DIVISORIAL LINEAR ALGEBRA OF NORMAL SEMIGROUP RINGS

WINFRIED BRUNS AND JOSEPH GUBELADZE

Abstract. We investigate the minimal number of generators μ and the depth of divisorial ideals over normal semigroup rings. Such ideals are defined by the inhomogeneous systems of linear inequalities associated with the support hyperplanes of the semigroup. The main result is that for every bound C there exist, up to isomorphism, only finitely divisorial ideals I such that μ(I) ≤ C. It follows that there exist only finitely many Cohen–Macaulay divisor classes. Moreover we determine the minimal depth of all divisorial ideals and the behaviour of μ and depth in “arithmetic progressions” in the divisor class group.

The results are generalized to more general systems of linear inequalities whose homogeneous versions define the semigroup in a not necessarily irredundant way. The ideals arising this way can also be considered as defined by the non-negative solutions of an inhomogeneous system of linear diophantine equations.

We also give a more ring-theoretic approach to the theorem on minimal number of generators of divisorial ideals: it turns out to be a special instance of a theorem on the growth of multigraded Hilbert functions.

1. Introduction

A normal semigroup S ⊂ ℤ^{n} can be described as the set of lattice points in a finitely generated rational cone. Equivalently, it is the set

S = {x ∈ ℤ^{n}: σ_{i}(x) ≥ 0, i = 1, …, s} (∗)

of lattice points satisfying a system of homogeneous inequalities given by linear forms σ_{i} with integral (or rational) coefficients. For a field K the K-algebra K[S] is a normal semigroup ring. In the introduction we always assume that S is positive, i.e. 0 is the only invertible element in S.

Let a_{1}, …, a_{s} be integers. Then the set

T = {x ∈ ℤ^{n}: σ_{i}(x) ≥ a_{i}, i = 1, …, s}

satisfies the condition S + T ⊂ T, and therefore the K-vector space KT ⊂ K[ℤ^{n}] is an R-module in a natural way.

It is not hard to show that such an R-module is a (fractional) ideal of R if the group gp(S) generated by S equals ℤ^{n}. Moreover, if the presentation (∗) of S is irredundant, then the R-modules KT are even divisorial ideals. These divisorial ideals represent the full divisor class group Cl(R). Therefore an irredundant system

1991 Mathematics Subject Classification. Primary 13C14, 13C20, 14M25, 20M25, 52B20, Secondary 11D75, 20G15.

The second author was partially supported by the Max-Planck-Institut für Mathematik at Bonn.

of homogeneous linear inequalities is the most interesting from the ring-theoretic point of view, but we will also treat the general case.

We are mainly interested in two invariants of divisorial ideals D, namely their number of generators μ(D) and their depth as R-modules, and in particular in the Cohen–Macaulay property. Our main result, based on combinatorial arguments, is that for each C ∈ ℤ_{+} there exist, up to isomorphism, only finitely many divisorial ideals D such that μ(D) ≤ C. It then follows by Serre’s numerical Cohen–Macaulay criterion that only finitely many divisor classes represent Cohen–Macaulay modules.

The second main result concerns the growth of Hilbert functions of certain multigraded modules and algebras. Roughly speaking, it says that the Hilbert function takes values ≤ C only at finitely many graded components, provided this holds along each arithmetic progression in the grading semigroup. The theorem on Hilbert functions can be applied to the minimal number of generators of divisorial ideals since R can be embedded into a polynomial ring P over K such that P is a Cl(R)-graded R-algebra in a natural way. This leads to a second proof of the result on number of generators mentioned above.

In Section 2 we study this “standard” embedding of R into a polynomial ring P over K. It has been used many times already, for example by Hochster [Ho] and Stanley [St1]. The standard embedding into a polynomial ring graded by the divisor class group, as described in Theorem 2.1(a), are exactly the homogeneous coordinate rings in the sense of [Cox] of the affine toric varieties Spec(k[S]). Moreover, Theorem 2.1(b) corresponds to the results in [Cox, Section 2] on representing (not necessarily affine) toric varieties as quotients of affine spaces by diagonalizable linear groups. We include the details because they are essential for the subsequent sections.

More generally we discuss so-called pure embeddings of normal affine semigroups into polynomial rings Q = K[X_{1} …, X_{n}], and the natural splitting of Q into the coset modules of R , i.e. those submodules of Q whose monomial basis is defined by a single residue class modulo gp(S) ⊂ ℤ^{n}. Such pure embeddings arise from linear actions of diagonalizable linear algebraic groups on polynomial rings over K, and, more generally, from systems of linear diophantine equations and congruences. In the setting of invariant theory the coset modules appear as modules of covariants, and in the context of diophantine equations they correspond to the sets of nonnegative solutions of the associated inhomogeneous systems. We describe all pure embeddings of a normal affine semigroup into finitely generated free semigroups, and characterize those embeddings for which all coset modules are divisorial ideals.

Section 3 contains results on the depth of divisorial ideals. We first show that a divisorial ideal whose class is a torsion element in Cl(R) is Cohen–Macaulay. This follows easily from Hochster’s theorem that normal semigroup rings are Cohen– Macaulay. Then we give a combinatorial description of the minimal depth of all divisorial ideals of R: it coincides with the minimal number of facets F_{1}, …, F_{u} of the cone generated by S such that F_{1} ∩ ⋯ ∩ F_{u} = {0}.

Section 4 contains our main result on number of generators. The crucial point in its proof is that the convex polyhedron C(D) naturally associated with a divisorial ideal D has a compact face of positive dimension if (and only if) the class of D is non-torsion. One can show that μ(D) ≥ Mλ where M is a positive constant only depending on the semigroup S and λ is the maximal length of a compact 1dimensional face of C(D). Moreover, since the compact 1-dimensional faces are in discrete positions and uniquely determine the divisor class, it follows that λ has to go to infinity in each infinite family of divisor classes.

The observation on compact faces of positive dimension is also crucial for our second approach to the number of generators via Hilbert functions. Their wellestablished theory allows us to prove quite precise results about the asymptotic behaviour of μ and depth along an arithmetic progression in the divisor class group.

From the viewpoint of applications it is quite unnatural to study only the sets of solutions T (a) of inhomogeneous systems of linear inequalities ξ_{i}(x) ≥ a_{i}, i = 1, …, n, whose associated homogeneous system defines its set of solutions S = T (0) irredundantly. Therefore we turn to the general case (but still under the assumption gp(S) = ℤ^{n}) in Section 5, replacing divisorial ideals by the ξ-convex ideals I = KT (a). The result on numbers of generators has a fully satisfactory generalization: up to isomorphism over R, there exist only finitely many ξ-convex ideals I such that μ(I) ≤ C for every constant C.

Section 6 finally contains the theorem on the growth of Hilbert functions outlined above. It is proved by an analysis of homomorphisms of affine semigroups and their “modules”.

The Cohen–Macaulay property of coset modules has been characterized by Stanley [St1, St3] in terms of local cohomology. Brion [Bri] has shown that the number of isomorphism classes of Cohen–Macaulay modules of covariants is finite for certain actions of linear algebraic groups; however, the hypotheses of his theorem exclude groups with infinitely many characters. Therefore our result is to some extent complementary to Brion’s.

For unexplained terminology we refer the reader to Bruns and Herzog [BH], Eisenbud [Ei] or Stanley [St3]. A standard reference for the theory of divisor class groups and divisorial ideals is Fossum [Fo]. Divisor class groups of normal semigroup rings are discussed in detail by Gubeladze [Gu2]. The theory of linear algebraic groups used in Section 2 can be found in Humphreys [Hu].

There also exists a considerable literature on the divisor theory of not necessarily finitely generated Krull monoids. See Halter-Koch [HK] for a comprehensive overview and further references.

2. Coset modules

Usually we write the operation in a semigroup additively, and in our terminology a semigroup is always commutative and contains a neutral element 0. Let S ⊂ T be semigroups. Then the integral closure (or saturation) of S in T is the subsemigroup

Ŝ = {x ∈ T: mx ∈ S for some m ∈ ℤ_{+}, m > 0}

of T; if S = Ŝ, then S is integrally closed in T. By definition, an affine semigroup S is a finitely generated subsemigroup of ℤ^{n}. The normalization \overline{S} of an affine semigroup S is its integral closure in its group of differences gp(S). We call an affine semigroup normal if S = \overline{S}. All this terminology is consistent with (and derived from) its use in commutative algebra: for a field K the semigroup algebra K[Ŝ] is the integral closure of K[S] in K[T] etc., at least if T can be linearly ordered and every element in K[T] has a leading monomial.

Let S ⊂ ℤ^{n} be an affine semigroup. One sees easily that its integral closure in ℤ^{n} is

Ŝ = C(S) ∩ ℤ^{n}

where C(S) ⊂ ℝ^{r}, the cone generated by S, is the set of all linear combinations of elements of S with non-negative real coefficients. The normalization \overline{S} of S is given by S̄ = C(S) ∩ gp(S). It follows from results of commutative algebra or Gordan’s lemma that Ŝ and \overline{S} are again affine semigroups.

Since S, and therefore the cone C(S), are finitely generated, C(S) is the intersection of finitely many rational halfspaces,

C(S) = ⋂_{i=1}^{s}{x ∈ ℝ^{r}: σ_{i}(x) ≥ 0}.

That the halfspaces are rational means that we can define them by linear forms σ_{i} ∈ (ℝ^{r})^{∗} with coprime integer coefficients, and if we require in addition that the representation is minimal, then the σ_{i} are uniquely determined (up to order). We call them as well as their restrictions to ℤ^{r} the support forms of S, and we set

supp(S) = {σ_{1}, …, σ_{s}}.

