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A realization theorem for sets of distances
Alfred Geroldinger, Wolfgang Schmid
To cite this version:
Alfred Geroldinger, Wolfgang Schmid. A realization theorem for sets of distances. Journal of Algebra,
Elsevier, 2017. �hal-01355253v2�
A REALIZATION THEOREM FOR SETS OF DISTANCES
ALFRED GEROLDINGER AND WOLFGANG A. SCHMID
Abstract. Let H be an atomic monoid. The set of distances ∆(H) ofH is the set of alld∈Nwith the following property: there are irreducible elements u1, . . . , uk, v1. . . , vk+d such thatu1·. . .·uk = v1·. . .·vk+dbutu1·. . .·ukcannot be written as a product ofℓirreducible elements for anyℓ∈Nwith k < ℓ < k+d. It is well-known (and easy to show) that, if ∆(H) is nonempty, then min ∆(H) = gcd ∆(H).
In this paper we show conversely that for every finite nonempty set ∆⊂Nwith min ∆ = gcd ∆ there is a finitely generated Krull monoidH such that ∆(H) = ∆.
1. Introduction
Sets of lengths (together with all invariants describing their structure, such as sets of distances and elasticities) are a well-studied means of describing the arithmetic structure of non-factorial monoids and domains. The first goal is to describe the arithmetical invariants in terms of algebraic invariants of the underlying structure. Then the question arises to which extent the achieved results are best possible, and answers can be given by providing monoids and domains with prescribed arithmetical invariants. For a sample of such realization results for sets of lengths, sets of elasticities, sets of catenary degrees we refer to [2, 8, 14, 15] and to various survey articles in [3].
In the present paper we focus on sets of distances, also called delta sets in the literature. The set of distances ∆(H) of an atomic monoid H is the set of all d ∈ N with the following property: there are atoms (irreducible elements)u1, . . . , uk, v1. . . , vk+d such thatu1·. . .·uk =v1·. . .·vk+d but u1·. . .·uk
cannot be written as a product of ℓ atoms for anyℓ ∈N with k < ℓ < k+d. The monoid H is called half-factorial if ∆(H) =∅(clearly, factorial monoids are half-factorial). IfH is not half-factorial, then a simple argument shows that min ∆(H) = gcd ∆(H) ([10, Proposition 1.4.4]). Sets of distances are finite for transfer Krull monoids of finite type (hence in particular for Krull domains with finite class group), weakly Krull domains with finite v-class group, finitely generated monoids, and others (see [9, Theorem 13], [10, Theorems 3.1.4 and 3.7.1], [13]). The question which finite sets can actually occur as a set of distances (of any monoid or domain) was open so far. It is easy to see that every singleton can be realized as a set of distances (we recall the argument in the proof of Theorem 1.1). One of the very few results beyond this is given in a recent paper where it is shown that every set ∆ with|∆|= 2 and min ∆ = gcd ∆ can be realized as the set of distances of a numerical monoid ([4]).
The arithmetic structure of a Krull monoid is completely determined by its class group and the distri- bution of prime divisors in the classes. LetH be a Krull monoid such that every class contains a prime divisor. If the class group is finite (e.g., for rings of integers in algebraic number fields), then ∆(H) is a finite interval and ifGis infinite, then ∆(H) =N([11]). The assumption on the prime divisors is crucial.
Indeed, in contrast to the above result we show in the present paper that every finite nonempty set ∆ with min ∆ = gcd ∆ can occur as the set of distances of a finitely generated Krull monoid.
2010Mathematics Subject Classification. 13A05, 13F05, 20M13.
Key words and phrases. Krull monoids, sets of lengths, sets of distances.
This work was supported by the Austrian Science Fund FWF, Project Number P26036-N26 and by the ANR Project Caesar, Project Number ANR-12-BS01-0011.
1
2 ALFRED GEROLDINGER AND WOLFGANG A. SCHMID
Theorem 1.1. Let ∆⊂Nbe a finite nonempty set of positive integers withmin ∆ = gcd ∆. Then there is a finitely generated Krull monoid H such that∆(H) = ∆.
