aa r X i v : . [ m a t h . A C ] F e b THE DERIVED SEQUENCE OF A PRE-JAFFARDFAMILY
DARIO SPIRITO
Abstract.
We introduce the concept of pre-Jaffard family , a gen-eralization of Jaffard families obtained by substituting the locallyfinite hypothesis with a much weaker compactness hypothesis. Fromany such family, we construct a sequence of overrings of the startingdomain that allows to decompose stable semistar operations andsingular length functions in more cases than what is allowed byJaffard families. We also apply the concept to one-dimensional do-mains, unifying the treatment of sharp and dull degree of a Pr¨uferdomain. Introduction A Jaffard family of an integral domain D is a family of flat overringsof D that satisfy some strong independence property, while simulta-neously respecting the structure of the whole ring (see Definition 3.7for a precise definition). This notion allows to extend several resultsof h -local domains to more general rings; in particular, it was usedto extend factorization properties from domains of Dedekind type toa wider class of domains [6, Chapter 6]. Jaffard families were subse-quently used to factorize the set of star operations [15, 18] and the setof length functions [17] on an integral domain as the product of theanalogous sets on the members of a Jaffard family.In this paper, we introduce two generalizations of Jaffard families,namely weak Jaffard families and pre-Jaffard families .Weak Jaffard families (see Section 5) are very similar to Jaffard fam-ilies, with the exception that we allow for a single member of the familyto behave “badly”.Pre-Jaffard families (see Section 4), on the other hand, need to sat-isfy weaker hypothesis, but for these reason are much more common;for example, the set of localizations at the maximal ideals of a domainof dimension 1 is always a pre-Jaffard family. We show in Section 6how every pre-Jaffard family Θ generates a sequence of weak Jaffardfamilies; this sequence is constructed very similarly to the sequence ofderived sets of a topological space, and for this reason we call it the Date : February 26, 2021.2010
Mathematics Subject Classification.
Key words and phrases.
Jaffard families; length functions; stable operations;sharp degree; dull degree; flat overrings. derived sequence of Θ. Indeed, when the dimension of the base ring D is 1, the members of the derived sequence of Θ correspond naturally tothe member of the sequence of derived sets of the maximal space of D ,endowed with the inverse topology. In particular, our construction is ageneralization of the study of sharp and dull primes tackled in [9, Sec-tion 6] for one-dimensional Pr¨ufer domains; moreover, our terminologysymmetrizes some of their results by unifying the concept of sharp anddull degree of a one-dimensional domain into the concept of Jaffarddegree of a pre-Jaffard family. See Section 8 for the discussion.In Section 7, we apply weak Jaffard families to the study of singularlength functions and of star operations: we show that, given a pre-Jaffard family, we can factorize their set through the derived sequence(Theorem 7.5) allowing a wide generalization of the results on Jaffardfamilies and of [17, Example 6.9].2.
Preliminaries
Throughout the paper, all rings will be commutative, unitary andwithout zero-divisors, i.e., integral domains; we denote such a ring by D , and we will always use K to denote its quotient field.We use Spec( D ) and Max( D ), respectively, to denote the spectrumand the maximal spectrum of D , and we denote by D ( I ) and V ( I ),respectively, the open and the closed set of Spec( D ) associated to theideal I . The inverse topology on Spec( D ) is the topology generatedby the V ( I ), as I ranges among the finitely generated ideals of D .We denote by ∆ inv a subset ∆ ⊆ Spec( D ) endowed with the inversetopology.The constructible topology is the topology generated by the D ( I )and the V ( J ) (as I ranges among all ideals and J among all finitelygenerated ideals); the constructible topology is still compact, but it isalso Hausdorff. We denote by ∆ cons a subset ∆ ⊆ Spec( D ) endowedwith the constructible topology. See [2, Chapter 1] for the constructionand properties of the inverse and the constructible topology.2.1. Overrings. An overring of D is a ring T such that D ⊆ T ⊆ K ;the set of all overrings of D is denoted by Over( D ). This set can beendowed with a topology (called the Zariski topology ) by taking as asubbasis the family of sets B ( x , . . . , x n ) := { T ∈ Over( D ) | x , . . . , x n ∈ T } , as x , . . . , x n range in K . Under this topology, Over( D ) is a compactspace that is not Hausdorff, and furthermore it is a spectral space in thesense of Hochster [10], i.e., there is a ring A (in general not determinedexplicitly) such that Spec( A ) (endowed with the Zariski topology) ishomeomorphic to Over( D ) (see e.g. [4, Proposition 3.5]). The name HE DERIVED SEQUENCE OF A PRE-JAFFARD FAMILY 3 “Zariski topology” is also due to the fact that the localization map λ : Spec( D ) −→ Over( D ) ,P D P is continuous and, indeed, a topological embedding [3, Lemma 2.4].The closure under generizations of a set Θ ⊆ Over( D ) isΘ ↑ := { T ∈ Over( D ) | T ⊇ S for some S ∈ Θ } ;a family Θ is closed by generizations if Θ = Θ ↑ . The family of allsets that are closed by generizations and compact with respect to theZariski topology is the family of closed sets of a topology, called the inverse topology of Over( D ); equivalently, the inverse topology is thetopology generated by the complements of the sets B ( x , . . . , x n ).The constructible topology on Over( D ) is the topology generated byboth the sets B ( x , . . . , x n ) and its complements. The space of all over-rings, under both the inverse and the constructible topology, is againcompact and a spectral space; moreover, under the constructible topol-ogy it is Hausdorff. Every set that is closed in the constructible topologyis compact with respect to the Zariski topology.2.2. Isolated points.
Let X be a topological space. A point x ∈ X is isolated if { x } is an open set; we denote the set of isolated pointsof X by I ( X ). The set of non-isolated (i.e., limit) points is called the derived set of X and is denoted by by D ( X ).We set D ( X ) := X and, for every ordinal α , we define: D α ( X ) := ( D ( D γ ( X )) if α = γ + 1 is a successor ordinal , T β<α D β ( X ) if α is a limit ordinal . The set D α ( X ) is called the α -th Cantor-Bendixson derivative of X ,and the smallest ordinal α such that D α ( X ) = D α +1 ( X ) is called the Cantor-Bendixson rank of X . If D α ( X ) = ∅ for some ordinal α , thespace X is said to be scattered ; equivalently, X is scattered if and onlyif every nonempty subspace has an isolated point.2.3. Semistar operations and length functions.
Let D be a do-main and let F D ( K ) be the set of D -submodules of K . A semistaroperation on D is a map ⋆ : F D ( K ) −→ F D ( K ), I I ⋆ , such that, forevery I, K ∈ F D ( K ) and every x ∈ K : • I ⊆ I ⋆ ; • if I ⊆ J , then I ⋆ ⊆ J ⋆ ; • ( I ⋆ ) ⋆ ) = I ⋆ ; • x · I ⋆ = ( xI ) ⋆ .If ( I ∩ J ) ⋆ = I ⋆ ∩ J ⋆ for every I, J , we say that ⋆ is stable . We denote thesets of semistar operations and of stable semistar operations, respec-tively, by SStar( D ) and SStar st ( D ). These two sets have a partial order,given by ⋆ ≤ ⋆ if and only if I ⋆ ⊆ I ⋆ for every ideal I ; the infimum DARIO SPIRITO of a family ∆ of semistar operations is the map ♯ : I T { I ⋆ | ⋆ ∈ ∆ } .If ∆ ⊆ SStar st ( D ), then also ♯ is stable.Let Mod( D ) be the category of D -modules. A length function on D is a function ℓ : Mod( D ) −→ R ≥ ∪ {∞} such that: • ℓ (0) = 0; • if 0 −→ N −→ M −→ P −→ ℓ ( M ) = ℓ ( P ) + ℓ ( N ); • for every module M , we have ℓ ( M ) = sup { ℓ ( N ) | N is a finitelygenerated submodule of M } .The sum of a family Λ of length functions is defined as the map suchthat X ℓ ∈ Λ ℓ ! ( M ) = sup { ℓ ( M ) + · · · + ℓ n ( M ) } , as { ℓ , . . . , ℓ n } ranges among the finite subsets of Λ.If T is a flat overring of D , then we can associate to any lengthfunction ℓ on D a length function ℓ D on T by restriction of scalars, i.e.,setting ℓ D ( M ) := ℓ ( M ) for all T -modules M . Moreover, we can defineda new length function ℓ ⊗ T on D by setting( ℓ ⊗ T )( M ) := ℓ ( M ⊗ T ) . for all D -modules M .By [17, Theorem 6.5] and the subsequent discussion, there is a bi-jection between the set L sing ( D ) of length functions such that ℓ ( M ) ∈{ , + ∞} for all M ∈ Mod( D ) and the set SStar st ( D ) of stable semistaroperations on D , and by [17, Proposition 6.6] the infimum of a familyof stable operations correspond to the sum of the corresponding lengthfunctions. Moreover, the passage from a length function ℓ on a flatoverring T to ℓ D correspond to the passage from the a stable operation ⋆ on T to the closure I ( IT ) ⋆ on D .3. Jaffard overrings
In this paper we will mostly use families consisting of flat overrings ,i.e., overrings of a domain D that are flat when considered as D -modules. However, in many case we will define rings by intersectinglocalizations; thus we need the following definition. Definition 3.1.
