Radicals of principal ideals and the class group of a Dedekind domain
aa r X i v : . [ m a t h . A C ] F e b RADICALS OF PRINCIPAL IDEALS AND THE CLASSGROUP OF A DEDEKIND DOMAIN
DARIO SPIRITO
Abstract.
For a Dedekind domain D , let P ( D ) be the set ofideals of D that are radical of a principal ideal. We show that, if D, D ′ are Dedekind domains and there is an order isomorphismbetween P ( D ) and P ( D ′ ), then the rank of the class groups of D and D ′ is the same. Introduction
The class group Cl( D ) of a Dedekind domain D is defined as thequotient between the group of the nonzero fractional ideals of D andthe subgroup of the principal ideals of D . Since Cl( D ) is trivial if andonly if D is a principal ideal domain (equivalently, if and only if it is aunique factorization domain), the class group can be seen as a way tomeasure how much unique factorization fails in D . For this reason, thestudy of the class group is an important part of the study of Dedekinddomains.It is a non-obvious fact that the class group of D actually dependsonly on the multiplicative structure of D , or, from another point ofview, depends only on the set of nonzero principal ideals of D . Indeed,the class group of D • := D \{ } as a monoid (where the operation is theproduct) is isomorphic to the class group of D as a Dedekind domain(see Chapter 2 – in particular, Section 2.10 – of [5]), and thus if D and D ′ are Dedekind domains whose sets of principal ideals are isomorphic(as monoids) then the class groups of D and D ′ are isomorphic too.In this paper, we show that the rank of Cl( D ) can be recoveredby considering only the set P ( D ) of the ideals that are radical of aprincipal ideal: that is, we show that if P ( D ) and P ( D ′ ) are isomorphicas partially ordered sets then the ranks of Cl( D ) and Cl( D ′ ) are equal.The proof of this result can be divided into two steps.In Section 3 we show that an order isomorphism between P ( D ) and P ( D ′ ) can always be extended to an isomorphism between the setsRad( D ) and Rad( D ′ ) of all radical ideals of D (Theorem 3.6): this is ac-complished by considering these sets as (non-cancellative) semigroups Date : February 19, 2020.2010
Mathematics Subject Classification.
Key words and phrases.
Dedekind domain; class group; radical ideals; principalideals. and characterizing coprimality in D through a version of coprimalityin P ( D ) (Proposition 3.3).In Section 4 we link the structure of P ( D ) and Rad( D ) with thestructure of the tensor product Cl( D ) ⊗ Q as an ordered topologi-cal space; in particular, we interpret the set of inverses of a set ∆ ⊆ Max( D ) with respect to P ( D ) (see Definition 4.1) as the negative conegenerated by the images of ∆ in Cl( D ) ⊗ Q (Proposition 4.2) and usethis connection to calculate the rank of Cl( D ) in function of some par-ticular partitions of an “inverse basis” of Max( D ) (Propositions 4.9 and4.10). As this construction is invariant with respect to isomorphism, weget the main theorem (Theorem 4.11).In Section 5 we give three examples, showing that some natural ex-tensions of the main result do not hold.2. Notation and preliminaries
Throughout the paper, D will denote a Dedekind domain , that is, aone-dimensional integrally closed Noetherian integral domain; equiva-lently, a one-dimensional Noetherian domain such that D P is a discretevaluation ring for all maximal ideals P . For general properties aboutDedekind domains, the reader may consult, for example, [2, Chapter7, § D • to indicate the set D \ { } . We denote by Max( D ) theset of maximal ideals of D . If I is an ideal of D , we set V ( I ) := { P ∈ Spec( D ) | I ⊆ D } . If I = xD is a principal ideal, we write V ( x ) for V ( xD ). If I = (0), theset V ( I ) is always a finite subset of Max( D ). We denote by rad( I ) theradical of the ideal I , and we say that I is a radical ideal (or simplythat I is radical ) if I = rad( I ).Every nonzero proper ideal I of D can be written uniquely as aproduct P e · · · P e n n = P e ∩ · · · ∩ P e n n , where P , . . . , P n are distinctmaximal ideals and e , . . . , e n ≥
1. In particular, in this case we have V ( I ) = { P , . . . , P n } , and rad( I ) = P · · · P n . An ideal is radical ifand only if e = · · · = e n = 1. If P is a maximal ideal, the P -adicvaluation of an element x is the exponent of P in the factorization of xD ; we denote it by v P ( x ). (If x / ∈ P , i.e., if P does not appear in thefactorization, then v P ( x ) = 0.)If P , . . . , P k are distinct maximal ideals and e , . . . , e k ∈ N , then bythe approximation theorem for Dedekind domains (see, e.g., [2, ChapterVII, §
