CHARACTERIZATIONS OF THE INTEGRAL DOMAINS WHOSE OVERRINGS ARE GOING-DOWN DOMAINS

Let R be a (commutative integral) domain with quotient K; let R′ be the integral closure of R (in K). Then each overring of R (inside K) is a going-down domain if and only if R′ is a locally pseudo-valuation domain, T ⊆ T ′ satisfies going-down for every overring T of R, and tr. deg[VR′ (M)/M(R )M : R′/M ] ≤ 1 for every maximal ideal M of R′ (where VR′ (M) denotes the valuation domain that is canonically associated to the pseudo-valuation domain (R)M ). Additional equivalences are given in case R is locally finitedimensional. Applications include the case where R is integrally closed or R is not a Jaffard domain or R[X] is catenarian. Mathematics Subject Classification (2010):Primary 13B02; Secondary


Introduction
All rings considered below are assumed to be commutative (integral) domains (with 1).Throughout, R denotes a domain with quotient field K, with R denoting the integral closure of R (in K).As usual, Spec(R) (resp., Max(R)) denotes the set of prime (resp., maximal) ideals of R; and, by an overring of R, we mean an R-subalgebra of K, that is, a ring T such that R ⊆ T ⊆ K.If an overring T of R is such that T = R (resp., T = R[u] for some u ∈ K), we say that T is a proper overring (resp., a simple overring) of R. We let dim(R) denote the (Krull) dimension of R, while dim v (R) denotes the valuative dimension of R (that is, the supremum, which may be ∞, of dim(T ) as T ranges over the set of overrings of R).
Also, for a (ring) extension R ⊆ S of domains with corresponding quotient fields F ⊆ L, we let tr.deg[S : R] denote the transcendence degree of S over R (that is, the transcendence degree of L over F ).
Let us next recall some basic definitions and facts.
As in [13] and [21], R is said to be a going-down domain if the extension R ⊆ T satisfies the going-down property for each domain T that contains R (as a subring); cf. also [32], [33], [15], [22], [17], [31].By [21, Theorem 1], the rings T that need to be tested (to check that R ⊆ T satisfies the going-down property) may be restricted to be either valuation overrings of R or simple overrings of R. The most natural examples of going-down domains are arbitrary Prüfer domains and domains of (Krull) dimension at most 1.
R is called a treed domain if Spec(R), when viewed as a poset via inclusion, is a tree; that is, if no prime ideal of R can contain incomparable prime ideals of R.
It was shown in [13,Theorem 2.2] that each going-down domain must be treed; however, a construction of W. J. Lewis, reported in [22,Example 2.2], showed that the converse is false.
Following Hedstrom and Houston ([28], [29]), R is called a pseudo-valuation domain (or, in short, a PVD) if each prime ideal P of R is strongly prime; that is, if xy ∈ P , with x ∈ K and y ∈ K, implies that either x ∈ P or y ∈ P .Each PVD is a quasi-local going-down domain.(In fact, more was shown in [14, page 560], namely, that each PVD is a divided domain in the sense of [15].)Recall from [29] (cf.also [1]) that R is a PVD if and only if there is a (uniquely determined) valuation overring V of R such that Spec(R) = Spec(V ) (as sets); in this case, V is called the canonically associated valuation overring of R. It is shown in [1,Proposition 2.6] that PVDs can be characterized as the pullbacks V × F k arising from a valuation domain (V, M ) and a field k ⊆ F := V /M .In order to globalize the PVD concept, Dobbs and Fontana [18] introduced the following definition: R is called a locally pseudo-valuation domain (in short, an LPVD) if R M is a PVD for each maximal ideal M of R. The class of LPVDs clearly contains all Prüfer domains, all PVDs (indeed, PVDs are the same as the quasi-local LPVDs), and an abundance of other semi-quasi-local domains arising from pullback constructions (see, for instance, [18,Example 2.5]).
Although each overring of a Prüfer domain must be a Prüfer domain, there exist one-dimensional domains with overrings of dimension greater than 1.By applying the classical D + M construction (as in [26]) to such examples, it was shown in [13] that an overring of a going-down domain need not be a going-down domain.In fact, by an iterated pullback construction, it was shown in [19] that an integral overring of a going-down domain need not be a going-down domain.(Earlier, it had been shown that each integral overring of a going-down domain is a going-down domain if dim v (R) ≤ 2 [14] or if R is both locally divided and locally finite-conductor [16, DOMAINS WITH GOING-DOWN OVERRINGS 101 Theorem 3.2].)The main goal of this paper is to characterize the domains which are such that each overring is treed (or is itself a going-down domain).
There have been several previous attempts at answering the above question or related questions, with success only under rather special additional hypotheses: cf.[13, Proposition 3.5], [4,Theorem 5.30], [12,Corollary 2.6], [6, Theorem 1], [7, Theorems 4.4 and 6.1] and [3, Theorem 10].An underlying difficulty in such studies is that the nature of R has very subtle influences on its overrings.In addressing the general case, we have found inspiration from two sources: Ayache's recent result [3] that if an integrally closed domain R is such that each overring of R is treed, then R must be an LPVD; and the role, in the above-cited result from [12], of the transcendence degree of R/M ⊆ V /M , in case (R.M ) is a PVD with canonically associated valuation overring V .Our main result, Theorem 2.8, provides the following characterization for the general case: each overring of (a domain) R is a going-down domain if and only if R is an LPVD, T ⊆ T satisfies going-down for every overring T of R, and tr.deg[V R (M )/M : R /M ] ≤ 1 for every maximal ideal M of R (where V R (M ) denotes the valuation domain that is canonically associated to the pseudo-valuation domain (R ) M ).Theorem 2.8 also shows that in case R is locally finite-dimensional, the above conditions are equivalent to: each overring T of R is treed and satisfies ht . Some other interesting applications are also given along these lines, including the case where R is integrally closed (Proposition 2.2) or R is not a Jaffard domain (Proposition 2.5) or R[X] is catenarian (Corollary 2.11).
If R is an LPVD and M is a maximal ideal of R, it will be convenient to let V R (M ) denote the valuation domain which is canonically associated to the pseudovaluation domain R M .Also, ⊂ will denote proper inclusion.Any unexplained material is standard, typically as in [26].