An overring T of D is a sublocalization of D if thereis a set ∆ ⊆ Spec( D ) such that T = T { D P | P ∈ ∆ } .If T is a sublocalization of D , we set: • σ ( T ) := { Q ∩ D | Q ∈ Spec( T ) } ; • Σ( T ) := { P ∈ Spec( D ) | T ⊆ D P } ; • T ⊥ := T { D P | P = (0) or P ∈ Spec( D ) \ Σ( T ) } . Note that, by definition, T ⊥ is a sublocalization too, and thus itmakes sense to consider σ ( T ⊥ ) and Σ( T ⊥ ). HE DERIVED SEQUENCE OF A PRE-JAFFARD FAMILY 5
Lemma 3.2.
For every sublocalization T of D , we have Σ( T ) ∪ Σ( T ⊥ ) =Spec( D ) .Proof. If P / ∈ Σ( T ), then by definition T ⊥ ⊆ D P , and thus P ∈ Σ( T ⊥ ). (cid:3) Every flat overring is a sublocalization [14, Corollary to Theorem 2],but the converse is not true (see [8] and [16, Example 6.3]). We cancharacterize when a sublocalization is flat.
Lemma 3.3.
Let T be a sublocalization of D . Then, T is flat over D if and only if Σ( T ) = σ ( T ) .Proof. The containment Σ( T ) ⊆ σ ( T ) holds for every sublocalization.If T is flat and P ∈ σ ( T ), then P T = T , and by [14, Theorem 1] wehave T ⊆ D P , i.e., P ∈ Σ( T ). Conversely, if σ ( T ) = Σ( T ) and P T = T ,let Q be a prime ideal of T above P T : then, P ′ := Q ∩ D ∈ σ ( T ) andthus T ⊆ D P ′ . Hence, P T ∩ D ⊆ P D P ′ ∩ D = P , and so P ′ = P , andin particular T ⊆ D P . Again by [14, Theorem 1], T is flat. (cid:3) Lemma 3.4.
Let A be a flat overring and B a sublocalization of D .Then, AB = K if and only if σ ( A ) ∩ σ ( B ) = { (0) } .Proof. Suppose that AB = K , and let P ∈ σ ( A ) ∩ σ ( B ). Since A isflat, P ∈ Σ( A ) and so A ⊆ D P ; on the other hand, if Q is a prime idealof B above P , then D P ⊆ B Q . Hence, K = ABD P = ( AD P )( BD P ) ⊆ D P B Q = B Q . It follows that Q = (0) and so P = (0) too.Conversely, if σ ( A ) ∩ σ ( B ) = { (0) } then the claim follows from [6,Lemma 6.2.1]. (cid:3) Definition 3.5.
Let D be an integral domain with quotient field K andlet Θ be a family of overrings of D . We say that Θ is: • complete if, for every ideal I of D , we have I = T { IT | T ∈ Θ } ; • independent if, for every A, B ∈ Θ , σ ( A ) ∩ σ ( B ) = { (0) } ; • strongly independent if, for every A ∈ Θ , we have A · \ B ∈ Θ B = A B = K • locally finite if every x ∈ K , x = 0 , is a nonunit in only finitelymany members of Θ . By Lemma 3.4, if Θ is a family of flat subsets, then Θ is independentif and only if AB = K for every A = B in Θ; in particular, a stronglyindependent set of flat overrings is always independent. We can provewhen the converse happens. Proposition 3.6.
Let Θ be a complete and independent set of flat over-ring of D . Then, Θ is strongly independent if and only if Θ is locallyfinite. DARIO SPIRITO
Proof.
If Θ is locally finite, the claim follows from [6, Theorem 6.3.1(4)](see below for the definitions used in the reference). Suppose that Θis strongly independent but not locally finite: then, there is a nonzero x ∈ D such that xT = T for an infinite family Θ ′ ⊆ Θ. Hence, foreach T ∈ Θ ′ there is a prime ideal P T ∈ Spec( D ) such that x ∈ P T and P T T = T ; let Λ be the family of such ideals. Then, Λ is an infinitesubset of the compact space Spec( D ) cons , and thus it has a limit point Q ; furthermore, Λ is contained in the clopen set V ( x ) of Spec( D ) cons and thus also Q ∈ V ( x ), i.e., x ∈ Q ; in particular, Q = (0).Since Θ is complete and independent, there is a unique S ∈ Θ suchthat QS = S ; let A := T { T ∈ Θ | T = S } . Then, A is a sublocalizationof D ; by [14, Theorem 1], σ ( A ) is the image of Spec( A ) under therestriction map Z Z ∩ D , and thus σ ( A ) is a closed set in theconstructible topology. Moreover, σ ( A ) contains all the elements of Λexcept one (the ideal P S ), and thus it must contain also the limit point Q of Λ \ { P S } . Therefore, D Q A = K . However, S ⊆ D Q since S is flat;therefore, AS ⊆ D Q S = K . This contradicts the fact that Θ is stronglyindependent: hence Θ must be locally finite. (cid:3) Note that the above result does not hold without the hypothesis thatΘ is complete: see Example 6.7 below.Families satisfying the hypothesis of the previous proposition havetheir own name.
Definition 3.7.
Let Θ be a family of overrings of D . We say that Θ is a Jaffard family of D if: • either K / ∈ Θ or Θ = { K } ; • every T ∈ Θ is flat; • Θ is complete; • Θ is independent; • Θ is locally finite.We say that an overring T of D is a Jaffard overring if it belongs to aJaffard family of D . Remark 3.8. (1) Definition 3.7 is not the original one of a Jaffard family, but itis the one most useful for our purpose; see [6, Section 6.3] and[15, Proposition 4.3].(2) By Proposition 3.6 (see also [6, Theorem 6.3.1(4)]) the two con-ditions “Θ is independent” and “Θ is locally finite” can be uni-fied into the single one “Θ is strongly independent”.(3) If P is a nonzero prime of D , then there is exactly one T ∈ Θsuch that
P T = T : indeed, such a T must exists since Θ isindependent, while there cannot be two of them due to in-dependence condition. In particular, Θ induces a partition onMax( D ), called a Matlis partition [6, Section 6.3].
HE DERIVED SEQUENCE OF A PRE-JAFFARD FAMILY 7 (4) If Θ ∈ K (i.e., Θ = { K } ) then since Θ must be complete wemust have also D = K , i.e., D must be a field. Proposition 3.9.
Let T be a flat overring of D . Then, the followingare equivalent:(i) T is a Jaffard overring of D ;(ii) T · T ⊥ = K ;(iii) { T, T ⊥ } is a Jaffard family of D ;(iv) σ ( T ) ∩ σ ( T ⊥ ) = { (0) } ;(v) for every nonzero P ∈ σ ( T ) , we have P T ⊥ = T ⊥ ;(vi) there is a sublocalization A of D such that { T, A } is completeand T A = K .Proof. (iii) = ⇒ (i) follows from the definitions.(i) = ⇒ (ii) Let Θ be a Jaffard family containing T , and let Θ ⊥ ( T ) := T { S ∈ Θ | S = T } . For every nonzero prime ideal P of D out of Σ( T ),there is a unique S ∈ Θ such that
P S = S ; since S is flat, we have S ⊆ D P , and soΘ ⊥ ( T ) ⊆ \ { P ∈ Spec( D ) | P / ∈ Σ( T ) } = T ⊥ . Since Θ is strongly independent, T · Θ ⊥ ( T ) = K , and so T T ⊥ = K .(ii) = ⇒ (iii) Let Θ := { T, T ⊥ } . Clearly, Θ is locally finite and com-plete, while it is independent by Lemma 3.4 (since T is flat by hypoth-esis). Since T ∩ T ⊥ = D , by [6, Theorem 6.2.2(1)] T ⊥ is also flat; hence,Θ is a Jaffard family.(ii) ⇐⇒ (iv) is exactly Lemma 3.4.(ii) = ⇒ (vi) is obvious. To show (vi) = ⇒ (i), it is enough to showthat Θ := { T, A } is a Jaffard family. By hypothesis, Θ is complete andlocally finite, while it is independent by Lemma 3.4. In particular, by[6, Theorem 6.2.2(1)] A is also flat, and thus Θ is a Jaffard family.(iii) = ⇒ (v) If P ∈ σ ( T ), P = (0), then P T = T ; but since { T, T ⊥ } is a Jaffard family, no nonzero prime can survive in both T and T ⊥ ,and so P T ⊥ = T ⊥ (v) = ⇒ (ii) Suppose that T · T ⊥ = K : then, there is a nonzero Q ∈ Spec( D ) such that QT T ⊥ = T T ⊥ , and so both QT = T and QT ⊥ = T ⊥ . However, the first condition implies that Q ∈ σ ( T ), contradictingthe hypothesis. (cid:3) Corollary 3.10.