We will often assume that S is positive, i.e. 0 is the only invertible element in S. Let K be a field and R = K[S]. By choosing a basis in gp(S), we identify it with ℤ^{r}

for some r ≥ 0. From the ring-theoretic point of view, S can then be considered as a semigroup of monomials of the Laurent polynomial ring K[X_{1}^{±1}, …, X_{r}^{±1}] ≅ K[ℤ^{r}]. The support forms, so far defined on gp(S) ⊂ K[X_{1}^{±1}, …, X_{r}^{±1}], can be extended to discrete valuations on the quotient field of K[S], which we also denote by σ_{i}. The prime ideals

𝔭_{i} = {x ∈ K[S]: σ_{i}(x) ≥ 1}

are exactly the divisorial prime ideals generated by monomials. By a theorem of Chouinard [Ch], the divisor class group Cl(R) is isomorphic to the quotient of the free abelian group generated by 𝔭_{1}, …, 𝔭_{s} modulo the subgroup generated by the principal divisors of the monomials x ∈ S. We define the linear map σ: ℤ^{r} → ℤ^{s} by

σ(x) = (σ_{1}(x), …, σ_{s}(x)).

Note that σ is injective if S is positive; in fact x ∈ S and −x ∈ S for any element x ∈ Ker σ. One has σ(S) ⊂ ℤ_{+}^{s}, and we can extend σ to an embedding

σ: K[S] → K[ℤ_{+}^{s}] ≅ K[Y_{1}, …, Y_{s}].

We call these embeddings the standard embeddings of S and K[S] respectively. Note that σ(S) (up to an automorphism of ℤ_{+}^{s}) only depends on the isomorphism class of S; in fact, according to Gubeladze [Gu3], it only depends on the isomorphism class of the K-algebra K[S].

Let K be an algebraically closed field, and S a positive normal affine semigroup. It is well-known that K[S] can be represented as the ring of invariants of a diagonal action of a suitable algebraic torus T_{n}(K) = (K^{×})^{n} on a polynomial ring K[X_{1}, …, X_{n}] for some n ∈ ℕ. Conversely, the ring of invariants of a linear action of a diagonalizable group D over K on such a polynomial ring has the structure of a normal affine semigroup ring K[S]. The diagonalizable groups are exactly those isomorphic to a direct product T_{n}(K) × A where A is a finite abelian group whose order is not divisible by char K. In the following theorem we will discuss the problem how D and K[S] are related. As we shall see, the connection is given through the divisor class group of K[S].

Theorem 2.1. Let K be a field, S a positive normal affine semigroup, R = K[S], and σ: R → P = K[Y_{1}, …, Y_{s}] the standard embedding.

(a) Then P decomposes as an R-module into a direct sum of rank 1 K[S]-modules M_{c}, c ∈ Cl(R), such that M_{c} is isomorphic to a divisorial ideal of class c.

(b) Suppose that K is algebraically closed and that Cl(R) does not contain an element of order char K. Then R = P^{D} where D = Hom_{ℤ}(Cl(R), K^{×}) ⊂ T_{s}(K) acts naturally on P.

Proof. To each divisorial ideal 𝔭_{1}^{(a_{1})}∩⋯∩𝔭_{s}^{(a_{s})}, a_{i} ∈ ℤ, we associate (a_{1}, …, a_{s}) ∈ ℤ^{s}. Under this assignment, the principal divisorial ideal generated by s ∈ gp(S) is mapped to σ(s). By Chouinard’s theorem this yields the isomorphism

Cl(R) ≅ ℤ^{s}/σ(ℤ^{r}).

For each c ∈ ℤ^{s}/σ(ℤ^{r}) we let M_{c} be the K-vector subspace of P generated by all monomials whose exponent vector in ℤ^{s}/σ(ℤ^{r}) has residue class −c. Then M_{c} is clearly an R-submodule of P. Moreover, by construction, P is the direct sum of these R-modules.

It remains to show for (a) that M_{c}, c ∈ Cl(R), is a divisorial ideal of class c (relative to the isomorphism above). We choose a representative a = (a_{1}, …, a_{s}) of c. Then a monomial s ∈ σ(ℤ^{r}) belongs to 𝔭_{1}^{(a_{1})} ∩ ⋯ ∩ 𝔭_{s}^{(a_{s})} if and only if s_{i} ≥ a_{i} for all i, and this is equivalent to s_{i} − a_{i} ∈ ℤ_{+} for all i. Hence the assignment s ↦ s − a, in ring-theoretic terms: multiplication by the monomial Y^{−a}, induces an R-isomorphism 𝔭_{1}^{(a_{1})} ∩ ⋯ ∩ 𝔭_{s}^{(a_{s})} ≅ M_{c}.

For (b) we consider the exact sequence

0 → σ(ℤ^{r}) → ℤ^{s} → Cl(R) → 0.

If K is algebraically closed, then this sequence induces an exact sequence

1 → Hom_{ℤ}(Cl(R), K^{×}) → Hom_{ℤ}(ℤ^{s}, K^{×}) → Hom_{ℤ}(σ(ℤ^{r}), K^{×}) → 1

since K^{×} is an injective ℤ-module.

The torus T_{s}(K) ≅ Hom_{ℤ}(ℤ^{s}, K^{×}) operates naturally on the Laurent polynomial ring L = K[Y_{1}^{±1}, …, Y_{s}^{±1}] through the substitution Y^{t} ↦ τ(t)Y^{t} for each τ ∈ T_{s}(K) and each monomial Y^{t} of P, t ∈ ℤ_{+}^{s}. In the following we identify t and Y^{t}.

The homomorphism

δ: ℤ^{s} → Hom_{alg}(T_{s}(K), K^{×}), δ(t)(τ) = τ(t),

is an isomorphism, and it induces an isomorphism

Hom_{alg} (Hom_{ℤ}(σ(ℤ^{r}), K^{×}), K^{×}) ≅ U

where U is the subgroup of all t ∈ ℤ^{s} such that δ(τ)(t) = 1 for all τ ∈ D. Thus the ring of invariants L^{D} is the K-vector space spanned by all monomials t ∈ U. There is a commutative diagram

[figure: 0; σ(ℤ^{r}); ℤ^{s}; Cl(R); 0]

[figure: 0; U; ℤ^{s}; ‖; Hom_{alg}(D, K^{×}); 0]

with exact rows and in which the vertical maps are the natural ones. The right hand vertical map is an isomorphism if and only if the characteristic of K does not divide the order of any element of Cl(R), and therefore U = σ(ℤ^{r}) in this case. Then the set of monomials generating P^{D} is exactly σ(ℤ^{r}) ∩ ℤ_{+}^{s}.

Let S and T be affine semigroups, S ⊂ T. Then S is called a pure subsemigroup if S = T ∩ gp(S) in gp(T) (in Hochster’s terminology [Ho] S is a full subsemigroup). It is not hard to check that S is a pure subsemigroup if and only if K[S] ⊂ K[T] is a pure extension of rings. More generally, we say that an injective homomorphism ξ: S → T is pure if ξ(S) is a pure subsemigroup of T. If T is normal, then every pure subsemigroup of T is also normal. Examples of pure embeddings are given by the standard embedding S → ℤ_{+}^{s}, s = # supp(S), and the embedding S → ℤ^{n} resulting from a representation of K[S] as a ring of invariants of a subgroup of the torus T_{n}(K) acting diagonally on K[Y_{1}, …, Y_{n}].

The most basic source of pure subsemigroups are systems composed of homogeneous linear diophantine equations

a_{i1}x_{1} + ⋯ + a_{in}x_{n} = 0, i = 1, …, u,

and homogeneous congruences

b_{j1}x_{1} + ⋯ + b_{jn}x_{n} ≡ 0 mod c_{j} j = 1, …, v.

The set of nonnegative solutions x ∈ ℤ_{+}^{n} of such a system evidently forms a pure subsemigroup of ℤ_{+}^{n}.

In module-theoretic terms the equation S = T ∩ gp(S) says that the monomial K-vector space complement of K[S] in K[T] is a K[S]-submodule of K[T], and this condition is actually equivalent to the requirement that K[S] is a direct summand of K[T] as a K[S]-module.

The next proposition describes all pure embeddings of S into a free affine semigroup:

Proposition 2.2. Let S be a positive normal affine semigroup, and ξ: gp(S) → ℤ_{+}^{n} an embedding. Then ξ is a pure embedding if and only if n ≥ s = # supp(S) and, up to renumbering, there exist e_{i} ∈ ℤ, e_{i} > 0, such that ξ_{i} = e_{i}σ_{i}, i = 1, …, s.

Moreover, each of ξ_{s+1}, …, ξ_{n} is a rational linear combination of σ_{1}, …, σ_{s} with non-negative coefficients.

Proof. Write gp(S) = ℤ^{r}. The embedding ξ can be extended to an ℝ-linear embedding ξ: ℝ^{r} → ℝ^{n}. Then ξ(C(S)) = C(ξ(S)). Suppose that ξ(S) is a pure subsemigroup of ℤ_{+}^{n}. Then the restrictions of the coordinate linear forms of ℝ^{n} to ξ(ℝ^{r}) cut out the cone ξ(C(S)) from ξ(ℝ^{r}). In other words, the inequalities ξ_{i}(s) ≥ 0, i = 1, …, n, define the cone C(S). Thus the ξ_{i} belong to the dual cone

C(S)^{∗} = {α ∈ Hom_{ℝ}(ℝ^{r}, ℝ): α(x) ≥ 0 for all x ∈ C(S)},

and they in fact generate it since C(S) = C(S)^{∗∗}. However, then we must have at least one linear form from each extreme ray of C(S^{∗}) among the ξ_{i}. The extreme rays of C(S)^{∗} correspond bijectively to the support forms σ_{i}, and each non-zero linear form α in the extreme ray through σ_{i} is of the form α = eσ_{i} with e ∈ ℤ_{+}. The rest is clear, as well as the converse implication.

