote this by ^{∗}I. We say I is weakly subintegrally closed in B if I is equal to _{B}^{∗}I. We say I is weakly subintegrally closed if I is equal to ^{∗}I.

One useful consequence of the notion of the weak subintegral closure of an ideal is that the direct sum of the weak subintegral closures of the powers of an ideal I in a ring A describes the weak normalization of the Rees ring A[It]. We present the precise statement now. For a proof one should consult [6, Theorem 3.3].

Theorem 2.9 Let A ⊆ B be an extension of rings, I an ideal of A, let R denote the Rees ring A[It], where t is an indeterminate, and let S = B[t]. Then,

_{S}^{∗}R = ⊕_{n≥0} _{B}^{∗}(I^{n})t^{n}.

In particular, _{B}^{∗}(I^{n}) contains each element of B that is weakly subintegral over _{B}^{∗}(I^{n}), for all n ≥ 0.

In the graded case we have the following result.

Corollary 2.10 Let H be a totally ordered abelian monoid, let A ⊆ B be an extension of H−graded rings, and I be a homogeneous ideal of A. Then, the weak subintegral closure of I in B is a homogeneous ideal of _{B}^{∗}A.

Proof. By Corollary 2.7 we know that _{B}^{∗}A is an H−graded subring of B. Let R := A[It] ⊆ S := B[t]. Consider

B[t] = ⊕_{(h,n)}B_{h}t^{n}

as an H × N− graded ring where we order H × N lexicographically. Now

_{S}^{∗}R = ⊕_{n≥0} _{B}^{∗}(I^{n})t^{n}

is an H × N−graded subring of S by Corollary 2.7. This implies that _{B}^{∗}I^{n} is an H−graded ideal of _{B}^{∗}A, that is, a homogeneous ideal of _{B}^{∗}A.

Notation 2.11 Let A = k[x_{1}, …, x_{n}] denote the polyonomial ring in n indeterminates over the field k. For α = (a_{1}, …, a_{n}) ∈ N^{n}, let x^{α} denote x_{1}^{a_{1}}x_{2}^{a_{2}} ⋯ x_{n}^{a_{n}}. Then, A is a N^{n}−graded ring where A_{α} = kx^{α} and we order N^{n} lexicographically. When we refer to A as a N^{n}−graded ring we mean with respect to this ordering.

Corollary 2.12 Let I be a monomial ideal in a polynomial ring k[x_{1}, …, x_{n}] in n indeterminates over a field k. Then, the weak subintegral closure of I is again a monomial ideal.

Proof. Let A = k[x_{1}, …, x_{n}]. Consider A as a N^{n}−graded ring as above. Consider the extension R := A[It] ⊆ S := A[t] of N^{n} × N−graded rings where we order N^{n} ×N lexicographically. By the preceding corollary, we know that ^{∗}I is a N^{n}−graded ideal of A. This is equivalent to saying that ^{∗}I is a monomial ideal.

3 The Weak Subintegral Closure of a Monomial Ideal: An Algebraic Description

The knowledge that the weak subintegral closure of a monomial ideal is again a monomial ideal leads to an algebraic characterization of the weak subintegral closure of a monomial ideal that is given in Proposition 3.3 to follow. First we need a couple of technical lemmas.

Lemma 3.1 Let f_{0}(t), f_{1}(t), …, f_{q}(t) be polynomials in Q[t] of degrees 0, 1, …, q, respectively, and let a_{j} denote the leading coefficient of f_{j} for j = 0, 1, …, q. Let Ω(t, q) denote the (q +1)×(q +1) matrix with (i, j) entry f_{j}(it) for 0 ≤ i, j ≤ q. Then, |Ω(t, q)| = a_{0}a_{1} ⋯ a_{q}1!2! ⋯ q!t^{\binom{q+1}{2}}.

Proof. We have

Ω(t, q) = [ \begin{matrix}a_{0} & a_{1}(0t) + b_{1} & ⋯ & a_{q}(0t)^{q} + ⋯ + w_{q} \\ a_{0} & a_{1}(1t) + b_{1} & ⋯ & a_{q}(1t)^{q} + ⋯ + w_{q} \\ ⋮ & ⋮ & & ⋮ \\ a_{0} & a_{1}(qt) + b_{1} & ⋯ & a_{q}(qt)^{q} + ⋯ + w_{q} \end{matrix} ] .

Thus we have

|Ω(t, q)| = a_{0} | \begin{matrix}1 & a_{1}(0t) + b_{1} & ⋯ \\ 1 & a_{1}(1t) + b_{1} & ⋯ \\ ⋮ & ⋮ \\ 1 & a_{1}(qt) + b_{1} & ⋯ \end{matrix} | = a_{0} | \begin{matrix}1 & a_{1}(0t) & ⋯ \\ 1 & a_{1}(1t) & ⋯ \\ ⋮ & ⋮ \\ 1 & a_{1}(qt) & ⋯ \end{matrix} | = a_{0}(a_{1}t) | \begin{matrix}1 & 0 & ⋯ \\ 1 & 1 & ⋯ \\ ⋮ & ⋮ \\ 1 & q & ⋯ \end{matrix} | .

These equalities were obtained by first subtracting b_{1} times the first column from the second column and subsequently factoring out a_{1}t from the new second column. The third column of the last matrix occurring in this string has entries

a_{2}(it)^{2} + b_{2}(it) + c_{2} (i = 0, …, q).

Subtracting b_{2}tcol2+c_{3}col1 from col3 we are left with a_{2}i^{2}t^{2} in the (i, 3) position. Factoring out a_{2}t^{2} from column 3 we are left with

|Ω(t, q)| = a_{0}(a_{1}t)(a_{2}t^{2}) | \begin{matrix}1 & 0 & 0^{2} & f_{3}(0t) & ⋯ \\ 1 & 1 & 1^{2} & f_{3}(1t) & ⋯ \\ ⋮ & ⋮ & ⋮ & ⋮ \\ 1 & q & q^{2} & f_{3}(qt) & ⋯ \end{matrix} | .

