Characterizations and constructions of h-local domains

Transcription

1 Contributions to Module Theory, 1 21 c de Gruyter 2007 Characterizations and constructions of h-local domains Bruce Olberding Abstract. H-local domains arise in a variety of module- and ideal-theoretic applications. In the first half of the paper, we survey some of these applications, and collect a number of characterizations and examples of h-local domains. In the second half, we show how diverse examples of h-local Prüfer domains arise as overrings of Noetherian domains and polynomial rings in finitely many variables. Key words. h-local domain, Prüfer domain, group of divisibility. AMS classification. 13F05, 13G05. 1 Introduction All rings in this article are commutative and have identity. Mostly we restrict to commutative integral domains, and when this is the case, we denote the quotient field of the domain by Q. A domain R has finite character if each nonzero element of R is contained in at most finitely many maximal ideals of R. The domain R is h-local if R has finite character and each nonzero prime ideal of R is contained in a unique maximal ideal. Thus every quasilocal domain (i.e., domain having a unique maximal ideal) is an h-local domain, and hence h-locality is uninteresting as a local property. As a global property however, it is a very desirable hypothesis under which to work because, although not immediately apparent from the definition, it allows one to pass with relative ease between a domain and its localizations. It manifests in a number of different guises, which we discuss in Section 2. In fact, one of the interesting aspects of this class of rings is how understanding of the h-local property evolved from close study of both non-noetherian ideal theory and module theory, especially with an eye toward capturing certain essential aspects of abelian group theory. One early such application was Matlis classification of D-rings. Matlis remarks in a discussion of indecomposable modules that an initial hypothesis, arising from a naive attempt to generalize the theory of finite-dimensional vector spaces, might be that rank 1 modules are the only indecomposable ones. Unfortunately, it can be shown that this hypotheses fails, even for abelian groups [36, p. 1]. Here Matlis surely has in mind the sort of pathological decompositions of torsion-free finite rank modules exemplified by Corner s constructions in [14]. Matlis continues, A more profitable approach is to turn the problem around and try to characterize the integral domains that do have this property. Matlis accomplishes such a characterization for both integrally closed domains and Noetherian domains, and he notes, It is an unexpected phenomenon that the theory of h-local rings provides a key link in the chain of solving the problem [36, p. 3]. The class of D-rings, those domains for which every indecomposable torsion-free finite rank module has rank one, is of course very narrow. However, the class of h-

2 2 Bruce Olberding local domains is, at least relative to the class of D-rings, or even to the ring of integers, quite large. We give some examples later to justify this assertion. H-local domains capture a fundamental aspect of the ring of integers: every torsion R-module T is isomorphic to the direct sum of primary components, that is, T = M T M, where M ranges over the maximal ideals of R (see Section 2). In classifying the D-rings, Matlis used this characterization, along with several others, to prove that such domains are necessarily h-local. A larger class of domains, the class of reflexive domains, those domains R for which Hom R (, R) induces a duality on submodules of finite rank free R-modules, were also shown by Matlis in his 1968 article [34] to be h-local. (Independently, Heinzer showed in this same year that the domains R for which every nonzero ideal is reflexive is h-local [25]). More recently, in this same vein Warfield s duality for torsion-free finite rank abelian groups has been extended to modules over domains and studied by several authors; see Fuchs-Salce [20, Chapter 15]. These domains are reflexive, and hence h-local. One other important and well-known application of h-locality having its origins in abelian group theory is the classification of FGC rings, those rings for which every finitely generated module is a direct sum of cyclic modules. One of the major obstacles in obtaining this classification was showing that FGC domains are h-local; see [10] and [56]. For the moment we confine ourselves to the above applications, and postpone till later some brief mentions of applications to module theory and multiplicative ideal theory. In any case, given the range of applications of h-locality to module theory and ideal theory, and the desirability of working under such a hypothesis, it is of interest to have a good stock of examples to illustrate the theory. To this end, we give several constructions that yield finite character and h-local domains. Sections 2 and 3 are mostly expository, while Sections 4 and 5 contain new examples, with an emphasis on examples that occur as overrings of Noetherian domains. Not surprisingly, given the strength of the hypotheses involved, non-obvious examples are not so easy to obtain. One of our main class of examples occurs as overrings of polynomial rings over fields, where the challenge in this context is that affine algebras of dimension > 1 are about as far from being h-local as a domain can be. Despite this, we are able to give a realization theorem for h-local Prüfer overrings of polynomial rings in terms of the group of invertible fractional ideals of the domain. I thank the referee for helpful comments. 2 Characterizations In this section we compile some characterizations of h-local domains from the literature. We postpone till after the statement of each theorem a discussion of the background and references for the characterizations. We require first some terminology. An ideal I of the domain R is unidirectional if R/I is a quasilocal ring. If X and Y are R-submodules of the quotient field Q, then [Y : X] = {q Q : qx Y }. If P is a prime ideal of R, we view X P as a submodule of Q.