2, Proposition 2]) there is an element x ∈ D such that v P i ( x ) = e i for i = 1 , . . . , k .A fractional ideal of D is a D -submodule I of the quotient field K of D such that xI ⊆ D (and thus xI is an ideal of D ) for some x ∈ D • . Theset F ( D ) of nonzero fractional ideals of D is a group under multiplica-tion; the inverse of an ideal I is I − := ( D : I ) := { x ∈ K | xI ⊆ D } . A ADICALS OF PRINCIPAL IDEALS AND THE CLASS GROUP 3 nonzero fractional ideal I can be written uniquely as P e · · · P e n n , where P , . . . , P n are distinct maximal ideals and e , . . . , e n ∈ Z \ { } (withthe empty product being equal to D ). Thus, F ( D ) is isomorphic to thefree abelian group over Max( D ). The quotient between this group andits subgroup formed by the principal fractional ideals is called the classgroup of D , and is denoted by Cl( D ).For a set S , we denote by P fin ( S ) the set of all finite and nonemptysubsets of S .3. The two semilattices P ( D ) and Rad( D )Let ( X, ≤ ) be a meet-semilattice , that is, a partially ordered set whereevery pair of elements has an infimum. Then, the operation x ∧ y as-sociating to x and y their infimum is associative, commutative andidempotent, and it has a unit if and only if X has a maximum. Theorder of X can also be recovered from the operation: x ≥ y if and onlyif x divides y in ( X, ∧ ), that is, if and only if there is a z ∈ X suchthat y = x ∧ z . A join-semilattice is defined in the same way, but usingthe supremum instead of the infimum.Let now D be a Dedekind domain. We will be interested in twostructures of this kind.The first one is the semilattice Rad( D ) of all nonzero radical idealsof D . In this case, the order ≤ is the usual containment order, whilethe product is equal to I ∧ J := I ∩ J = rad( IJ ) . The second one is the semilattice P ( D ) of the ideals of D that areradical of a nonzero, principal ideal of D . This is a subsemilattice ofRad( D ) since rad( aD ) ∧ rad( bD ) = rad( abD ) , i.e., the product of two elements of P ( D ) remains inside P ( D ).A nonzero radical ideal I is characterized by the finite set V ( I ).Hence, the map from Rad( D ) to P fin (Max( D )) sending I to V ( I ) is anorder-reversing isomorphism of partially ordered sets, which becomesan order-reversing isomorphism of semilattices if the operation on thepower set is the union. We denote by V ( D ) the image of P ( D ) underthis isomorphism; that is, V ( D ) := { V ( x ) | x ∈ D • } . The inverse ofthis map is the one sending a set Z to the intersection of the primeideals contained in Z .Those semilattices have neither an absorbing element (which wouldbe the zero ideal) nor a unit (which should be D itself). Lemma 3.1.
Let
X, Y ∈ P fin (Max( D )) (resp., X, Y ∈ V ( D ) ). Then, X | Y in P fin (Max( D )) (resp., X | Y in V ( D ) ) if and only if X ⊆ Y . DARIO SPIRITO
Proof. If X | Y , then Y = X ∪ Z for some Z ∈ V ( D ), and thus X ⊆ Y .If X ⊆ Y , then Y = Y ∪ X and thus X | Y . (This works both in P fin (Max( D )) and in V ( D ).) (cid:3) Definition 3.2.
Let M be a commutative semigroup. We say that a , . . . , a n ∈ M are product-coprime if, whenever there is an x ∈ M such that x = a b = a b = · · · = a n b n , then for every j the element a j divides Q i = j b i . Proposition 3.3.
Let D be a Dedekind domain, and let a , . . . , a n ∈ D • . Then, a , . . . , a n are coprime in D if and only if V ( a ) , . . . , V ( a n ) are product-coprime in V ( D ) .Proof. Suppose that a , . . . , a n are coprime, and let X ∈ V ( D ) be suchthat X = V ( a ) ∪ B = · · · = V ( a ) ∪ B n for some B , . . . , B n ∈ V ( D ).By symmetry, it is enough to prove that V ( a ) divides B ∪ · · · ∪ B n in V ( D ), i.e., that V ( a ) ⊆ B ∪ · · ·∪ B n . Take any prime ideal P ∈ V ( a ):since a , . . . , a n are coprime there is a j such that P / ∈ V ( a j ). However, P ∈ V ( a j ) ∪ B j , and thus P ∈ B j . Therefore, V ( a i ) ⊆ B ∪ · · · ∪ B n ,as claimed.Conversely, suppose V ( a ) , . . . , V ( a n ) are product-coprime, and sup-pose that a , . . . , a n are not coprime. Then, there is a prime ideal P containing all a i ; passing to powers, without loss of generality we cansuppose that the P -adic valuation of the a i is the same, say v P ( a i ) = t for every i . By prime avoidance, there is a b ∈ D \ P such that v Q ( b ) ≥ v Q ( a i ) for all i > Q = P . Let x := a b . Byconstruction, a i | x for each i , and thus we can find b , . . . , b n ∈ D suchthat x = a i b i . Therefore, V ( x ) = V ( a i ) ∪ V ( b i ) for every i ; by hy-pothesis, it follows that V ( a ) divides V ( b ) ∪ · · · ∪ V ( b n ), i.e., that V ( a i ) ⊆ V ( b ) ∪ · · · ∪ V ( b n ). However, v P ( x ) = v P ( a ) + v P ( b ) = t ,and thus v P ( b i ) = 0 for every i ; in particular, P / ∈ V ( b i ) for every i .This is a contradiction, and thus a , . . . , a n are coprime. (cid:3) Definition 3.4.