Continuing in this manner we deduce that

|Ω(t, q)| = a_{0}a_{1} ⋯ a_{q}t^{0}t^{1} ⋯ t^{q}|V |,

where V is the Vandermonde matrix (i^{j}) with 0 ≤ i, j ≤ q. Since

| \begin{matrix}1 & b_{0} & ⋯ & b_{0}^{n} \\ 1 & b_{1} & ⋯ & b_{1}^{n} \\ ⋮ & ⋮ & & ⋮ \\ 1 & b_{n} & ⋯ & b_{n}^{n} \end{matrix} | = ∏_{i<j}(b_{j} − b_{i}),

we have

|V | = ∏_{0≤i<1}(1 − i) ∏_{0≤i<2}(2 − i) ⋯ ∏_{0≤i<q}(q − i) ⋯ = 1!2! ⋯ q!.

Hence |Ω(t, q)| = a_{0}a_{1} ⋯ a_{q}1!2! ⋯ q!t^{\binom{q+1}{2}}.

Corollary 3.2 Let Ω(t, q) be the (q + 1) × (q + 1) matrix over Q[t] with (i, j) entry \binom{it+1}{j} for 0 ≤ i, j ≤ q. Then, det(Ω(t, q)) = t^{\binom{q+1}{2}}.

Proof. Let f_{j}(t) = \binom{t+1}{j} = \frac{(t+1)t···(t+1−j+1)}{j!} for 0 ≤ j ≤ q. Then, Ω(t, q) = (f_{j}(it)). The leading coefficient of f_{j}(t) is \frac{1}{j!}. Thus

det(Ω(t, q)) = \frac{1}{0!} \frac{1}{1!} ⋯ \frac{1}{q!}1!2! ⋯ q!t^{\binom{q+1}{2}} = t^{\binom{q+1}{2}},

by the preceding lemma.

Proposition 3.3 Let I be a monomial ideal in a polynomial ring k[x_{1}, …, x_{n}] in n indeterminates over a field k.

(a) Ifchar(k) = 0, then ^{∗}I is the monomial ideal generated by all monomials x^{γ} such that x^{mγ} ∈ I^{m} for all sufficiently large positive integers m.

(b) Ifchar(k) = p > 0, then^{∗}I is the monomial ideal generated by all monomials x^{γ} such that x^{p^{m}γ} ∈ I^{p^{m}} for some nonnegative integer m.

Proof. By Corollary 2.10, ^{∗}I is again a monomial ideal and hence is generated by monomials. Both assertions follow from the characterizations of those monomials that lie in ^{∗}I.

(a) First assume that char(k) = 0. Suppose that there exists q ∈ N such that x^{mγ} ∈ I^{m} for all m > q. Define a_{1} = ⋯ = a_{q} = 0, a_{q+1} = −x^{(q+1)γ} ∈ I^{q+1}. Suppose that the elements a_{q+1}, ⋯, a_{r} have been defined recursively so that a_{j} = e_{j}x^{jγ} ∈ I^{j} for all j ≥ 1, where e_{j} ∈ k and

x^{nγ} + ∑_{i=1}^{n} \binom{n}{i}a_{i}x^{(n−i)γ} = 0 (q + 1 ≤ n ≤ r).

We want

x^{(r+1)γ} + ∑_{i=1}^{r} \binom{r + 1}{i}a_{i}x^{(r+1−i)γ} + a_{r+1} = 0.

Just define

a_{r+1} = − (1 + ∑_{i=1}^{r} \binom{r + 1}{i}e_{i}) x^{(r+1)γ}.

Thus x^{γ} ∈ ^{∗}I. Conversely assume a monomial x^{γ} ∈ ^{∗}I. Then, x^{γ} is integral over I. By the characterization of the integral closure of a monomial ideal [3, Exercise 4.23] there exists a positive integer M such that x^{Mγ} ∈ I^{M}. Just suppose there does not exist a positive integer l such that x^{mγ} ∈ I^{m} for all m ≥ l. Then, x^{iM+1} ∉ I^{iM+1} for all i ≥ 0, since the semigroup generated by M and iM + 1 contains {l, l + 1, …} for all sufficiently large positive integers l. Now x^{γ} ∈ ^{∗}I implies that there exists a nonnegative integer q and c_{1}, …, c_{q} ∈ A such that

x^{nγ} + ∑_{i=1}^{q} \binom{n}{i}c_{i}x^{(n−i)γ} ∈ I^{n} (n ≥ 1).

Let e_{i} denote the coefficient of x^{iγ} in c_{i} (i = 1, …, q). Then,

(1 + ∑_{i=1}^{q} \binom{n}{i}e_{i}) x^{nγ} ∈ I^{n} (n ≥ 1),

since I is a monomial ideal. Letting e_{0} = 1 we must have

∑_{i=0}^{q} \binom{n}{i}e_{i} = 0 whenever x^{nγ} ∉ I^{n}.

Letting n = 0M + 1, 1M + 1, …, qM + 1 we get a system of q + 1 homogeneous equations in e_{0}, …, e_{q} with coefficient matrix

Ω(M, q) = \binom{iM + 1}{j}_{0≤i,j≤q}.

By the preceding corollary we have

det(Ω(M, q)) = M^{\binom{q+1}{2}} ≠ 0.

Thus there is no nonzero solution to Ω(M, q)x = 0, contradicting the existence of e_{0}, e_{1}, …, e_{q}.

(b) Now assume that char(k) = p > 0. By [12, Theorem 4.3], x^{γ}t is weakly subintegral over A[It] if and only if (x^{γ}t)^{p^{m}} ∈ A[It] for some m ≥ 0. Thus by (2.9) x^{γ} ∈ ^{∗}I if and only if x^{p^{m}γ} ∈ I^{p^{m}} for some m ≥ 0.