3 Characterizations and constructions of h-local domains 3 Theorem 2.1. The following conditions are equivalent for a domain R. (1) R is an h-local domain. (2) R/I is a finite product of quasilocal rings for every nonzero proper ideal I of R. (3) Every nonzero proper ideal of R is a finite intersection (product) of unidirectional ideals. (4) Every nonzero prime ideal of R contains an invertible unidirectional ideal. (5) Every nonzero prime ideal of R contains a nonzero principal ideal that is a product of unidirectional ideals. (6) For every maximal ideal M and 0 x M, xr M R is an invertible unidirectional ideal. (7) Every nonzero prime ideal of R is contained in a unique maximal ideal, and for every ideal I, if I M is finitely generated for each maximal ideal M, then I is a finitely generated ideal. (8) If {I α } is a collection of ideals of R having nontrivial intersection and M is a maximal ideal of R such that α I α M, then I α M for some α. (9) ( α X α) M = α (X α) M for every maximal ideal M of R and collection {X α } of R-submodules of Q having nontrivial intersection. (10) ( N M R N)R M = Q for each maximal ideal M of R, where N ranges over the maximal ideals of R distinct from M. (11) Q/R = M Q/R M, where M ranges over the maximal ideals of R. (12) R = M R M, where R denotes the completion of R in the R-topology and M ranges over the maximal ideals of R. (13) T = M T M for every torsion R-module T, where M ranges over the maximal ideals of R. (14) T = M T M for every cyclic torsion R-module T, where M ranges over the maximal ideals of R. (15) C = M Hom R(R M, C) for each cotorsion R-module C, where M ranges over the maximal ideals of R. (16) For all torsion R-modules T and R-submodules T 1 of T such that T/T 1 is a finitely generated R-module, T 1 is a supplemented R-submodule of T. (17) For all nonzero ideals I of R, I is a supplemented R-submodule of R. (18) For all R-modules A, A/t(A) does not contain a maximal R-submodule if and only if for all B A such that A/B is a finitely generated R-module, B is a supplemented R-submodule of A. (19) For all cyclic R-modules A, t(a) is the sum of all local submodules B of A. (20) Every finitely generated torsion R-module has a strongly flat cover.

4 4 Bruce Olberding Notes on Theorem 2.1. The characterization in (2) has been proved independently by several authors; see for example [2, Lemma 4.5] and [5, Theorem 4.9]. Characterization (3) was obtained by Jaffard in Théorème 6 of [29] and Matlis in Theorem 2.3 of [35] 1. Characterizations (4) (7) appear in Corollary 3.4 of [4]; for (4) see also Remark 5.4 of [44]. In [4], Anderson and Zafrullah place these characterizations in a more general setting involving star operations. Characterizations (8) and (9) are taken from Proposition 3.1 of [51]. Characterizations (10) (13) can be found in Theorem 22 of [36]. A very general view on the fundamental characterization in (13) was obtained by McAdam in [39]: Let T be an R-module. Then T = M T M if and only if R/Ann R (t) is a finite direct product of quasilocal rings for each nonzero t T. For some issues closely related to this characterization, see also the articles of Fuchs and Lee [18, 19]. Characterization (14) is proved by Brandal in Proposition 1 of [9]. Characterization (15) appears in Theorem 2.3 of [35]; recall that an R-module C is cotorsion if Hom R (Q, C) = Ext 1 R(Q, C) = 0. Characterizations (16) (19) are proved in Theorem 4.6 of [2]. The notation t(a) denotes the torsion submodule of A. If A B are R-submodules, then A has a supplement C in B if C is minimal in the collection of R-submodules D of B such that A + D = B. An R-module B is local if it is not a sum of its proper submodules. Bazzoni and Salce classify in [5] the domains for which the strongly flat modules form a covering class, and they prove in Theorem 4.9 of this article the characterization in (20). Regarding the terminology here, an R-module is strongly flat if it is a direct summand of an extension of a free module by a torsion-free divisible module. Let A be an R-module and X be a strongly flat R-module. An R-homomorphism φ : X A is a strongly flat precover of A if for every R-homomorphism φ : X A, with X a strongly flat module, there exists a homomorphism f : X X such that φ = φf. The precover φ : X A is a strongly flat cover of A if every endomorphism f of X such that φ = φf is an automorphism. Although we do not include it here, the interesting case of h-local domains such