Topological properties of localizations, flat overrings and sublocalizations
aa r X i v : . [ m a t h . A C ] M a y TOPOLOGICAL PROPERTIES OF LOCALIZATIONS,FLAT OVERRINGS AND SUBLOCALIZATIONS
DARIO SPIRITO
Abstract.
We study the set of localizations of an integral domainfrom a topological point of view, showing that it is always a spectralspace and characterizing when it is a proconstructible subspace ofthe space of all overrings. We then study the same problems in thecase of quotient rings, flat overrings and sublocalizations. Introduction
The Zariski topology on the set Over( D ) of overrings of an integraldomain was introduced as a natural generalization of the Zariski topol-ogy on the space Zar( D ) of valuation overrings of D (called the Zariskispace of D ), which in turn was introduced by Zariski in order to tacklethe problem of resolution of singularities [35, 36].It has been proved that Over( D ), like Zar( D ), is a spectral space ,meaning that it is homeomorphic to the prime spectrum of a ring [10,Proposition 3.5]. There are other subspaces of Over( D ) that are alwaysspectral: for example, this happens for the space of integrally closedoverrings [10, Proposition 3.6] and the space of local overrings [12,Corollary 2.14].In the last two cases, the role of D in the definition of the space ismerely to provide a setting (Over( D )): that is, for an overring, beingintegrally closed or local (or a valuation domain, for the case of Zar( D ))is a property completely independent from D . Indeed, with very similarproofs it is possible to generalize these results to the case of the spacesof rings comprised between two fixed rings (see e.g. [10, Propositions3.5 and 3.6] and [12, Example 2.13]), as well as using these methods tostudy spaces of modules [31, Example 2.2].In this paper, we study four subspaces of Over( D ) that are muchmore closely related to D ; more precisely, such that, given an overring T , the belonging of T to the space depends not on the properties of T but rather on the relation between D and T . In Section 3 we shall Date : September 24, 2018.2010
Mathematics Subject Classification.
Key words and phrases.
Localizations; flat overrings; spectral spaces; con-structible topology.This work was partially supported by
GNSAGA of Istituto Nazionale di AltaMatematica . start from the space of localizations (at prime ideals); then we willconsider the space of quotient rings (Section 4), sublocalizations of D (i.e., intersection of localizations of D ; Section 5) and flat overrings(Section 6).In each case, we will study two questions: under which conditionsthey are spectral spaces and under which condition they are closedin the constructible topology of Over( D ). We shall answer completelythese questions in the case of localizations (Theorem 3.2) and quo-tient rings (Corollary 4.3 and Theorem 4.4); for sublocalizations wewill find a sufficient condition (Theorem 5.5), while for flat overringswe will prove a characterization that is, however, very difficult to use(Proposition 6.1). We shall also study the space of flat submodules ofan R -module (for rings R that are not necessarily integral domains)and the possibility of representing the space of sublocalizations of D ina more topological way. 2. Preliminaries
Spectral spaces. A spectral space is a topological space homeo-morphic to the prime spectrum of a (commutative, unitary) ring (en-dowed with the Zariski topology). Spectral spaces can be characterizedtopologically as those spaces that are T (i.e., such that for every pairof points at least one of them is contained in an open set not containingthe other), compact, with a basis of open and compact subsets closedby finite intersections, and such that every nonempty irreducible closedsubset has a generic point (i.e., it is the closure of a single point) [25,Proposition 4].If X is a spectral space, the constructible topology (or patch topology )on X (which we denote by X cons ) is the coarsest topology such thatthe open and compact subspaces of the original topology are both openand closed. The space X cons is always a spectral space, that is moreoverHausdorff and totally disconnected [25, Theorem 1].A subset Y ⊆ X is said to be proconstructible if it is closed, withrespect to the constructible topology; in this case, the constructibletopology on Y coincides with the topology induced by the constructibletopology on X , and Y (with the original topology) is a spectral space(this follows from [6, 1.9.5(vi-vii)]). The converse does not hold, i.e.,a subspace Y of a spectral space X may be spectral but not procon-structible; however, the following result holds. Lemma 2.1.
Let Y ⊆ X be spectral spaces. Suppose that there is asubbasis B of X such that, for every B ∈ B , both B and B ∩ Y arecompact. Then, Y is a proconstructible subset of X .Proof. The hypothesis on B implies that the inclusion map Y ֒ → X isa spectral map; by [6, 1.9.5(vii)], it follows that Y is a proconstructiblesubset of X . (cid:3) OPOLOGICAL PROPERTIES OF LOCALIZATIONS 3
For further results about the constructible topology and the relationbetween ultrafilters and the constructible topology, see [19, 11, 10, 12].2.2.
The space X ( X ) . Let X be a spectral space. The inverse topol-ogy on X is the space X inv having, as a basis of closed sets, the openand compact subspaces of X ; equivalently, it is the topology havingas closed sets the subsets of X that are compact and closed by gener-izations. The space X inv is again a spectral space. Following [15], wedenote by X ( X ) the space of nonempty subsets of X that are closedin the inverse topology; this space can be endowed with a topologyhaving, as a basis of open sets, the sets of the form U (Ω) := { Y ∈ X ( X ) | Y ⊆ Ω } , as Ω ranges among the open and compact subspaces of X . Under thistopology, X ( X ) is again a spectral space [15, Theorem 3.2(1)].If X = Spec( R ) for some ring R , we set X ( R ) := X (Spec( R )).2.3. Semistar operations.