4 The Weak Subintegral Closure of a Monomial Ideal: A Geometric Description

In order to calculate the integral closure of a monomial ideal in a polynomial ring in a small number of indeterminates one uses the geometric description of the integral closure of the ideal that involves the convex hull of a certain subset of N^{n} that is associated with the monomial ideal. This set consists of all exponents of monomials in the ideal. In this section we will present a similar description of the weak subintegral closure of a monomial ideal in a polynomial ring over a field of characteristic 0. Once this characterization is established we will be able to calculate some examples and show that over a field of characteristic zero the (2,3)-closure of a monomial ideal as defined by Oda and Yoshida [7] may be a proper subset of the weak subintegral closure of the ideal. In particular, the graded ring formed by taking the direct sum of the (2,3)-closures of powers of the ideal is not, in general, the seminormalization of the Rees ring of the ideal. Before we can describe the geometric characterization of the subintegral closure of a monomial ideal we must introduce some additional notation and preliminary results.

Notation 4.1 For an arbitrary subset Γ of N^{n} we let

m · Γ = {γ_{1} + ⋯ + γ_{m} |γ_{i} ∈ Γ (i = 1, …, m)}.

To a monomial ideal I in a polynomial ring k[x_{1}, …, x_{n}] over a field k we associate various subsets of N^{n} as follows. We let

Γ(I) = {γ ∈ N^{n} |x^{γ} ∈ I};
Γ_{m}(I) = {γ ∈ N^{n} |mγ ∈ m · Γ(I)};
\overline{Γ}(I) = ⋃_{m≥1} Γ_{m}(I); and
∗Γ(I) = ⋃_{M≥1}( ⋂_{m≥M} Γ_{m}(I)).

We say that Γ(I) is the exponent set for I. We suppress the ideal I in the notation for each of these subsets if we are working with a fixed monomial ideal.

Notice that n · Γ is the exponent set of I^{n},

Γ(I) = Γ_{1}(I), and Γ_{i}(I) ⊆ Γ_{j}(I) whenever i divides j.

Finally,

∗Γ(I) = ⋃_{(l,m)=1}(Γ_{l}(I) ∩ Γ_{m}(I)).

Notation 4.2 For a subset S of R^{n} we let conv(S), aff(S), and pos(S) denote the convex hull, affine hull, and positive cone of S, respectively. When we write α ≤ β for α, β ∈ R^{n}, we mean α_{i} ≤ β_{i} for all i; that is, ≤ is the product order on R^{n}.

We remind the reader that we have

conv(S) = {∑_{i=1}^{r} c_{i}α_{i} | {α_{1}, …, α_{r}} ⊆ S, c_{1}, …, c_{r} ∈ R_{+}, ∑_{i=1}^{r} c_{i} = 1};

aff (S) = {∑_{i=1}^{r} c_{i}α_{i} | {α_{1}, …, α_{r}} ⊆ S, ∑_{i=1}^{r} c_{i} = 1}; and

pos(S) = {∑_{i=1}^{r} c_{i}α_{i} | {α_{1}, …, α_{r}} ⊆ S, c_{1}, …, c_{r} ∈ R_{+}} .

Since the exponent set Γ(I) of a monomial ideal can be regarded as a subset of R^{n} we can talk about the convex hull of Γ(I). Arguing as in the proof of (3.3) one may deduce that the integral closure of a monomial ideal I is the monomial ideal with exponent set \overline{Γ}(I). It is well known (for example, [3, Exercise 4.23]) that \overline{Γ}(I) = conv(Γ) ∩ N^{n}.

We now present some elementary results about these sets.

Lemma 4.3 Let I = (x^{γ_{1}}, …, x^{γ_{r}}) be a monomial ideal in the polynomial ring k[x_{1}, ⋯, x_{n}] over a field k and let Γ = Γ(I). Then,

(a) Γ = {γ_{1}, …, γ_{r}} + N^{n};

(b) conv(Γ) = conv(Γ + R_{+}^{n});

(c) conv(Γ) = conv(Γ) + R_{+}^{n} = conv({γ_{1}, …, γ_{r}}) + R_{+}^{n};

(d) Γ_{m} = Γ_{m} + N^{n}, \overline{Γ} = \overline{Γ} + N^{n}, and ^{∗}Γ = ^{∗}Γ + N^{n}; and

(e) if σ ∈ N^{n} ∩ conv(Γ), then there exist ρ_{i} ∈ Γ, b_{i} ∈ Q_{+} with ∑ b_{i} = 1 and σ = ∑ b_{i}ρ_{i}.

Proof. (a) follows from the fact that a monomial x^{γ} ∈ I if and only if x^{γ_{i}}|x^{γ} for some i if and only if γ = γ_{i} +ρ for some i between 1 and r and some ρ ∈ N^{n}.

(b) Clearly conv(Γ) ⊆ conv(Γ + R_{+}^{n}). Thus it suffices to establish the opposite containment. Since conv(Γ) is closed under convex linear combinations it suffices to show that Γ+R_{+}^{n} ⊆ conv(Γ). Suppose γ ∈ Γ and ρ ∈ R_{+}^{n}. For each i between 1 and n there exist a_{i}, b_{i} ∈ N with a_{i} ≤ ρ_{i} ≤ b_{i}. So ρ is in the convex hypercube [a_{1}, b_{1}] × ⋯ × [a_{n}, b_{n}]. Thus there are nonnegative real numbers c_{i} and vectors μ_{i} ∈ N^{n} (i = 1, …, n) with ∑ c_{i} = 1 and ρ = ∑ c_{i}μ_{i}. Thus

γ + ρ = ∑ c_{i}γ + ∑ c_{i}μ_{i}
= ∑ c_{i} (γ + μ_{i}) ∈ conv(Γ),

since Γ + N^{n} = Γ.