that Pic(R) = 0 is treated in Theorem 2.1 of [3] and Corollary 3.6 of [4]. Recall that an integral domain R is a Prüfer domain if every nonzero finitely generated ideal of R is invertible; equivalently, R M is a valuation domain for each maximal ideal M of R. The classes of finite character and h-local Prüfer domains arise in many ideal- and module-theoretic contexts; see for example the monograph of Fuchs and Salce [20], where these classes of rings figure prominently in a number of applications. Theorem 2.2. The following conditions are equivalent for a domain R. (1) R is an h-local Prüfer domain. (2) For every maximal ideal M of R and nonzero x, y M, (x, y)r M R is an invertible unidirectional ideal. 1 As discussed in [20, p. 148], Jaffard, in his 1952 article [29] and its sequel [30], appears to have been the first author to recognize the usefulness of pairing the two defining properties of h-local domains. He called the rings satisfying Theorem 2.1(3) rings of Dedekind type, and showed that (3) was equivalent to the domain being, in our terminology, h-local. In the sixties, Matlis, evidently unaware of Jaffard s work, came at the concept from the module-theoretic point of view of Theorem 2.1(13), and according to his student Brandal, he introduced in 1964 the terminology h-local, where h referred to homological [10, p. 17].

5 Characterizations and constructions of h-local domains 5 (3) For all ideals I, J 1, J 2,..., J n of R, n n n n [I : [I : J i ] ] = [I : [I : J i ] ] and [I : [I : J i ] ] = [I : [I : J i ] ]. i=1 i=1 i=1 i=1 (4) [I : J K] = [I : J] + [I : K] for all ideals I, J and K of R. (5) [M : J K] = [M : J] + [M : K] for all ideals J and K of R and maximal ideals M. (6) [X : Y W ] = [X : Y ] + [X : W ] for all R-submodules X, Y and W of Q such that Y + W X. (7) [I + J : K] = [I : K] + [J : K] for all ideals I, J and K of R. (8) [Y + W : X] = [Y : X] + [W : X] for all R-submodules X, Y and W of Q such that X Y W. (9) [R : I J] = [R : I] + [R : J] for all ideals I and J of R; R is integrally closed; and each maximal ideal M contains a finitely generated unidirectional ideal. (10) [R : I J] = [R : I] + [R : J] for all ideals I and J of R; R is integrally closed; and no subspace of Max R is homeomorphic to the Stone-Ĉech compactification βn of N. (11) Every short exact sequence of torsion-free modules, where I 1,..., I n are ideals of R, splits. 0 A I 1 I 2 I n B 0, (12) For all (maximal) ideals I of R, I is injective with respect to every short exact sequence of torsion-free modules, where J 1,..., J n are ideals of R. 0 A J 1 J 2 J n B 0, Notes on Theorem 2.2. Characterization (2) is quoted from Proposition 4.5 in [4]. Characterizations (3) (5) and (9) (12) are taken from Theorems 3.2 and 6.10 of [44]; (6) (8) can be found in Theorem 2.6 and Corollary 3.5 of [49]. In the article [44], an ideal I of R is defined to be colon-splitting if [I : J K] = [I : J] + [I : K] for all ideals J and K of R. This notion is motivated by the fact that every ideal of injective dimension 1 is a colon-splitting ideal [44, Lemma 7.1]. Thus from characterization (5) we deduce that if every maximal ideal of a domain R has injective dimension 1, then R is an h-local Prüfer domain. It follows that a domain R is an almost maximal Prüfer domain if and only if every maximal ideal has injective dimension 1; see [37] and [44]. A number of characterizations and decomposition results for colon-splitting ideals are given in [44]. With regards to (9) and (10), there does exist an integrally closed domain R such that R is a colon-splitting R-ideal but R is not an h-local domain. Such an example is discussed in Example 1.1 of [44]; it has its origins in an article of Brandal

6 6 Bruce Olberding [9, Example 2.1], where he proves that a certain overring of the ring of entire functions has injective dimension 1 as a module over itself. To state (5) of Theorem 2.3, we introduce some ideas developed by Brewer and Klinger in [11] that will also be important in Sections 4 and 5. For a Prüfer domain R, let I(R) denote the group of invertible fractional nonzero ideals of R. Then I(R) can be partially ordered by A B if and only if B A for all A, B I(R). In fact, since R is a Prüfer domain, then I(R) is a lattice-ordered group [11, Theorem 2]. Write the set of maximal ideals of R as {M i }. For each i, let v i denote the valuation corresponding to the valuation ring R Mi, and let G i denote the value group of R Mi. The group i I G i is