Let Θ be a complete and independent family of flatoverrings of D . Then, Θ is a Jaffard family if and only if each T ∈ Θ is a Jaffard overring.Proof. If Θ is a Jaffard family then every T ∈ Θ is a Jaffard overringby definition. Conversely, suppose each T ∈ Θ is a Jaffard overring; byProposition 3.6 we only need to show that Θ is strongly independent.
DARIO SPIRITO
Fix T ∈ Θ. If S ∈ Θ \ { T } , then σ ( S ) ∩ σ ( T ) = Σ( S ) ∩ Σ( T ) = ∅ ,and thus T ⊥ ⊆ S . Therefore, T \ S ∈ Θ \{ T } S ⊇ T T ⊥ = K using Proposition 3.9. Hence, Θ is strongly independent and thus aJaffard family. (cid:3) We conclude this section with a lemma that will be useful later.
Lemma 3.11.
Let Θ be a complete and independent family of flat over-rings of D and let P = (0) be a prime ideal of D . For every S ∈ Θ ,either P S = S or D P S = K .Proof. Suppose that
P S = S . Since Θ is complete, there is a S ′ ∈ Θsuch that
P S ′ = S ′ ; since S ′ is flat, by Lemma 3.3 S ′ ⊆ D P . Hence, SS ′ ⊆ SD P ; however, since Θ is independent SS ′ = K . Hence K = SD P . (cid:3) Pre-Jaffard families
The hypothesis that a family is locally finite is usually very strong.To expand our reach beyond Jaffard families, we define a new class offamilies by weakening this condition.
Definition 4.1.
Let Θ be a family of overrings of D . We say that Θ is a pre-Jaffard family of D if: • either K / ∈ Θ or Θ = { K } ; • every element of Θ is flat over D ; • Θ is independent; • Θ is complete; • Θ is compact in the Zariski topology. Remark 4.2.
We do not know any example of a family of overringsatisfying the first three conditions of Definition 4.1 but that is notcompact; it is possible that the compactness condition is actually re-dundant.
Proposition 4.3.
A pre-Jaffard family is Hausdorff, with respect tothe inverse topology.Proof.
Without loss of generality we can suppose that
K / ∈ Θ. Fix twodistinct overrings
T, S ∈ Θ. Let C := {B ( x ) | x ∈ T \ S } ∪ {B ( y ) | y ∈ S \ T } ; we claim that C is a cover of Θ.Since ST = K , we have S ( T and T ( S , and thus T \ S and S \ T are both nonempty; it follows that S, T belong to some member of C .Let A ∈ Θ \ { S, T } : then, A ∩ ( T \ S ) = A ∩ T ∩ ( K \ S ) , HE DERIVED SEQUENCE OF A PRE-JAFFARD FAMILY 9 and thus if A ∩ ( T \ S ) = ∅ then A ∩ T ⊆ S . However, ( A ∩ T ) S = AS ∩ T S = K ; hence, A ∩ ( T \ S ) = ∅ and so there is an x ∈ A ∩ ( T \ S ),i.e., A ∈ B ( x ). Thus, any such A belong to some member of C , and C is a cover of Θ.Since Θ is compact in the Zariski topology, we can find x , . . . , x n , y , . . . , y m such that {B ( x ) , . . . , B ( x n ) , B ( y ) , . . . , B ( y m ) } is a finite subcover. LetΩ := n \ i =1 B ( x i ) c and Ω := m \ j =1 B ( y j ) c ;then, Ω and Ω are both open in the inverse topology, since they are afinite intersection of subbasic open sets. Moreover, S ∈ Ω since x i / ∈ S for every i , while T ∈ Ω since y j / ∈ T for every j . Finally,Ω ∩ Ω = n \ i =1 B ( x i ) c ∩ m \ j =1 B ( y j ) c = n [ i =1 B ( x i ) ∪ m [ j =1 B ( y j ) ! c ⊆ Θ c , that is, Ω ∩ Ω does not intersect Θ. Hence, Ω ∩ Θ and Ω ∩ Θ aredisjoint neighborhood of S and T in Θ, with respect to the inversetopology. Thus, Θ is Hausdorff in the inverse topology. (cid:3) Proposition 4.4.
A Jaffard family of D is a pre-Jaffard family.Proof. By definition, any Jaffard family of D is independent, completeand composed of flat overrings. Furthermore, any locally finite family ofoverrings is compact, and thus a Jaffard family is also pre-Jaffard. (cid:3) Proposition 4.5.
Let Θ be a pre-Jaffard family of D , and let T ∈ Θ .Then, T is a Jaffard overring of D if and only if Θ \ { T } is compactin the Zariski topology. Furthermore, if this happens, then T is isolatedin Θ , with respect to the inverse topology.Proof. If T is a Jaffard overring, by Proposition 3.9 we have T T ⊥ = K ,and in particular no overring of D different from K contains both T ⊥ and T . Then, Θ ↑ ∩ { T ⊥ } ↑ = (Θ \ { T } ) ↑ : however, since Θ ↑ and { T ⊥ } ↑ are closed in the inverse topology (the former since Θ is compact byhypothesis), then also (Θ \ { T } ) ↑ is inverse-closed. Therefore, Θ \ { T } ,which is the set of minimal elements of (Θ \ { T } ) ↑ , is compact withrespect to the Zariski topology. This also shows that T is isolated inΘ, with respect to the inverse topology.Suppose Θ \ { T } is compact, and let A := T { S | S ∈ Θ , S = T } .Then, A is a sublocalization of D ; moreover, since T is flat and Θ \ { T } is compact, by [5, Corollary 5] we have T A = T \ S ∈ Θ S = \ S ∈ Θ T S = K. Hence, T is a Jaffard overring by Proposition 3.9. (cid:3) Remark 4.6.
Proposition 4.5 cannot be improved to a full equivalencebetween being a Jaffard overring and being and isolated point of Θ inv .Consider the ring D defined in [7, Example 2]. Then, D is a two-dimensional domain such that: • all its finitely generated ideals are principal (i.e., D is a B´ezoutdomain); in particular, D M is a valuation domain for all maxi-mal ideals M ; • all its maximal ideals, except for one (say M ∞ ), have height 1; • M ∞ is the radical of a principal ideal; • the unique nonzero, nonmaximal prime ideal P is containedin a unique maximal ideal ( M ∞ ), but also in the union of allmaximal ideals distinct from M ∞ .Let Θ := { D M | M ∈ Max( D ) } . Then, every T ∈ Θ is flat and Θ iscomplete; furthermore, Θ is independent (if D M D N = K , then M ∩ N should contain a nonzero prime ideal, a contradiction) and compactin the Zariski topology (since the localization map is continuous by[3, Lemma 2.4] and so Θ is homeomorphic to Max( D )); thus, Θ is apre-Jaffard family.Let V := D M ∞ : then, T is not a Jaffard overring of D . Indeed, V ⊥ = T { D M | M ∈ Max( D ) , M = M ∞ } is such that P V ⊥ = V ⊥ : otherwise,since D is B´ezout, there would be an a ∈ P such that aV ⊥ = V ⊥ .However, there is also a maximal ideal M = M ∞ such that a ∈ M :hence, M V ⊥ = V ⊥ , against the fact that V ⊥ ⊆ D M by construction.Hence, P V ⊥ = V ⊥ and so V ⊥ ⊆ D P , so that V V ⊥ ⊆ V D P = D P (since V = D M ∞ ⊆ D P ). By Proposition 3.9, V is not a Jaffard overring of D .We claim that V is isolated in Θ, with respect to the inverse topology.Indeed, M ∞ is the radical of a principal ideal, say bD ; hence, M ∞ is theunique T ∈ Θ such that b − / ∈ T , i.e., B ( b − ) c ∩ Θ = { M ∞ } . However, B ( b − ) c is the complement of an open and compact subset of Over( D ),and thus it is open in the inverse topology; hence, V is isolated in Θ inv .The following two results show how to construct pre-Jaffard familiesfrom other such families by taking intersections. Lemma 4.7.