(c) Suppose σ ∈ conv(Γ) and ρ ∈ R_{+}^{n}. As in part (b) we may write ρ = ∑ c_{i}μ_{i}, where μ_{i} ∈ N^{n}, c_{i} ≥ 0 (i = 1, …, n) and ∑ c_{i} = 1. Then,

σ + ρ = ∑ c_{i} (σ + μ_{i}) ,

and it suffices to show that each vector σ + μ_{i} ∈ conv(Γ). But if σ = ∑ a_{j}β_{j} is written as a convex linear combination of vectors β_{j} ∈ Γ, then

σ + μ_{i} = ∑_{j} a_{j} (β_{j} + μ_{j}) ∈ conv(Γ),

since Γ + N^{n} = Γ. Thus conv({γ_{1}, …, γ_{r}}) + R_{+}^{n} ⊆ conv(Γ) + R_{+}^{n} ⊆ conv(Γ). Since conv({γ_{1}, …, γ_{r}}) + R_{+}^{n} is a convex set containing {γ_{1}, …, γ_{r}} + R_{+}^{n} and conv(Γ) is the smallest convex set containing Γ = {γ_{1}, …, γ_{r}} + N^{n} we must have conv(Γ) = conv(Γ) + R_{+}^{n} = conv({γ_{1}, …, γ_{r}}) + R_{+}^{n}.

(d) Suppose that γ ∈ Γ_{m} and μ ∈ N^{n}. Write

mγ = β_{1} + ⋯ + β_{m} with β_{i} ∈ Γ (i = 1, …, m).

Then

m(γ + μ) = (β_{1} + μ) + ⋯ + (β_{m} + μ) ∈ m · Γ,

since Γ + N^{n} = Γ. The remaining assertions in part (d) follow immediately.

(e) Suppose σ ∈ N^{n} ∩ conv(Γ). Write

σ = ∑_{i=1}^{t} c_{i}ρ_{i}, ρ_{i} ∈ Γ, c_{i} ∈ R_{+}, ∑ c_{i} = 1.

By Caratheodory’s Theorem [18, Proposition 1.15], there exists an affinely independent subset {ρ_{l(1)}, …, ρ_{l(q)}} of {ρ_{1}, …, ρ_{t}} such that σ is a convex linear combination of the vectors ρ_{l(j)}:

σ = ∑_{j=1}^{q} b_{j}ρ_{l(j)}.

Letting A denote the matrix whose columns are ρ_{l(1)}, …, ρ_{l(q)} and b = (b_{1}, …, b_{q}), then b is a solution to Ax = σ in R^{q}. Let σ′, A′ and b′ denote σ, A, and b augmented by a row of ones, respectively. Then b′ is a solution to A′x = σ′. Since A′ and σ′ have entries in Q there must be a solution to A′x = σ′ in Q^{q+1}. Since the ρ_{l(j)} are affinely independent that solution must be b′.

One consequence of the preceding lemma is that if σ ∈ N^{n} and σ ≥ η ∈ ^{∗}Γ, then σ ∈ ^{∗}Γ. The same assertion holds if we replace ^{∗}Γ by \overline{Γ}.

We now remind the reader of some definitions and notions from the theory of convex polyhedra. The reader can consult [14] or [18] for a fuller discussion of convex polyhedra.

A polyhedron is a finite intersection of closed half-spaces in R^{n}. Any polyhedron P can be written as P = {x ∈ R^{n}|A · x ≥ b}, where A is a matrix with n columns. Given a polyhedron P in R^{n} and a vector ω ∈ R^{n}, the face of P determined by ω is given by

face_{ω}(P) = {x ∈ P | ω · x ≤ ω · z for all z ∈ P }.

Every subset F of P that has this form, that is, which minimizes some linear functional on P, is called a face of P. Notice that P = face_{0}(P). The relation “is a face of” is transitive. The dimension of a face F of a polyhedron is the dimension of its affine span. Faces of dimension 0 and 1 are called vertices and edges, respectively. A face F of dimension 1 less than the dimension of P is called a facet of P. Notice that the vertices of any face F of the polyhedron P are vertices of P. Every set that is the sum of convex hull of a finite set of points in R^{n} and the positive cone of a finite subset of R^{n} is a polyhedron and vice versa [18, Theorem 1.2].

We now offer a few of useful facts about the faces of the convex hull of the exponent set of a monomial ideal.

Notation 4.4 For a face face_{ω}(Σ) of the convex hull Σ of the exponent set of a monomial ideal I = (x^{γ_{1}}, …, x^{γ_{r}}) in a polynomial ring k[x_{1}, …, x_{n}] let

J_{ω} = {j |ω · e_{j} = 0};
S_{ω} = {i |ω · γ_{i} ≤ ω · x ∀x ∈ Σ}; and
Γ_{ω} = conv ({γ_{i} |i ∈ S_{ω}}) + ∑_{j∈J_{ω}} Ne_{j}.

Lemma 4.5 Let Σ denote the convex hull of the exponent set Γ of an ideal I = (x^{γ_{1}}, …, x^{γ_{r}}) in a polynomial ring k[x_{1}, …, x_{n}] over a field k. Consider a nonempty face F = face_{ω}(Σ) of Σ. Then, the following statements are true.

(a) There exists a vector η ∈ Q_{+}^{n} such that F = face_{η}(Σ).

(b) F = face_{ω}(Σ) = conv({γ_{i} |i ∈ S_{ω}}) + pos({e_{j} |j ∈ J_{ω}}); and

(c) F = conv(Γ_{ω}).

Proof. Suppose F = face_{ω}(Σ) is a nonempty face of Σ. Let c = ω · x, for some x ∈ F.

(a) We first show that ω ∈ R_{+}^{n}. Consider any vector ρ ∈ R^{n} all of whose components are positive. Then, mρ ≥ γ_{1} for m >> 0 and consequently, mρ ∈ Σ for m >> 0. Thus

m(ω · ρ) = ω · mρ ≥ c, for all m ≫ 0.