lattice-ordered with respect to the pointwise ordering, and, when equipped with this ordering, is called the cardinal product of the G i (cf. [20, p. 108]). Then the mapping Φ : I(R) i I G i : A (v i (A)) i I is an embedding of lattice-ordered groups [11, Theorem 2]. (Here v i (A) = min {v i (a) : a A}. Since A is finitely generated such a minimum exists.) Moreover, the Prüfer domain R has finite character if and only if the image of Φ is a subgroup of the cardinal sum i I G i [11, Theorem 2]. Theorem 2.3. The following statements are equivalent for a Prüfer domain R. (1) R is an h-local domain. (2) For each maximal ideal M of R, N M R N is not a fractional ideal of R, where N ranges over the maximal ideals of R distinct from M. (3) For each maximal ideal M and nonzero prime ideal P contained in M, N M R N R P, where N ranges over the maximal ideals of R distinct from M. (4) For every finite collection {M 1,..., M n } of maximal ideals of R and every choice of nonnegative elements g i G i, there is a finitely generated ideal A of R such that v i (A) = g i for 1 i n, and v j (A) = 0 for all other maximal ideals M j of R. (5) Φ maps onto i I G i. (6) [J : I] M = [J M : I M ] for all ideals I and J of R and maximal ideals M. (7) [R : I] M = [R M : I M ] for all ideals I of R and maximal ideals M. (8) For all torsion-free R-modules G and H such that the canonical homomorphism Hom R (G, H) R Q Hom R (G, Q R H) is an isomorphism, the canonical homomorphism Hom R (G, H) R R M Hom RM (G M, H M ) is an isomorphism for each maximal ideal M of R. (9) There exists an R-submodule X of Q such that [X : [X : Y ]] = Y for all R- submodules Y of X (equivalently, for all ideals Y of R). (10) For each nonzero ideal I of R, I is divisorial if and only if I M is a divisorial ideal of R M for each maximal ideal M of R.

7 Characterizations and constructions of h-local domains 7 (11) For each nonzero ideal I of R, I = (I 1 ) 1 M 1 M n, where M 1,..., M n are precisely the nondivisorial maximal ideals of R which contain I for which I M is nondivisorial and this factorization is unique in the sense that no M i can be omitted. (12) Max R is a Noetherian space, and every nonzero prime ideal of R is contained in a unique maximal ideal. (13) Every nonzero ideal of R has finitely many minimal prime ideals, each of which is contained in a unique maximal ideal of R. Notes on Theorem 2.3. Characterizations (2) and (3) appear in Corollary 3.2 of [51]; (6) and (7) are taken from Theorem 3.10 of this same article. The characterization in (6) has been placed in a more general setting by Anderson and Zafrullah in [4]. Characterizations (4) and (5) are proved by Brewer and Klingler in Proposition 4 and Theorem 5 of [11]. For another point of view on (5), see [41, Proposition 4.10], where McGovern characterizes among Bézout domains those that are h-local using the group of divisibility of the domain. Characterization (8) is taken from Lemma 4.4 in [48]; it is deduced there from an argument similar to the one Goeters gives in Theorem 1.4 of [22]. For (9), see Lemmas 4.4 and 4.6 of [50]. Characterizations (10) and (11) are proved by Fontana, Huckaba and Lucas in Theorem 1.12 of [17], where it is shown in fact that (10) is also a valid characterization of h-local Prüfer domains if it is only assumed that locally divisorial ideals are divisorial. (Recall that an ideal I is divisorial if I = (I 1 ) 1.) The characterization in (13) appears in Proposition 3.4 of [51]. The characterization in (12) follows from this same proposition and Corollaries 1.3 and 1.5 of [53]; the topology here is the subspace topology on Max R induced by the Zariski topology on Spec R. Regardless of whether R is a Prüfer domain, statements (2), (3), (6), (7), (8), (10) and (12) always hold for an h-local domain; see the above-mentioned references. 3 Examples We collect next some examples of finite character and h-local domains. Example 3.1. If R is a Noetherian domain of Krull dimension 1, then R is an h-local domain. This is because R/I is an Artinian ring for each nonzero ideal I of R, so that every nonzero ideal of R is contained in at most finitely many maximal ideals of R. More generally, a domain R is almost perfect if for every nonzero ideal I of R, R/I has the descending chain condition for principal ideals. Bazzoni and Salce prove in Theorem 4.5 of [5] that a domain is almost perfect if and only if it is h-local and locally almost perfect. Interesting examples of almost perfect, hence h-local, non-noetherian domains can be found in [7], [54] and [58]. Example 3.2. There exist non-local Noetherian domains D such that D is h-local and every maximal ideal of D has height 2. In Theorem 2 of [57], S. Wiegand proves the following realization theorem for posets: Let X be a countable partially ordered set