Let D be an integral domain with quo-tient field K , and let F ( D ) be the set of D -submodules of K . A semis-tar operation on D is a map ⋆ : F ( D ) −→ F ( D ) such that, for every I, J ∈ F ( D ) and every x ∈ K ,(1) I ⊆ I ⋆ ;(2) ( I ⋆ ) ⋆ = I ⋆ ;(3) if I ⊆ J then I ⋆ ⊆ J ⋆ ;(4) x · I ⋆ = ( xI ) ⋆ .A semistar operation is called spectral if it is in the form s ∆ for some∆ ⊆ Spec( D ), where I s ∆ := \ { ID P | P ∈ ∆ } for every I ∈ F ( D ). If ⋆ is spectral, then ( I ∩ J ) ⋆ = I ⋆ ∩ J ⋆ for every I, J ∈ F ( D ).Starting from any semistar operations ⋆ , we can define two maps ⋆ f and e ⋆ by putting, for every I ∈ F ( D ), I ⋆ f = [ { J ⋆ | J ⊆ I, J is finitely generated } and I e ⋆ := [ { ( I : E ) | ∈ E ⋆ , E is finitely generated } . Both ⋆ f and e ⋆ are semistar operations, and we always have ( ⋆ f ) f = ⋆ f and ee ⋆ = e ⋆ . If ⋆ = ⋆ f then ⋆ is said to be of finite type ; on the otherhand, ⋆ = e ⋆ if and only if ⋆ is spectral and of finite type.If ⋆ = s ∆ is a spectral operation, then ⋆ is of finite type if and onlyif ∆ is compact [16, Corollary 4.4].The space SStar( D ) of semistar operations on D can be endowedwith a topology having, as a basis of open sets, the sets of the form V I := { ⋆ ∈ SStar( D ) | ∈ I ⋆ } , DARIO SPIRITO as I ranges in F ( D ). In the induced topology, both the space SStar f ( D )of finite-type operations and the space SStar f,sp ( D ) of finite-type spec-tral operations are spectral (see [16, Theorem 2.13] for the former and[13, Theorem 4.6] for the latter). Moreover, SStar f,sp ( D ) is homeomor-phic to X ( D ) [15, Proposition 5.2].2.4. The t -operation. Let D be an integral domain with quotientfield K , and let ⋆ be a semistar operation on D . If D ⋆ = D , then therestriction of ⋆ to the set F ( D ) of fractional ideals of D is said to bea star operation on D . A classical example of a star operation is the divisorial closure (or v -operation), which is defined by I v := ( D : ( D : I )), where ( I : J ) := { x ∈ K | xJ ⊆ I } ; the divisorial closure is thebiggest star operation on D , in the sense that I ⋆ ⊆ I v for every staroperation ⋆ and every I ∈ F ( D ).The t -operation is the finite-type operation associated to the v -operation; that is, t := v f . The t -operation is the biggest finite-typestar operation. The w -operation , defined by w := e t = e v , is the biggestspectral star operation of finite type.If ⋆ is a star operation on D , a prime ideal P of D such that P = P ⋆ is said to be a ⋆ -prime ; the set of all ⋆ -primes is called the ⋆ -spectrum and is denoted by QSpec ⋆ ( D ). If ⋆ = s ∆ is a spectral star operation,then QSpec ⋆ ( D ) = ∆ ↓ = { Q ∈ Spec( D ) | Q ⊆ P for some P ∈ ∆ } .We always have D = T { D P | P ∈ QSpec t ( D ) } .See [20, Chapter 32] for more properties of star operations.2.5. Overrings.
Let D be an integral domain with quotient field K .An overring of D is a ring comprised between D and K . The spaceOver( D ) of the overrings of D can be endowed with a topology having,as a basis of open sets, the sets of the form B ( x , . . . , x n ) := { T ∈ Over( D ) | x , . . . , x n ∈ T } = Over( D [ x , . . . , x n ]) , as x , . . . , x n range in K . Under this topology, Over( D ) is a spectralspace [10, Proposition 3.5].3. Localizations
The first space we analyze is the space of localizations of an integraldomain D at its primes ideals, which we denote by Loc( D ); that is,Loc( D ) := { D P | P ∈ Spec( D ) } . Definition 3.1.
Let D be an integral domain. We say that D is rad-colon coherent if, for every x ∈ K \ D , there is a finitely generated ideal I such that rad( I ) = rad(( D : D x )) , i.e., if and only if D (( D : D x )) iscompact in Spec( D ) for every x ∈ K . Obvious examples of rad-colon coherent domains are Noetherian do-mains or, more generally, domains with Noetherian spectrum. Another
OPOLOGICAL PROPERTIES OF LOCALIZATIONS 5 large class of such domains is the class of coherent domains , i.e., do-mains where the intersection of two finitely generated ideals is stillfinitely generated; this follows from the fact that ( D : D x ) = D ∩ x − D .In particular, this class contains all Pr¨ufer domains [20, Proposition25.4(1)], or more generally the polynomial rings in finitely many vari-ables over Pr¨ufer domains [22, Corollary 7.3.4]. See the following Ex-ample 3.3 for a domain that is not rad-colon coherent. Theorem 3.2.
Let D be an integral domain.(a) Loc( D ) is a spectral space.(b) Loc( D ) is proconstructible in Over( D ) if and only if D is rad-colon coherent.Proof. (a) By [7, Lemma 2.4], the map λ : Spec( D ) −→ Over( D ) P D P . is a topological embedding whose image is exactly Loc( D ). In particu-lar, since Spec( D ) is a spectral space, so is Loc( D ).(b) We first note that B ( x ) ∩ Loc( D ) = { D P ∈ Loc( D ) | x ∈ D P } = { D P ∈ Loc( D ) | ∈ ( D P : x ) ∩ D } == { D P ∈ Loc( D ) | ∈ ( D : D : x ) D P } == { D P ∈ Loc( D ) | ( D : D : x ) ( P } = λ ( D (( D : D x ))) . Suppose Loc( D ) is proconstructible in Over( D ). Since, for any x ∈ K , B ( x ) is also a proconstructible subspace of Over( D ), then B ( x ) ∩ Loc( D ) is closed in Over( D ) cons ; since the Zariski topology is weakerthan the constructible topology, B ( x ) ∩ Loc( D ) must be compact inthe Zariski topology. By the previous calculation, B ( x ) ∩ Loc( D ) = λ ( D ( D : D x )), and thus D (( D : D x )) must be compact. Hence, D israd-colon coherent.Conversely, suppose D is rad-colon coherent. Then, each B ( x ) ∩ Loc( D ) is compact, and thus {B ( x ) ∩ Loc( D ) | x ∈ K } is a subba-sis of compact subsets for Loc( D ); applying Lemma 2.1 we see thatLoc( D ) is a proconstructible subset of Over( D ). (cid:3) As a first use of this theorem, we give an example of a domain thatis not rad-colon coherent.