Hence

ω · ρ ≥ \frac{c}{m}, for all m ≫ 0.

Thus for any vector ρ ∈ R^{n} all of whose components are positive we have

ω · ρ ≥ 0.

Therefore ω ∈ R_{+}^{n} and necessarily c ≥ 0. We now prove (b) before concluding the proof of (a).

(b) Let F_{ω} = conv ({γ_{i} |ω · γ_{i} = c}) + pos ({e_{j} |ω · e_{j} = 0}). Clearly,

F_{ω} ⊆ face_{ω}(Σ) ⊆ Σ = conv ({γ_{1}, …, γ_{r}}) + pos ({e_{1}, …, e_{n}}) .

On the other hand, suppose that σ = ∑_{i=1}^{r} a_{i}γ_{i} + ∑_{j=1}^{n} b_{j}e_{j} ∈ F, where a_{i}, b_{j} ≥ 0, ∑ a_{i} = 1. Let ω = (w_{1}, …, w_{n}). Then,

c = ω · σ = ( ∑_{i:ω·γ_{i}=c} a_{i}) c + ∑_{i:ω·γ_{i}>c} a_{i} (ω · γ_{i}) + ∑_{j:w_{j}≠0} b_{j}w_{j}.

Since all nonzero components of ω are positive, the only way this sum can be equal to c is if ω · γ_{i} = c whenever a_{i} > 0 and b_{j} = 0 whenever w_{j} ≠ 0. Thus σ ∈ F_{ω}, as asserted.

(a) (concluded) We now know that ω ∈ R_{+}^{n} and

F = face_{ω}(Σ) = conv ({γ_{i} |ω · γ_{i} = c}) + pos ({e_{j} |ω · e_{j} = 0}) .

Reindexing the generators γ_{i} of I and permuting the coordinates on R^{n} we may and shall assume that

F = conv({γ_{1}, …, γ_{q}}) + pos({e_{s+1}, …, e_{n}}).

Thus face_{ω}(Σ) = face_{η}(Σ) for η ∈ R_{+}^{n} provided that

η · γ_{i} = η · γ_{1} (i = 1, …, q),
η · γ_{i} > η · γ_{1} (i = q + 1, …, r), and
η · e_{j} = 0 ⇔ j ∈ {s + 1, …, n}.

Projecting onto the first s coordinates of R^{n} we may assume that each component of ω is positive. Then η = (n_{1}, …, n_{s}), ω ∈ R_{+}^{s} define the same face of Σ provided that

n_{i} > 0, (i = 1, …, s);
η · (γ_{i} − γ_{1}) = 0, (i = 1, …, q); and
η · (γ_{i} − γ_{1}) > 0 (i = q + 1, …, r).

Let A be the matrix with rows γ_{2} − γ_{1}, …, γ_{q} − γ_{1} and let k_{1}, …, k_{t} be the leading columns of the reduced echelon form B of A. Say the rows of B are

e_{k_{i}} − ∑_{j} a_{ij}e_{l_{j}} (i = 1, …, t),

where e_{l_{1}}, …, e_{l_{s−t}} are the nonleading columns of B. Note that each coefficient a_{ij} ∈ Q since every γ_{i} ∈ N^{s}. So η, ω ∈ R_{+}^{s} define the same face of Σ provided that

η_{k_{i}} = ∑_{j} a_{ij}η_{l_{j}} (i = 1, …, t),
∑_{j} a_{ij}η_{l_{j}} > 0 (i = 1, …, t), and
(γ_{i} − γ_{1}) · η > 0 (i = q + 1, …, r).

Rewriting the last system of inequalities in terms of the nonleading variables x_{l_{1}}, …, x_{l_{s−t}} and letting x′ denoting the projection of a vector x ∈ R^{s} onto the corresponding components we get a combined system of inequalities with coefficient matrix C having rational entries that has a solution x′ if and only if each entry of Cx′ is positive. Since every entry of Cω′ is positive and the map x ↦ Cx′ is continuous we may choose a vector η ∈ Q_{+}^{s} very close to ω so that every component of η is positive and every entry of Cη′ is positive. This concludes the proof of part (a).

(c) Clearly

conv(Γ_{ω}) ⊆ F = face_{ω}(Σ) = conv({γ_{i} |ω · γ_{i} = c}) + pos({e_{j} |ω · e_{j} = 0}).

Suppose σ ∈ conv({γ_{i} |i ∈ S_{ω}}), ρ ∈ pos({e_{j} |j ∈ J_{ω}}). Say

σ = ∑_{i∈S_{ω}} c_{i}γ_{i}, c_{i} ≥ 0 (i ∈ S_{ω}), ∑ c_{i} = 1.

Then,

σ + ρ = ∑_{i∈S_{ω}} c_{i}(γ_{i} + ρ).

As in the proof of (4.3)(b) ρ is a convex linear combination of vectors in ∑_{j∈J_{ω}} Ne_{j} and hence γ_{i} + ρ ∈ conv(Γ_{ω}). Since σ + ρ is a convex linear combination of the vectors γ_{i} + ρ, we have σ + ρ ∈ conv(Γ_{ω}). Therefore we may conclude that F = conv(Γ_{ω}).

Henceforth whenever we write a nonempty face F of the convex hull Σ of the exponent set of a monomial ideal in the form face_{ω}(Σ) we will assume that ω ∈ Q_{+}^{n}.

Definition 4.6 Let W be an affine subspace of R^{n}. We say a point σ ∈ W is a W −internal point of a subset Σ of W if for every τ ∈ W, there exists τ′ ∈ (σ, τ] such that the segment [σ, τ′] ⊆ Σ.

Putting τ = σ + μ we see that σ ∈ Σ is a W −internal point if and only if for every μ ∈ W there is an 0 < ϵ ≤ 1 such that [σ, σ + ϵμ] ⊆ Σ.