8 8 Bruce Olberding with unique minimal element 0 and finitely many maximal elements m 1, m 2,..., m n, all of height 2, and suppose that each element of X is contained in a maximal element. For each i n, let G(m i ) = {p X : p m i }. Then X is order isomorphic to Spec R for some countable semilocal Noetherian domain R if and only if for each i n, the set G(m i ) \ j i G(m j) is infinite. By choosing X so that n > 1 and the sets G(m i ) \ {0} are pairwise disjoint, we obtain from Wiegand s theorem an h-local Noetherian domain of Krull dimension 2 that is not local. If R is a domain and F is a collection of prime ideals of R, then F has finite character if every nonzero element of R is contained in at most finitely many prime ideals in F. Example 3.3. For each n > 0, there exists a Noetherian h-local domain having infinitely many maximal ideals and Krull dimension n. In Theorem 5.3 and Corollary 5.4 of [24], Gulliksen gives an example of a finite character Noetherian domain R whose regular locus and Cohen-Macaulay locus coincide but are not constructible in Spec R. He does so by providing a general method of constructing Noetherian domains as localizations of polynomial rings in infinitely many variables. Let K be a field, and let D = K[X i : i I] be a polynomial ring over a set {X i : i I} of indeterminates for K. Suppose that F is a partition of {X i : i I} into finite sets. For each A F, define p A = AD. Observe that each 0 f D is in at most finitely many of the p A, A F, so that {p A } is a finite character collection of prime ideals of D. Define S = D\( A F p A) and R = D S. Then by Theorems 3.6 and 4.1 of [24], R is a finite character Noetherian domain whose set of maximal ideals is {AR : A F}. Thus if F has infinitely many members, and all but one of these members, say A 1, is a singleton, then R is an h-local Noetherian domain having infinitely many maximal ideals, and the Krull dimension of R is equal to the number of elements in A 1. This is because each maximal ideal of R other than A 1 R is a height 1 ideal of R. If R is a domain with quotient field Q, then the group of divisibility of R is the group of nonzero principal fractional ideals of R. This is a partially ordered group with respect to the ordering defined by xr yr if and only if yr xr for all nonzero x, y R. The group of divisibility of the domain R encodes a good deal of the ideal-theoretic structure of R; see for example Chapter III of [20]. Recall that an integral domain R is a Bézout domain if every finitely generated ideal of R is a principal ideal, and note that if R is a Beźout domain, then the group of divisibility is lattice-ordered and coincides with I(R). The Jaffard-Kaplansky-Ohm Theorem asserts that every lattice-ordered group can be realized as the group of divisibility of a Bézout domain [20, Theorem 5.3]. This is the key idea behind the next example. Example 3.4. Every cardinal sum of totally ordered abelian groups occurs as the group of divisibility of an h-local Bézout domain. Let {G i } be a collection of totally ordered abelian groups, and define G to be the cardinal sum i G i. Then G is a lattice-ordered abelian group, so there exists an h-local Bézout domain whose group of divisibility is isomorphic to G [20, Example III.5.4]. A Prüfer domain is strongly discrete if P P 2 for every nonzero prime ideal P of R. It is implicit in the article [6] of Bazzoni and Salce that a Prüfer domain R is h-local and strongly discrete if and only if every nonzero ideal of every overring S of R is a