Example 3.3.
Let D be an essential domain that is not a P v MD; thatis, suppose that D is the intersection of a family of valuation rings,each of which is a localization of D , but suppose that there is a t -primeideal P such that D P is not a valuation ring. Such a ring does indeedexists – see [23].Let E be the set of prime ideals P of D such that D P is a valuationdomain. Since D is not a P v MD, not all t -primes are in E . Since E ⊆
DARIO SPIRITO
QSpec t ( D ) [27, Lemma 3.17], we thus have E ( QSpec t ( D ). If E iscompact, then s E is a semistar operation of finite type on D ; however,since D is essential (and thus, by definition, T { D P | P ∈ E } = D ) wehave D s E = D , and thus the restriction of s E to the fractional ideals of D is a spectral star operation of finite type, which implies that I s E ⊆ I w for every finite-type operation. In particular, E = QSpec s E ( D ) ⊇ QSpec w ( D ) ⊇ QSpec t ( D ) , and thus E = QSpec t ( D ), a contradiction. Therefore, E is not compact.However, λ ( E ) = Loc( D ) ∩ Zar( D ); if Loc( D ) were to be procon-structible in Over( D ), so would be λ ( E ) (since Zar( D ) is always procon-structible). But this would imply that λ ( E ) is, in particular, compact,a contradiction. Hence Loc( D ) is not proconstructible in Over( D ), and D is not rad-colon coherent.There are at least three natural ways to extend Loc( D ) to non-localoverrings of D .The first is by considering general localizations of D (which we willcall, for clarity, quotient rings ), that is, overrings in the form S − D for some multiplicatively closed subsets S of D . We denote this set byOver qr ( D ).The second is through the set of flat overrings of D (that is, overringsthat are flat when considered as D -modules). We denote this set byOver flat ( D ).The third is by considering sublocalizations of D , i.e., overrings thatare intersection of localizations (or, equivalently, quotient rings) of D .We denote this set by Over sloc ( D ).It is well-known that Over qr ( D ) ⊆ Over flat ( D ) ⊆ Over sloc ( D ), andthat both inclusions may be strict. For example, any overring of aPr¨ufer domain is flat, but it need not be a quotient ring: in the caseof Dedekind domains, this happens if and only if the class group of D is torsion [21, Corollary 2.6] (more generally, a Pr¨ufer domain D suchthat Over qr ( D ) = Over flat ( D ) is said to be a QR-domain – see [20,Section 27] or [18, Section 3.2]). As for sublocalizations that are notflat, we shall give an example later (Example 6.3); see also [24].In all three cases, a natural question is to ask if (or when) the spacesare spectral, and if (or when) they are proconstructible in Over( D );moreover, we could ask if there is some construction through which wecan represent them. We shall treat the case of quotient rings in Section4, the case of sublocalizations in Section 5 and the case of flat overringsin Section 6.A first result is a relation between their proconstructibility and theproconstructibility of Loc( D ). Proposition 3.4.
Let D be an integral domain. If Over qr ( D ) or Over flat ( D ) is proconstructible, then D is rad-colon coherent. OPOLOGICAL PROPERTIES OF LOCALIZATIONS 7
Proof.
Let X be either Over qr ( D ) or Over flat ( D ), and let LocOver( D )be the space of local overrings of D . Then, X ∩ LocOver( D ) = Loc( D );since LocOver( D ) is always proconstructible [12, Corollary 2.14], if X is proconstructible so is Loc( D ). By Theorem 3.2(b), it follows that D is rad-colon coherent. (cid:3) Note that Over sloc ( D ) ∩ LocOver( D ) may not be equal to Loc( D ) –see Example 6.3. 4. Quotient rings
As localizations at prime ideals of D can be represented throughSpec( D ), we can represent quotient rings by multiplicatively closedsubsets; more precisely, there is a one-to-one correspondence betweenOver qr ( D ) and the set of multiplicatively closed subsets that are sat-urated. For technical reasons, it is more convenient to work with thecomplements of multiplicatively closed subsets. Definition 4.1.
Let R be a ring (not necessarily a domain). A semi-group prime on R is a nonempty subset Q ⊆ R such that:(1) for each r ∈ R and for each π ∈ Q , rπ ∈ Q ;(2) for all σ, τ ∈ R \ Q , στ ∈ R \ Q ;(3) Q = R . By [30, (2.3)], a nonempty Q ⊆ R is a semigroup prime of R if andonly if it is a union of prime ideals, if and only if R \ Q is a saturatedmultiplicatively closed subset.Let S ( R ) denote the set of semigroup primes of a ring R . As in[30] and in [14], we endow S ( R ) with the topology (which we call the Zariski topology ) whose subbasic closed sets have the form V S ( x , . . . , x n ) := { Q ∈ S ( R ) | x , . . . , x n ∈ Q } , as x , . . . , x n ranges in R ; equivalently, we can consider the subbasis ofopen sets D S ( x , . . . , x n ) := S ( R ) \V S ( x , . . . , x n ) = { Q ∈ S ( R ) | x i / ∈ Q for some i } . We collect the properties of this topology of our interest in the nextproposition.
Proposition 4.2. [14, Propositions 2.3 and 3.1]
Let R be a ring andendow S ( R ) with the Zariski topology.(a) The family {D S ( x ) | x ∈ R } is a basis of compact and opensubsets of S ( R ) , which is closed by intersections.(b) The set-theoretic inclusion Spec( R ) ֒ → S ( R ) is a topologicalembedding.(c) S ( R ) is a spectral space. DARIO SPIRITO (d) Suppose D is an integral domain. The map λ qr : S ( D ) −→ Over( D ) Q ( R \ Q ) − D. is a topological embedding whose image is Over qr ( D ) . In particular, by points (c) and (d) of the previous proposition weget immediately the following result.