Let Θ be a family of overrings that is compact in theZariski topology. Let Θ , . . . , Θ n be subsets of Θ , and for each i let S i := T { T | T ∈ Θ i } . Then, the family Θ ′ := Θ \ n [ i =1 Θ i ! ∪ { S , . . . , S n } is compact in the Zariski topology.Proof. Let Ω := { Ω α } α ∈ A be an open cover of Θ ′ . If S i ∈ Ω α , thenΘ i ⊆ Ω α (since this holds for every subbasic open set B ( x )); hence, Ω is also an open cover of Θ. Since Θ is compact, we can find a finite HE DERIVED SEQUENCE OF A PRE-JAFFARD FAMILY 11 subcover { Ω α , . . . , Ω α k } of Θ; furthermore, for every i we can find a β i ∈ A such that S i ∈ Ω β i . Then, { Ω α , . . . , Ω α k , Ω β , . . . , Ω β n } is afinite subcover of Θ ′ . Thus, Θ ′ is compact. (cid:3) Proposition 4.8.
Let Θ be a pre-Jaffard family, and let Θ , . . . , Θ n bepairwise disjoint subsets of Θ that are compact in the Zariski topology.For every i , let S i := T { T | T ∈ Θ i } . Then, the family Θ ′ := Θ \ n [ i =1 Θ i ! ∪ { S , . . . , S n } is a pre-Jaffard family.Proof. By induction, it is enough to prove the claim for n = 1; let S := S .By construction, Θ ′ is complete, and by Lemma 4.7 it is compactin the Zariski topology. It is independent: indeed, take T , T ∈ Θ ′ . If T = S = T then T T = K since Θ is independent, while if S = T then T S = T \ T ∈ Θ T = \ T ∈ Θ T T = K by [5, Corollary 5], since Θ is compact and every T ∈ Θ is in Θ andis different from T . Thus, Θ is independent.We only need to prove that S is flat. By construction, S is a sublocal-ization. If P ∈ σ (Spec( S )) is a nonzero prime, then P T = T for every T ∈ Θ \ { S } (otherwise K = T S ⊆ D P , a contradiction); on the otherhand, there is a S ′ ∈ Θ such that P ∈ Σ( S ′ ), and thus P ∈ Σ( S ).Hence, σ (Spec( S )) = Σ( S ), and S is flat by Lemma 3.3. (cid:3) Weak Jaffard families
Let Θ be a pre-Jaffard family. In general, we cannot expect the prop-erties of a Jaffard family to hold also for Θ; however, we want to showthat at least some properties hold also under the weaker pre-Jaffardhypothesis. To do so, we want to proceed “step-by-step”, isolating firstthe Jaffard overring belonging to Θ; from a technical point of view, weneed the following definition.
Definition 5.1.
Let Θ be a family of overrings of D and let T ∞ ∈ Θ .We say that Θ is a weak Jaffard family of D pointed at T ∞ if: • either K / ∈ Θ or Θ = { K } ; • Θ is complete and independent; • every T ∈ Θ \ { T ∞ } is a Jaffard overring of D ; • T ∞ is flat over D . Lemma 5.2.
Let Θ be a family of flat overrings of D , and let B be anoverring of D . Let Θ B := { T B | T ∈ Θ } (a) If Θ is independent, Θ B is independent. (b) If Θ is complete with respect to D , then Θ B is complete withrespect to B .(c) If every T ∈ Θ is flat as a D -module, every T B ∈ Θ B is flat asa B -module.(d) If T is a Jaffard overring of B and T = K , then T B is a Jaffardoverring of B .(e) If Θ is a Jaffard family of B , then Θ B \ { K } is a Jaffard familyof B .Proof. (a) If T B = T ′ B with T = T in Θ, then ( T B )( T ′ B ) = ( T T ′ ) B = K .(b) Let I be a B -submodule of the quotient field K . Then, IB = B ,and thus I = IB = \ T ∈ Θ ( IB ) T = \ T ∈ Θ I ( BT ) = \ S ∈ Θ B IS so that Θ B is complete with respect to B .(c) Since B is an overring, the extension A ⊆ B is an epimorphism,and thus T B ≃ T ⊗ B [11, Lemma 1.0]. The claim follows.(d) If T is a Jaffard overring, then by Proposition 3.9 Θ := { T, T ⊥ } is a Jaffard family of D . By the previous points, Θ B = { T B, T ⊥ B } iscomplete, independent and formed by flat overrings; since it is clearlylocally finite, it is a Jaffard family, and thus T B is a Jaffard overring.(e) follows from the previous points and from Corollary 3.10. (cid:3)
A Jaffard family is always a weak Jaffard family, pointed to any ofits elements. Analogously, a weak Jaffard family is pre-Jaffard, as weshow next.
Proposition 5.3.
Let Θ be a weak Jaffard family of D pointed at S .(a) If J is a proper ideal of D and J S = S , then J T = T for onlyfinitely many T ∈ Θ .(b) Θ is a pre-Jaffard family of D .Proof. (a) Let B := T { D M | M ∈ V ( J ) } : we claim that BS = K .Indeed, V ( J ) is a compact subset of Spec( D ), and thus BS = \ M ∈V ( J ) D M S = \ M ∈V ( J ) D M S = K since if J ⊆ M then J T = T for some T ∈ Θ and thus D M S ⊇ T S = K .Consider the family Θ B := { BT | T ∈ Θ } \ { K } : by Lemma 5.2,Θ B is a complete and independent set of flat overrings of B , and all itselement except BS are Jaffard overrings of B . By Corollary 3.10, Θ B is a Jaffard family of B , and thus it is locally finite. For every T ∈ Θsuch that
J T = T , also J T B = T B ; hence, there are at most finitelymany elements of Θ such that
J T = T , as claimed. HE DERIVED SEQUENCE OF A PRE-JAFFARD FAMILY 13 (b) We need only to show that Θ is compact, with respect to theZariski topology. Let {B ( x α ) } α ∈I be an open cover of Θ, and suppose S ∈ B ( x ). Let J := ( D : D x ): then, J S = ( S : S x ) = S (using theflatness of S ), and thus there are only finitely many T ∈ Θ such that
J T = T ; call them T , . . . , T n . Therefore, if A ∈ Θ \ { T , . . . , T n } then J A = A and ( A : A x ) = A , i.e., x ∈ A ; thus B ( x ) \ Θ is finite. It fol-lows that we can find a subcover {B ( x ) , B ( x ) , . . . , B ( x n ) } by choosing x , . . . , x n such that x i ∈ T i . Since the cover was arbitrary, Θ ∪ { S } iscompact. (cid:3) Weak Jaffard families are much more ubiquitous than Jaffard fami-lies; the main reason is that a weak Jaffard family has a place to “hidethe singularities” of D (namely, the ring T ∞ to which the family ispointed), while a Jaffard family does not have such a luxury. A firstway in which weak Jaffard families arise is from a set of Jaffard over-rings. Proposition 5.4.
Let Θ be an independent set of Jaffard overrings of D , and let S := \ { D P | P T = T for every T ∈ Θ } . Then, Θ ∪ { S } is a weak Jaffard family of D pointed at S .Proof. Let P ∈ Spec( D ). If P T = T for some T ∈ Θ, then T ⊆ D P since T is flat [14, Theorem 1]; if P T = T for every T ∈ Θ, then S ⊆ D P . Therefore, every localization D P of D contains at least oneelement of Θ ∪ { S } . Hence, for every ideal I of D , I = \ M ∈ Max( D ) ID M ⊇ \ T ∈ Θ ∪{ S } IT ⊇ I. Thus Θ ∪ { S } is complete.If T, T ′ ∈ Θ, T = T ′ , then T T ′ = K by hypothesis. To show that Θis independent, let Σ := { P ∈ Spec( T ) | P T = T for every T ∈ Θ } .We claim that Σ = T { Spec( D ) \ Σ( T ) | T ∈ Θ } . Indeed, if P ∈ Σthen
P T = T for every T ∈ Θ, and thus P is in the intersection;conversely, if P / ∈ Σ( T ) for every T ∈ Θ, then (since each T is flat) wehave by Lemma 3.3 P / ∈ σ ( T ), and thus P T = T , so that P ∈ Σ. ByProposition 3.9, Spec( D ) \ Σ( T ) = Σ( T ⊥ ) \ { (0) } ; hence,Σ ∪ { (0) } = \ T ∈ Θ (Σ( T ⊥ ) \ { (0) } ) ∪ { (0) } = \ T ∈ Θ Σ( T ⊥ )Again by Proposition 3.9, T ⊥ is a Jaffard overring of D , and thus itis flat; hence, Σ( T ⊥ ) = σ ( T ⊥ ) is closed in the constructible topology,and thus also Σ ∪ { (0) } is closed in the constructible topology, and inparticular it is compact. Fix now a T ∈ Θ. Using the flatness of T , we have T S = T \ P ∈ Σ ∪{ (0) } D P = K ∩ \ P ∈ Σ T D P = K, since D P ⊇ T ⊥ for every nonzero P ∈ Σ ⊇ Σ( T ⊥ ). Hence, Θ ∪ { S } isindependent.Since every T ∈ Θ is a Jaffard overring, we only need to show that S is flat. Suppose that P ∈ σ ( S ): since Θ ∪ { S } is complete and in-dependent, we must have P T = T for every T ∈ Θ, and thus P ∈ Σ,so that, by definition S ⊆ D P . Thus P ∈ Σ( S ) and σ ( S ) = Σ( S ); byLemma 3.3, S is flat. (cid:3) The previous two propositions provide a way to pass from a pre-Jaffard family to a weak Jaffard family.