9 Characterizations and constructions of h-local domains 9 divisorial ideal of S. In another direction, it is shown in [46] (but see also [47]) that a domain is an h-local strongly discrete Prüfer domain if and only if every proper ideal is a product of prime and finitely generated ideals. Example 3.5. There exist h-local strongly discrete Bézout domains of arbitrarily large Krull dimension, and having arbitrarily large numbers of maximal ideals. Let (X, ) be a Noetherian tree having a least element 0, and suppose that for each 0 p X there exists a unique maximal element m of X such that p m. Then there exists an h-local Bézout domain R such that Spec R is isomorphic as a partially ordered set to X. This is proved in [51, Proposition 5.5], using a result of Facchini [16, Theorem 5.3] that depends ultimately on the Jaffard-Kaplansky-Ohm Theorem. Let D be a domain, and let Σ be a collection of valuation overrings of D. Then Σ has finite character if each nonzero element of D is a unit in all but at most finitely many members of Σ. The next examples show how certain finite character collections of valuation rings give rise to finite character and h-local Prüfer domains. If R is the intersection of finitely many independent valuation rings all sharing the same quotient field, then R is an h-local Prüfer domain; this is a consequence of a theorem of Nagata [42, Theorem 11.11]. The next example discusses a situation where independent is redundant. Recall the notion of a reflexive domain discussed in the introduction. A valuation domain V is reflexive if and only if V has a principal maximal ideal and Q/V is an injective V -module; see for example [20, Theorem XV.9.5]. Example 3.6. If V 1,..., V n are reflexive valuation domains, all of which have the same quotient field, then R = V 1 V n is an h-local domain. In fact, R is reflexive domain [50, Corollary 4.3]. The content of this assertion is that such valuation rings must be independent, a fact that can be traced through a result of Facchini [16, Theorem 4.3] back to Vamos [55]. In general, a finite character intersection of reflexive valuation domains need not be h-local; consider for example the ring K[X, Y ], where K is a field and X and Y are indeterminates. Here K[X, Y ] is a finite character intersection of DVRs (i.e., quasilocal PIDs), but K[X, Y ] is neither h-local nor a Prüfer domain. However, working with power series in infinitely many variables, it is possible to construct h-local Prüfer overrings as finite character intersections of infinitely many reflexive valuation domains. This is done in [45]. Example 3.7. A small finite character intersection of valuation rings is a finite character Bézout domain. Let D be a domain containing a field of cardinality α, and let Σ be a collection of valuation overrings of D. If Σ has cardinality < α, then R := V Σ V is a Bézout domain [52, Theorem 6.6]. If in addition Σ has finite character, then R is a finite character domain. This follows from the fact that a Prüfer domain that is a finite character intersection of valuation overrings is necessarily of finite character [12, Corollary 2]. Note that R is h-local if the valuation rings in Σ are pairwise independent. Example 3.8. A finite character intersection of valuation overrings of a domain containing a non-algebraically closed field K is a finite character Prüfer domain if each valuation ring has residue field K. If also these valuation rings are independent, then R is an h-local domain. Let D be a domain containing a non-algebraically closed field

10 10 Bruce Olberding K. Suppose that Σ is a finite character collection of valuation overrings of D, each having residue field K. Then R = V Σ V is a Prüfer domain, a fact attributable to several authors; see [33] and [43] for discussion and references for this result. For simplicity we have stated only a special case, and by consulting [33] or [43] the reader may easily generalize our example. As in Example 3.7, we see that if Σ has finite character, then R is a finite character domain. If in addition the valuation rings in Σ are independent, then R is h-local. For the constructions in the previous two examples to be useful, it is necessary to be able to locate interesting collections of finite character collections of valuation overrings. One straightforward way to do this is the following. Let D be a domain, let Σ be a collection of valuation overrings of D and let F be the collection of prime ideals of D on which the valuation rings in Σ are centered. Then Σ has finite character if and only if F has finite character and for each prime ideal P in F, there are at most finitely many members of Σ centered on P. Thus, given a finite character collection F of prime ideals, a finite character collection of valuation overrings can be constructed by selecting for each prime ideal P in F, finitely many valuations overrings centered on P. (This is in fact what we do in Sections 4 and 5.) We give in the following proposition a simple criterion for when a countable domain has an infinite finite character