Corollary 4.3.
Over qr ( D ) is a spectral space for every integral domain D . On the other hand, proconstructibility holds less frequently for Over qr ( D )than it does for Loc( D ). Theorem 4.4.
Let D be an integral domain with quotient field K .Then, Over qr ( D ) is proconstructible in Over( D ) if and only if, for every x ∈ K , the ideal rad(( D : D x )) is the radical of a principal ideal.Proof. As in the proof of Theorem 3.2, we see that an overring T is in B ( x ) ∩ Over qr ( D ) if and only if T = λ qr ( Q ) for some semigroup prime Q not containing ( D : D x ). Moreover, we note that a semigroup primecontains an ideal I if and only if it contains the radical of I .Therefore, if each rad(( D : D x )) is the radical of a principal ideal,then each B ( x ) ∩ Over qr ( D ) is equal to λ qr ( D S ( y )) for some y ∈ D .However, by Proposition 4.2(a), D S ( y ) is compact, and thus so is B ( x ) ∩ Over qr ( D ); by Lemma 2.1, Over qr ( D ) is proconstructible in Over( D ).Conversely, suppose there is a x ∈ K be such that I := rad(( D : D x ))is not the radical of a principal ideal. Claim 1 : let y ∈ D . Then, D [ y − ] ∈ B ( x ) if and only if y ∈ I .If x ∈ D [ y − ], then(1) 1 ∈ (cid:0) D (cid:2) y − (cid:3) : D [ y − ] x (cid:1) = ( D : D x ) D (cid:2) y − (cid:3) , since D [ y − ] is flat over D .If now P ∈ V ( I ) (i.e., I ⊆ P ), then in particular ( D : D x ) ⊆ P ,and so P D [ y − ] = D [ y − ]; it follows that y ∈ P . Since this happens forevery P ∈ V ( I ) and I is a radical ideal, y ∈ I .Suppose now that y ∈ I . Then, every prime ideal containing I ex-plodes in D [ y − ], and thus ID [ y − ] = D [ y − ]. Therefore, the samehappens to ( D : D x ), and so x ∈ D [ y − ] (with the same calculation of(1), just backwards).Let now U := {B ( z − ) | z ∈ I } . Claim 2 : U is an open cover of B ( x ) ∩ Over qr ( D ).Let T ∈ B ( x ) ∩ Over qr ( D ): then, 1 ∈ ( T : T x ) = ( D : D x ) T , andthus there are d , . . . , d n ∈ ( D : D x ), t , . . . , t n ∈ T such that 1 = d t + · · · + d n t n . For every i , there is a w i ∈ D such that w − i ∈ T OPOLOGICAL PROPERTIES OF LOCALIZATIONS 9 and w i t i ∈ D ; let w := w · · · w n . Then, w is invertible in T , and thus D [ w − ] ⊆ T , that is, T ∈ B ( w − ); moreover, w = d wt + · · · + d n wt n ∈ d D + · · · + d n D ⊆ ( D : D x ) ⊆ I, and so B ( w − ) ∈ U . Therefore, U is a cover of B ( x ) ∩ Over qr ( D ). Claim 3 : there are no finite subsets of U that cover B ( x ) ∩ Over qr ( D ).Consider a finite subset U := {B ( z − ) , . . . , B ( z − n ) } of U , for some z , . . . , z n ∈ I . In particular, rad( z i D ) ⊆ I for every I ; moreover,rad( z i D ) = I since I is not the radical of any principal ideal. It fol-lows that for every i there is a prime ideal P i containing z i but not I .By prime avoidance, there is an y ∈ I \ ( P ∪ · · · ∪ P n ); in particular, D [ y − ] ∈ B ( x ) ∩ Over qr ( D ).We claim that D [ y − ] / ∈ B ( z − i ) for every i : indeed, z i ∈ P i , and P i D [ y − ] = D [ y − ]. Therefore, z i is not invertible in D [ y − ], and z − i / ∈ D [ y − ]. Hence, D [ y − ] is an element of B ( x ) ∩ Over qr ( D ) not containedin any element of U , which thus is not a cover.Therefore, B ( x ) ∩ Over qr ( D ) is not compact; it follows that Over qr ( D )is not proconstructible, as claimed. (cid:3) We remark that the first implication of the previous theorem followsalso from [24, Theorem 2.5] and the following Theorem 5.5.
Corollary 4.5.
Let D be a Noetherian domain, and let X ( D ) be theset of height-1 prime ideals of D . The following are equivalent:(i) Over qr ( D ) is proconstructible in Over( D ) ;(ii) D = T { D P | P ∈ X ( D ) } and every P ∈ X ( D ) is the radicalof a principal ideal.Proof. (i = ⇒ ii) Suppose that Over qr ( D ) is proconstructible.Let Q be a prime t -ideal, and consider A := T { D P | P ∈ D ( Q ) } . Weclaim that A = D : indeed, if A = D , then the map ⋆ : I T { ID P | P ∈ D ( Q ) } would be a star operation of finite type (since D ( Q ) iscompact) such that Q ⋆ = D * Q = Q t , i.e., it would not be smallerthan the t -operation, an absurdity. Hence, there is an x ∈ A \ D , andrad(( D : D x )) = Q . By Theorem 4.4, Q = rad( yD ) for some y ∈ D .If Q has not height 1, then this contradicts the Principal Ideal The-orem; thus, QSpec t ( D ) = X ( D ), and D = T { D P | P ∈ X ( D ) } .(ii = ⇒ i) Conversely, suppose that the two conditions hold; the firstone implies that QSpec t ( D ) = X ( D ) (since X ( D ) is a compact sub-space of Spec( D )). For every x ∈ K \ D , ( D : D x ) is a proper t -ideal,and thus its minimal primes are t -ideals, i.e., have height 1. However,( D : D x ) has only finitely many minimal primes, say P , . . . , P n , and byhypothesis P i = rad( y i D ) for some y i ∈ D ; hence, rad(( D : D x )) is theradical of the principal ideal y · · · y n D . By Theorem 4.4, Over qr ( D ) isproconstructible. (cid:3) Corollary 4.6.