Definition 5.5. If Θ is a pre-Jaffard family of D , we denote by Θ J the set of Jaffard overrings contained in Θ . Proposition 5.6.
Let Θ be a pre-Jaffard family of D , and suppose that Θ J = Θ . Let S := T { T | T ∈ Θ \ Θ J } . Then:(a) Θ J ∪ { S } is a weak Jaffard family of D pointed at S ;(b) Θ \ Θ J is a pre-Jaffard family of S .Proof. (a) The set Θ J is a set of Jaffard overrings of D ; we claim that S = T { D P | P T = T for every T ∈ Θ } . Indeed, since every T ∈ Θ isflat we have S = \ T ∈ Θ \ Θ J \ P ∈ Σ( T ) D P = \ P ∈ Σ D P where Σ := S { Σ( T ) | T ∈ Θ \ Θ J } . Hence, Θ J ∪ { S } is a weak Jaffardfamily by Proposition 5.4.(b) Let Θ ′ := Θ \ Θ J . Then, Θ ′ is an independent family of flatoverrings of S , and is complete with respect to S , since if I is an idealof S then I = IS = \ T ∈ Θ IST = \ T ∈ Θ J IST ∩ \ T ∈ Θ ′ IST = \ T ∈ Θ ′ IS as S ⊆ T if T ∈ Θ ′ while ST = K if T ∈ Θ J . We thus need to showthat Θ ′ is compact.For every T ∈ Θ J , by Proposition 4.5 Θ \ { T } is compact in theZariski topology, and thus Λ T := (Θ \ { T } ) ↑ is closed in the inversetopology. Thus, also the intersection Λ := T { Λ T | T ∈ Θ J } is closedin the inverse topology. However, since { S } ↑ ∩ { S ′ } ↑ = { K } for every S = S ′ in Θ, the set of minimal elements of Λ is exactly Θ \ Θ J = Θ ′ ;hence, Θ ′ is compact, as claimed. (cid:3) HE DERIVED SEQUENCE OF A PRE-JAFFARD FAMILY 15 The derived sequence
Let Θ be a pre-Jaffard family of D , and let Θ J be the set of Jaffardoverrings inside Θ. If Θ = Θ J , then by Corollary 3.10 Θ is a Jaffardfamily; on the other hand, if Θ = Θ J then by Proposition 5.6 we candefine an overring T of D such that: • Θ J ∪ { T } is a weak Jaffard family of D ; • Θ := Θ \ Θ J is a pre-Jaffard family of T .In particular, we can repeat the same construction on Θ : either Θ is a Jaffard family of T or we can find an overring T of T (andso of D ) such that (Θ ) J ∪ { T } is a weak Jaffard family of T andΘ := Θ \ (Θ ) J is a pre-Jaffard family of T ; then we can use thesame construction on T , and so on. We now want to define rings T α and subfamilies Θ α for every ordinal α .To start, define T := D and Θ := Θ.Suppose that for every ordinal β < α we have defined a ring T β anda subset Θ β ⊆ Θ that is a pre-Jaffard family of T β . Then: • if α = γ + 1 is a successor ordinal, we defineΘ α := Θ γ \ (Θ γ ) J ; • if α is a limit ordinal, we defineΘ α := \ β<α Θ β . In both cases, we define T α := \ { S | S ∈ Θ α } with the convention that T α := K if Θ α = ∅ . Proposition 6.1.
Preserve the notation above. Then:(a) Θ α is a pre-Jaffard family of T α ;(b) if α = γ + 1 is a successor ordinal, then Θ α ∪ { T α } is a weakJaffard family of T γ pointed at T α .Proof. We proceed by transfinite induction: the claim is true by hy-pothesis for α = 0. Suppose that it holds for every β < α . If α = γ + 1is a successor ordinal, then the two statements are exactly Proposition5.6. Suppose thus that α is a limit ordinal.Each T ∈ Θ α is flat over D and thus over T α ; furthermore, Θ α isindependent since it is contained in the independent set Θ. Since alsoevery Θ β (for β < α ) is independent, as in the proof of Proposition 5.6we have Θ ↑ α = \ β<α Θ β ! ↑ = \ β<α Θ ↑ β which is closed in the inverse topology since each Θ β is compact withrespect to the Zariski topology (being a pre-Jaffard family by inductive hypothesis), and so also Θ α , which is the set of minimal elements of Θ ↑ α ,is compact in the Zariski topology. Thus, we only need to show thatΘ α is complete. Let P be a nonzero prime ideal of T α , and supposethat P S = S for some S ∈ Θ α : then, by Lemma 3.11, SD P = K .Therefore, if P S = S for all S ∈ Θ α then, by the flatness of D P andthe compactness of Θ α , by [5, Corollary 5] we have D P = D P T α = D P \ S ∈ Θ α S = \ S ∈ Θ α D P S = K, a contradiction. Hence Θ α is complete and thus it is a pre-Jaffard fam-ily. (cid:3) Definition 6.2.
We call the family { T α } defined in this way the derivedsequence with respect to Θ . By construction, the derived sequence of Θ is an ascending chain ofrings: D = T ⊆ T ⊆ T ⊆ · · · ⊆ T ω ⊆ · · · , which corresponds to a descending chain of sets of overrings:Θ = Θ ⊇ Θ ⊇ Θ ⊇ · · · ⊇ Θ ω ⊇ · · · . Proposition 6.3.
Preserve the notation above. There is an ordinal α such that T α = T α ′ for all α ′ > α (equivalently, such that Θ α = Θ α ′ forall α ′ > α ).Proof. Note that, if T α = T α +1 , then T α = T α ′ for all α ′ > α ; thus,suppose by contradiction that T α ) T α +1 for all α . Then, T α +1 \ T α isnonempty for all α ; but since all the T α are contained inside K , thisis impossible if the cardinality of α is larger than the cardinality of K . (cid:3) Definition 6.4.
We call the minimal ordinal α such that T α = T α ′ the Jaffard degree of the family Θ , and we call T α the dull limit of Θ . Wesay that Θ is: • a sharp family if T α = K ; • a dull family if T α = K . Equivalently, Θ is a sharp family if Θ α = ∅ for some α , while it is adull family otherwise.The terminology sharp/dull family is chosen in analogy with [13]and [9], where sharp and dull domains (and, in correspondence, sharpand dull degrees) are defined, respectively, for almost Dedekind do-mains and for one-dimensional Pr¨ufer domains; our definition can beseen as a wide generalization of their concept. However, we do not usethe terminology “sharp degree” and “dull degree”, both because thedefinition of Jaffard degree unifies them and because there is actuallya small difference in the sharp case. See Section 8 for a more detaileddiscussion. HE DERIVED SEQUENCE OF A PRE-JAFFARD FAMILY 17
Example 6.5.
Let Θ be a Jaffard family of D , with D = K . Then,Θ = Θ J , and so Θ = ∅ ; thus, T = K = T α for all ordinals α > Example 6.6.
Let Θ be a weak Jaffard family of D pointed at S .Then, Θ = { S } , and so T = S ; on the other hand, Θ = ∅ and so T = K . Thus Θ is sharp with Jaffard degree 2. Example 6.7.