collection of prime ideals. More interesting examples are given in Section 5. Proposition 3.9. Let D be a countable domain. Suppose that F is a collection of prime ideals with p F p = 0. Then there exists an infinite finite character collection of prime ideals in F. Proof. For each d D, let V (d) denote the set of prime ideals in F that contain d. Enumerate the nonzero nonunits of D as d 1, d 2, d 3,.... We claim: ( ) for all N > 0, there exists t > N such that V (d t ) V (d 1 d 2 d t 1 ). For otherwise there is N > 0 such that for all i > 0, V (d N+i ) V (d 1 d 2 d N+i 1 ). But then for each i > 0, V (d N+i ) V (d 1 d 2 d N ) V (d N+1 ) V (d N+i 1 ) = V (d 1 d 2 d N ), and it follows that F = V (d 1 ) V (d 2 ) V (d N ), so that d 1 d 2 d N p F p = (0), contrary to the assumption that the d i are nonzero elements of D. This proves ( ). Using the claim ( ), we now define a sequence of positive numbers t 1 < t 2 < < t n < in the following way. Let t 1 be the smallest positive number such that V (d t1 ) V (d 1 d t1 1). For each k > 0, define (via ( )) t k to be the smallest number such that t k > t k 1 and V (d tk ) V (d 1 d 2 d tk 1), and let p k V (d tk ) \ V (d 1 d 2 d tk 1). Then for each i > 0, d i p k only if t k i. In particular, d i is contained in at most finitely many of the p k. There is an interesting connection here to a couple of topics, one from module theory and the other from ideal theory. Slender rings. In Theorem 2 of [32], E. L. Lady proves that if a commutative ring R has an infinite finite character collection of maximal ideals, then R is a slender ring, meaning that for every countable family {A i } of R-modules and every homomorphism φ : i=1 A i R, there exists n > 0 such that φ( i n A i) = 0. Lady shows in Examples 5 and 6 of [32] that every domain that is finitely generated as an algebra over a field has a countably infinite finite character collection of maximal ideals. We

11 Characterizations and constructions of h-local domains 11 derive this same fact in Corollary 5.3, but from a somewhat more general point of view (namely, Proposition 5.1). Conforming spectra. Motivated by applications in [28] and [38, Chapters 3,7 and 14], McAdam considers in the unpublished manuscript [40] a stronger version of what we consider here 2. Let R be a commutative ring, let P Spec R and let X Spec R such that every prime ideal in X properly contains P. Then (P, X) is a conforming pair if X is infinite and {L/P : L X} is a finite character collection of prime ideals of R/P. Thus a domain R has finite character and infinitely many maximal ideals if and only if (0, Max R) is a conforming pair. The ring R has a conforming spectrum if for all prime ideals P of R and X Spec R such that every prime ideal in X properly contains P and such that L X L = P, there exists Y X such that (P, Y ) is a conforming pair. Thus if a domain R has Jacobson radical 0 and a conforming spectrum, then R has an infinite finite character collection of maximal ideals. Among a number of interesting results in [40], McAdam proves: If a ring R is finitely or countably generated over a field or R is an integral extension of a ring with a conforming spectrum, then R has a conforming spectrum. He asks whether every Noetherian ring has a conforming spectrum, and he indicates that he suspects the answer is no. However, in the other direction, he shows that to prove that every Noetherian ring of finite Krull dimension has a conforming spectrum, it suffices to show that every Noetherian domain R of Krull dimension 2 has the property that whenever Y is a set of nonzero prime ideals of R with P Y P = 0, then there exists an infinite finite character collection X Y. 4 H-local Prüfer overrings of Noetherian rings Our goal in this section and the next is to construct h-local Prüfer overrings of Noetherian domains. Recall from the discussion preceding Theorem 2.3 the lattice-ordered group I(R) of invertible fractional ideals of a Prüfer domain R. In this section we identify precisely which lattice-ordered groups occur as the group of invertible fractional ideals of an h-local Prüfer overring of a Noetherian domain. As indicated by Theorem 2.3 and Example 3.4, a lattice-ordered abelian group is the group of invertible fractional ideals of an h-local Prüfer domain if and only if it is a cardinal sum i I G i, where {G i : i I} is a collection of totally ordered abelian groups. As explained in Example 3.4, one way that a given cardinal sum i I G i