Corollary 2.3. Cl(K[S]) is isomorphic to a subquotient of ℤ^{n}/ξ(gp(S)) if ξ is pure.

Proof. By the proposition we can assume that ξ_{i} = e_{i}s_{i}, e_{i} ∈ ℤ_{+}, e_{i} > 0, i = 1, …, s. We define ζ: ℤ^{n} → ℤ^{s} as the restriction to the first s coordinates. Then ϑ = ζ ∘ ξ is injective, and we have a chain of subgroups

ϑ(gp(S)) ⊂ ℤe_{1} ⊕ ⋯ ⊕ ℤe_{s} ⊂ ℤ^{s}.

The corollary follows since ℤ^{s}/ϑ(gp(S)) is a quotient of ℤ^{n}/ξ(gp(S))), and (ℤe_{1} ⊕ ⋯ ⊕ ℤe_{s})/ϑ(gp(S)) is isomorphic to Cl(K[S]).

Theorem 2.1 and Corollary 2.3 show that the divisor class group sets the lower limit to all diagonalizable groups D having K[S] as their ring of invariants. In fact, if D ⊂ T_{n}(K) acts on Q = K[X_{1}, …, X_{n}], then Q^{D} is generated by a pure (normal) subsemigroup of the semigroup ℤ_{+}^{n} of monomials of Q, and as in the proof of Theorem 2.1 one sees that ℤ^{n}/ gp(S) ≅ Hom_{ℤ}(D, K^{×}).

If S is a pure subsemigroup of ℤ_{+}^{n}, then (as in the special case of the standard embedding)

Q = K[X_{1}, …, X_{n}] = ⨁_{z∈ℤ^{n}/ gp(S)}Q_{z}

where Q_{z} is the K-vector space generated by all monomials that have residue class −z. Clearly Q_{z} is an K[S]-submodule, and Q is a ℤ^{n}/ gp(S)-graded K[S]-algebra.

Evidently Q_{z} ≠ 0 for all z ∈ ℤ^{n}/ gp(S) if and only if S contains an element of ℤ_{+}^{n} all of whose components are positive. Every non-zero K[S]-module Q_{z} is finitely generated and of rank 1. In fact, multiplication by the monomial X^{z} maps Q_{z} into K[gp(S)] and if the image contains a monomial with a negative exponent, then S contains an element whose corresponding component is positive. Since the exponents of the monomials in Q_{z} are bounded below by z we can embed Q_{z} into K[S]. The modules Q_{z} are called the coset modules of S.

In the invariant-theoretic context discussed above, the Q_{z} are simply the modules of covariants associated with the character z ∈ ℤ^{n}/ gp(S) ≅ Hom_{alg}(D, K^{×}). In fact, an element x ∈ Q = K[X_{1}, …, x_{n}] belongs to Q_{z} if and only if δ(x) = δ(z)x for all δ ∈ D.

In the context of systems of linear equations and congruences the coset modules correspond to inhomogeneous such systems. In fact, the monomials forming the K-basis of Q_{z} are the set of non-negative solutions of an inhomogeneous system of linear diophantine equations and inequalities whose associated homogeneous system defines S.

Proposition 2.4. Suppose that S is a positive normal affine semigroup and ξ: S → ℤ_{+}^{n} a pure embedding. Let R = K[S], s = # supp(S), and Q = K[X_{1}, …, X_{n}]. With the notation as in Proposition 2.2, the following are equivalent:

(a) every non-zero coset module Q_{z} of ξ(S) is divisorial;

(b) for each i = 1, …, n there exists j_{i} ∈ {1, …, s} and e_{i} ∈ ℤ_{+} with ξ_{i} = e_{i}σ_{j_{i}}.

Proof. The implication (b) ⟹ (a) is proved in the same way as Theorem 2.1. In fact,

Q_{z} ≅ ⋂_{i=1}^{n} 𝔭_{j_{i}}^{(z′_{i})}, z′_{i} = ⌈z_{i}/e_{i}⌉.

For the converse implication we may assume that ξ_{n} is not a multiple of any σ_{j}. Then we write ξ_{n} as a non-negative linear combination of σ_{1}, …, σ_{s} with as few non-zero coefficients as possible. By Carathédory’s theorem on convex cones and after reordering the σ_{j} we can assume that ξ_{n} = a_{1}σ_{1} + ⋯ + a_{p}σ_{p} where p ≥ 2, σ_{1}, …, σ_{p} are linearly independent, and a_{1}, …, a_{p} > 0 are rational numbers.

It is not hard to check that gp(S) contains elements t_{j} with σ_{j}(t_{j}) < 0 and σ_{k}(t_{j}) ≥ 0 for all k = 1, …, s, j ≠ k. Furthermore S contains elements u_{1}, …, u_{s} with σ_{j}(u_{j}) = 0 and σ_{k}(u_{j}) > 0 for j ≠ k. Given q ∈ ℤ_{+} and j ∈ {1, …, s}, we can therefore find an element v_{jq} ∈ gp(S) such that σ_{j}(v_{jq}) ≤ −q, but ξ_{n}(v_{jq}) ≥ 0.

Set

z_{m} = (−m, …, −m, 0) mod ξ(gp(S)).

Then Q_{z_{m}} is isomorphic to the fractional ideal I(m) spanned by all the monomials x ∈ S with ξ_{j}(x) ≥ −m for j = 1, …, n − 1 and ξ_{n}(x) ≥ 0. Let

J(m) = 𝔭_{1}^{(b_{1}(m))} ∩ ⋯ ∩ 𝔭_{s}^{(b_{s}(m))}

be the smallest divisorial ideal containing I(m). Then the existence of the elements v_{jq} implies b_{i}(m) → −∞ with m → ∞ for all i = 1, …, s.

On the other hand we can find an element w ∈ gp(S) with σ_{1}(w) < 0 and σ_{j}(w) = 0 for j = 2, …, p. In particular w ∈ J(m) for m ≫ 0. However ξ_{n}(w) < 0. Therefore I(m) ≠ J(m) for m ≫ 0, and Q_{z_{m}} ≅ I(m) is not divisorial.

Condition (b) of Proposition 2.4 can easily be checked algorithmically. For example, if a system of generators E = {x_{1}, …, x_{m}} of S is known, then one forms the sets E_{i} = {x_{j}: ξ_{i}(x_{j}) = 0} for each coordinate i = 1, …, n. Condition (b) is satisfied if and only if none of the E_{i} contains some E_{k} properly, and one has n = # supp(S) if and only if none of the E_{i} contains some E_{k}, k ≠ i.

3. Depth of divisor classes

Divisorial ideals of normal semigroup rings whose class is a torsion element are Cohen–Macaulay. More generally, the following holds:

Theorem 3.1. Let S ⊂ ℤ_{+}^{n} be a pure subsemigroup and set R = K[S], Q = K[ℤ_{+}^{n}]. Furthermore let Ŝ = {z ∈ ℤ_{+}^{n}: mz ∈ S for some m ∈ ℕ} be the integral closure of S in ℤ_{+}^{n}.

(a) For all z ∈ gp(Ŝ)/ gp(S) the coset module Q_{z} is a Cohen–Macaulay R-module, provided Q_{z} ≠ 0.

(b) In particular every divisorial ideal I of R whose class in Cl(R) is a torsion element is a Cohen–Macaulay R-module.

Proof. (a) Since K[Ŝ] is a normal affine semigroup, the ring K[Ŝ] is Cohen–Macaulay by Hochster’s theorem [Ho]. It decomposes into the direct sum of the finitely many and finitely generated R-modules Q_{z} considered in (a). Therefore it is a Cohen– Macaulay ring if and only if all the Q_{z} are Cohen–Macaulay R-modules.

(b) This follows from (a) if one chooses S ⊂ ℤ_{+}^{s} as the standard embedding of R.

In general a normal semigroup ring R may very well have Cohen–Macaulay divisorial ideals whose classes are not torsion. This fact and several other aspects of our discussion are illustrated by the following

Example 3.2. Consider the Segre product

R_{mn} = K[X_{i}Y_{j}: 1 ≤ i ≤ m, 1 ≤ j ≤ n]

of the polynomial rings K[X_{1}, …, X_{m}] and K[Y_{1}, …, Y_{n}]. It has divisor class group isomorphic to ℤ, and the two generators of Cl(R) correspond to the prime ideals 𝔭 = (X_{1}Y_{1}, …, X_{1}Y_{n}) and 𝔮 = (X_{1}Y_{1}, …, X_{m}Y_{1}). Their symbolic powers coincide with the ordinary ones: 𝔭^{(i)} = 𝔭^{i} and 𝔮^{(i)} = 𝔮^{i} for all i ≥ 0 (see [BV, (9.18)]). Therefore

μ(𝔭^{(i)}) = μ(𝔭^{i}) = \binom{n + i − 1}{n − 1}, μ(𝔮^{(i)}) = μ(𝔮^{i}) = \binom{m + i − 1}{m − 1}.

The Cohen–Macaulay divisorial ideals are represented by

𝔮^{(n−1)}, …, 𝔮, R_{mn}, 𝔭, …, 𝔭^{(m−1)}

(see Bruns and Guerrieri [BGu]), and in particular, their number is finite. However, the finiteness of the number of Cohen–Macaulay classes is not a peculiar property of R_{mn}: it holds for all normal semigroup rings, as we will see in Corollary 4.2.

Moreover, one has

inf_{i} depth 𝔭^{(i)} = m,  inf_{i} depth 𝔮^{(i)} = n,

and for i ≫ 0 the minimal values are attained (see [BV, (9.27)]). Set p(i) = μ(𝔭^{(i)}).It follows that the degree of the polynomial p and inf_{i} depth 𝔭^{(i)} add up to m + n − 1 = dim R. This is another instance of a general fact (see Theorem 4.5).