Let M be a commutative semigroup. We say that I ( M is product-proper if no finite subset of I is product-coprime. Wedenote the set of maximal product-proper subsets of M by M ( M ) . Proposition 3.5.
Let D be a Dedekind domain. The maps ν : Max( D ) −→ M ( V ( D )) ,P V ( x ) | x ∈ P } and θ : M ( V ( D )) −→ Max( D ) , Y 7−→ { x ∈ D | V ( x ) ∈ Y } are bijections, inverse one of each other.Proof. We first show that ν and θ are well-defined. ADICALS OF PRINCIPAL IDEALS AND THE CLASS GROUP 5 If P is a maximal ideal of D , then P ∈ X for every X ∈ ν ( P ); thus,if V ( a ) ∈ ν ( P ) then a ∈ P and ν ( P ) is product-proper. If ν ( P ) ( Y ⊆V ( D ), take Y ∈ Y \ ν ( P ): then, Y = V ( b ) for some b / ∈ P . If Y = { Q , . . . , Q k } , by prime avoidance we can find a ∈ P \ ( Q ∪ · · · ∪ Q k );then, a and b are coprime and thus V ( a ) and V ( b ) are product-coprime.Hence, ν ( P ) is a maximal product-proper subset of V ( D ).Conversely, let Y ∈ M ( V ( D )). If θ ( Y ) is contained in some primeideal P , then Y ⊆ ν ( P ), and thus we must have Y = ν ( P ); in par-ticular, θ ( Y ) = P ∈ Max( D ). If θ ( Y ) is not contained in any primeideal, let V ( a ) = { Q , . . . , Q k } ∈ Y . Since θ ( Y ) * Q i , for every i wecan find b i / ∈ Q i such that V ( a i ) ∈ Y ; then, a, b , . . . , b n are coprimeand thus V ( a ) , V ( b ) , . . . , V ( b n ) are a product-coprime subset of Y , acontradiction. Hence Y = ν ( P ).The fact that they are inverses one of each other follows similarly. (cid:3) Theorem 3.6.
Let
D, D ′ be Dedekind domains. If there is an orderisomorphism ψ : P ( D ) −→ P ( D ) , then there is an order isomorphism Ψ : Rad( D ) −→ Rad( D ′ ) extending ψ .Proof. The statement is equivalent to saying that any isomorphism φ : V ( D ) −→ V ( D ′ ) can be extended to an isomorphism Φ : P fin (Max( D )) −→ P fin (Max( D ′ )). For simplicity, let P := P fin (Max( D )) and P ′ := P fin (Max( D ′ )).If φ is an isomorphism, then it sends product-proper sets into product-proper sets, and thus φ induces a bijective map η : M ( V ( D )) −→ M ( V ( D ′ )). Using the map θ of Proposition 3.5, η induces a bijection η : Max( D ) −→ Max( D ′ ), such that the diagram M ( V ( D )) Max( D ) M ( V ( D ′ )) Max( D ) η θ ηθ ′ commutes (explicitly, η = θ ′ ◦ η ◦ θ − ). In particular, η induces anorder isomorphism Φ between P and P ′ , sending X ⊆ Max( D ) to η ( X ) ⊆ Max( D ′ ). To conclude the proof, we need to show that Φextends φ .Let X = { P , . . . , P k } ∈ V ( D ). Then, by definition, Φ( X ) = η ( X ) = { η ( P ) , . . . , η ( P k ) } . The maximal product-proper subsets of V ( D ) con-taining X are Y i := ν ( P i ), for i = 1 , . . . , k ; since φ is an isomorphism,the maximal product-proper subsets of V ( D ′ ) containing φ ( X ) are thesets φ ( Y i ). By construction, φ ( Y i ) = η ( Y i ); however, θ ′ ( η ( Y i )) = η ( P i ), and thus η ( X ) = { φ ( Y ) , . . . , φ ( Y k ) } = φ ( X ). Thus, Φ extends φ , as claimed. (cid:3) The following corollary was obtained, with a more ad hoc reasoning,in the proof of [9, Theorem 2.6].