Let D be an almost Dedekind domain with only finitelymany maximal ideals that are not finitely generated, say M , . . . , M n .(Those rings do indeed exists: see [12].) Let Θ := { D M | M ∈ Max( D ) } :then, Θ is a pre-Jaffard family of D (see Proposition 8.1 below).If P is a maximal ideal of D different from the M i , then D P is aJaffard overring of D , since Max( D ) \{ P } is compact and thus D P D ⊥ P = K . On the other hand, each D M i is not a Jaffard overring, since D ⊥ Q = T { D Q | Q ∈ Max( D ) \ { M i }} = D ; in particular, there is no weakJaffard family of D that can contain at the same time every D M i .Furthermore, the family Θ ′ := { D M | M ∈ Max( D ) , M = M , . . . , M n } is strongly independent (since every D M is a Jaffard overring and D ⊥ M contains the intersection of all T ∈ Θ ′ \ { D M } ), but it is not locallyfinite, since otherwise the whole Θ = Θ ′ ∪ { D M , . . . , D M n } would belocally finite and thus a Jaffard family.The set Θ is equal to { M , . . . , M n } and thus it is finite; moreover, T = D M ∩ · · · ∩ D M n is a semilocal almost Dedekind domain, and thusa PID. Therefore, Θ = ∅ and T = K , so that Θ is sharp with Jaffarddegree 2. Example 6.8.
Let D be the ring of all algebraic integers, i.e., theintegral closure of Z in Q . Then, D is a one-dimensional B´ezout (inparticular, Pr¨ufer) domain such that none of its maximal ideals arefinitely generated, nor any nonzero primary ideal is finitely generated.Therefore, Θ := { D M | M ∈ Max( D ) } is a pre-Jaffard family of D (as in the previous example), but none of its elements are Jaffardoverrings: hence Θ = ∅ and T = D = T . Therefore, Θ is dull withJaffard degree 0 and its dull limit is D itself.Let now Θ ′ := { D M | M ∈ Max( D ) , ∈ M } ∪ { D [1 / } . Then, Θ ′ is obtained from Θ with the construction of Proposition 4.8 appliedto Θ := { S ∈ Θ | / ∈ S } = B (1 / ∩ Θ (which is compact),and thus is a pre-Jaffard family. The ring D [1 /
2] is a Jaffard overringof D , since it belongs to the Jaffard family { D [1 /p ] | p is a primenumber } , while no other element of Θ ′ is a Jaffard overring; hence,(Θ ′ ) = { D M | M ∈ Max( D ) , ∈ M } , while (Θ ′ ) = (Θ ′ ) . Hence Θ ′ is dull with Jaffard degree 1, and its dull limit is T = \ M ∈ Max( D )2 ∈ M D M = D [1 / , / , . . . ] = D [1 /p | p = 2 is a prime number] . Remark 6.9.
Note that, if D is not a field and Θ is sharp, then theJaffard degree of D cannot be 0.7. Stable operations
Let T be an overring of D . Then, F D ( K ) ⊆ F T ( K ), and the imageof any T -submodule of K by any semistar operation on D is still a T -module. Then, we have two ways to relate the semistar operationson D and T : the first one is the restriction map ψ T : SStar( D ) −→ SStar( T ) ,⋆ ⋆ | F T ( K ) , while the second is the extension map φ T : SStar( T ) −→ SStar( D ) ,⋆ φ T ( ⋆ ) : F D ( K ) −→ F D ( K ) ,I ( IT ) ⋆ . If now we have a family Θ of overrings of D , then we can put togetherthe maps relative to each member of the family: we obtain a restrictionmap Ψ Θ : SStar( D ) −→ Y T ∈ Θ SStar( T ) ,⋆ (Ψ T ( ⋆ ))and an extension mapΦ Θ : Y T ∈ Θ SStar st ( T ) −→ SStar st ( D ) , ( ⋆ ( T ) ) T ∈ Θ inf T ∈ Θ Φ T ( ⋆ ( T ) ) . All these maps are order-preserving when SStar( D ) and SStar( T ) areendowed with the natural order, and when the product is endowed withthe product order. Proposition 7.1.
Let Θ be a complete and independent family of over-rings of D . Then, Ψ Θ ◦ Φ Θ is the identity on Q T ∈ Θ SStar( T ) .Proof. For every T , let ⋆ ( T ) ∈ SStar( T ), and let ⋆ := Φ Θ (( ⋆ ( T ) ) T ∈ Θ ).Fix S ∈ Θ and let I ∈ F S ( D ). Then, I = IS = (0); since I is complete,we have I ⋆ S = I ⋆ = \ T ∈ Θ ( IT ) ⋆ ( T ) = ( IS ) ⋆ ( S ) ∩ \ T ∈ Θ \{ S } ( IST ) ⋆ ( T ) = I ⋆ ( S ) as ST = K for every T ∈ Θ \ { S } (since Θ is independent). The claimis proved. (cid:3) HE DERIVED SEQUENCE OF A PRE-JAFFARD FAMILY 19
Definition 7.2.
Let Θ be a family of overrings of D . We say that Θ is stable-preserving if, for every ⋆ ∈ SStar st ( D ) and every I ∈ F D ( K ) ,we have I ⋆ = \ T ∈ Θ ( IT ) ⋆ . A stable semistar operation is uniquely determined by its action onproper ideals of D . Hence, if ⋆ is a stable semistar operation fixing D ,then the notion of extension of a star operation studied in [15] can beused to show that if Θ is a Jaffard family then I ⋆ = T T ∈ Θ ( IT ) ⋆ (see,in particular, [15, Theorems 5.4 and 5.6]); a similar result, without thehypothesis D = D ⋆ , can be shown joining the results in Sections 3 and6 of [17] (passing through length functions), so that any Jaffard familyis stable-preserving. We want to generalize this case, but we first pointout why stable-preserving properties are useful. Proposition 7.3.
Let Θ be a stable-preserving family of flat overringsof D . Then, Ψ Θ and Φ Θ establish an order-preserving isomorphismbetween SStar st ( D ) and Q { SStar st ( T ) | T ∈ Θ } .Proof. If ⋆ is a stable semistar operation, then the restriction Ψ T ( ⋆ )is stable for every overring T ; conversely, the infimum of a family ofrestriction of stable operations is still stable, since( I ∩ J ) Φ Θ ( ⋆ ( T ) ) = \ T ∈ Θ (( I ∩ J ) T ) ⋆ ( T ) = \ T ∈ Θ (( IT ∩ J T )) ⋆ ( T ) == \ T ∈ Θ ( IT ) ⋆ ( T ) ∩ ( J T ) ⋆ ( T ) = I Φ Θ ( ⋆ ( T ) ) ∩ J Φ Θ ( ⋆ ( T ) ) , using the flatness of the members of Θ. Hence, the maps Φ Θ and Ψ Θ restrict to maps from SStar st ( D ) to Q T ∈ Θ SStar st ( T ).By Proposition 7.1, Ψ Θ ◦ Φ Θ is the identity. Let now ⋆ ∈ SStar st ( D ).Then, I Φ Θ ◦ Ψ Θ ( ⋆ ) = \ T ∈ Θ ( IT ) ⋆ = I ⋆ since Θ is stable-preserving. Hence, Φ Θ ◦ Ψ Θ is the identity of SStar st ( D ),and thus Φ Θ and Ψ Θ are isomorphism. (cid:3) Proposition 7.4.
A weak Jaffard family is stable-preserving.Proof.
Let Θ be a weak Jaffard family pointed at T ∞ . Fix any ⋆ ∈ SStar st ( D ), and let ♯ be the map I T T ∈ Θ ( IT ) ⋆ . Then, ♯ is stable,and ⋆ ≤ ♯ ; in particular, if 1 ∈ I ⋆ then 1 ∈ I ♯ .Conversely, let I ⊆ D be such that 1 ∈ I ♯ ; without loss of generality,we can suppose that I = I ⋆ . Let T ∈ Θ \ { T ∞ } : then, then, T is aJaffard overring of D , and thus { T, T ⊥ } is a Jaffard family of D byProposition 3.9. Hence, IT = I ⋆ T = ( IT ∩ IT ⊥ ) ⋆ T = (( IT ) ⋆ ∩ ( IT ⊥ ) ⋆ ) T = ( IT ) ⋆ ∩ ( IT ⊥ ) ⋆ T. By definition, I ♯ ⊆ ( IT ) ⋆ , and thus 1 ∈ ( IT ) ⋆ ; on the other hand, T T ⊥ = K and thus ( IT ⊥ ) ⋆ T = K . Thus, 1 ∈ IT and IT = T .Since Θ is complete, we thus have I = ( IT ∞ ∩ D ), and so IT ∞ = I ⋆ T ∞ = ( IT ∞ ∩ D ) ⋆ T ∞ = ( IT ∞ ) ⋆ ∩ D ⋆ T ∞ . Again, by construction 1 belongs to both ( IT ∞ ) ⋆ and D ⋆ T ∞ , and thus1 ∈ IT ∞ , so that IT ∞ = T ∞ . Hence, IT = T for every T ∈ Θ, and thus I = D . Therefore, for every I ⊆ D we have 1 ∈ I ⋆ if and only if 1 ∈ I ♯ ;since ⋆ and ♯ are stable, it follows that ⋆ = ♯ , as claimed. Thus, Θ isstable-preserving, as claimed. (cid:3) Theorem 7.5.