That divisorial ideals I of non-torsion class are in general not Cohen-Macaulay, follows already from the asymptotic behaviour of depth I described in the next theorem. (See the proof of Theorem 2.1 for the definition of the modules M_{c} representing the divisor classes.) By depth_{R} M or simply depth M we denote the length of a maximal M-sequence for a finitely generated R-module M. If S is positive, R = K[S] and M is gp(S)-graded, then depth_{R} M = depth_{R_{𝔪}} M_{𝔪} where 𝔪 is the maximal ideal generated by all the non-unit monomials of R. (See [BH, 1.5.15] for the proof of an analogous statement about ℤ-graded rings and modules.) For an ideal I in a noetherian ring R we let grade I denote the length of a maximal R-sequence in I; one always has grade I ≤ height I. But if R is Cohen–Macaulay, then grade I = height I.

Theorem 3.3. Let K be a field, S a positive normal affine semigroup, R = K[S], and σ: R → P = K[X_{1}, …, X_{s}] the standard embedding. Furthermore let 𝔪 be the irrelevant maximal ideal of R generated by all non-unit monomials, and λ the maximal length of a monomial R-sequence. Then

λ ≤ grade 𝔪P = min{depth M_{c}: c ∈ Cl(R)}.

Proof. For the inequality it is enough to show that a monomial R-sequence is also a P -sequence. (It is irrelevant whether we consider P as an R-module or a P -module if elements from R are concerned.) Let μ_{1}, …, μ_{u} be monomials in R forming an R-sequence. Then the subsets A_{i} = Ass_{R}(R/(μ_{i})) are certainly pairwise disjoint. On the other hand, A_{i} consists only of monomial prime ideals of height 1 in R, since R is normal. So A_{i} = {𝔭_{j}: σ_{j}(μ_{i}) > 0}, and the sets of indeterminates of P that divide σ(μ_{i}) in P, i = 1, …, u, are pairwise disjoint. It follows that μ_{1}, …, μ_{u} form a P -sequence.

In order to prove the equality we first extend the field K to an uncountable one. This is harmless, since all data are preserved by base field extension. Then we can form a maximal P -sequence in 𝔪P by elements from the K-vector subspace 𝔪. Such a P -sequence of elements in R then has length equal to grade 𝔪P and is clearly an M-sequence for every R-direct summand M of P, and in particular for each of the modules M_{c} representing the divisor classes. Thus depth M_{c} ≥ grade 𝔪P.

Whereas this argument needs only finite prime avoidance, we have to use countable prime avoidance for the converse inequality. Suppose that u < min{depth M_{c}: c ∈ Cl(R)} and that x_{1}, …, x_{u} is a P -sequence in 𝔪. Then the set

A = ⋃_{c∈Cl(R)} Ass(M_{c}/(x_{1}, …, x_{u})M_{c})

is a countable set of K-vector subspaces of 𝔪. Each prime ideal associated to M_{c}/(x_{1}, …, x_{u})M_{c} is a proper subspace of 𝔪 because of u < depth M_{c}. Hence A cannot exhaust 𝔪, as follows from elementary arguments. So we can choose an element x_{u+1} in 𝔪 not contained in a prime ideal associated to any of the M_{c}/(x_{1}, …, x_{u})M_{c}. So x_{u+1} extends x_{1}, …, x_{u} to an M_{c}-sequence simultaneously for all c ∈ Cl(R).

Both the numbers λ and grade 𝔪P can be characterized combinatorially:

Proposition 3.4. With the notation of the previous theorem, the following hold:

(a) grade 𝔪P is the minimal number u of facets F_{i_{1}}, …, F_{i_{u}} of C(S) such that F_{i_{1}} ∩ ⋯ ∩ F_{i_{u}} = {0}.

(b) λ is the maximal number ℓ of subsets F_{1}, …, F_{ℓ} of F = {F_{1}, …, F_{s}} with the following properties:

(i) F_{i} ∩ F_{j} = ∅, (ii) ⋂_{F ∈F\F_{i}} F ≠ {0}

for all i, j such that i ≠ j.

Proof. (a) The ideal 𝔪P of P is generated by monomials. Therefore all its minimal prime ideals are generated by indeterminates of P. The ideal generated by X_{i_{1}}, …, X_{i_{u}} contains 𝔪P if and only if for each monomial μ ∈ 𝔪 there exists a σ_{i_{j}} such that σ_{i_{j}}(μ) > 0. The monomials for which none such inequality holds are precisely those in F_{i_{1}} ∩ ⋯ ∩ F_{i_{u}}.

(b) Let μ_{1}, …, μ_{ℓ} be a monomial R-sequence. Then the sets F_{i} = {F: 𝔭_{F} ∈ Ass(R/(μ_{i}))} are pairwise disjoint, and moreover μ_{i} ∈ ⋂_{F ∈F\F_{i}} F. Thus conditions (i) and (ii) are both satisfied.

For the converse one chooses monomials μ_{i} ∈ ⋂_{F ∈F\F_{i}} F. Then {F: 𝔭_{F} ∈ Ass(R/(μ_{i}))} ⊂ F_{i}, and since the F_{i} are pairwise disjoint, the μ_{i} form even a P -sequence as observed above.

Remark 3.5. The following are equivalent:

(a) S is simplicial;

(b) every divisorial ideal of R is a Cohen–Macaulay module.

If S is simplicial, then Cl(R) is finite, and so every divisorial ideal is Cohen– Macaulay by Theorem 3.1. Conversely, if every divisorial ideal is Cohen–Macaulay, then it is impossible that facets F_{1}, …, F_{u}, u < rank S, of C(S) intersect only in 0, and this property characterizes simplicial cones. However, the implication (b) ⟹ (a) follows also from Theorem 4.1 below.

It is an amusing consequence that a rank 2 affine semigroup is simplicial – an evident geometric fact. In fact, if dim R ≥ 2, then depth I ≥ 2 for every divisorial ideal.

The results of this section can be partially generalized to arbitrary pure embeddings; we leave the details to the reader.

4. The number of generators

In this section we first prove our main result on the number of generators of divisorial ideals of normal semigroup rings R = K[S]. In its second part we then show that it can also be understood and proved as an assertion about the growth of the Hilbert function of a certain multigraded K-algebra.

Theorem 4.1. Let R be a positive normal semigroup ring over the field K, and m ∈ ℤ_{+}. Then there exist only finitely many c ∈ Cl(R) such that a divisorial ideal D of class c has μ(D) ≤ m.

As a consequence of Theorem 4.1, the number of Cohen–Macaulay classes is also finite:

Corollary 4.2. There exist only finitely many c ∈ Cl(R) for which a divisorial ideal of class c is a Cohen–Macaulay module.

Remark 4.3. (a) One should note that μ(D) is a purely combinatorial invariant. If S is the underlying semigroup and T is a monomial basis of a monomial representative of D, then μ(D) is the smallest number g such that there exists x_{1}, …, x_{g} ∈ T with T = (S + x_{1}) ∪ ⋯ ∪ (S + x_{g}). Therefore Theorem 4.1 can very well be interpreted as a result on the generation of the sets of solutions to inhomogeneous linear diophantine equations and congruences (with fixed associated homogeneous system).

(b) Both the theorem and the corollary hold for all normal affine semigroups S, and not only for positive ones. It is easily seen that a normal semigroup S splits into a direct summand of its largest subgroup S_{0} and a positive normal semigroup S′ Thus one can write R = K[S] as a Laurent polynomial extension of the K-algebra R′ = K[S′]. Each divisor class of R has a representative D′ ⊗_{R′} R. Furthermore μ_{R′}(D′) = μ_{R}(D′ ⊗_{R′} R) and the Cohen–Macaulay property is invariant under Laurent polynomial extensions.

We first derive the corollary from the theorem. Let 𝔪 be the irrelevant maximal ideal of R. If M_{c} is a Cohen–Macaulay module, then (M_{c})_{𝔪} is a Cohen–Macaulay module (and conversely). Furthermore

e((M_{c})_{𝔪}) ≥ μ((M_{c})_{𝔪}) = μ(M_{c}).

By Serre’s numerical Cohen–Macaulay criterion (for example, see [BH, 4.7.11]), the rank 1 R_{𝔪}-module (M_{c})_{𝔪} is Cohen–Macaulay if and only if its multiplicity e((M_{c})_{𝔪}) coincides with e(R_{𝔪}).

Proof of Theorem 4.1. Let D be a monomial divisorial ideal of R. As pointed out already, there exist integers a_{1}, …, a_{s} such that the lattice points in the set

C(D) = {x ∈ S: σ_{i}(x) ≥ a_{i}, i = 1, …, s}

give a K-basis of D (again s = # supp(S), and the σ_{i} are the suppoprt forms). The polyhedron C(D) is uniquely determined by its extreme points since each of its facets is parallel to one of the facets of C(S) and passes through such an extreme point. (Otherwise C(D) would contain a full line, and this is impossible if S is positive.)

Moreover, D is of torsion class if and only if C(D) has a single extreme point. This has been proved in [Gu2], but since it is the crucial point (sic!) we include the argument.

Suppose first that D is of torsion class. Then there exists m ∈ ℤ_{+}, m > 0, such that D^{(m)} is a principal ideal, D^{(m)} = xR with a monomial x. It follows that C(D^{(m)}) = mC(D) has a single extreme point in (the lattice point corresponding to) x, and therefore C(D) has a single extreme point.

Conversely, suppose that C(D) has a single extreme point. The extreme point has rational coordinates. After multiplication with a suitable m ∈ ℤ_{+}, m > 0, we obtain that C(D^{(m)}) = mC(D) has a single extreme point x which is even a lattice point. All the facets of C(D^{(m)}) are parallel to those of S and must pass through the single extreme point. Therefore C(D^{(m)}) has the same facets as C(S) + x. Hence C(D^{(m)}) = C(S) + x. This implies D^{(m)} = Rx (in multiplicative notation), and so m annihilates the divisor class of D.