It is well known that for a polyhedron P ⊆ R^{n}, a point σ ∈ P is an aff (P)−internal point of P if and only if σ is an interior point of P with respect to the embedding of P in its affine hull [18, Lemma 2.9].

Notation 4.7 For a polyhedron P we let

relint(P) := P − ∪E,

where the union is taken over all facets E of P

We now offer another characterization of the set of W −internal points of Σ.

Lemma 4.8 Let Σ = conv(Γ) ⊆ R^{n} for the exponent set Γ of some monomial ideal, let F be a face of Σ and let W = aff (F). Then, relint(F) is the set of W −internal points of F.

Proof. Let E_{1}, … E_{s} be the facets of F. Then E_{1}, …, E_{s} are faces of Σ so there exist vectors ω_{1}, …, ω_{s} ∈ Q_{+}^{n} with E_{i} = face_{ω_{i}}(Σ) (i = 1, …, s). We know that [18, Theorem 2.15]

x ∈ F ⇔ x ∈ aff (F) and ω_{i} · x ≥ c_{i} (i = 1, …, s)

and

x ∈ relint(F) ⇔ x ∈ aff (F) and ω_{i} · x > c_{i} (i = 1, …, s).

First assume that σ is a W −internal point of F. For each i between 1 and s choose a point z_{i} ∈ F with ω_{i}·z_{i} ≠ 0 Then we must have ω_{i}·z_{i} > 0 (i = 1, …, s). Choose a positive number ϵ such that the segment [σ, σ−ϵz_{i}] ⊆ F (i = 1, …, s). Thus

ω_{i} · σ − ϵω_{i} · z_{i} ≥ c_{i} (i = 1, …, s).

Since ω_{i} · z_{i} > 0 (i = 1, …, s), we must have ω_{i} · σ > c_{i} (i = 1, …, s). Thus σ is not in any facet of F.

Conversely, assume that σ ∈ F is not in any facet of F. We must have

d_{i} := σ · ω_{i} > c_{i} (i = 1, …, s).

Let η ∈ W be arbitrary. Then,

ω_{i} · (σ + t(η − σ)) =
ω_{i} · σ + tω_{i} · (η − σ) =
d_{i} + tω_{i} · (η − σ) ≥ c_{i},

provided that

d_{i} − c_{i} ≥ tω_{i} · (σ − η).

The latter inequality holds for all nonnegative values of t if ω_{i} · (σ − η) ≤ 0. If ω_{i} · (σ − η) > 0 just take t ≤ (d_{i} − c_{i})/ω_{i} · (σ − η) := ϵ_{i}. Thus taking

ϵ = min \{\frac{d_{i} − c_{i}}{ω_{i} · (σ − η)} | ω_{i} · (σ − η) > 0\} ,

we have

σ + t(η − σ) ∈ F whenever 0 ≤ t ≤ ϵ.

Therefore σ is a W −internal point of F.

Notation 4.9 For a face F of the convex hull conv(Γ) of the exponent set Γ of a monomial ideal, define

∗F = {x ∈ relint(F)|x = ∑ n_{i}γ_{i}, γ_{i} ∈ F ∩ Γ , n_{i} ∈ Z, and ∑ n_{i} = 1} .

That is, ^{∗}F is the set of lattice points generated by F ∩Γ which lie in the relative interior of F.

Example 4.10 Let Γ denote the exponent set of the ideal (x^{5}, x^{2}y^{3}, y^{5}) in k[x, y] and E the segment connecting (0,5) to (5,0). Observe that

(2, 3) = (0, 5) + 2(1, −1)

and

(5, 0) = (0, 5) + 5(1, −1).

Thus

(1, 4) = (0, 5) + (1, −1)
= 2(0, 5) − 2(2, 3) + (5, 0);
(3, 2) = (0, 5) + 3(1, −1)
= (5, 0) − (2, 3) + (5, 0); and
(4, 1) = (0, 5) + 4(1, −1)
= −(0, 5) + 2(2, 3).

Thus the points comprising E ∩ Γ and ^{∗}E are as follows.

[figure: (0,5); (5,0); (2,3); E ∩ Γ; (1,4); (2,3); (3,2); (4,1); ^{∗}E]

In the set ^{∗}E the spacing between the lattice points is uniformized and the points of E ∩ Γ not in the relative interior of E are omitted.

We are finally ready to state the main theorem of this section.

Theorem 4.11 Let Γ = Γ(I) be the exponent set of a monomial ideal I in a polynomial ring k[x_{1}, …, x_{n}] over a field k of characteristic zero. Then, an integral point γ ∈ conv(Γ) is in the exponent set of ^{∗}I if and only if γ ∈ ⋃^{∗}F, where the union is taken over all faces F of conv(Γ).

Proof. Let Σ = conv(Γ) and consider a point γ ∈ Σ ∩ N^{n}. First assume that γ ∈ ^{∗}Γ. By (3.3) there exists a positive integer m such that

mγ ∈ m · Γ and (m + 1)γ ∈ (m + 1) · Γ.

Thus there exist vectors γ_{i} ∈ Γ, nonnegative integers a_{i}, b_{i} with ∑ a_{i} = m, ∑ b_{i} = m + 1 such that

mγ = ∑ a_{i}γ_{i} and
(m + 1)γ = ∑ b_{i}γ_{i}.