The above realization theorem is not restricted to abstract Krull monoids, but any such set ∆ can be realized as the set of distances of a Dedekind domain or as the set of distances of a monoid of modules.
To formulate this precisely, let R be a ring, C a class of right R-modules which is closed under finite direct sums, under direct summands, and under isomorphisms, and suppose that C has a set V(C) of representatives (this means that every moduleM inCis isomorphic to a unique [M] inV(C)). ThenV(C) becomes a commutative semigroup with operation given by [M]+[N] = [M⊕N], and it encodes all possible information about direct sum decompositions of modules inC. In particular, the Krull-Remak-Schmidt- Azumaya Theorem holds forC if and only if V(C) is factorial. This semigroup-theoretical approach to the study of direct-sum decompositions of modules was pushed forward by Facchini and Wiegand, and we refer to the surveys [1, 6].
Corollary 1.2. Let∆⊂Nbe a finite nonempty set of positive integers withmin ∆ = gcd ∆.
1. There is a Dedekind domain R with finitely generated class group such that ∆(R) = ∆.
2. There is a ring R and a class of right R-modules C with the above properties such that for the monoid of modules V(C)we have∆ V(C)
= ∆.
We gather some background and fix notation in Section 2. The proofs of Theorem 1.1 and of Corollary 1.2 are given in Section 3.
2. Background on Krull monoids and factorizations
By a monoid, we mean a commutative semigroup which has a unit element and which satisfies the cancelation laws. For any setP, letF(P) be the free abelian monoid with basisP and for an element
a=p1·. . .·pℓ= Y
p∈P
pvp(a)∈ F(P), where ℓ∈N0 andp1, . . . , pℓ∈P , we denote by|a|=ℓ=P
p∈Pvp(a)∈N0 the length ofa.
Let H be a monoid. We denote by H× the group of units, by q(H) the quotient group of H, and by A(H) the set of atoms of H. The monoid H is called reduced if H× = {1}, and we denote by Hred=H/H× the associated reduced monoid ofH. We say thatH is
• atomicif every nonunit can be written as a finite product of atoms;
• root-closed ifx∈q(H) andxm∈H for somem∈Nimplies thatx∈H.
LetZ(H) = F(A(Hred)) be the factorization monoid ofH and letπH:Z(H)→ Hred be the canonical epimorphism. Fora∈H, we denote by
• ZH(a) =π−1H (aH×)⊂Z(H) the set of factorizationsofa, and by
• LH(a) ={|z|:z∈ZH(a)} ⊂N0 theset of lengthsofa.
ThusH is atomic if and only ifZH(a)6=∅for alla∈H. A monoidH is aKrull monoidif it satisfies one of the following equivalent conditions ([10, Chapter 2]) :
(a) H is completely integrally closed and satisfies the ascending chain condition on divisorial ideals.
(b) There is a free abelian monoidF and a homomorphism ϕ:H →F such that for alla, b∈H we have thata|b inH if and only ifϕ(a)|ϕ(b) inF.
Every Krull monoid is atomic, its sets of lengths are all finite, and since it is completely integrally closed, it is root-closed. LetRbe a domain. Then its multiplicative semigroup of nonzero elementsR•=R\ {0}
is a monoid, andRis a Krull domain if and only ifR• is a Krull monoid. Thus Property (a) reveals that every integrally closed noetherian domain is a Krull domain.
We briefly discuss a Krull monoid having a combinatorial flavor which plays a crucial role in all arith- metic studies of Krull monoids and which we will need in the proofs of Theorem 1.1 and of Corol- lary 1.2. Let G be an additively written abelian group and let G0 ⊂ G be a subset. For an element S=g1·. . .·gℓ=Q
g∈G0gvg(S)∈ F(G0) let σ(S) =
Xℓ
i=1
gi= X
g∈G0
vg(S)g∈G denote itssum.
ThenB(G0) ={S∈ F(G0) :σ(S) = 0}is a submonoid ofF(G0), called themonoid of zero-sum sequences overG0. Since the inclusionB(G0)֒→ F(G0) satisfies Property (b), B(G0) is a Krull monoid. We refer the reader to the monographs [10, 12] for detailed expositions on Krull monoids.