Suppose that D is not of torsion class. We form the line complex \overline{L} consisting of all 1-dimensional faces of the polyhedron C(D). Then \overline{L} is connected, and each extreme point is an endpoint of a 1-dimensional face. Since there are more than one extreme points, all extreme points are endpoints of compact 1-dimensional faces, and the line complex L(D) formed by the compact 1-dimensional faces is also connected. Since each facet passes through an extreme point, D is uniquely determined by L(D) (as a subset of ℝ^{s}).

Let C be an infinite family of divisor classes and choose a divisorial ideal D_{c} of class c for each c ∈ C. Assume that the minimal number of generators μ(D_{c}), c ∈ C, is bounded above by a constant C. By Lemma 4.4 below the Euclidean length of all the line segments ℓ ∈ L(D_{c}), c ∈ C, is then bounded by a constant C′.

It is now crucial to observe that the endpoints of all the line segments under consideration lie in an overlattice L = ℤ^{n}[1/d] of ℤ^{n}. In fact each such point is the unique solution of a certain system of linear equations composed of equations σ_{i}(x) = a_{i}, and therefore can be solved over ℤ[1/d] where d ∈ ℤ is a suitable common denominator. (Again we have denoted the support forms of S by σ_{i}.)

Let us consider two line segments ℓ and ℓ′ in ℝ^{n} as equivalent if there exists z ∈ ℤ^{n} such that ℓ′ = ℓ + z. Since the length of all the line segments under consideration is bounded and their endpoints lie in ℤ^{n}[1/d], there are only finitely many equivalence classes of line segments ℓ ∈ L(D_{c}), c ∈ C.

Similarly we consider two line complexes L(D) and L(D′) as equivalent if L(D′) = L(D) + z, z ∈ ℤ^{n}. However, this equation holds if and only if C(D′) = C(D) + z, or, in other words, the divisor classes of D and D′ coincide.

Since there are only finitely many equivalence classes of line segments and the number of lines that can appear in a complex L(D) is globally bounded (for example, by 2^{s}), one can only construct finitely many connected line complexes that appear as L(D), up to equivalence of line complexes. This contradicts the infinity of the family C.

Lemma 4.4. Let S be a positive normal semigroup, K a field, D a monomial divisorial ideal whose class is not torsion. Then there exists a constant M > 0, which is independent of K, such that μ(D) ≥ Mλ where λ is the maximal Euclidean length of a compact 1-dimensional face of C(D).

Proof. We assume that ℤ^{n} = gp(S) so that the cone C(S) and the polyhedron C(D) are subsets of ℝ^{n}. Let ℓ be a 1-dimensional compact face of C(D). Suppose D is given by the inequalities

σ_{i}(x) ≥ a_{i}, i = 1, …, s (s = # supp(C(S))).

There exists ε > 0 such that U_{ε}(x)∩C(D) contains a lattice point for each x ∈ C(D). (In fact, C(S) contains a unit cube, and x+C(S) ⊂ C(D) for x ∈ C(D).) Let x ∈ ℓ.

We can assume that

σ_{i}(x) \begin{cases}= a_{i}, & i = 1, …, m, \\ > a_{i}, & i > m. \end{cases}

Let τ = σ_{1} + ⋯ + σ_{m}. There exists C > 0 such that τ(y) < C for all y ∈ ℝ^{n} with |y| < ε.

Furthermore we have τ(z) > 0 for all z ∈ C(S), z ≠ 0. Otherwise the facets F_{1}, …, F_{m} would meet in a line contained in C(S), and this is impossible if ℓ is compact. In particular there are only finitely many lattice points z in S such that τ(z) < C, and so there exists δ > 0 such that τ(z) < C for z ∈ S is only possible with |z| < δ.

Now suppose that D is generated by x_{1}, …, x_{q}. For x ∈ ℓ we choose a lattice point p ∈ U_{ε}(x) ∩ C(D). By assumption there exists z ∈ C(S) such that p = x_{i} + z. Then

τ(z) = τ(p) − τ(x_{i}) ≤ τ(p) − τ(x) = τ(p − x) < C.

Thus |z| < δ, and therefore |x − x_{i}| < δ + ε.

It follows that the Euclidean length of ℓ is bounded by 2q(δ + ε). Of course δ depends on τ, but there exist only finitely many choices for τ if one varies ℓ.

As pointed out, the polyhedron C(D) contains a 1-dimensional compact face if D is not of torsion class, but in general one cannot expect anything stronger. On the other hand, there exist examples for which C(D) for every non-torsion D has a compact face of arbitrarily high dimension; see Example 3.2.

If C(D) has a d-dimensional face F, then the argument in the proof of Lemma 4.4 immediately yields that μ(D^{(j)}) ≥ Mj^{d} for a constant M > 0: one has only to replace the length of the line segment by the d-dimensional volume of F. We now give another proof of a slightly more general statement. As we will see, it leads to a quite different proof of Theorem 4.1.

Let S be a positive normal affine semigroup. Recall that the polynomial ring P of the standard embedding σ: R → P decomposes into the direct sum of modules M_{c}, c ∈ Cl(R). In the following C(M_{c}) stands for any of the congruent polyhedra C(D) where D is a divisorial ideal of class c.

Theorem 4.5. Let c, d ∈ Cl(R) and suppose that c is not a torsion element.

(a) Then lim_{c→∞} μ(M_{jc+d}) = ∞.

(b) More precisely, let d be the maximal dimension of the compact faces of C(M_{c}). Then there exists e ∈ ℕ such that

lim_{c→∞} \frac{μ(M_{(ej+k)c+d})}{c^{d}}

is a positive real number for each k = 0, …, e − 1.

(c) One has inf_{j} depth M_{jc} = dim R − d and inf_{j} depth M_{cj+d} ≤ dim R − d.

Proof. Let

D = ⨁_{j=0}^{∞} M_{jc} and M = ⨁_{j=0}^{∞} M_{jc+d}.

Then D is a finitely generated K-algebra. This follows for general reasons from Theorem 6.1 below: D is the direct sum of graded components of the Cl(R)-graded R-algebra P, taken over a finitely generated subsemigroup of Cl(R). Theorem 6.1 also shows that M is a finitely generated D-module. However, these assertions will be proved directly in the following. In particular we will see that D is a normal semigroup ring over K.

By definition D is a ℤ_{+}-graded R-algebra with D_{0} = M_{0} = R, and M is a graded D-module if we assign degree j to the elements of M_{jc+d}. There exists e > 0 such that D is a finitely generated module over its R-subalgebra generated by elements of degree e; for example, we can take e to be the least common multiple of the degrees of the generators of D as an R-algebra. Let E be the eth Veronese subalgebra of D. We decompose M into the direct sum of its E-submodules

M_{k} = ⨁_{j=0}^{∞} M_{(ej+k)c+d}, k = 0, …, e − 1.

In view of what has to be proved, we can replace D by E and M by M_{k}. Then we have reached a situation in which D is a finitely generated module over the subalgebra generated by its degree 1 elements.

Note that M is isomorphic to an ideal of D: multiplication by a monomial X^{a} such that a has residue class −(d + k) in Cl(R) ≅ ℤ^{s}/σ(gp(S)) maps M into D. Since M is not zero (and D is an integral domain), we see that Supp M = Spec D.

Let 𝔪 be the irrelevant maximal ideal of R; it is generated by all elements x ∈ S, x ≠ 1 (in multiplicative notation). Then clearly \overline{M} = M/𝔪M is a finitely generated D = D/𝔪D-module with Supp \overline{M} = Spec D. Note that D is a K-algebra with D_{0} = K in a natural way. Furthermore it is a finitely generated module over its subalgebra D′ generated by its degree 1 elements. In particular \overline{M} is a finitely generated D′-module. By construction (and Nakayama’s lemma) we have

μ(M_{(ej+k)c+d}) = dim_{K} M_{(ej+k)c+d}/𝔪M_{(ej+k)c+d} = H(\overline{M}, j)

where H denotes the Hilbert function of \overline{M} as a ℤ_{+}-graded D- or D′-module. For j ≫ 0 the Hilbert function is given by the Hilbert polynomial. It is a polynomial of degree δ − 1 where δ is the Krull dimension of \overline{M}. Note that Supp \overline{M} = Spec D, since Supp M = Spec D; in particular one has dim \overline{M} = dim D. Moreover the leading coefficient of the Hilbert polynomial is e(\overline{M})/(δ − 1)! and so all the claims for M follow if d + 1 = δ > 1.

At this point we have to clarify the structure of D as a normal semigroup ring over K. For convenience we choose a divisorial ideal I ⊂ R of class c generated by monomials. Then there exists an R-module isomorphism M_{c} → I mapping monomials to monomials, and such an isomorphism induces a K-algebra isomorphism from D to

R = ⨁_{j=0}^{∞} I^{(j)}T^{j} ⊂ R[T] = K[S ⊕ ℤ_{+}].

There exist a_{1}, …, a_{s} ≥ 0 such that I = 𝔭_{1}^{(a_{1})} ∩ ⋯ ∩ 𝔭_{s}^{(a_{s})}. The monomial corresponding to (s, z) ∈ gp(S) ⊕ ℤ belongs to R if and only if

z ≥ 0,  σ_{i}(s) ≥ 0,  σ_{i}(s) − za_{i} ≥ 0,  i = 1, …, s.

It follows immediately that R is a normal semigroup ring over K. Let S be its semigroup of monomials. One has gp(S) = gp(S) ⊕ ℤ, and the elements with last component j give the monomials of I^{(j)}.