Letting c_{i} = b_{i} − a_{i} we have

γ = (m + 1)γ − mγ = ∑ c_{i}γ_{i},

where the coefficients c_{i} are integers with ∑ c_{i} = 1. If γ ∈ relint(Σ), then γ ∈ ^{∗}Σ, by definition. So suppose γ ∈ relint(F) for some proper face F of Σ, where F = face_{ω}(Σ), ω ∈ Q_{+}^{n}, ω ·x = c for all x ∈ F, and ω ·z ≥ c for all z ∈ Σ. Now γ = ∑ \frac{a_{i}}{m} γ_{i} implies for each γ_{i} occurring with a nonzero coefficient we must have ω · γ_{i} = c (otherwise, ω · γ = ∑ \frac{a_{i}}{m} (ω · γ_{i}) > c). Similarly each γ_{i} occurring with a nonzero coefficient b_{i} is in F. Thus we may assume each γ_{i} occurring with a nonzero coefficient c_{i} is in F. This implies γ ∈ ^{∗}F, as asserted.

Conversely, let us assume that γ ∈ ^{∗}F for some face F of Σ. Then,

γ = ∑ c_{i}γ_{i}, where γ_{i} ∈ Γ ∩ F, c_{i} ∈ Z, and ∑ c_{i} = 1.

Now γ ∈ ^{∗}F ⊆ conv(Γ) ∩ N^{n} implies

γ = ∑ \frac{a_{i}}{m}η_{i},

where m is a positive integer, each a_{i} ∈ N, each η_{i} ∈ Γ and ∑ a_{i} = m by (4.3)(e). As above, each η_{i} occurring with a positive coefficient must be in F. Taking the union of the sets {γ_{i}} and {η_{j}} we may assume that

γ = ∑ c_{i}γ_{i}
= ∑ \frac{a_{i}}{m}γ_{i},

where the vectors γ_{i} ∈ Γ ∩ F and the integers m, c_{i}, and a_{i} are as above.

If γ ≥ γ_{i} for some i then γ ∈ Γ ⊆ ^{∗}Γ and we are done. So we may assume that γ ≱ γ_{i} for all i. Consider a fixed i such that c_{i} < 0. We know that some component of γ −γ_{i} is negative. Hence the ray {γ_{i}+t(γ −γ_{i}) |t ≥ 0} is contained in aff (F) but is not contained in F ⊆ R_{+}^{n}. Let

t_{0} = sup {t ≥ 0 | γ_{i} + t(γ − γ_{i}) ∈ F } .

Then 1 < t_{0} and μ := γ_{i}+t_{0}(γ−γ_{i}) ∈ F, since F is closed. Since γ_{i}+t(γ−γ_{i}) ∉ F for all t > t_{0} we must have μ ∈ E for some facet E of F by (4.8). If γ_{i} ∈ E then γ ∈ [γ_{i}, μ] implies γ ∈ E, contradicting the assumption that γ ∈ relint(F). Thus

γ = (1 − \frac{1}{t_{0}} ) γ_{i} + \frac{1}{t_{0}} μ = bγ_{i} + (1 − b)μ.

Notice that b ≠ 0 since γ ∉ E. Since every face of F is a face of Σ, we may write E = face_{ω}(Σ) = conv(Γ_{ω}) and express μ as a convex linear combination of vectors in Γ_{ω}. We thus obtain

γ = bγ_{i} + (1 − b) ∑_{j=1}^{t} b_{j}ν_{j},

where each ν_{j} ∈ Γ_{ω} ⊆ Γ ∩ F. Let ν_{0} = γ_{i} and write γ = ∑_{j=0}^{t} d_{j}ν_{j} where each d_{j} ≥ 0 and ∑ d_{j} = 1. By Caratheodory’s Theorem [18, Proposition 1.15] we may rewrite γ as a convex linear combination of an affinely independent subset of {ν_{0}, …, ν_{t}}. Furthermore ν_{0} must occur in this subset since γ ∉ E. As in the proof of (4.3)(b), reindexing ν_{1}, …, ν_{t} we may assume

γ = ∑_{j=0}^{s} d_{j}ν_{j},

where ν_{0} = γ_{i}, 0 < d_{j} = a_{j}′/m′ ∈ Q_{+} for each j, and ∑ d_{j} = 1. Again enlarging the set of {γ_{l}} so the new set contains {ν_{j}} we obtain

m′γ = ∑ a′_{l}γ_{l},

and now the coefficient a′_{i} of γ_{i} is positive. Repeating this for each i such that c_{i} < 0 we may assume that

γ = ∑ c_{i}γ_{i}
= ∑ \frac{a_{i}}{m}γ_{i},

where a_{i} > 0 whenever c_{i} < 0. Then,

mγ = ∑ a_{i}γ_{i}.

Multiplying the latter equation by a positive integer M such that Ma_{i} + c_{i} ≥ 0 for all i and replacing m by Mm, a_{i} by Ma_{i} we may and shall assume that a_{i} + c_{i} > 0 for all i. Then

(m + 1)γ = ∑(a_{i} + c_{i})γ_{i},

where each γ_{i} ∈ Γ∩F, each (a_{i} +c_{i}) ∈ Z and ∑(a_{i} +c_{i}) = ∑ a_{i} +∑ c_{i} = m+1. Hence γ ∈ Γ_{m} ∩ Γ_{m+1}, which implies that γ ∈ ^{∗}Γ.

We illustrate the ideas of the proof in the following example.

Example 4.12 Let I = (x^{6}, x^{2}y^{4}, y^{6}) ⊆ k[x, y], Γ = Γ(I), and Σ = conv(Γ). Observe that Σ is the sole 2-dimensional face of Σ, E_{1} = {(0, 6)} + pos({e_{2}}), E_{2} = [(0, 6), (6, 0)], and E_{3} = {(6, 0)}+pos({e_{1}}) are the edges and V_{1} = {(0, 6)} and V_{2} = {(6, 0)} are the vertices of Σ. The sets Γ, Σ, E_{1}, E_{2}, E_{3}, V_{1}, V_{2} are depicted below. The set Γ is bounded by the thick-lined staircase figure and Σ is the 1st quadrant with the left-hand corner clipped. The points depicted by open circles are in Σ \ Γ.