Let D be a Krull domain, and let X ( D ) be the set ofheight-1 prime ideals of D . Then, the following are equivalent:(i) Over qr ( D ) is proconstructible in Over( D ) ;(ii) each P ∈ X ( D ) is the radical of a principal ideal;(iii) the class group of D is a torsion group.Proof. The equivalence between (i) and (ii) follows as in the previouscorollary, noting that D = T { D P | P ∈ X ( D ) } holds for every Krulldomain; the equivalence of (ii) and (iii) follows from the proof of The-orem 1 of [32]. (cid:3) Sublocalizations
Our first result about Over sloc ( D ) shows a striking difference be-tween the space of sublocalizations and the spaces we considered in theprevious sections. Proposition 5.1.
Let D be an integral domain. Then, Over sloc ( D ) isa spectral space if and only if it is proconstructible in Over( D ) .Proof. If Over sloc ( D ) is proconstructible, then it is spectral. On theother hand, for every x , . . . , x n ∈ K , the intersection B ( x , . . . , x n ) ∩ Over sloc ( D ) is compact, since it has a minimum, namely the intersectionof the localizations of D that contain x , . . . , x n . Since {B ( x , . . . , x n ) ∩ Over sloc ( D ) | x , . . . , x n ∈ K } is a subbasis of Over sloc ( D ), by Lemma2.1 if Over sloc ( D ) is spectral then it is also proconstructible in Over( D ). (cid:3) We are now tasked to study the spectrality of Over sloc ( D ). To thisend, we use spectral semistar operations; more precisely, we use thefact that there is a map π : SStar sp ( D ) −→ Over sloc ( D ) ⋆ D ⋆ that is continuous [12, Proposition 3.2(2)] and surjective (by definitionof Over sloc ( D )). We shall use the following topological lemma. Lemma 5.2.
Let φ : X −→ Y be a continuous surjective map betweentwo topological spaces. Suppose that:(a) X is spectral;(b) Y is T ;(c) there is a subbasis C of Y such that, for every C ∈ C , φ − ( C ) is compact.Then, Y is a spectral space and φ is a spectral map.Proof. Let Ω := O ∩ . . . ∩ O m be a finite intersection of elements of C .Then, φ − (Ω) = T i φ − ( O i ) is compact, since X is spectral and each φ − ( O i ) is compact by hypothesis; moreover, since φ is surjective, alsoΩ i = φ ( φ − (Ω)) is compact. Therefore, the set C of finite intersections OPOLOGICAL PROPERTIES OF LOCALIZATIONS 11 of elements of C is a basis of compact subsets. If now Ω ′ is any openand compact subset of Y , then Ω is a finite union of elements of C ,and thus φ − (Ω ′ ) is also compact.The claim now follows from [8, Proposition 9]. (cid:3) Proposition 5.3.
Let D be an integral domain. If SStar sp ( D ) is aspectral space, then so is Over sloc ( D ) .Proof. Let B := {B ( x ) ∩ Over sloc ( D ) | x ∈ K } be the canonical subbasisof Over sloc ( D ). Then, π − ( B ( x ) ∩ Over sloc ( D )) = { ⋆ ∈ SStar sp ( D ) | x ∈ D ⋆ } == { ⋆ ∈ SStar sp ( D ) | ∈ x − D ⋆ } == { ⋆ ∈ SStar sp ( D ) | ∈ ( x − D ) ⋆ } == { ⋆ ∈ SStar sp ( D ) | ∈ ( x − D ∩ D ) ⋆ } == V x − D ∩ D ∩ SStar sp ( D ) = V ( D : D x ) ∩ SStar sp ( D ) . However, V ( D : D x ) ∩ SStar sp ( D ) is compact since it has a minimum (ex-plicitly, s D (( D : D x )) ). Hence, the map π : SStar sp ( D ) −→ Over sloc ( D )satisfies the hypothesis of Lemma 5.2, and thus Over sloc ( D ) is a spec-tral space. (cid:3) However, SStar sp ( D ) is not, in general, a spectral space. To avoidthis problem, we restrict π to the space SStar f,sp ( D ) (which is alwaysspectral [13, Theorem 4.6]), obtaining the map π s : SStar f,sp ( D ) −→ Over sloc ( D ); analogously to the previous proof, we need to show that π s is surjective and that π − s ( B ( x ) ∩ Over sloc ( D )) is compact. We claimthat D being rad-colon coherent is a sufficient condition for this tohappen; we need a lemma. Lemma 5.4.
Let D be an integral domain, and let ⋆ be a spectral semis-tar operation on D .(a) If D ( F ∩ D ) is a compact subset of Spec( D ) for every finitelygenerated fractional ideal F of D , then ⋆ f = e ⋆ .(b) If D is rad-colon coherent, then D ⋆ f = D e ⋆ . Note that the equality ⋆ f = e ⋆ may actually fail; see [2, p.2466]. Proof. (a) Since ⋆ f and e ⋆ are of finite type, it is enough to show that F ⋆ f = F e ⋆ if F is finitely generated. The containment F e ⋆ ⊆ F ⋆ f alwaysholds; suppose x ∈ F ⋆ f . Then, since F ⋆ f ⊆ F ⋆ , we have x ∈ F ⋆ .Consider I := x − F ∩ D . Then, xI = F ∩ xD ⊆ F . Moreover, I ⋆ = ( x − F ∩ D ) ⋆ = x − F ⋆ ∩ D ⋆ since ⋆ is spectral, and thus 1 ∈ I ⋆ . Since x − F is finitely generated,by hypothesis D ( I ) is compact, and thus there is a finitely generatedideal J of D such that rad( I ) = rad( J ); passing, if needed, to a powerof J , we can suppose J ⊆ I , so that xJ ⊆ xI ⊆ F . For any spectraloperation ♯ , rad( A ) = rad( B ) implies that 1 ∈ A ♯ if and only if 1 ∈ B ♯ ; therefore, 1 ∈ J ⋆ , and thus x ∈ ( F : J ) ⊆ F e ⋆ , and x ∈ F e ⋆ . Hence, ⋆ f = e ⋆ , as requested.(b) It is enough to repeat the proof of the previous point by using F = D , and noting that D ( x − D ∩ D ) is compact since D is rad-coloncoherent. (cid:3) Theorem 5.5.