It is not hard to show that the faces ⊄ C(S) of C(S) are the closed envelopes of the ℝ_{+}-envelopes of the faces of C(I)′ = {(x, 1): x ∈ C(I)}, and that exactly those faces that do not contain an element from 𝔪 intersect C(I)′ in a compact face. Since the dimension of R/𝔪R is just the maximal dimension of a face F of C(S) not containing an element of 𝔪, we see that dim R/𝔪R = d + 1. In fact, the largest dimension of a compact face of C(I)′ is d, and it extends to a d + 1-dimensional face of C(S).

Thus δ = d + 1 and δ > 1, since C(I) has at least a 1-dimensional compact face: by hypothesis I is not of torsion class.

For part (c) we note that height 𝔪R = grade 𝔪R = dim R−dim R/𝔪R = dim R− d since R is Cohen–Macaulay by Hochster’s theorem (and all the invariants involved are stable under localization with respect to the maximal ideal of R generated by monomials). Moreover grade 𝔪R = inf_{j} depth M_{cj}, as follows by arguments analogous to those in the proof of Theorem 3.3.

By similar arguments the inequality for inf_{j} depth M_{cj+d} results from height 𝔪R = dim R − d.

Remark 4.6. The limits in Theorem 4.5(b) coincide if and only if the D′-modules M_{k}/𝔪M_{k} all have the same multiplicity. However, in general this is not the case. As an example one can take the semigroup ring R = K[U^{2}, UV, V^{2}, XW, Y W, XZ, Y Z]. It has divisor class group Cl(R) = ℤ/(2) ⊕ ℤ; the class of 𝔭 = (U^{2}, UV) is non-zero, but a 2-torsion element, and the class of 𝔮 = (XW, Y W) generates the free direct summand. Let c ∈ Cl(R) be the class of 𝔭𝔮. It is not hard to show that jc is represented by the divisorial ideal 𝔮𝔭^{j} for j odd, and by 𝔭^{j} for j even. The limits for k = 0 and k = 1 therefore differ by a factor of 2 (d = 0, e = 2).

Second proof of Theorem 4.1. Let P be the polynomial ring of the standard embedding of R. Then P is a Cl(R)-graded R-algebra whose graded component P_{c}, c ∈ Cl(R) is the module M_{c}. Passing to residue classes modulo 𝔪 converts the assertion of the theorem into a statement about the Hilbert function (with respect to K) of the Cl(R)-graded K-algebra P/𝔪P; note that (P/𝔪P)_{0} = R/𝔪 = K. By Theorem 4.5 the Hilbert function goes to infinity along each arithmetic progression in Cl(R). Therefore we are in a position to apply Theorem 6.3 below. It says that there are only finitely many c ∈ Cl(R) where μ(M_{c}) = H(P/𝔪P, c) does not exceed a given bound m.

This deduction of Theorem 4.1 uses the combinatorial hypotheses on R only at a single point in the proof of Theorem 4.5, namely where we show that dim D/𝔪D ≥ 2. Thus the whole argument can be transferred into a more general setting, provided an analogous condition on dimension holds.

5. Convex ideals

Let ξ_{1}, …, ξ_{n} be ℤ-linear forms on ℤ^{r}. Then

S = {x ∈ ℤ^{r}: ξ_{i}(x) ≥ 0, i = 1, …, n} (∗)

is a normal semigroup. For a ∈ ℤ^{n} we set

T (a; ξ) = {x ∈ ℤ^{r}: ξ_{i}(x) ≥ a_{i}, i = 1, …, n}.

Let K be a field. Then I(a; ξ) = KT (a; ξ) is an R = K[S]-module in a natural way.

In the following we will always assume that S is positive and that gp(S) = ℤ^{r}; furthermore we will assume that ξ is non-degenerate, i.e. ξ_{i} ≠ 0 for all i; this assumption is certainly no restriction of generality. Then it follows easily that I(a; ξ) is a non-zero fractional ideal of R for every a. These ideals are called ξ-convex.

If σ = (σ_{1}, …, σ_{s}) consists of the support forms of S, then the σ-convex ideals are just the monomial divisorial ideals of R studied in the previous sections.

The linear map ξ: gp(S) → ℤ^{n} evidently restricts to a pure embedding of S into ℤ^{n} (if S is positive), and, conversely, a pure embedding induces a presentation (∗) of S. As in the proof of Theorem 2.1 one sees that each coset modules of ξ(S) is isomorphic to a ξ-convex ideal, and vice versa. Proposition 2.4 therefore describes the condition under which all ξ-convex ideals are divisorial (and in its proof we have actually switched from a coset module to the corresponding ξ-convex ideal).

However, in contrast to the standard embedding, different coset modules can now be isomorphic. Proposition 5.1 below clarifies this fact. For T ⊂ gp(S) we define the “effective ξ-bounds” eff(T; ξ) ∈ ℤ^{n} component wise by

eff(T; ξ)_{i} = inf{ξ_{i}(t): t ∈ T };

for a fractional monomial ideal I we set eff(I; ξ) = eff(T; ξ) where T is the monomial basis of I; and for a ∈ ℤ^{n} we set eff(a; ξ) = eff(I(a; ξ); ξ). The convex polyhedron C(I; ξ) ⊂ ℝ^{r} associated with I is given by

C(I; ξ) = {x ∈ ℝ^{r}: ξ_{i}(x) ≥ eff(I; ξ)_{i}, i = 1, …, n}.

Proposition 5.1. (a) The assignment I ↦ C(I; ξ) is injective on the set of ξ-

convex ideals I.

(b) The following are equivalent for ξ-convex ideals I and J:

(i) I and J are isomorphic R-modules.

(ii) There exists y ∈ gp(S) such that eff(J; ξ) = eff(I, ξ) + ξ(y). (iii) There exists y ∈ gp(S) such that C(I; ξ) = C(J; ξ) + ξ(y).

(c) If C(I; ξ) has a single extreme point for a ξ-convex ideal, then I is a divisorial ideal of torsion class.

(d) The coset modules Q_{a}, Q_{b} of ξ(S) associated with the residue classes of a, b ∈ ℤ^{n} are isomorphic if and only if eff(a; ξ) ≡ eff(b; ξ) mod ξ(gp(S)).

Proof. (a) One can recover I from C(I; ξ).

(b) Fractional monomial ideals I and J of K[S] are isomorphic if and only if there exists a monomial x ∈ K[gp(S)] such that J = Ix. If one converts this into additive notation, one immediately obtains the equivalence of (i), (ii), and (iii).

(c) Set a_{i} = eff(I; ξ_{i}), i = 1, …, n. Each of the hyperplanes given by the equations ξ_{i}(x) = a_{i} passes through an extreme point of C(I; ξ). Since there is a single extreme point z, all the parallels to the support hyperplanes pass through this point. Thus C(I; ξ) = C(S) + z, and C(I; ξ) is cut out by parallels of the support hyperplanes of S. Then I is a divisorial ideal. Divisorial ideals I for which C(I) has a single extreme point are of torsion class as observed in the proof of Theorem 4.1.

(d) Since Q_{a} ≅ I(eff(a; ξ); ξ) and Q_{b} ≅ I(eff(b; ξ); ξ), this follows from (b).

It is now clear that Lemma 4.4 generalizes to the case of ξ-convex ideals, and since the isomorphism classes of ξ-convex ideals are again parameterized by the equivalence classes of the polyhedra C(I; ξ) modulo translations by vectors from gp(S), we obtain a complete generalization of Theorem 4.1:

Theorem 5.2. For every m ∈ ℤ_{+} there exist only finitely many isomorphism classes of ξ-convex ideals such that a representative I has μ(I) ≤ m.

There is no need for a corollary regarding the Cohen–Macaulay property, since Cohen–Macaulay fractional ideals are automatically divisorial. As outlined in Remark 4.3, Theorem 5.2 has a purely combinatorial statement (and, by the way, has been proved by purely combinatorial arguments).

Remark 5.3. (a) Let us set Eff(ℤ^{n}; ξ) = {eff(a; ξ): a ∈ ℤ^{n}}. Then it is not hard to show that Eff(ℤ^{n}; ξ) is a subsemigroup of ℤ^{n} containing ξ(gp(S)), and it follows immediately from Proposition 5.1 that the semigroup Cl(R; ξ) = Eff(ℤ^{n}; ξ)/ξ(gp(S)) parametrizes the isomorphism classes of ξ-convex ideals.

For the converse of Proposition 5.1(c) one has to replace “torsion class” (in Cl(R)) by “torsion class in Cl(R; ξ)”.

Theorem 4.5 generalizes to arithmetic progressions in Cl(R; ξ). We leave the details to the reader. If Cl(R; ξ) is always finitely generated, then the second proof of Theorem 4.1 can also be generalized.

(b) Theorem 5.2 can be generalized to arbitrary normal affine semigroups in the same way as Theorem 4.1 was generalized in Remark 4.3.

(c) It is necessary to fix ξ in Theorem 5.2. Already for S = ℤ_{+}^{2} one can easily find infinitely many pairwise non-isomorphic ideals generated by 2 elements such that each of them is ξ-convex for a suitable ξ.

6. On the growth of Hilbert functions

We introduce some terminology: if S is a subsemigroup of an abelian group G, then T ⊂ G is an S-module if S + T ⊂ T (the case T = ∅ is not excluded). If S is finitely generated and T is a finitely generated S-module, then every S-module T′ ⊂ T is also finitely generated. For example, this follows by “linearization” with coefficients in a field K: M = KT ⊂ K[G] is a finitely generated module over the noetherian ring R = K[S], and so all its submodules are finitely generated over R. For KT′ this implies the finite generation of T′ over S.

First we note a result on the finite generation of certain subalgebras of graded algebras and submodules of graded modules. We do not know of a reference covering it in the generality of Theorem 6.1.