[figure: V_{1}; Γ; Σ; E_{1}; E_{2}; E_{3}; V_{2}]

Observe that

∗Σ = (Γ \ (E_{1} ∪ E_{2} ∪ E_{3}) ∩ Γ) ∪ {(4, 3), (5, 2), (5, 3)};
∗E_{1} = {(0, 7), (0, 8), (0, 9), …};
∗E_{2} = {(2, 4), (4, 2)};
∗E_{3} = {(7, 0), (8, 0), (9, 0), …};
∗V_{i} = V_{i} (i = 1, 2);
∗Γ = Γ ∪ {(4, 2), (4, 3), (5, 2), (5, 3)}; and
\overline{Γ} = Γ ∪ \{(1, 5), (3, 3), (4, 2), (4, 3), (5, 1), (5, 2), (5, 3)\}.

Thus we have Γ ⊂ ^{∗}Γ ⊂ \overline{Γ}.

Recently Oda and Yoshida [7] defined the (2,3)-closure I′ of an ideal I of a noetherian domain A with finite normalization and showed that I′ is an ideal in the seminormalization of A. Their construction can be generalized to the non-noetherian case by using transfinite induction. One can also generalize to the case where A is a reduced ring with finitely many minimal primes. When A is a noetherian weakly normal domain the Oda-Yoshida (2,3)-closure of I can be described as follows. Let ^{#}I = (a ∈ A |a^{2} ∈ I^{2}, a^{3} ∈ I^{3}). Inductively define I_{0} = I, …, I_{n+1} = ^{#}(I_{n}). The (2,3)-closure I′ of I is defined by

I′ = ⋃_{n≥0} I_{n}.

It is clear that I′ is (2,3)-closed, that is, (I′)′ = I′. Oda and Yoshida showed that for ideals I and J of A we have I′J′ ⊆ (IJ)′. Thus for a fixed ideal I of A we may define a graded subring R′ of a polynomial ring A[t] in one indeterminate over A by

R′ = ⊕_{n≥0}(I^{n})′t^{n}.

We have the following chain of subrings of A[t]

R′ ⊆ A[It] ⊆ ^{∗}R = ⊕_{n≥0}^{∗}(I^{n})t^{n},

where A[It] denotes the Rees ring of I and ^{∗}R denotes the weak normalization of R; the latter equality follows from (2.9) . In their original paper Oda and Yoshida asserted that R′ is the seminormalization of the Rees ring. This is not the case in general as the example below illustrates.

First notice that if I is a monomial ideal in a polynomial ring A = k[x_{1}, …, x_{n}] over a field k, then I′ is again a monomial ideal. Since we are assuming char(k) = 0 throughout this section, the seminormalization and the weak normalization of the Rees ring A[It] coincide; both of these rings are equal to the weak normalization of A[It] in the normal domain A[t]. In what follows let Γ = Γ(I), ^{#}Γ = Γ(^{#}I),^{#}Γ_{n} = Γ(I_{n}), and Γ′ = Γ(I′). Notice that ^{#}Γ = Γ_{2} ∩Γ_{3}, where Γ_{n} is defined in (4.1).

Example 4.13 Let I = (x^{5}, x^{2}y^{3}, y^{5}) ⊆ k[x, y]. Notice that

Γ = {(5, 0), (2, 3), (0, 5)} + N^{2} and
\overline{Γ} = conv(\{(5, 0), (0, 5)\} + R_{+}^{2}) ∩ N^{2}
= Γ ∪ {(1, 4), (3, 2), (4, 1), (4, 2)}.

In the figure below Γ consists of the lattice points in the the first quadrant bounded by the thick-lined staircase structure whereas \overline{Γ} picks up the circled points in the convex hull of Γ, which is the first quadrant with the lower lefthand corner clipped.

[figure: Γ; E]

Notice that

2(4, 2) > (2, 3) + (5, 0) ⇒ (4, 2) ∈ Γ_{2} and
3(4, 2) > 2(2, 3) + (5, 0) ⇒ (4, 2) ∈ Γ_{3}.

Hence (4, 2) ∈ Γ_{2} ∩ Γ_{3} = ^{#}Γ ⊆ ^{∗}Γ. Consider the integral points (0,5), (1,4), (2,3),(3,2), (4,1), (5,1) along the edge E = [(0, 5), (5, 0)] of conv(Γ). By (4.10) the points (1,4), (3,2), and (4,1) are in ^{∗}E and hence in ^{∗}Γ. Therefore ^{∗}Γ = \overline{Γ} by (4.11). Now let us finish the calculation of I′.

The only way (1,4) can belong to Γ_{n} is if n(1, 4) ∈ n · (Γ ∩ E). Looking at the figure above we see that 3(1, 4) ∉ 3 · (Γ ∩ E). Thus (1, 4) ∉ Γ_{3} and hence (1, 4) ∉ ^{#}Γ. Similarly, 2(3, 2) ∉ 2 · (Γ ∩ E), which implies (3, 2) ∉ ^{#}Γ and 2(4, 1) ∉ 2 · (Γ ∩ E) implies (4, 1) ∉ ^{#}Γ. Thus ^{#}Γ = Γ ∪ {(4, 2)}. The same type of analysis implies that ^{#}Γ := ^{#}Γ_{1} = ^{#}Γ_{2} = ⋯ = Γ′ and hence Γ′ = Γ ∪ {(4, 2)}. Therefore,

I′ = (x^{5}, x^{2}y^{3}, y^{5}, x^{4}y^{2}) ⊂ ^{∗}I = (x^{5}, x^{4}y, x^{3}y^{2}, x^{2}y^{3}, xy^{4}, y^{5}).

Notice that the ideal I′ above is (2,3)-closed but not weakly subintegrally closed. Thus R′ ⊂ ^{∗}R.

In a later paper [8], Oda and Yoshida stated that for ideals I such that (I′)^{n} = (I^{n})′, the graded ring R′ does give the seminormalization of the Rees ring. The example above shows that this additional condition is, in fact, necessary.

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