3. Proof of the Main Results
Before starting the proof of Theorem 1.1, we recall some basic facts that are needed in the proof. A submonoidH′ ⊂H of a monoidH is called divisor-closed ifa∈H′ andb∈H with b|a(in H) implies thatb∈H′. IfH′ ⊂H is a divisor-closed submonoid, then for each a∈H′ one has ZH′(a) =ZH(a), in particularLH′(a) =LH(a) and ∆(H′)⊂∆(H).
IfH =H1×H2 is the product of two reduced atomic monoids H1, H2, thenH1 andH2 are divisor- closed submonoids whence ∆(H1)∪∆(H2) ⊂∆(H). In general, this inclusion is strict. This is why a direct product construction (as used in a realization result for sets of catenary degrees and others, [8, Proposition 3.2]) cannot be used to construct arbitrary sets of distances.
Proof of Theorem 1.1. Let ∆⊂Nbe a finite subset with min ∆ = gcd ∆. We proceed by induction on|∆|.
Suppose that ∆ ={d}. LetGbe an additive abelian group having an elementg∈Gwith ord(g) =d+ 2, and letH =B({g,−g}). We setv1=gd+2,v2= (−g)d+2, andv3= (−g)g. ThenA(B(G0)) ={v1, v2, v3}, v1v2=vd+23 , and H is a reduced finitely generated Krull monoid with ∆(H) ={d}.
Now suppose that |∆| > 1 and that the assertion holds for all sets under consideration which are strictly smaller than |∆|. We set d1 = min ∆, d2 = max ∆, ∆1 = ∆\ {d2}, and ∆2 = {d2}. Then min ∆1= gcd ∆1=d1and min ∆2= gcd ∆2. Thus by the induction hypothesis there are finitely generated Krull monoidsH1andH2such that ∆(Hi) = ∆ifori∈[1,2]. Without restriction we may suppose thatH1
andH2 are reduced and thatA(H2) ={v1, v2, v3}withv1v2=v3d2+2 as constructed for the case|∆|= 1.
We setA(H1) ={u1, . . . , uk} and observe thatA(H1×H2) =A(H1)⊎ A(H2) ={u1, . . . , uk, v1, v2, v3}.
We define
π:Nk+30 −→∼ Z(H1×H2) −→H1×H2
m7→z=um11·. . .·umkkv1n1vn22v3n3 7→πH1×H2(z) and
Ω ={m= (m1, . . . , mk, n1, n2, n3)∈Nk+30 : (n1, n2, n3)6= (0,0,0),(m1, . . . , mk)6= (0, . . . ,0), and
|m|<maxLH1×H2(π(m))−d1}
We note that this set is in fact nonempty, as for example LH1×H2(u1v1v2) = {3,3 +d2}. The set of minimal elements Min(Ω)⊂Ω (with respect to the usual partial order) is finite by Dickson’s Theorem ([10, Theorem 1.5.3]). For eachm∈Min(Ω) and each
ℓ∈[|m|+d1,maxLH1×H2(π(m))−d1]∩(|m|+d1N)
we consider a set Pℓm ={pmℓ,2, . . . , pmℓ,ℓ}. We define Ω′ ⊂ Min(Ω)×N to be the set of all (m, ℓ) where m∈Min(Ω) andℓas above, and note that Ω′ is finite too. For each (m, ℓ)∈Ω′, we define
pmℓ,1=π(m)(pmℓ,2·. . .·pmℓ,ℓ)−1∈q(H1×H2)× Y
(m,ℓ)∈Ω′
q(F(Pℓm)),
4 ALFRED GEROLDINGER AND WOLFGANG A. SCHMID
and setPℓm,∗={pmℓ,1, . . . , pmℓ,ℓ}. Let
H ⊂q(H1×H2)× Y
(m,ℓ)∈Ω′
q(F(Pℓm)) be the submonoid generated byA=A(H1×H2)⊎U
(m,ℓ)∈Ω′Pℓm,∗. ThusH is a reduced finitely generated monoid with quotient groupq(H) =q(H1×H2)×Q
(m,ℓ)∈Ω′q(F(Pℓm)). Since Ais a minimal generating set (with respect to inclusion), it follows by [10, Proposition 1.1.7] that A is the set of atoms of H. If i∈[1,2], thenHi⊂H is a divisor-closed submonoid and hence ifa∈Hi, thenZHi(a) =ZH(a) and thus LHi(a) =LH(a). Clearly,H1×H2⊂H is not divisor-closed, and hence ifa∈H1×H2\(H1∪H2), then LH(a) andLH1×H2(a) need not be equal. Consider a product of the form
a= Yk
ν=1
urνν Y3
ν=1
vsνν Y
(m,ℓ)∈Ω′
Yℓ
ν=1
(pmℓ,ν)tm,ℓ,ν,
where all exponentsrν, sν, andtm,ℓ,ν are non-negative integers. Then, by construction, we have (3.1) a∈H1×H2 if and only if tm,ℓ,1=. . .=tm,ℓ,ℓ for all (m, ℓ)∈Ω′.