DARIO SPIRITO
Corollary 3.7.
Let
D, D ′ be Dedekind domains such that P ( D ) and P ( D ′ ) are order-isomorphic. Then, Cl( D ) is torsion if and only if Cl( D ′ ) is torsion.Proof. The class group of D is torsion if and only if every prime idealhas a principal power [6, Theorem 3.1], and thus if and only if P ( D ) =Rad( D ).If P ( D ) and P ( D ′ ) are isomorphic, then by Theorem 3.6 there is anisomorphism Φ : Rad( D ) −→ Rad( D ′ ) sending P ( D ) to P ( D ′ ); hence,Rad( D ) = P ( D ) if and only if Rad( D ′ ) = P ( D ′ ). Therefore, Cl( D ) istorsion if and only if Cl( D ′ ) is torsion. (cid:3) Remark 3.8.
Let Princ( D ) be the set of principal ideals of D and I ( D ) be the set of all ideals of D .The method used in this section can also be applied to prove theanalogous result for non-radical ideals, i.e., to prove that an isomor-phism φ : Princ( D ) −→ Princ( D ′ ) can be extended to an isomorphismΦ : I ( D ) −→ I ( D ′ ).The most obvious analogue of Proposition 3.3 does not hold, sincethe ideals ( a ) , . . . , ( a n ) may be product-coprime in Princ( D ) without a , . . . , a n being coprime (for example, take a = y , a = y and a = y , where y is a prime element of D ). However, this can be repaired: a , . . . , a n ∈ D • are coprime if and only if the ideals ( a ) k , . . . , ( a n ) k n are product-coprime in Princ( D ) for every k , . . . , k n ∈ N . The proof isessentially analogous to the one given for Proposition 3.3.Proposition 3.5 carries over without significant changes: the maxi-mal product-proper subsets of Princ( D ) are in bijective correspondencewith the maximal ideals of D . Theorem 3.6 carries over as well: the onlydifference is that, instead of the restricted power set P fin (Max( D )) itis necessary to use the free abelian group generated by Max( D ).In particular, this result directly implies that if Princ( D ) and Princ( D ′ )are isomorphic as partially ordered sets then the class groups Cl( D ) andCl( D ′ ) are isomorphic as groups, since the class group depends exactlyon which ideals are principal. This result is also a consequence of thetheory of monoid factorization (see [5]), of which this reasoning can beseen as a more direct (but less general) version.4. Calculating the rank
The rank rk G of an abelian group G is the dimension of the tensorproduct G ⊗ Q as a vector space over Q . In particular, the rank of G is 0 if and only if G is a torsion group; therefore, Corollary 3.7 canbe rephrased by saying that, if P ( D ) and P ( D ′ ) are order-isomorphic,then the rank of Cl( D ) is 0 if and only if the rank of Cl( D ′ ) is 0. Inthis section, we want to generalize this result by showing that rk Cl( D )is actually determined by P ( D ) in every case. ADICALS OF PRINCIPAL IDEALS AND THE CLASS GROUP 7
Let D be a Dedekind domain. If I ( D ) is the set of proper ideals of D ,then the quotient from F ( D ) to Cl( D ) restricts to a map π : I ( D ) −→ Cl( D ), which is a monoid homomorphism (i.e., π ( IJ ) = π ( I ) · π ( J )).Moreover, π is surjective since the class of I coincide with the class of dI for every d ∈ D • .There is also a natural map ψ : Cl( D ) −→ Cl( D ) ⊗ Q , g g ⊗ Q -vector space Cl( D ) ⊗ Q ; the map ψ is a group homomorphism, and its kernel is the torsion subgroup T of Cl( D ). By construction, the image C of ψ spans Cl( D ) ⊗ Q as a Q -vector space.Thus, we have a chain of maps I ( D ) π −−−→ Cl( D ) ψ −−−→ Cl( D ) ⊗ Q ;we denote by ψ the composition ψ ◦ π . Definition 4.1.
Let ∆ ⊆ Max( D ) . A maximal ideal Q is an almost in-verse of ∆ if there is a set { P , . . . , P k } ⊆ ∆ (not necessarily nonempty)such that Q ∧ P ∧· · ·∧ P n belongs to P ( D ) . We denote the set of almostinverses of ∆ as Inv(∆) . Our aim is to characterize Inv(∆) in terms of the map ψ ; to do so,we use the terminology of ordered topological spaces (for which we referthe reader to, e.g., [4]). Given a Q -vector space V and a set S ⊆ V ,the positive cone spanned by S ispos( S ) := ( k X i =1 λ i v i | λ i ∈ Q ≥ , v i ∈ S ) ;if C = pos( S ), we say that C is positively spanned by S . Symmetrically,the negative cone is neg( S ) := − pos( S ). Proposition 4.2.