Let Θ be a Jaffard family, α an ordinal, and let Θ ′ :=(Θ \ Θ α ) ∪ { T α } . Then, Θ ′ is stable-preserving.Proof. For every β ≤ α , let Λ β := (Θ \ Θ β ) ∪ { T β } : we want to show byinduction that Λ β is stable-preserving. Note that each Λ β is completeand, by definition, Λ α = Θ ′ .If β = 0 then Λ = (Θ \ Θ) ∪ { T } = { T } = { D } is clearly stable-preserving; suppose thus β >
0, and suppose that the claim holds forevery γ < β ; we distinguish two cases.If β = γ + 1 is a successor ordinal, then Θ β = Θ γ \ (Θ γ ) J and thusΛ β = (Θ \ (Θ γ \ (Θ γ ) J )) ∪ { T β } = (Θ \ Θ γ ) ∪ (Θ γ ) J ∪ { T β } . Let Λ ′ := Θ \ Θ γ , and take a stable semistar operation on D . Byinductive hypothesis, Λ γ = Λ ′ ∪ { T γ } is stable-preserving, and thus I ⋆ = \ A ∈ Λ ′ ( IA ) ⋆ ∩ ( IT γ ) ⋆ . Moreover, by construction, (Θ γ ) J ∪ { T β } is a weak Jaffard family of T γ pointed at T β ; by Proposition 7.4, it is stable-preserving on T β , andthus ( IT γ ) ⋆ = \ A ∈ Θ γ ∪{ T β } ( IT γ A ) ⋆ = \ A ∈ Θ γ ∪{ T β } ( IA ) ⋆ , so that I ⋆ = \ A ∈ Λ ′ ( IA ) ⋆ ∩ \ A ∈ Θ γ ∪{ T β } ( IA ) ⋆ = \ A ∈ Λ β ( IA ) ⋆ . Hence, Λ β is stable-preserving.Suppose now that β is a limit ordinal: then, Θ β = T γ<β Θ γ , and thusΛ β = Θ \ \ γ<β Θ γ ! ∪ { T β } = [ γ<β (Θ \ Θ γ ) ∪ { T β } . Let ⋆ be a stable semistar operation, and let ♯ be the map ♯ : I \ A ∈ Λ β ( IA ) ⋆ = \ γ<β \ A ∈ Θ γ ( IA ) ⋆ ∩ ( IT β ) ⋆ . HE DERIVED SEQUENCE OF A PRE-JAFFARD FAMILY 21
Then, ♯ is a stable semistar operation, and I ⋆ ⊆ I ♯ for all ideals I (as I ⋆ is contained in all ( IA ) ⋆ and in ( IT β ) ⋆ ). We claim that it is equalto ⋆ , and to do so it is enough to show that if 1 ∈ I ♯ then also 1 ∈ I ⋆ ,where I is a proper ideal of D (this follows from condition (4) of [1,Theorem 2.6]).Take thus a proper ideal I such that 1 ∈ I ♯ , and let Γ := { γ < β | IT = T for some T ∈ Θ γ \ Θ γ +1 } .Suppose that Γ is nonempty: then, it has a minimum γ . Since Λ γ iscomplete and IS = S for all S ∈ Θ δ with δ < γ , we must have I = IT γ ∩ D ; as above, it follows that I ⋆ = ( IT γ ) ⋆ ∩ D ⋆ Let T ∈ Θ γ \ Θ γ +1 :then, T is a Jaffard overring of T γ . Let A := \ { ( T γ ) P | P ∈ Spec( T γ ) , P T = T } , that is, A = T ⊥ with respect to T γ . Then, T A = K and J = J T ∩ J A for all T γ -submodules J of K . Thus,( IT γ ) ⋆ T = ( IT γ T ∩ IT γ A ) ⋆ T = ( IT ) ⋆ ∩ (( IA ) ⋆ ) T = ( IT ) ⋆ . Therefore, I ⋆ T = (( IT γ ) ⋆ ∩ D ⋆ ) T = ( IT ) ⋆ ∩ D ⋆ T contains 1 since it contains I ♯ . Since T was arbitrary in Θ γ \ Θ γ +1 , thisis a contradiction.Therefore, Γ must be empty. Since β is a limit ordinal, Λ β is also equalto S γ<β (Θ γ \ Θ γ +1 ); therefore, since Λ β is complete and I is proper,we must have I = IT β ∩ D ; therefore, I ⋆ = ( IT β ∩ D ) ⋆ = ( IT β ) ⋆ ∩ D ⋆ since ⋆ is stable. However, 1 ∈ ( IT β ) ⋆ since ( IT β ) ⋆ contains I ♯ , whileobviously 1 ∈ D ⋆ ; hence, 1 ∈ I ⋆ .By induction, it follows that Λ α = Θ ′ is stable-preserving, as claimed. (cid:3) Corollary 7.6.
Let Θ be a Jaffard family, α an ordinal, and let Θ ′ :=(Θ \ Θ α ) ∪ { T α } . Then:(1) for every stable semistar operation ⋆ on D , we have I ⋆ = T { ( IT ) ⋆ | T ∈ Θ ′ } ;(2) SStar st ( D ) ≃ Q { SStar st ( T ) | T ∈ Θ ′ } .Proof. The first part follows by joining Theorem 7.5 with Definition7.2, the second part from Theorem 7.5 and Proposition 7.3. (cid:3)
From the correspondence between stable semistar operations andlength functions we have the following.
Proposition 7.7.
Let D be an integral domain and let Θ be a stable-preserving family of flat overrings of D . Then, for every ℓ ∈ L sing ( D ) ,we have ℓ = X T ∈ Θ ℓ ⊗ T. In particular, this holds for Jaffard families, as was proved in [17,Theorem 3.10]; likewise, the analogue of Corollary 7.6 holds.
Corollary 7.8.
Let Θ be a Jaffard family, α an ordinal, and let Θ ′ :=(Θ \ Θ α ) ∪ { T α } . Then:(1) for every length function ℓ on D , we have ℓ = P { ℓ ⊗ T | T ∈ Θ ′ } ;(2) L sing ( D ) ≃ Q {L sing ( T ) | T ∈ Θ ′ } . Obviously, the previous results are at their strongest when α is theJaffard degree of Θ, so that T α is the dull limit of Θ.8. The dimension case In this section, we specialize the results of the previous sections todomains of dimension 1. In this case, there is a natural pre-Jaffardfamily to consider.
Proposition 8.1.
Let D be a domain of dimension , and let Θ := { D M | M ∈ Max( D ) } . Then, Θ is a pre-Jaffard family of D .Proof. The family Θ is clearly complete, independent (no nonzero primesurvives in D M and in D N for M = N ) and composed of flat over-rings. Furthermore, the localization map λ : Spec( D ) −→ Over( D ) isa homeomorphism between Spec( D ) and λ (Spec( D )) when the spacesare endowed with the respective Zariski topologies [3, Lemma 2.4]; inparticular, Θ = λ (Max( D )) is compact. Hence, Θ is a pre-Jaffard fam-ily. (cid:3) Definition 8.2.
Let D be a one-dimensional integral domain, and let Θ := { D M | M ∈ Max( D ) } . We say that D is: • ultimately sharp if Θ is sharp; • ultimately dull if Θ is dull. The second reason why the dimension 1 hypothesis is powerful isthat we can improve Proposition 4.5.
Proposition 8.3.
Let D be a domain of dimension , and let Θ := { D M | M ∈ Max( D ) } . Let M ∈ Max( D ) . Then, the following areequivalent:(i) D M is a Jaffard overring of D ;(ii) Θ \ { D M } is compact, with respect to the Zariski topology;(iii) Max( D ) \ { M } is compact, with respect to the Zariski topology;(iv) M is isolated in Max( D ) , with respect to the inverse topology.Proof. The equivalence of (i) and (ii) follows from Proposition 4.5 (andProposition 8.1), while the equivalence of (ii) and (iii) from the factthat Θ \ { D M } ≃ Max( D ) \ { M } via the localization map. Again byProposition 4.5, (i) implies (iv). HE DERIVED SEQUENCE OF A PRE-JAFFARD FAMILY 23
Suppose (iv) holds. Then, Max( D ) \ { M } is closed in Max( D ), withrespect to the inverse topology. Since D has dimension 1, it followsthat Spec( D ) \ { M } is closed in the inverse topology, and thus thatMax( D ) \ { M } is compact. Hence, (iii) holds and all the conditions areequivalent. (cid:3) Recall that, for a topological space X , I ( X ) and D ( X ) are, respec-tively, the set of isolated points and the set of limit points of X . Propo-sition 8.3 allows to describe the derived sequence in a purely topologicalway. Theorem 8.4.