In the remainder of the proof we show the following three assertions. The first one verifies a condition that is important for technical reasons and the latter two then establish what we intended to show.
A1. Ifa1∈H1 anda2∈H2, then maxLH(a1a2) = maxLH1×H2(a1a2).
A2. ∆(H) = ∆.
A3. H is a Krull monoid.
Proof of A1. Leta1 ∈H1,a2∈H2, andz∈ZH(a1a2) with|z|= maxLH(a1a2). We need to show that there exists somez′ ∈ZH1×H2(a1a2) such that|z|=|z′|. Suppose that
z= Yk
ν=1
urνν Y3
ν=1
vνsν Y
(m,ℓ)∈Ω′
Yℓ
ν=1
(pmℓ,ν)tm,ℓ,ν,
where all exponents rν, sν, and tm,ℓ,ν are non-negative integers. If all tm,ℓ,ν are equal to 0, then z ∈ ZH1×H2(a1a2), and the claim holds withz′=z. Assume to the contrary that there are (m′, ℓ′)∈Ω′ and ν′∈[1, ℓ′] such thattm′,ℓ′,ν′ >0. Since we consider a factorization of the elementa1a2∈H1×H2, we infer by (3.1) thattm′,ℓ′,ν′ =tm′,ℓ′,1=. . .=tm′,ℓ′,ℓ′. By definition of Ω′ we have maxLH1×H2(π(m′))≥ℓ′+d1. Ify∈ZH1×H2(π(m′)) is a factorization of maximal length, then
ℓ′
Y
ν=1
(pmℓ′,ν′)−1z y∈ZH(a1a2)
is a factorization ofa1a2of length greater than or equal to|z|+d1, a contradiction to|z|= maxLH(a1a2).
Proof of A2. Since H is finitely generated, [10, Theorem 3.1.4] implies that ∆(H) is finite and by construction we have ∆(H)⊂d1N. Since, fori∈[1,2],Hi⊂H is a divisor-closed submonoid, it follows that ∆i = ∆(Hi)⊂∆(H) and hence ∆ = ∆1∪∆2⊂∆(H). To verify the reverse inclusion, leta∈H be given. We choose a factorizationz∈ZH(a) with|z|<maxLH(a) and show that the distance from|z|to the next largest element inLH(a) lies in ∆. More formally, letd= min{k− |z|: k∈LH(a) with k >|z|}
and we assert thatd ∈∆. We write z in the form z =z1z2z0, where zi ∈ F(A(Hi)) for i ∈[1,2] and z0∈ F(U
(m,ℓ)∈Ω′Pℓm,∗), and distinguish two cases.
CASE 1: There exists (m, ℓ)∈Ω′ such thatQℓ
ν=1pmℓ,ν|z inZ(H).