Let ∆ ⊆ Max( D ) . Then, Inv(∆) = ψ − (neg( ψ (∆))) .Proof. Let Q ∈ Inv(∆), and let P , . . . , P n ∈ ∆ be such that L := Q ∧ P ∧ · · · ∧ P n ∈ P ( D ). Then, there is a principal ideal I = aD with radical L ; thus, there are positive integers e, f , . . . , f n > I = Q e P f · · · P f n n (this holds also if Q = P i for some i ). Since I is principal, ψ ( I ) = ; hence, = ψ ( I ) = ψ ( Q e P f · · · P f n n ) = eψ ( Q ) + n X i =1 f i ψ ( P i ) . Solving in ψ ( Q ), we see that ψ ( Q ) = P i − f i e ψ ( P i ) ∈ neg( ψ (∆)), asclaimed.Conversely, suppose that ψ ( Q ) is in the negative cone. Then, either ψ ( Q ) = (in which case Q ∈ Inv(∆) by taking no P ∈ ∆ in the def-inition) or we can find P , . . . , P n ∈ ∆ and negative rational numbers DARIO SPIRITO q , . . . , q n such that ψ ( Q ) = P i q i ψ ( P i ). By multiplying for the mini-mum common multiple of the denominators of the q i we obtain a rela-tion eψ ( Q ) + P i f i ψ ( P i ) = , with e, f i ∈ N + . If I := Q e P f · · · P f n n , itfollows that π ( I ) is torsion in the class group, i.e., there is an n > I n is principal; thus rad( I n ) = rad( I ) = Q ∧ P ∧ · · · ∧ P n ∈ P ( D ),as claimed. (cid:3) Corollary 4.3.
Let ∆ ⊆ Max( D ) . Then, Inv(∆) = Max( D ) if andonly if ψ (∆) positively spans Cl( D ) ⊗ Q .Proof. Suppose Inv(∆) = Max( D ), and let q ∈ Cl( D ) ⊗ Q . Since theimage C of ψ generates Cl( D ) ⊗ Q as a Q -vector space and is a subgroup,there is a d ∈ N + such that d q ∈ C . Hence, d q = ψ ( I ) for some I ∈ I ( D ); factorize I as P e · · · P e n n , with P i ∈ Max( D ) and e i >
0. ByProposition 4.2, we have ψ ( I ) = X i e i ψ ( P i ) ∈ X i e i neg( ψ (∆)) = neg( ψ (∆)) , and thus also q = d ψ ( I ) ∈ neg( ψ (∆)). Hence, ψ (∆) negatively spansCl( D ) ⊗ Q , and thus it also positively spans Cl( D ) ⊗ Q .Conversely, suppose ψ (∆) positively spans Cl( D ) ⊗ Q ; thus, it alsonegatively spans Cl( D ) ⊗ Q . Let Q ∈ Max( D ): then, ψ ( Q ) ∈ neg( ψ (∆)),so that Q ∈ Inv(∆) by Proposition 4.2. Hence, Inv(∆) = Max( D ). (cid:3) We can now characterize when the rank of Cl( D ) is finite. Proposition 4.4.
Let D be a Dedekind domain. Then, rk Cl( D ) < ∞ if and only if there is a finite set ∆ ⊆ Max( D ) such that Inv(∆) =Max( D ) .Proof. Suppose first that rk Cl( D ) = n < ∞ . Then, Inv(Max( D )) =Max( D ), and thus ψ (Max( D )) positively spans Cl( D ) ⊗ Q , by Corollary4.3. Let { e , . . . , e n } be a basis of Cl( D ) ⊗ Q : then, each e i belongs tothe positive cone spanned by a finite subset Λ i of ψ (Max( D )). Thus,the union Λ of the Λ i is a finite set positively spanning Cl( D ) ⊗ Q , sothe corresponding subset ∆ of Max( D ) is finite and Inv(∆) = Max( D )by Corollary 4.3.Conversely, suppose there is a finite set ∆ = { P , . . . , P k } ⊆ Max( D )such that Inv(∆) = Max( D ). For every Q ∈ Max( D ), there are i , . . . , i r such that Q ∧ P i ∧ · · · ∧ P i r ∈ P ( D ); as in the proof of Proposition4.2, it follows that there are e, f , . . . , f r > Q e P f i · · · P f r i r is principal. It follows that [ Q ] ⊗ Q -vector subspaceof Cl( D ) ⊗ Q generated by P i ⊗ , . . . , P i r ⊗
1. Since Q was arbitrary,the set { P ⊗ , . . . , P k ⊗ } is a basis of Cl( D ) ⊗ Q . In particular,rk Cl( D ) = dim Q Cl( D ) ⊗ Q ≤ k < ∞ . (cid:3) We will also need a criterion to understand when Inv(∆) correspondto a linear subspace.