Gordan’s lemma says that a subsemigroup S ⊂ ℤ^{n} is finitely generated if and only if the cone C(S) ⊂ ℝ^{n} is finitely generated. It implies that the integral closure Ŝ of an affine subsemigroup S in ℤ^{n} is finitely generated, since C(S) = C(Ŝ). The reader may check that we do not use Gordan’s lemma in the proof of Theorem 6.1. We will however use that a finitely generated cone has only finitely many support hyperplanes and that Ŝ = C(S) ∩ ℤ^{n}.

Theorem 6.1. Let G be a finitely generated abelian group, S a finitely generated subsemigroup of G, and T ⊂ G a finitely generated S-module. Furthermore let R be a noetherian G-graded ring and M a G-graded finitely generated R-module. Then the following hold:

(a) R_{0} is noetherian ring, and each graded component M_{g}, g ∈ G, of M is a finitely generated R_{0}–module.

(b) A = ⨁_{s∈S} R_{s} is a finitely generated R_{0}-algebra.

(c) N = ⨁_{t∈T} M_{t} is a finitely generated A-module.

Proof. (a) One easily checks that M′R∩M_{g} = M′ for each R_{0}-submodule M′ of M_{g}. Therefore ascending chains of such submodules M′ of M_{g} are stationery.

(b) First we do the case in which G is torsionfree, G = ℤ^{m}, and S is an integrally closed subsemigroup of ℤ^{m}.

Let φ: ℤ^{m} → ℤ be a non-zero linear form. It induces a ℤ-grading on R with deg_{ℤ}(a) = φ(deg_{ℤ^{M}}(a)) for each non-zero ℤ^{m}-homogeneous element of R. Let R′ denote R with this ℤ-grading. Set R′_{−} = ⨁_{k≤0} R′_{k} and define R′_{+} is analogously. By [BH, 1.5.5] the R′_{0}-algebras R′_{+} and R′_{−} are finitely generated R′_{0}-algebras, and R′_{0} is a noetherian ring (by (a)). On the other hand, R′_{0} is a (Ker φ)-graded ring in a natural way, and by induction we can conclude that R′_{0} is a finitely generated R_{0}-algebra.

If S = ℤ^{m}, then it follows immediately that R, the sum of R_{−} and R_{+} as an R_{0}-algebra, is again a finitely generated R_{0}-algebra.

Otherwise S = ℤ^{m} ∩ C(S), and C(S) has at least one support hyperplane:

S = {s ∈ ℤ^{m}: α_{i}(s) ≥ 0, i = 1, …, v}

with v ≥ 1. We use induction on v, and the induction hypothesis applies to R′ = ⨁_{s∈S′} R_{s},

S′ = {s ∈ ℤ^{m}: α_{i}(s) ≥ 0, i = 1, …, v − 1}.

Applying the argument above with φ = α_{v}, one concludes that A = R′_{+} is a finitely generated R_{0}-algebra.

In the general case for G and S we set G′ = G/H where H is the torsion subgroup of G, and denote the natural surjection by π: G → G′. Let R′ be R with the G′-grading induced by π (its homogeneous components are the direct sums of the components R_{g} where g is in a fixed fiber of π). Let S′ be the integral closure of π(S) in G′. Then A′ = ⨁_{s′∈S′} R′_{s′} is a finitely generated algebra over the noetherian ring R′_{0}, as we have already shown. But R′_{0} is a finitely generated module over R_{0} by (a), and so A′ is a finitely generated R_{0}-algebra. In particular, R itself is finitely generated over R_{0}.

It is not hard to check that A′ is integral over A; in fact, each element s ∈ π^{−1}(S′) has a power s^{n} ∈ S for suitable n ∈ ℕ. Furthermore it is a finitely generated Aalgebra, and so a finitely generated A-module. But then a lemma of Artin and Tate (see Eisenbud [Ei, p. 143]) implies that A is noetherian. As shown above, noetherian G-graded rings are finitely generated R_{0}-algebras.

(c) By hypothesis, T is the union of finitely many translates S + t. Therefore we can assume that T = S + t. Passing to the shifted module M(−t) (given by M(−t)_{g} = M_{g−t}), we can even assume that S = T. Now the proof follows the same pattern as that of (b). In order to deal with an integrally closed subsemigroup of a free abelian group G = ℤ^{m}, one notes that M_{+} is a finitely generated module over R_{+} where M_{+} is the positive part of M with respect to a ℤ-grading (induced by a linear form φ: ℤ^{m} → ℤ). This is shown as follows: the extended module RM_{+} is finitely generated over R , and every of its generating systems E ⊂ M_{+} together with finitely many components M_{i}, i ≥ 0, generate M_{+} over R_{+}; furthermore the M_{i} are finitely generated over R_{0} by (a).

For the general situation we consider N′ defined analogously as A′. It is a finitely generated A′-module by the previous argument. Since A′ is a finitely generated A-module, N′ is finitely generated over A, and so is its submodule N.

We note a purely combinatorial consequence.

Corollary 6.2. Let S and S′ be affine subsemigroups of ℤ^{m}, T ⊂ ℤ^{m} a finitely generated S-module, and T′ ⊂ ℤ^{m} a finitely generated T -module. Then S ∩ S′ is an affine semigroup, and T ∩ T′ is a finitely generated S ∩ S′-module.

Proof. We choose a field K of coefficients and set R = K[S′], M = KT′. Then the hypotheses of the theorem are satisfied, and it therefore implies the finite generation of

A = ⨁_{s∈S} R_{s} = K[S ∩ S′], and N = ⨁_{t∈T} M_{t} = K(T ∩ T′).

as a K = R_{0}-algebra and an A-module respectively. However, finite generation of the “linearized” objects is equivalent to that of the combinatorial ones.

The next theorem is our main result on the growth of Hilbert functions. Note that we do not assume that R_{0} = K; the graded components of R and M may even have infinite K-dimension.

Theorem 6.3. Let K be a field, G a finitely generated abelian group, R a noetherian G-graded K-algebra, and M a finitely generated G-graded R-module. Define S to be the subsemigroup of G generated by the elements deg r, r ∈ R \ {0} homogeneous, and define T to be the S-submodule of G generated by the elements deg x, x ∈ M \ {0} homogeneous. Furthermore let H be the G-graded Hilbert function, H(M, t) = dim_{K} M_{t} for all t ∈ G.

Suppose lim_{k→∞} H(M, kc + d) = ∞ for all choices of c ∈ T, c not a torsion element of G, and d ∈ T. Then

#{t ∈ T : H(M, t) ≤ C} < ∞

for all C ∈ ℤ_{+}.

Proof. Note that S is a finitely generated subsemigroup of G, and T is a finitely generated S-module, generated by the degrees of the elements in a finite generating system of homogeneous elements of R as an R_{0}-algebra and M as an R-module respectively. (The finite generation follows from Theorem 6.1.)

We split G as a direct sum of a torsionfree subgroup L and its torsion subgroup G_{tor}. Let R′ = ⨁_{ℓ∈L} R_{ℓ}, and split M into the direct sum

M = ⨁_{h∈G_{tor}} M′_{h}, M′_{h} = ⨁_{ℓ∈L} M_{(ℓ,h)}.

By Theorem 6.1 R′ is a finitely generated R_{0}-algebra and M′_{h} is a finitely generated L-graded R′-module for all h ∈ G_{tor}, and since the hypothesis on the Hilbert function is inherited by M′_{h}, it is enough to do the case G = L = ℤ^{m}.

We use induction on m. In the case m = 1 the hypothesis contains the conclusion. As a first step we want to improve the hypothesis on the Hilbert function from

a “1-dimensional ” condition to a “1-codimensional” condition by an application of the induction hypothesis.

Let U be a proper subgroup of L and u ∈ L. Then U is finitely generated as a subsemigroup. We set

R′ = ⨁_{s∈U} R_{s} and M′ = ⨁_{t∈U+u} M_{t}.

Theorem 6.1 implies that R′ is a finitely generated K-algebra, and M′ is a finitely generated R′-module.

After fixing an origin in U + u we can identify it with U. Therefore we can apply the induction hypothesis to R′ and M′. It follows that

#{t ∈ T ∩ (U + u) : H(t, M) ≤ C} < ∞. (∗)

Now we represent R as the residue class ring of an L-graded polynomial ring P over K in a natural way (in particular the monomials in P are homogeneous in the L-grading). Thus we may assume that R itself is generated by algebraically independent elements as a K-algebra.

Obviously M has a filtration

0 = M_{0} ⊂ M_{1} ⊂ ⋯ ⊂ M_{n} = M

where each successive quotient M_{i+1}/M_{i} is a cyclic L-graded R-module, that is

M_{i+1}/M_{i} ≅ (R/I_{i+1})(−s_{i})

with an L-graded ideal I_{i+1} in R and a shift s_{i} ∈ L. As far as the Hilbert function is concerned, we can replace M by the direct sum of these cyclic modules. After the introduction of a term order we can replace R/(I_{i+1})(−s_{i}) by R/(in(I_{i+1}))(−s_{i}) where in(I_{i+1}) is the initial ideal. It is well known that R/ in(I_{i+1}) has a filtration whose successive quotients are of the form R/𝔭 with a prime ideal 𝔭 generated by monomials, and therefore by indeterminates of R. (For example, see the proof of [BH, 4.1.3], and use that associated prime ideals of multigraded modules are generated by indeterminates if the multigrading is that induced by the semigroup of all monomials in R.)

Altogether this reduces the problem to the case in which the K-vector space M is isomorphic to the direct sum of vector spaces P_{i}(−s_{i}) where P_{i} a polynomial ring generated by indeterminates with degrees in L, and s_{i} ∈ L. Furthermore we can use that the Hilbert function of M satisfies condition (∗). The Hilbert function now counts the total number of monomials in each degree. Replacing the monomials by their exponent vectors, we can deduce the theorem from the next one.