We show thatd=d1. Ifℓ <maxLH1×H2(π(m))−d1, then (m, ℓ+d1)∈Ω′, and (
Yℓ
ν=1
pmℓ,ν)−1z(
ℓ+d1
Y
ν=1
pmℓ+d1,ν) ∈ZH(a)
is a factorization ofaof length|z|+ (ℓ+d1)−ℓ=|z|+d1. Since ∆(H)⊂d1N, it follows that d=d1. Ifℓ= maxLH1×H2(π(m))−d1, then we choosey∈ZH1×H2(π(m)) with|y|= maxLH1×H2(π(m)). Then
( Yℓ
ν=1
pmℓ,ν)−1zy ∈ZH(a)
is a factorization ofaof length|z|+d1. Since ∆(H)⊂d1N, it follows thatd=d1. CASE 2: There exists no (m, ℓ)∈Ω′ such thatQℓ
ν=1pmℓ,ν|z inZ(H).
Fori∈[1,2] we setai=πHi(zi), and we start with the following assertion.
A4. ZH(a1a2)z0=ZH(a).
Proof of A4. Obviously, we haveZH(a1a2)z0⊂ZH(a). To show the reverse inclusion, letz′∈ZH(a). We writez′ in the formz′=z1′z′2z0′ wherezi′∈ F(A(Hi)) fori∈[1,2] andz0′ ∈ F(U
(m,ℓ)∈Ω′Pℓm,∗). Clearly, it is sufficient to show thatz0|z0′. For each (m, ℓ)∈Ω′ we define
z(m, ℓ) = Yℓ
ν=1
(pmℓ,ν)kν resp. z′(m, ℓ) = Yℓ
ν=1
(pmℓ,ν)k′ν
wherekν resp. kν′ is the multiplicity ofpmℓ,ν in z0 resp. z0′ for all ν ∈ [1, ℓ]. The assumption of CASE 2 implies that there is aν ∈[1, ℓ] such thatkν = 0. We claim thatz(m, ℓ)|z′(m, ℓ) for each (m, ℓ)∈Ω′. Once this is done, then it follows thatz0|z′0.
Let (m, ℓ) ∈ Ω′ and let φmℓ : q(H) → q(F(Pℓm)) denote the canonical projection. We note that φmℓ (u) = 1 for u ∈ A(H)\Pℓm,∗ and φmℓ (pmℓ,ν) = pmℓ,ν for ν ∈ [2, ℓ] and φmℓ (pmℓ,1) = Qℓ′
ν=2(pmℓ,ν)−1. It follows that
φmℓ (a) = Yℓ
ν=2
(pmℓ,ν)kν−k1 and also φmℓ (a) = Yℓ
ν=2
(pmℓ,ν)k′ν−k′1.
Consequentlykν−k1=k′ν−k1′ for eachν∈[2, ℓ]. It follows thatk′ν =kν+ (k′1−k1) for eachν∈[1, ℓ].
Since allkν, k′ν are non-negative and at least one of the kν is equal to 0, it follows thatk1′ −k1 ≥0, and
hencez(m, ℓ)|z′(m, ℓ). (Proof ofA4)
Ifa2 = 1, thena1a2∈H1. As H1⊂H is divisor-closed, it follows that ∆(LH(a)) = ∆(LH1(a1)) and henced∈∆1⊂∆. Ifa1= 1, then the analogous argument yieldsd∈∆2⊂∆. Now we assume thata16=
1,a26= 1, and assert thatd=d1. If|z| − |z0|=|z1z2|= maxLH(a1a2), thenZH(a) =ZH(a1a2)z0andA1 imply that|z|= maxLH(a), a contradiction. If|z|− |z0|=|z1z2| ∈[maxLH(a1a2)−d1,maxLH(a1a2)−1], thend=d1 by the minimality ofd1.
It remains to consider the case where|z1z2|<maxLH(a1a2)−d1. Letm= (m1, . . . , mk, n1, n2, n3)∈ Nk+3
0 such thatz1z2=um11·. . .·umkkvn11v2n2v3n3. Sincea16= 1,a26= 1, and (byA1)
|m|=|z1z2|<maxLH(a1a2)−d1= maxLH1×H2(a1a2)−d1,
it follows thatm ∈ Ω. Letm′ ∈Min(Ω) with m′ ≤m, and let z′ denote the respective factorization.