ADICALS OF PRINCIPAL IDEALS AND THE CLASS GROUP 9
Proposition 4.5.
Let ∆ ⊆ Max( D ) . Then, neg( ψ (∆)) is a linear sub-space of Cl( D ) ⊗ Q if and only if ∆ ⊆ Inv(∆) .Proof.
Suppose neg( ψ (∆)) is a linear subspace, and let Q ∈ ∆. Then,there are P i ∈ ∆, λ i ∈ Q − such that ψ ( Q ) = P i λ i ψ ( P i ); multiplyingby the minimum common multiple of the denominators we get an equal-ity eψ ( Q ) + P i f i ψ ( P i ) = where e, f i ∈ N + . Let I := Q e P f · · · P f n n :then, ψ ( I ) = , so that π ( I ) is torsion in Cl( D ), i.e., I n is principal forsome n . Thus, Q ∧ P ∧ · · · ∧ P n ∈ P ( D ), and Q ∈ Inv(∆).Conversely, suppose ∆ ⊆ Inv(∆), and let q be an element of thelinear subspace generated by ψ (∆). Then, there are P i , Q j ∈ ∆, θ i ∈ Q + and µ j ∈ Q − such that q = X i θ i ψ ( P i ) + X i µ j ψ ( Q j ) . By construction, each θ i ψ ( P i ) belongs to pos( ψ (∆)). Furthermore, each ψ ( Q j ) is in neg( ψ (∆)) by Proposition 4.2, and thus µ j ψ ( Q j ) ∈ pos( ψ (∆))for every j . Therefore, q ∈ pos( ψ (∆)), so the positive cone of ψ (∆) isa linear subspace and neg( ψ (∆)) = pos( ψ (∆)) is a subspace too. (cid:3) Proposition 4.4 can be interpreted by saying that rk Cl( D ) is finiteif and only if Max( D ) is “negatively generated” by a finite set. In thecase of finite rank, we need a way to link the dimension of Cl( D ) ⊗ Q with the cardinality of the sets spanning it as a positive cone; that is,we need to consider a notion analogue to the basis of a vector space.Since we need only to consider the case of finite rank, from now onwe suppose that n := rk Cl( D ) < ∞ , and we identify Cl( D ) ⊗ Q with Q n . Definition 4.6.
A set X ⊆ Q n is positive basis of Q n if pos( X ) = Q n and if pos( X \ { x } ) = Q n for every x ∈ X . Definition 4.7.
A subset ∆ ⊆ Max( D ) is an inverse basis of Max( D ) if Inv(∆) = Max( D ) and Inv(∆ ′ ) = Max( D ) for every ∆ ′ ( ∆ . These two notions are naturally connected.
Proposition 4.8.
Let ∆ ⊆ Max( D ) . Then, ∆ is an inverse basis of Max( D ) if and only if ψ (∆) is a positive basis of Q n .Proof. If ∆ is an inverse basis, then ψ (∆) positively spans Q n by Corol-lary 4.3, while ψ (∆ ′ ) does not for every ∆ ′ ( ∆ (again by the corol-lary). Hence, ψ (∆) is a positive basis. The converse follows in the sameway. (cid:3) Given a positive basis X of Q n , we call a partition { X , . . . , X s } of X a weak Reay partition if, for every j , the positive cone of X ∪ · · · ∪ X i is a linear subspace of Q n . The following is a variant of [8, Theorem 2]. Proposition 4.9.
Let X be a positive basis of Q n . Then: (a) every weak Reay partition of X has cardinality at most | X | − n ;(b) there is a weak Reay partition of cardinality | X | − n .Proof. Let { X , . . . , X s } be a weak Reay partition, and let V i be thelinear space spanned by X , . . . , X i (with V := (0)). We claim thatdim V i − dim V i − ≤ | X i | −
1. Indeed, let X i := { z , . . . , z t } : then, − z t belongs to the positive cone generated by V i − and X i , and thus wecan write − z t = y + P j λ j z j for some y ∈ V i − and λ j ≥
0. Thus, − (1 + λ t ) z t = y + λ z + · · · + λ t − z t − , and since λ t = − z t is linearly dependent from X ∪ · · · ∪ X i − ∪ { z , . . . , z t − } . Hence,dim V i ≤ dim V i − + t −
1, as claimed.Therefore, n = dim Q n =(dim V s − dim V s − ) + · · · + dim V ≤≤ ( | X s | −
1) + · · · + ( | X | −
1) = | X | − s, and thus s ≤ | X | − n , and (a) is proved. (b) is a direct consequence of[8, Theorem 2]. (cid:3) Similarly, if ∆ ⊆ Max( D ) is an inverse basis of Max( D ), we calla partition { ∆ , . . . , ∆ s } a weak Reay partition if ∆ ∪ · · · ∪ ∆ i ⊆ Inv(∆ ∪ · · · ∪ ∆ i ) for every i . Proposition 4.10.