Results
For reference purposes, we begin by stating the above-mentioned recent result of the first-named author.To appreciate the significance of the hypothesis in Proposition 2.2, note the following consequence of [17,Example 2.3]: the implication (ii) ⇒ (i) in Proposition 2.2 would fail if the domain R is not integrally closed.Also, for the proof of Proposition 2.2 (and later), we need to recall that a domain R is called a Jaffard Proposition 2.2.Let R be an integrally closed domain.Then the following three conditions are equivalent: If, in addition, R is locally finite-dimensional, then the above three conditions are also equivalent to the following three conditions: Assume henceforth that R is locally finite-dimensional.
(iii) ⇔ (iv): Let M be a maximal ideal of R. Then R M is a PVD.So, if [8,Theorem 3.1]); that valuative dimension can be found by taking the supremum of the dimensions of valuation overrings; and that the supremum of the dimensions of valuation domains that are intermediate between two given fields is the transcendence degree of the field extension (cf.[26,Theorem 20.9]).Therefore, it follows from [23, Proposition (v) ⇒ (iv): Clear (as any finite-dimensional valuation domain is a Jaffard domain).
(iv) ⇒ (v): Let M be a maximal ideal of R. Then R M is a PVD such that and we can then immediately deduce the following result from Proposition 2.2.
Corollary 2.3.Let R be an integrally closed domain such that dim v (R) ≤ 2. Then the following conditions are equivalent:  2)] (noting that the latter can be used to show that dim v (R) = sup M ∈Max(R) dim v (R M ) for all domains R), this easily leads to the following result.
Corollary 2.4.Let R be an integrally closed coequidimensional domain.Then the following conditions are equivalent: Let R be a finite-dimensional quasi-local integrally closed domain.It is known that if R is a Jaffard domain, then each overring of R is a going-down domain (or treed) if and only if R is a valuation domain [6, Lemma 1].Our next proposition concerns the case where R is not a Jaffard domain (where it will turn out that, for a finite-dimensional quasi-local integrally closed domain which is not a Jaffard domain, having each overring being a going-down domain is equivalent to being a certain kind of PVD).This result points out that the domains that are quasilocal and integrally closed with all their overrings being going-down domains (or treed) are intimately connected to the domains for which each proper overring is (or satisfies) ℘, where ℘ is a ring-theoretic property, which means "(locally) Jaffard domain" or "(stably) strong S-domain" or "universally catenarian domain" or "satisfying the altitude formula".We will assume familiarity with results on all these concepts, as in [2], [5], [30], [10], but pause to collect the relevant definitions.
R is said to be catenarian in case, for each pair P ⊂ Q of prime ideals of R, all saturated chains of prime ideals going from P to Q have a common finite length; R is called universally catenarian if the polynomial rings R[X 1 , X 2 , . . ., X n ] are catenarian for each positive integer n.
R is said to be a strong S-domain if, for each pair of consecutive prime ideals With the above definitions recorded, we can give the following interesting characterizations.
Proposition 2.5.Let (R, M ) be a finite-dimensional quasi-local integrally closed domain.If R is not a Jaffard domain, then the following conditions are equivalent: (i) Each overring of R is a going-down domain; (ii) Each overring of R is treed; (iii) Each proper overring of R is a Jaffard domain; (iv) Each proper overring of R is a locally Jaffard domain; (v) Each proper overring of R is a strong S-domain; (vi) Each proper overring of R is a stably strong S-domain; (vii) Each proper overring of R is a universally catenarian domain; (viii) Each proper overring of R satisfies the altitude formula; Proof.The assertions (iii), (iv), (v), (vi), (vii), (viii), (ix), and (x) are equivalent by combining [9, Theorem 1.4, Theorem 1.8 and Proposition 2.2].(x) ⇒ (i) ⇔ (ii): These assertions follow directly from Proposition 2.2.
(ii) ⇒ (ix): Assume (ii).Then R is a PVD by Theorem 2.1.As R is not a Jaffard domain, then R is not a valuation domain.Thus, dim v (R) = dim(R) + 1, by Proposition 2.2.
We turn next to the case where R is not necessarily integrally closed.First, it will be convenient to provide a characterization of the integral extensions R ⊆ S such that each intermediate ring between R and S is treed.Notice that the next result improves [7,Theorem 6.4] since the extension R ⊂ S in it is not assumed to satisfy the going-down property.
Lemma 2.6.Let R ⊂ S be an integral extension of domains.Then the following conditions are equivalent: (ii) R is treed and every non-maximal prime ideal of R is unibranched in S.
Proof.[7,Lemma 6.3] handles the implication (i) ⇒ (ii).For the converse, assume, by way of contradiction, that there exists a non-treed intermediate ring T between R and S. It is easy to see, by the lying-over property and (ii), that each non-maximal prime ideal of R is unibranched in T .Choose q and q to be incomparable prime ideals of T that are contained in some maximal ideal m of T , and let p := q ∩ R and p := q ∩ R be their contractions to R. Then p and p are each contained in the maximal ideal m := m ∩ R of R. As R is treed, then p and p must be comparable, say p ⊆ p ⊂ m.If p = p , then q = q (since p is unibranched in T ), a contradiction.Now, suppose that p ⊂ p .As R ⊆ T enjoys the going-up property, there is a prime ideal q of T such that q ⊂ q and q ∩ R = p .Again, since p is unibranched in T , we conclude that q = q , and so q ⊂ q , another contradiction.Hence, T is treed, as desired.
Lemma 2.7.Let R be a domain such that each overring of R is treed.Then the following conditions are equivalent: (i) R is a going-down domain; (ii) R ⊆ R satisfies going-down; If, in addition, R is locally finite-dimensional, then the above two conditions are equivalent to the following condition: R is treed, by [7,Theorem 5.5].Let T be an overring of R. Set T o := R ∩ T .By hypothesis, T o is a going-down domain; and the extension T o ⊆ T satisfies the incomparable property, by the above comments.Since, in addition, T is treed, a standard argument (which, for the sake of completeness, is included in the next paragraph) shows that T is a going-down domain, as desired.
By [21,Theorem], it is enough to prove that T ⊆ V satisfies going-down for each valuation overring (V, P) of T .Put J := P ∩ R .As R is a Prüfer domain,