Let D be an integral domain. If D is rad-colon coherent,then Over sloc ( D ) is a spectral space.Proof. Suppose D is rad-colon coherent. If T ∈ Over sloc ( D ), then thereis a ♯ ∈ SStar sp ( D ) such that T = D ♯ ; since D is D -finitely generated,moreover, we have D ♯ = D ♯ f . By Lemma 5.4(b), D ♯ f = D e ♯ ; but e ♯ ∈ SStar f,sp ( D ), and thus π s is surjective.As in the proof of Proposition 5.3, π − s ( B ( x ) ∩ Over sloc ( D )) = V ( D : D x ) ∩ SStar f,sp ( D ) , which is compact since it has a minimum ( s D (( D : D x )) ). Since SStar f,sp ( D )is a spectral space [13, Theorem 4.6], by Lemma 5.2 Over sloc ( D ) is spec-tral. (cid:3) Corollary 5.6. If D is a domain with Noetherian spectrum (in partic-ular, if D is Noetherian) then Over sloc ( D ) is a spectral space. Note that it is not hard to see that, if D ( J ) is not compact inSpec( D ), then V J ∩ SStar f,sp ( D ) is actually not compact; therefore,the proof of Theorem 5.5 cannot easily be further generalized.Another natural question is whether π s is injective; however, thisis usually false. For example, if ∆ is any subset of Spec( D ) contain-ing the t -spectrum, then π s ( s ∆ ) = D . Thus, π s does not give a wayto “represent” Over sloc ( D ) like Spec( D ) does for Loc( D ) and S ( D )for Over qr ( D ). To circumvent this problem, we shall use, instead ofthe whole spectrum, the t -spectrum; note that QSpec t ( D ) is a procon-structible subspace of Spec( D ) [5, Proposition 2.5], so a spectral space,and thus the space X (QSpec t ( D )) is defined and spectral.Consider the map π t : X (QSpec t ( D )) −→ Over sloc ( D )∆ D s ∆ . Note that, if D is rad-colon coherent, π t is continuous and spectral,since it is the composition of the spectral inclusion X (QSpec t ( D )) ֒ → X ( D ) ([15, Proposition 4.1], noting the inclusion QSpec t ( S ) ֒ → Spec( D )is spectral since QSpec t ( D ) is proconstructible), the homeomorphism X ( D ) −→ SStar f,sp ( D ) and the map π s : SStar f,sp ( D ) −→ Over( D )(which is spectral by Lemma 5.2 and the proof of Theorem 5.5).We first show that, using π t , we do not lose anything. Proposition 5.7.
Let D be an integral domain. Then: OPOLOGICAL PROPERTIES OF LOCALIZATIONS 13 (a) for any ∆ , Λ ∈ X ( D ) , if ∆ ∩ QSpec t ( D ) = Λ ∩ QSpec t ( D ) then π s ( s ∆ ) = π s ( s Λ ) ;(b) π s (SStar f,sp ( D )) = π t ( X (QSpec t ( D ))) .Proof. It is enough to show that, for every ∆ ∈ X ( D ), π s (∆) = π s (∆ ),where ∆ := ∆ ∩ QSpec t ( D ). Let T := π s ( s ∆ ); then, since ∆ is a procon-structible subset of Spec( D ), also ∆ is proconstructible. In particular,∆ is compact and closed by generizations relative to QSpec t ( D ), andso it belongs to X (QSpec t ( D )). We claim that T = π t (∆ ).Indeed, let P ∈ ∆. Then, t P : ID P I t D P is a star operation offinite type on D P (see [26]), and QD P is a maximal t P -ideal if andonly if Q is maximal among the t -prime ideals contained in P . Hence, D P = T { D Q | Q ⊆ P, Q = Q t } , and T = \ { D Q | Q = Q t , Q ⊆ P for some P ∈ ∆ } . The set of primes on the right hand side is exactly ∆ . Therefore, T = π t (∆ ) ∈ π t ( X (QSpec t ( D ))), and (a) is proved.Moreover, this also shows that π s (SStar f,sp ( D )) ⊆ π t ( X (QSpec t ( D )));since the other inclusion is obvious, (b) holds. (cid:3) The t -spectrum is much less redundant than Spec( D ): indeed, if D = T { D P | P ∈ ∆ } for some compact ∆ ⊆ QSpec t ( D ), then ∆ mustcontain the t -maximal ideals, since t is the biggest star operation offinite type. In general, π t is not always injective; however, when thishappens then π t is also a homeomorphism, as the next propositionshows. Proposition 5.8.
Let D be a rad-colon coherent domain. Then, thefollowing are equivalent:(i) π t is a homeomorphism;(ii) π t is injective;(iii) if ∆ , Λ ∈ X ( D ) are such that π s ( s ∆ ) = π s ( s Λ ) , then ∆ ∩ QSpec t ( D ) = Λ ∩ QSpec t ( D ) .Proof. The implication (i = ⇒ ii) is obvious; the equivalence between(ii) and (iii) follows from Proposition 5.7.Suppose now that π t is injective; then, π t is bijective (since it is alsosurjective by Theorem 5.5, being D rad-colon coherent), continuous andspectral. Clearly, if ∆ ⊇ Λ then π t (∆) ⊆ π t (Λ). Conversely, suppose π t (∆) ⊆ π t (Λ): then, T := T { D P | P ∈ ∆ } ⊆ T { D Q | Q ∈ Λ } , andthus T ⊆ D Q for every Q ∈ Λ. Hence, π t (∆) = π t (∆ ∪ Λ), and bythe injectivity of π t is must be ∆ = ∆ ∪ Λ, i.e., Λ ⊆ ∆. Therefore, π t is also an order isomorphism (in the order induced by the respectivetopologies of X (QSpec t ( D )) and Over sloc ( D )); by [25, Proposition 15], π t is a homeomorphism. (cid:3) A prime ideal P of D is well-behaved if P D P is t -closed in D P [34];this is equivalent to D P being a DW-domain, i.e., to the fact that, on D P , the w -operation coincides with the identity (this follows from[29, Proposition 2.2]). A domain is called well-behaved if every t -primeideal is well-behaved; examples of well-behaved domains are Noetheriandomains, Krull domains and domains where every t -prime ideal hasheight 1. Proposition 5.9.