Thenz′ |z1z2in Z(H), (m′,|z′|+d1)∈Ω′, and z′−1z1z2(
|z′|+d1
Y
ν=1
pm|z′|+d1,ν) ∈ZH(a1a2)
is a factorization ofa1a2of length|z1z2|+d1. Thus we obtain a factorization ofaof length|z|+d1 which impliesd=d1.
6 ALFRED GEROLDINGER AND WOLFGANG A. SCHMID
Proof of A3. Since H is a reduced and finitely generated monoid, it is sufficient to verify that H is root-closed by [10, Theorem 2.7.14]. Letx∈q(H), say
x=y Y
(m,ℓ)∈Ω′
Yℓ
ν=2
(pmℓ,ν)rm,ℓ,ν ∈q(H)
wherey∈q(H1×H2) and all exponentsrm,ℓ,ν ∈Z. Suppose there is anm∈Nsuch thatxm∈H, say xm=b Y
(m,ℓ)∈Ω′
Yℓ
ν=1
(pmℓ,ν)sm,ℓ,ν ∈H
whereb∈H1×H2 and all exponents sm,ℓ,ν ∈N0; note that here we usepmℓ,1, too. We have to show that x∈H. Clearly, we have, inq(H),
ym Y
(m,ℓ)∈Ω′
Yℓ
ν=2
(pmℓ,ν)mrm,ℓ,ν = b Y
(m,ℓ)∈Ω′
Yℓ
ν=1
(pmℓ,ν)sm,ℓ,ν
=b Y
(m,ℓ)∈Ω′
π(m)sm,ℓ,1 Y
(m,ℓ)∈Ω′
Yℓ
ν=2
(pmℓ,ν)sm,ℓ,ν−sm,ℓ,1,
where the second equation follows from the fact thatpmℓ,1=π(m)(Qℓ
ν=2pmℓ,ν)−1 for all (m, ℓ)∈Ω′. Since b∗:=bQ
(m,ℓ)∈Ω′π(m)sm,ℓ,1 ∈H1×H2⊂q(H1×H2), it follows that
(3.2) mrm,ℓ,ν=sm,ℓ,ν−sm,ℓ,1 for all indices and ym=b∗∈H1×H2.
SinceH1×H2 is a Krull monoid and hence root-closed, we infer thaty ∈H1×H2. For each (m, ℓ)∈Ω′, we choose an indexν′ ∈[1, ℓ] such that
rm,ℓ,ν′ = min{rm,ℓ,ν:ν ∈[1, ℓ]},
and let Ω′′⊂Ω′ be the the subset of all (m, ℓ)∈Ω′ for whichrm,ℓ,ν′ <0. We writex as a product of four factors and show that each factor lies inH which impliesx∈H. Indeed we have
x=y Y
(m,ℓ)∈Ω′
Yℓ
ν=2
(pmℓ,ν)rm,ℓ,ν
= (y Y
(m,ℓ)∈Ω′′
π(m)rm,ℓ,ν′)( Y
(m,ℓ)∈Ω′′
π(m)−rm,ℓ,ν′) Y
(m,ℓ)∈Ω′
Yℓ
ν=2
(pmℓ,ν)rm,ℓ,ν
= (y Y
(m,ℓ)∈Ω′′
π(m)rm,ℓ,ν′)( Y
(m,ℓ)∈Ω′′
Yℓ
ν=1
(pmℓ,ν)−rm,ℓ,ν′) Y
(m,ℓ)∈Ω′
Yℓ
ν=2
(pmℓ,ν)rm,ℓ,ν
= (y Y
(m,ℓ)∈Ω′′
π(m)rm,ℓ,ν′)
| {z }
(i)
( Y
(m,ℓ)∈Ω′′
(pmℓ,1)−rm,ℓ,ν′)
| {z }
(ii)
Y
(m,ℓ)∈Ω′′
Yℓ
ν=2
(pmℓ,ν)rm,ℓ,ν−rm,ℓ,ν′
| {z }
(iii)
Y
(m,ℓ)∈Ω′\Ω′′
Yℓ
ν=2
(pmℓ,ν)rm,ℓ,ν
| {z }
(iv)
We consider each of the four products individually:
(i) Since for all indicessm,ℓ,1+mrm,ℓ,ν =sm,ℓ,ν ≥0 by (3.2), we obtain that
(y Y
(m,ℓ)∈Ω′′
π(m)rm,ℓ,ν′)m=b∗ Y
(m,ℓ)∈Ω′′
π(m)mrm,ℓ,ν′
=b Y
(m,ℓ)∈Ω′
π(m)sm,ℓ,1 Y
(m,ℓ)∈Ω′′
π(m)mrm,ℓ,ν′
=b Y
(m,ℓ)∈Ω′\Ω′′
π(m)sm,ℓ,1 Y
(m,ℓ)∈Ω′′
π(m)sm,ℓ,1+mrm,ℓ,ν′ ∈H1×H2. SinceH1×H2 is a Krull monoid and hence root-closed, it follows that
y Y
(m,ℓ)∈Ω′′
π(m)rm,ℓ,ν′ ∈H1×H2⊂H .