By replacing
V with V Q2 , we may suppose that Q 2 = P.Put p i := P i ∩ T o , for i = 1, 2. Note that p 1 ⊂ p 2 by the incomparable property of the extension T o ⊆ T .Also note that J ∩ T o = p 2 .Since (vi) ensures that T o ⊆ R satisfies going-down, we therefore get I ∈ Spec(R ) such that I ∩ T o = p 1 and I ⊂ J. Consider P 3 := IV ∩ T ∈ Spec(T ).
We have that P 3 ∩ T o = I ∩ T o = p 1 and P 3 ⊆ P 2 .Then, since the extension T o ⊆ T satisfies the incomparable property, either P 1 = P 3 or P 1 and P 3 are incomparable.
But P 1 and P 3 are not incomparable, since T is treed.Hence P 1 = P 3 .Then
a valuation domain.By the above reasoning, the assumed condition gives that tr.deg[V R (M )/M R M : R/M ] = 0, and so are strong Sdomains for each positive integer n.R is said to satisfy the altitude formula if, for each finite-type R-algebra S containing R and each prime ideal Q of S that lies over P , we have ht S (Q) + tr.deg[S/Q : R/P ] = ht R (P ) + tr.deg[S : R].

Q 1 : 1 and 2 . 3 ]
= I(R ) J = IV has the desired properties, which completes the proof.Remark 2.10.(a) We point out how the quasi-local QQR-domain (D, M ) of [27, Example 4.3] relates to conditions (i) and (ii) in Corollary 2.9.Note that D is not a pseudo-valuation domain, by [28, Theorem 1.7], since its integral closure D is not quasi-local.In fact, D is a Prüfer domain with exactly two maximal ideals, and D is the only integral proper overring of D.Moreover, D ⊂ D is the special kind of minimal (over)ring extension that was studied in [27], namely, the type of ring extension where every proper overring of D contains D .Consequently, each proper overring of D is a Prüfer domain (hence a going-down domain, hence a treed domain).However, D fails to be a treed domain (and so D is the only overring of D which is not a going-down domain, thus showing that D "narrowly" fails to satisfy condition (i) in Corollary 2.9).Indeed, [25, Remark 3.3] established, i.a., the following more general fact: if (A, N ) is a quasi-local QQR-domain and A ⊂ A is (an integral minimal ring extension which is) decomposed (in the sense that A /N ∼ = A/N × A/N as algebras over A/N ), then A is not a treed domain.(In contrast, note that each of the base rings denoted by R in [17, Examples 2.is a treed domain.)Similarly, D "narrowly" fails to satisfy condition (ii) in Corollary 2.9, since D is the only overring T of D such that T ⊆ T does not satisfy going-down.