Let ∆ ⊆ Max( D ) be an inverse basis of Max( D ) ,and let { ∆ , . . . , ∆ s } be a partition of ∆ . Then, { ∆ , . . . , ∆ s } is a weakReay partition of ∆ if and only if { ψ (∆ ) , . . . , ψ (∆ s ) } is a weak Reaypartition of ψ (∆) .Proof. By Proposition 4.5, ∆ ∪ · · · ∪ ∆ i ⊆ Inv(∆ ∪ · · · ∪ ∆ i ) if andonly if the positive cone of ψ (∆ ∪ · · · ∪ ∆ i ) = ψ (∆ ) ∪ · · · ∪ ψ (∆ i ) is alinear subspace of Q n . The claim now follows from the definition. (cid:3) Theorem 4.11.
Let
D, D ′ be Dedekind domains such that P ( D ) and P ( D ′ ) are isomorphic. Then, rk Cl( D ) = rk Cl( D ′ ) .Proof. Let φ : P ( D ) −→ P ( D ′ ) be an isomorphism; by Theorem 3.6,we can find an isomorphism Φ : Rad( D ) −→ Rad( D ′ ) sending P ( D )to P ( D ′ ). In particular, Φ(Max( D )) = Max( D ′ ).Since Inv(∆) is defined only through P ( D ) and Rad( D ), Φ respectsthe inverse construction, in the sense that Φ(Inv(∆)) = Inv(Φ(∆)) forevery ∆ ⊆ Max( D ). In particular, Inv(∆) = Max( D ) if and only ifInv(Φ(∆)) = Max( D ′ ); by Proposition 4.4, it follows that rk Cl( D ) = ∞ if and only if rk Cl( D ′ ) = ∞ .Suppose now that the two ranks are finite, say equal to n and n ′ respectively. Let ∆ ⊆ Max( D ) be an inverse basis of Max( D ). Let { ∆ , . . . , ∆ s } be a weak Reay partition of ∆ of maximum cardinality;by Propositions 4.10 and 4.9, s = | ∆ | − n .Every weak Reay partition of ∆ gets mapped by Φ into a weakReay partition of ∆ ′ := ψ (∆), and conversely; therefore, the maximum ADICALS OF PRINCIPAL IDEALS AND THE CLASS GROUP 11 cardinality of the weak Reay partitions of ∆ ′ is again | ∆ | − n . However,applying Propositions 4.10 and 4.9 to ∆ ′ we see that this quantity is | ∆ ′ | − n ′ ; since | ∆ | = | ∆ ′ | , we get n = n ′ , as claimed. (cid:3) Corollary 4.12.
Let
D, D ′ be Dedekind domains, and let T ( D ) (re-spectively, T ( D ′ ) ) be the torsion subgroup of Cl( D ) (resp., Cl( D ′ ) ). If P ( D ) and P ( D ′ ) are isomorphic and if Cl( D ) and Cl( D ′ ) are finitelygenerated, then Cl( D ) /T ( D ) ≃ Cl( D ′ ) /T ( D ′ ) .Proof. Since Cl( D ) is finitely generated, it has finite rank n and Cl( D ) /T ( D ) ≃ Z n ; analogously, Cl( D ′ ) /T ( D ′ ) ≃ Z m , where m := rk Cl( D ′ ). By Theo-rem 4.11, n = m , and in particular Cl( D ) /T ( D ) ≃ Cl( D ′ ) /T ( D ′ ). (cid:3) Counterexamples
In this section, we collect some examples showing that Theorem 4.11is, in many ways, the best possible.
Example 5.1.
It is not possible to improve the conclusion of Theorem4.11 from “rk Cl( D ) = rk Cl( D ′ )” to “Cl( D ) ≃ Cl( D ′ )”. Indeed, ifrk Cl( D ) = 0 (i.e., if Cl( D ) is torsion) then P ( D ) = Rad( D ), and thuswhenever rk Cl( D ) = rk Cl( D ′ ) = 0 the posets P ( D ) and P ( D ′ ) areisomorphic.For the next examples, we need to use a construction of Claborn [3].Let G := P i x i Z be the free abelian group on the countable set { x i } i ∈ N . Let I be a subset of G satisfying the following two properties: • all coefficients of the elements of I (with respect to the x i ) arenonnegative; • for every finite set x i , . . . , x i k and every n , . . . , n k ∈ N there isan element y of I such that the component of y relative to x i t is n t .Then, [3, Theorem 2.1] says that there is an integral domain D withcountably many maximal ideals { P i } i ∈ N such that the map sending theideal P n · · · P n k k to n x + · · · + n k x k sends principal ideals to elementsof the subgroup H generated by I . In particular, Cl( D ) ≃ G/H . Example 5.2.