Let D be an integral domain. Then, D is well-behavedif and only if the map π t : X (QSpec t ( D )) −→ Over sloc ( D ) is injective.Proof. Suppose π t is injective, and let P ∈ QSpec t ( D ) and ∆ :=QSpec t ( D P ). Then, ∆ is compact (being proconstructible in Spec( D P )),and thus ∆ ∩ D := { Q ∩ D | P ∈ ∆ } is a compact subspace ofQSpec t ( D ), since it is the continuous image of ∆ under the canonicalmap Spec( D P ) −→ Spec( D ). If P D P / ∈ ∆, then P / ∈ ∆ ∩ D ; however, π t (∆ ∩ D ) = \ { D Q ∩ D | Q ∈ ∆ } = \ { ( D P ) Q | Q ∈ ∆ } = D P , with the last equality coming from the properties of the t -spectrum.If we denote by Λ the closure in the inverse topology of QSpec t ( D )of ∆ ∩ D , and by Λ the closure of (∆ ∩ D ) ∪ { P } , we have thus π t (Λ ) = π t (Λ ) while Λ = Λ , against the injectivity of π t .On the other hand, suppose D is well-behaved. Suppose π t (∆) = π t (Λ) =: T for some ∆ , Λ ∈ X (QSpec t ( D )), ∆ = Λ, and let P ∈ ∆ \ Λ.By [7, Lemma 2.4], the subspace { D Q | Q ∈ Λ } ⊆ Over( D ) is compact;then, D P = D P T = D P \ Q ∈ Λ D Q = \ Q ∈ Λ D P D Q , with the last equality coming from [17, Corollary 5]. The family { D P D Q | Q ∈ Λ } is again compact [17, Lemma 4]; thus, ⋆ : I T Q ∈ Λ ID P D Q is a finite-type spectral semistar operation such that D ⋆ = D P , andthus it restricts to a finite-type star operation ⋆ ′ on D P . Since P D P is t -closed, and ⋆ ′ is of finite type, ( P D P ) ⋆ ′ must be equal to P D P ;however, P ⋆ ′ = P ⋆ = \ Q ∈ Λ P D Q D P = \ Q ∈ Λ D Q D P = D P , since P * Q for every Q ∈ Λ. This is a contradiction, and π t is injective. (cid:3) Remark 5.10. (1) There are examples of integral domains that are not well-behaved(see [34, Section 2] or [1, Example 1.4]), and thus π t is not al-ways injective.(2) It would be tempting to substitute the space X (QSpec t ( D ))with X (∆), where ∆ is the set of well-behaved t -prime ideals of D . However, ∆ may not be compact and thus, a fortiori , maynot be a spectral space. For example, consider a domain D and OPOLOGICAL PROPERTIES OF LOCALIZATIONS 15 a prime ideal Q that is a maximal t -ideal (that is, P is maximalamong the ideals I such that I = I t ) but not well-behaved.(An explicit example is E + XE S [ X ], where E is the ring ofentire functions, X is an indeterminate and S is the set of finiteproducts of elements of the form Z − α , as α ranges in C ; see[33, Example 2.6, Section 4.1 and Proposition 4.3].) Let Λ bethe set of prime ideals that are associated to some principalideal; then, P ∈ Λ if and only if P is minimal over the ideal( bD : D aD ), for some a, b ∈ D .Since a principal ideal is t -closed, so is ( bD : D aD ) = ba D ∩ D ;moreover, a minimal prime over a t -ideal is again a t -ideal, andthus Λ ⊆ QSpec t ( D ). Moreover, if P ∈ Λ then
P D P will beassociated to a principal ideal of D P (if P is minimal over ( bD : D aD ), then P D P is minimal over ( bD : D aD ) D P = ( bD P : D P aD P )). Hence, each prime of Λ is well-behaved, and Λ ⊆ ∆.By [4], we have D = T { D P | P ∈ Λ } , and thus also D = T { D P | P ∈ ∆ } . If ∆ were compact, it would define a finite-type star operation ⋆ : I T { ID P | P ∈ ∆ } such that Q ⋆ = D . On the other hand, we should have ⋆ ≤ t and thus Q ⋆ ⊆ Q t = Q , a contradiction. Hence, ∆ is not compact.Recall that a domain is v -coherent if, for any ideal I , ( D : I ) = ( D : J ) for some finitely generated ideal J . Corollary 5.11.
Let D be a v -coherent domain. Then, π t is injective.Proof. Since D is v -coherent, ( ID Q ) t = I t D Q for every ideal I of D [26, proof of Proposition 4.6] and every Q ∈ Spec( D ); thus, if P ∈ QSpec t ( D ) then ( P D P ) t = P t D P = P D P . By Proposition 5.9, π t isinjective. (cid:3) Flat overrings
The space Over flat ( D ) of flat overrings of D is much more mysteriousthan Over qr ( D ) and Over sloc ( D ), and we are not able to characterizewhen it is spectral or proconstructible. The main theorem of this sectionis the following partial result. Proposition 6.1.