Theorem 6.4. Let G be a finitely generated abelian group, S a finitely generated subsemigroup of G, and T a finitely generated S-submodule of G. Consider maps

ψ_{i}: A_{i} → T, ψ_{i}(x) = φ_{i}(x) + t_{i} for all x ∈ A_{i}

where A_{i} is an affine semigroup, φ_{i} : A_{i} → S is a homomorphism of semigroups, and t_{i} ∈ T, i = 1, …, v. Furthermore let

Ψ: A_{1} ∪ ⋯ ∪ A_{v} → T, Ψ|_{A_{i}} = ψ_{i},

be the map defined on the disjoint union of the A_{i} by all the ψ_{i}. For t ∈ G set

H(t) = #{x ∈ A_{1} ∪ ⋯ ∪ A_{v} : Ψ(x) = t}.

Suppose that lim_{k→∞} H(kc + d) = ∞ for all c ∈ S, c not a torsion element of G, and all d ∈ T. Then

#{t ∈ T : H(t) ≤ C} < ∞

for all C ∈ ℤ_{+}.

Proof. In step (a) we prove the theorem under the assumption that G = L = ℤ^{m} for some m and that

#{t ∈ T ∩ (U + u) : H(t) ≤ C} < ∞.

for all proper subgroups U of L. This is enough to complete the proof of Theorem 6.3. In step (b) we can then use Theorem 6.3.

(a) The first observation is that we can omit all the maps ψ_{i} that are injective. This reduces the function H in each degree by at most v, and has therefore no influence on the hypothesis or the desired conclusion.

The difficult case is C = 0, and we postpone it. So suppose that we have already shown that the number of “gaps” (elements in T with no preimage at all) is finite. Then we can restrict ourselves to Im Ψ if we want to show that there are only finitely many elements with at most C > 0 preimages.

It is enough to show that the elements in Im ψ_{i} with at most C preimages are contained in the union of finitely many sets of the form U + u where U is a proper direct summand of L. Then we can use the hypothesis on the sets {x ∈ T ∩(U +u) : H(x) ≤ C}. We can certainly assume that v = 1 and t_{1} = 0, and have only to consider a non-injective, surjective homomorphism φ : A → S.

For an ideal (i. e. S-submodule) I ≠ ∅ of S we have that S \ I is contained in finitely many sets U + u. In fact, I contains an ideal J ≠ ∅ of the normalization S̄ of S, namely the conductor ideal {s ∈ S: S̄ + s ⊂ S}. (This follows from the corresponding theorem of commutative algebra; see [Gu2, Lemma 5.3] where the assertion has been proved under the superfluous condition that S is positive.) Therefore S contains a set S̄ + s with s ∈ S. It follows that (S \ I) ⊂ (S̄ \ (S̄ + s)).

The latter set is contained in finitely many parallels to the facets of S through points of S (in gp(S)). Each of these parallels is itself contained in a set of type U + u. To sum up, it is enough to find an ideal I in S such that each element of I has at least C + 1 preimages.

Now we go to A and choose a ∈ A such that Ā + a ⊂ A where Ā is again the normalization. The homomorphism φ has a unique extension to a group homomorphism gp(A) → L, also denoted by φ. By assumption Ker φ ≠ 0. A sufficiently large ball B in gp(A) ⊗ ℝ with center 0 therefore contains C + 1 elements of Ker φ, and there exists b ∈ Ā for which B +b is contained in the cone ℝ_{+}A. Thus (B ∩Ker φ)+b ⊂ Ā. It follows that each element in I = φ(Ā + a + b) has at least C + 1 preimages. Since Ā + a + b is an ideal in A and φ is surjective, I is an ideal in S.

(b) By linearization we now derive Theorem 6.4 from Theorem 6.3. Let K be a field. Then we set R_{i} = K[A_{i}], and the homomorphism φ_{i} allows us to consider R_{i} as a G-graded K-algebra. Next we choose a polynomial ring P_{i} whose indeterminates are mapped to a finite monomial system of generators of A_{i}, and so P_{i} is also Ggraded. Set

R = P_{1} ⊗_{K} ⋯ ⊗_{K} P_{v} and M = R_{1}(−s_{1}) ⊕ ⋯ ⊕ R_{v}(−s_{v})

Evidently R is a finitely generated G-graded K-algebra; in particular it is noetherian. Moreover R_{i} is residue class ring of R in a natural way, and therefore R_{i}(−s_{i}) can be considered a G-graded R-module. Therefore M is a G-graded R-module whose Hilbert function is the function H of the theorem.

It remains to do the case C = 0. For simplicity we only formulate it under the special assumptions of step (a) in the proof of Theorem 6.4. We leave the general as well as the commutative algebra version to the reader. The semigroups A_{i} of Theorem 6.4 can now be replaced by their images.

Proposition 6.5. Let L = ℤ^{m}, S an affine subsemigroup of L, T a finitely generated S-submodule of L. Consider subsemigroups A_{1}, …, A_{v} of L and elements t_{1}, …, t_{v} ∈ T such that the set

G = T \ ((A_{1} + t_{1}) ∪ ⋯ ∪ (A_{v} + t_{v}))

of “gaps” satisfies the following condition: for each proper subgroup U of L and each u ∈ L the intersection (U + u) ∩ G is finite. Then G is finite.

Proof. Note that T is contained in finitely many residue classes modulo gp(S). Therefore we can replace each A_{i} by A_{i} ∩ gp(S): the intersection of A_{i} + t_{i} with a residue class modulo gp(S) is a finitely generated A_{i} ∩ gp(S)-module by Corollary 6.2.

We order the A_{i} in such a way that A_{1}, …, A_{w} have the same rank as S, and A_{w+1}, …, A_{v} have lower rank. Let W be the intersection of gp(A_{i}), i = 1, …, w. Since gp(S)/W is a finite group, we can replace all the semigroups involved by their intersections with W, and split the modules into their intersection with the residue classes modulo W. We have now reached a situation where A_{i} ⊂ gp(S) for all i, and gp(A_{i}) = gp(S), unless rank A_{i} < rank S.

Next one can replace the A_{1}, …, A_{w} by their normalizations. In this way we fill the gaps in only finitely many U +u (compare the argument in the proof of Theorem 6.4), and therefore we fill only finitely many gaps.

At this point we can assume that A_{1}, …, A_{w} are integrally closed in L. Furthermore we must have C(S) ⊂ C(A_{1}) ∪ ⋯ ∪ C(A_{w}) – otherwise an open subcone of C(S) would remain uncovered, and this would remain so in T: the lower rank semigroups cannot fill it, and neither can it be filled by finitely many translates U + u where U is a proper subsemigroup of S. In fact, (A_{i} + t_{i}) \ A_{i} is contained in the union of finitely many such translates, and the same holds for (C(S) ∩ L) \ S. Since A_{1}, …, A_{w} are integrally closed, we have S ⊂ A_{1} ∪ ⋯ ∪ A_{w}.

Now we choose a system of generators u_{1}, …, u_{q} of T over S. We have

T ⊂ ⋃_{i,j} A_{i} + u_{j}.

But A_{i} + u_{j} and A_{i} + t_{i} only differ in finitely many translates of proper direct subgroups of L parallel to the support hyperplanes of A_{i}. So in the last step we have filled only finitely many gaps. Since no gaps remain, their number must have been finite from the beginning.

References

[Bri] M. Brion, Sur les modules de covariants, Ann. scient. Éc. Norm. Sup., 4^{e} série, 26 (1993), 1–21.

[BGu] W. Bruns and A. Guerrieri, The Dedekind–Mertens lemma and determinantal rings, Proc. Amer. Math. Soc. 127, (1999), 657–663.

[BH] W. Bruns and J. Herzog. Cohen–Macaulay rings (Rev. Ed.). Cambridge University Press, 1998.

[BV] W. Bruns and U. Vetter, Determinantal rings, Springer Lecture Notes 1327 (1988).

[Ch] L. Chouinard, Krull semigroups and divisor class groups, Canad. J. Math. 33 (1981), 1459– 1468.

[Cox] D. A. Cox, The homogeneous coordinate ring of a toric variety, J. Algebr. Geom. 4 (1995), 17–50.

[Ei] D. Eisenbud, Commutative algebra with a view towards algebraic geometry, Springer 1995.

[Fo] R. M. Fossum, The divisor class group of a Krull domain, Springer 1973.

[Gu1] J. Gubeladze, The elementary action on unimodular rows over a monoid ring, J. Algebra 148 (1992), 135–161.

[Gu2] J. Gubeladze, Geometric and algebraic representations of commutative cancellative monoids, Proc. A. Razmadze Math. Inst., Georg. Acad. Sci. 113 (1995), 31-81.

[Gu3] J. Gubeladze, The isomorphism problem for commutative monoid rings, J. Pure Appl. Algebra 129 (1998), 35–65.

[HK] F. Halter-Koch, Finitely generated monoids, finitely primary monoids and factorization properties of integral domains, pp. 31–72 in: D. D. Anderson (Ed.), Factorization in integral domains, M. Dekker 1997.

[Ho] M. Hochster, Rings of invariants of tori, Cohen–Macaulay rings generated by monomials, and polytopes, Ann. Math. 96 (1972), 318–337.

[Hu] J. E. Humphreys, Linear algebraic groups, Springer 1975.

[St1] R. P. Stanley, Hilbert functions of graded algebras, Adv. in Math. 28 (1978), 57–83.

[St2] R. P. Stanley, Linear diophantine equations and local cohomology, Invent. math. 68 (1982), 175–193.

[St3] R. P. Stanley, Combinatorics and commutative algebra (2nd ed.), Birkhäuser 1996.

Universität Osnabrück, FB Mathematik/Informatik, 49069 Osnabrück, Germany

E-mail address: Winfried.Bruns@mathematik.uni-osnabrueck.de

A. Razmadze Mathematical Institute, Alexidze St. 1, 380093 Tbilisi, Georgia

E-mail address: gubel@rmi.acnet.ge