Let D be a one-dimensional domain, let Θ := { D M | M ∈ Max( D ) } and let X := Max( D ) inv . Let { T α } be the derived se-quence of Θ and let { Θ α } be the corresponding chain of subsets of Θ .Then, for every ordinal α and every M ∈ Max( D ) , we have M T α = T α if and only if M ∈ D α ( X ) , and Θ α = { D M | M ∈ D α ( X ) } .Proof. Let Λ α := { M ∈ Max( D ) | M T α = T α } ; then, { Λ α } is adescending chain of subsets of Max( D ), and we need to show thatΛ α = D α ( X ). By definition, Θ α is a pre-Jaffard family of T α ; it followsthat M ∈ Λ α if and only if D M ∈ Θ α .We proceed by transfinite induction. For α = 0, Θ = Max( D ) and D ( X ) = X , so the claim is proved. Suppose that the claim holds forevery β < α ; we distinguish two cases.Suppose first that α = γ + 1 is a successor ordinal. Then, by hy-pothesis, { M ∈ Max( D ) | M T γ = T γ } = D γ ( X ); hence, the restrictionmap ρ : Max( T γ ) −→ Max( D ), P P ∩ D establishes a homeomor-phism between Max( T γ ) and its image D γ ( X ) = Λ γ ( X ) both in theZariski and in the inverse topology. By definition and by Proposition8.3, Θ α = Θ γ +1 = { ( T γ ) P | P ∈ (Max( T γ )) inv } , i.e., given a maximalideal M of D , we have D M ∈ Θ α if and only if M T γ is a limit point of(Max( T γ )) inv . Since ρ is a homeomorphism in the inverse topology, thisis equivalent to saying that M ∈ D γ +1 ( X ) = D α ( X ); that is, D M ∈ Θ α if and only if M ∈ D α ( X ). By the remark at the beginning of the proof,we have our claim.Suppose now that α is a limit ordinal. If M ∈ Λ α , then M ∈ Λ β for all β < α , and thus by induction M ∈ D β ( X ) for all β < α ; by definition,this is exactly the condition M ∈ D α ( X ). Conversely, if M ∈ D α ( X )then M ∈ D β ( X ) for all β < α , and thus by induction D M ∈ Θ β for all β < α ; therefore, D M ∈ Θ α by definition and M ∈ Λ α . Thus M ∈ Λ α if and only if M ∈ D α ( X ), and the claim is proved. (cid:3) Corollary 8.5.
Let D be a one-dimensional integral domain. Then, theJaffard degree of Θ is equal to the Cantor-Bendixson rank of Max( D ) inv .Proof. By definition, the Cantor-Bendixson rank of X is the least or-dinal α such that D α ( X ) = D α +1 ( X ). By Theorem 8.4, when X = Max( D ) inv the latter condition is equivalent to Θ α = Θ α +1 , and thus α is also the Jaffard degree of Θ. (cid:3) Corollary 8.6.
Let D be a one-dimensional integral domain, and let X := Max( D ) inv . Then, D is ultimately sharp if and only if X is ascattered space.Proof. By Theorem 8.4, D α ( X ) becomes empty if and only if there isan α such that M T α = T α for all maximal ideal M , where T α is the α -thelement of the derived sequence of Θ. However, the latter condition isequivalent to T α = K , i.e., to the fact that D is ultimately sharp. (cid:3) When D is a Pr¨ufer domain, a similar construction has been given in[9, Section 6], following ideas introduced in [13]. In this case, a maximalideal M is said to be sharp if T { D N | N ∈ Max( D ) , N = M } * D M ,while dull otherwise; by [9, Lemma 6.3(2)] and Proposition 8.3 (or bydirect proof) we have that M is sharp if and only if D M is a Jaffardoverring of D . If Max ♯ ( D ) is the set of sharp maximal ideals of D , theydefine recursively an ascending sequence of rings by D α := (T { ( D γ ) M | M ∈ Max ♯ ( D γ ) } if α = γ + 1 is a successor ordinal, S { D γ | γ < α } if α is a limit ordinal , and they show [9, Lemma 6.5(2)] that Max( D α ) = { M D α | M ∈D α (Max( D ) inv ) } , so that D α actually coincides with our T α , the α -thelement of the derived sequence of Θ := { D M | M ∈ Max( D ) } (thisalso, a fortiori , for limit ordinals α , for which the definitions of T α and D α do not coincide). Then, they say that D has sharp degree α if D α = K while D α +1 = K , and that D has dull degree α if D α = D α +1 = K and D β = D α for all β < α .In the case of dull degree, our definition agrees with theirs: it isstraightforward to see (using D α = T α ) that D has a dull degree if andonly if D is ultimately dull, and that the dull degree of D coincideswith the Jaffard degree of Θ.On the other hand, for sharp degree, there is a difference: indeed, if D has sharp degree α then Θ has Jaffard degree α + 1, and converselyif the Jaffard degree of Θ is a successor ordinal α + 1 then D has sharpdegree α . However, if the Jaffard degree of Θ is a limit ordinal α , thenthe sharp degree of D does not exist, because the definition requiresthat the first β such that D β = K is a successor ordinal. Thus, D has asharp degree if and only if D is ultimately sharp and the Jaffard degreeof Θ is a successor ordinal.Our approach allows to give a simpler form to some of their results.Indeed, Corollary 8.6 above is a more symmetric version of [9, Theorem6.6], because it gives a complete equivalence between a topological fact(Max( D ) inv is scattered) and the sharpness of D , without requiring thatthe Jacobson radical of D is nonzero (as in part (2) of the reference) HE DERIVED SEQUENCE OF A PRE-JAFFARD FAMILY 25 and we do not need to separate the dull and the sharp case (as in parts(1) and (3)).To conclude the paper, we apply the results of Section 7 to one-dimensional domain.
Proposition 8.7.
Let D be a one-dimensional domain. Let T α be thedull limit of Θ := { D M | M ∈ Max( D ) } . Then, the family Θ ′ := { D M | M ∈ Max( D ) , M T ∞ = T α } ∪ { T α } is stable-preserving.Proof. The claim is a direct consequence of Theorem 7.5. (cid:3)
Proposition 8.8.
Let D be a one-dimensional Pr¨ufer domain. Then,the family Θ := { D M | M ∈ Max( D ) } is stable-preserving if and onlyif D is ultimately sharp.Proof. If D is ultimately sharp, then its dull limit T α is equal to K ,and thus the family Θ ′ = { D M | M ∈ Max( D ) , M T ∞ = T α } ∪ { T α } ofΘ coincides with Θ ∪ { K } . Hence, Θ ∪ { K } is stable-preserving and sois Θ.Suppose D is ultimately dull, and let T := T α be its dull limit.Consider the set Λ := { N | N T = T } . For every N ∈ Λ, D N = T NT is not a Jaffard overring of T (by construction of T α ), and thusMax( T ) \ { N T } is not compact; hence, T N ′ = N T N ′ = T . For every N ∈ Λ, let ⋆ N be the stable semistar operation ⋆ N : I \ N ′ ∈ Λ N ′ = N ID N . Then, ⋆ N fixes T , but I ⋆ N = K for all I that are D N -modules or D M -modules for some M / ∈ Λ. Let ⋆ be the supremum of all the ⋆ N ; then, ⋆ fixes T . However, ( T D M ) ⋆ N = K for every M ∈ Max( D ), and thus K = \ M ∈ Max( D ) ( T D M ) ⋆ N = T ⋆ N . Hence, Θ is not stable-preserving. (cid:3)
Corollary 8.9.
Let D be an almost Dedekind domain that is ultimatelysharp. Then, there is a natural bijection between SStar st ( D ) and thepower set of Max( D ) .Proof. By Propositions 7.3 and 8.8, SStar st ( D ) is isomorphic to theproduct Q { SStar st ( D M ) | M ∈ Max( D ) } . However, each D M is adiscrete valuation ring, and thus SStar st ( D M ) contains exactly two op-erations (the identity and the one sending everything to K ). The claimfollows. (cid:3) References [1] D. D. Anderson and Sylvia J. Cook. Two star-operations and their inducedlattices.
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Dipartimento di Matematica “Tullio Levi-Civita”, Universit`a degliStudi di Padova, Padova, Italy
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