(ii) This product lies inH since allpmℓ,1are in H and the exponent −rm,ℓ,ν′ is positive.
(iii) This product lies inH since all allpmℓ,ν are inH and all exponentsrm,ℓ,ν−rm,ℓ,ν′ are non-negative by the minimality ofrm,ℓ,ν′.
(iv) This product lies in H since all pmℓ,ν are in H and the exponentsrm,ℓ,ν are non-negative for all (m, ℓ)∈Ω′\Ω′′ by definition of Ω′′.
Proof of Corollary 1.2. 1. Let ∆⊂Nbe a finite nonempty set of positive integers with min ∆ = gcd ∆.
By Theorem 1.1 there exists a finitely generated Krull monoid H such that ∆(H) = ∆. Let Gdenote the class group of H and let G0 ⊂ G be the set of classes containing prime divisors. Since H is a finitely generated monoid, Gis a finitely generated abelian group. By [10, Theorem 3.4.10], there is a homomorphismβ:H → B(G0) such that ∆(H) = ∆ B(G0)
. By Claborn’s Realization Theorem ([10, Theorem 3.7.8]) there is a Dedekind domainRand an isomorphism Φ :G→ C(R) such thatG0is mapped onto the subsetGP of the class groupC(R) which contains prime ideals. SinceR is a Dedekind domain, its multiplicative monoidR• =R\ {0} is a Krull monoid and again by [10, Theorem 3.4.10] there is a homomorphismθ:R•→ B(GP) such that ∆(R•) = ∆ B(GP)
. Thus we obtain that
∆ = ∆(H) = ∆ B(G0)
= ∆ B(GP)
= ∆(R•).
2. Let R be a ring, C a class of right R-modules which is closed under finite direct sums, under direct summands, and under isomorphisms, and suppose thatC has a setV(C) of representatives. If the endomorphism ring EndR(M) is semilocal for all modulesM in C, then the monoid V(C) is a reduced Krull monoid by a theorem of Facchini ([5, Theorem 3.4]. Conversely, every reduced Krull monoid is isomorphic to a monoid of modulesV(C) by a realization theorem of Facchini and Wiegand ([7, Theorem 2.1]). Their result together with Theorem 1.1 implies the assertion.
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8 ALFRED GEROLDINGER AND WOLFGANG A. SCHMID
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Institute for Mathematics and Scientific Computing, University of Graz, NAWI Graz, Heinrichstraße 36, 8010 Graz, Austria
E-mail address: alfred.geroldinger@uni-graz.at URL:http://imsc.uni-graz.at/geroldinger
Universit´e Paris 13, Sorbonne Paris Cit´e, LAGA, CNRS, UMR 7539, Universit´e Paris 8, F-93430, Villetaneuse, France, and, Laboratoire Analyse, G´eom´etrie et Applications (LAGA, UMR 7539), COMUE Universit´e Paris Lumi`eres, Universit´e Paris 8, CNRS, 93526 Saint-Denis cedex, France
E-mail address: schmid@math.univ-paris13.fr