Corollary 4.12 does not hold without the hypothesisthat Cl( D ) and Cl( D ′ ) are finitely generated.For example, let H be the subgroup of G generated by x n + x n +1 ,as n ranges in N , and I to be the subset of the elements of H havingall coefficients nonnegative. Then, I satisfies the above conditions; thecorresponding domain D has a class group isomorphic to Z , and itsprime ideals are concentrated in two classes: if n is even P n is equivalentto P , if n is odd P n is equivalent to P , and P P is principal. (Thisis exactly Example 3-2 of [3].) In particular, P ( D ) is equal to themembers of Rad( D ) that are contained both in some P n with n evenand in some P m with m odd. Let now H to be the subgroup of G generated by x n + 2 x n +1 , as n ranges in N , and let I be the subset of the elements of H having allcoefficients nonnegative. Then, I too satisfies the condition above. Let D be the corresponding Dedekind domain. Then, Cl( D ) is isomorphicto the quotient G/H , which is isomorphic to the subgroup Z (2 ∞ ) of Q generated by 1 , , , . . . , n , . . . (that is, to the Pr¨ufer 2-group): thiscan be seen by noting that the map G −→ Q ,P n ( − n n is a group homomorphism with kernel H and range Z (2 ∞ ). In thisisomorphism, the prime ideals Q n with n even are mapped to positiveelements of Z (2 ∞ ), while the prime ideals Q m with m odd are mappedto the negative elements. Hence, P ( D ) is equal to the member ofRad( D ) that are contained in both an “even” and an “odd” prime.Therefore, the map Rad( D ) −→ Rad( D ) sending P i ∩ · · · ∩ P i k to Q i ∩· · ·∩ Q i k is an isomorphism sending P ( D ) to P ( D ). However, theclass groups of D and D are both torsionfree (i.e., T ( D ) = T ( D ) =0) but not isomorphic. Example 5.3.
The converse of Theorem 4.11 does not hold; that is, itis possible that rk Cl( D ) = rk Cl( D ′ ) even if P ( D ) and P ( D ′ ) are notisomorphic.Take H and D as in the previous example.Take H to the the subgroup of G generated by x and by x n + x n +1 for n >
0, and let I be the subset of the elements of H having allcoefficients nonnegative. Then, I satisfies Claborn’s conditions, andthe corresponding domain D satisfies Cl( D ) ≃ Z (in particular,rk Cl( D ) = 1), so Cl( D ) and Cl( D ) are isomorphic.However, D has a principal maximal ideal (the one correspond-ing to x ), while D does not. Therefore, there is no isomorphismRad( D ) −→ Rad( D ) sending P ( D ) to P ( D ); by Theorem 3.6, itfollows that P ( D ) and P ( D ) cannot be isomorphic. References [1] M. F. Atiyah and I. G. Macdonald.
Introduction to Commutative Algebra .Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969.[2] Nicolas Bourbaki.
Commutative Algebra. Chapters 1–7 . Elements of Mathemat-ics (Berlin). Springer-Verlag, Berlin, 1989. Translated from the French, Reprintof the 1972 edition.[3] Luther Claborn. Specified relations in the ideal group.
Michigan Math. J. ,15:249–255, 1968.[4] Chandler Davis. Theory of positive linear dependence.
Amer. J. Math. , 76:733–746, 1954.[5] Alfred Geroldinger and Franz Halter-Koch.
Non-unique factorizations , volume278 of
Pure and Applied Mathematics (Boca Raton) . Chapman & Hall/CRC,Boca Raton, FL, 2006. Algebraic, combinatorial and analytic theory.
ADICALS OF PRINCIPAL IDEALS AND THE CLASS GROUP 13 [6] Robert Gilmer and Jack Ohm. Integral domains with quotient overrings.
Math.Ann. , 153:97–103, 1964.[7] J¨urgen Neukirch.
Algebraic Number Theory , volume 322 of
Grundlehren derMathematischen Wissenschaften [Fundamental Principles of Mathematical Sci-ences] . Springer-Verlag, Berlin, 1999. Translated from the 1992 German originaland with a note by Norbert Schappacher, With a foreword by G. Harder.[8] John R. Reay. A new proof of the Bonice-Klee theorem.
Proc. Amer. Math.Soc. , 16:585–587, 1965.[9] Dario Spirito. The Golomb topology on a Dedekind domain and the group ofunits of its quotients.
Topology and its Applications , 273:107101, 2020.
Dipartimento di Matematica e Fisica, Universit`a degli Studi “RomaTre”, Roma, Italy
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