Let D be an integral domain. Then, Over flat ( D ) isa proconstructible subspace of Over( D ) if and only if Over flat ( D ) ∩B ( x , . . . , x n ) is compact for every x , . . . , x n ∈ K .Proof. If Over flat ( D ) is proconstructible, the compactness of Over flat ( D ) ∩B ( x , . . . , x n ) follows like in the proof of Proposition 5.1.Suppose that the compactness property holds, and let x , . . . , x n ∈ K . Consider the canonical subbasis S := {B ( x ) ∩ X | x ∈ K } of X := Over flat ( D ). By [10, Proposition 3.3] and [19, Theorem 8] (or [10,Corollary 2.17]), we need to show that, for every ultrafilter U on X ,the ring A U := { x ∈ K | B ( x ) ∩ X ∈ U } is flat. Take a , . . . , a n ∈ D , x , . . . , x n ∈ A U such that a x + · · · + a n x n = 0.For all C ∈ Over flat ( D ) ∩ B ( x , . . . , x n ), by the equational characteri-zation of flatness (see e.g. [28, Theorem 7.6] or [9, Corollary 6.5]) thereare b ( C ) jk ∈ D , y ( C ) k ∈ C such that(2) ( a b ( C )1 k + · · · + a n b ( C ) nk for all kx i = b ( C ) i y ( C )1 + · · · + b ( C ) iN y ( C ) N for all i. Let Ω( C ) := B ( y ( C )1 , . . . , y ( C ) n C ). Then, the family of the Ω( C ) is an opencover of Over flat ( D ) ∩ B ( x , . . . , x n ). Hence, there is a finite subcover { Ω( C ) , . . . , Ω( C n ) } ; by the properties of ultrafilters, it follows thatΩ( C j ) ∈ U for some j . Thus, y ( C j ) i ∈ A U for all i ; then, (2) holds in A U . Hence, applying again the equational criterion, A U is flat. (cid:3) Corollary 6.2.
Let D be an integral domain such that Over flat ( D ) =Over sloc ( D ) . Then, Over flat ( D ) is a proconstructible subset of Over( D ) .In particular, D is rad-colon coherent.Proof. It is enough to note that Over sloc ( D ) ∩ B ( x , . . . , x n ) has alwaysa minimum, and apply Proposition 6.1. (cid:3) Example 6.3.
The space of flat overrings can be spectral even if it isnot proconstructible.Let K be a field, and let D := K [[ X , X , XY, Y ]]; that is, D is theset of the power series in two variables over K without the monomialcorresponding to X . Then, D is a two-dimensional local Noetherian do-main; its integral closure is A := K [[ X, Y ]] = D [ X ], which is also equalto the intersection of the localizations at the height-1 primes of D . (Inparticular, A is a local sublocalization of D that is not a localization.)By Corollary 5.11, it is easy to see that the sublocalizations of D are D itself and the intersections T (∆) := T { D P | P ∈ ∆ } , as ∆ rangesamong the subsets of X ( D ) := { P ∈ Spec( D ) | P has height 1 } .A power series φ := P i,j ≥ a ij X i Y j is invertible in A if and only if a = 0; hence, if φ ∈ A is not invertible then φ ∈ D . Since everyheight-1 prime ideal of A is principal (being A a unique factorizationdomain) and the canonical map Spec( A ) −→ Spec( D ) is surjective,every height-1 prime ideal of D is the radical of a principal ideal (if P = Q ∩ D , for Q ∈ Spec( A ), Q = φA , then P is the radical of φ D ). Hence, T (∆) is a quotient ring of D for every ∆ ( X ( D ); inparticular, they are all flat. Hence, Over qr ( D ) = Over flat ( D ) is spectral;however, ( D : D X ) is equal to the maximal ideal of D , which cannotbe the radical of a principal ideal since it is of height 2. By Theorem4.4, Over qr ( D ) (and so Over flat ( D )) is not proconstructible.The space Over flat ( D ) is, however, amenable to generalizations. In-deed, if R is a ring and M is an R -module, then the set SMod R ( M ) of R -submodules of M can be endowed with a topology (called the Zariski
OPOLOGICAL PROPERTIES OF LOCALIZATIONS 17 topology ) whose basic open sets are of the form D ( x , . . . , x n ) := { N ∈ SMod R ( M ) | x , . . . , x n ∈ N } , as x , . . . , x n vary in M . Under this topology, SMod R ( M ) is a spectralspace [31, Example 2.2(2)]; moreover, if D is an integral domain withquotient field K , then the Zariski topology on Over( D ) is exactly therestriction of the Zariski topology on SMod D ( K ) = F ( D ), and Over( D )is proconstructible in F ( D ).We can consider on SMod R ( M ) the subspace SModFlat R ( M ) con-sisting of all flat R -submodules of M . Surprisingly, in many cases spec-trality and proconstructibility of SModFlat R ( M ) are equivalent. Proposition 6.4.
Let R be a ring and M be an R -module; suppose that R is an integral domain or that M is torsion-free. Then, SModFlat R ( M ) is a spectral space if and only if it is proconstructible in SMod R ( M ) .Proof. Clearly if SModFlat R ( M ) is proconstructible in SMod R ( M ) thenit is spectral.Conversely, suppose that Y := SModFlat R ( M ) is spectral. By Lemma2.1, Y is proconstructible if and only if Ω ∩ Y is compact for every Ω insome subbasis of SMod R ( M ); since D ( x , . . . , x n ) = D ( x ) ∩ · · ·∩ D ( x n )for every x , . . . , x n ∈ M , we can consider the subbasis {D ( x ) ∩ Y | x ∈ M } . By definition, D ( x ) ∩ Y := { N ∈ Y | x ∈ Y } .Let x ∈ M . If x has no torsion (so, in particular, if M is torsion-free),then the principal submodule h x i is isomorphic to R , which is flat; thus, D ( x ) ∩ Y has a minimum, namely h x i , and D ( x ) ∩ Y is compact. Onthe other hand, if R is an integral domain, then every flat R -moduleis torsion-free [3, I.2, Proposition 3]; thus, if x has torsion then nomodule containing x can be flat, and so D ( x ) ∩ Y must be empty (andin particular compact).In all the cases considered, it follows that SModFlat R ( M ) is procon-structible in SMod R ( M ). (cid:3) Corollary 6.5.
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Dipartimento di Matematica e Fisica, Universit`a degli Studi “RomaTre”, Roma, Italy
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