can be realized as the group of invertible fractional ideals of a domain R is via the Jaffard-Kaplansky- Ohm Theorem. However, the ring R arising from this construction is very large it occurs as an overring of a polynomial ring in infinitely many variables. In this section we show that a similar realization theorem can be proved for overrings of Noetherian domains, and thus one obtains a diverse set of examples of h-local Prüfer overrings of rings of classical interest. Our method of construction is via intersections of valuation rings hidden over finite character collections of maximal ideals. We establish in Lemma 4.2 that such intersections are indeed h-local and Prüfer. This is a consequence of a technical lemma: topics. 2 I thank Steve McAdam for showing me this manuscript, and for some very helpful conversations on these

12 12 Bruce Olberding Lemma 4.1. Let D be a domain, let Σ be a finite character collection of valuation overrings of D and let X = {P D : P is a nonzero prime ideal of some member V of Σ}. If X satisfies the descending chain condition and any two incomparable prime ideals in X are comaximal in D, then R := V Σ V is a finite character Prüfer domain. Proof. We show first that R is a Prüfer domain by proving that R M is a valuation domain for each maximal ideal M of R. Let M be a maximal ideal of R, and set m = M D. Since Σ has finite character and localization commutes with finite character intersections, we have R m = V Σ V m [26, Lemma 1]. Let Γ = {V Σ : V m Q}. We claim that Γ is a finite set. For once this is established, it follows that R m is a finite intersection of valuation overrings, and hence R m is a Prüfer domain [42, Theorem 11.11]. Since then R M is a quasilocal overring of the Prüfer domain R m, it must be that R M is a valuation domain, proving the claim. Define Y = {P D : P is a nonzero prime ideal of some member of Γ such that P D m}. Since by assumption incomparable elements of X are comaximal, Y is a chain, and hence, by assumption, has a smallest element, say p. However, since Σ has finite character, p survives in only finitely many members of Σ, and this then implies that Γ is finite. For suppose that V Γ. Then there exists a nonzero prime P of V such that P D m, and hence P D Y. But then p P, so that p survives in V, and since p survives in only finitely many members of Σ, we conclude that Γ is a finite set. This proves that R is a Prüfer domain. Moreover, as a Prüfer domain that is a finite character intersection of valuation domains, R is a finite character domain [12, Corollary 2]. Let V be a valuation overring of the domain D. Then V is a hidden valuation overring of D, if for every nonzero prime ideal P of V, P D is a maximal ideal of D. Examples of hidden valuation overrings of Noetherian domains abound. For example, suppose that (x 1,..., x n )D is a maximal ideal of the Noetherian domain D with x 1 0, and set R = D[x 1, x 2 x 1 1,..., x nx 1 1 ]. If V is a valuation overring of R such that x 1 is in every nonzero prime ideal of V, then V is a hidden valuation overring of D; indeed, it is hidden over (x 1,..., x n )D. In this way, one can construct diverse examples of hidden valuation overrings; see Lemma 4.3 and the discussion that precedes it. Lemma 4.2. Let D be a domain, and let Σ be a finite character collection of hidden valuation overrings of D. Then R := V Σ V is a finite character Prüfer domain. If also distinct valuation rings in Σ are centered on distinct maximal ideals of D, then R is an h-local domain and Σ = {R M : M is a maximal ideal of R}. Proof. The set X (where X is as in Lemma 4.1) consists of maximal ideals of D, and hence meets the requirements of the lemma. Thus in view of the lemma, R is a finite character Prüfer domain. Suppose also that distinct valuation rings in Σ are centered on distinct maximal ideals of R, and that M and N are distinct maximal ideals of R containing a nonzero prime ideal P of R. Then since R is a Prüfer domain and Σ has finite character, we have R M, R N Σ [12, Theorem 1]. Thus, since R M and R N are valuation domains, we have P R M = P R P = P R N. But then P R M D = P R P D =

13 Characterizations and constructions of h-local domains 13 P R N D, contrary to the assumption that distinct members of Σ are centered on distinct maximal ideals of D. We conclude that R is an h-local Prüfer domain. We have already observed that {R M : M is a maximal ideal of R} Σ. To see that the reverse inclusion holds, let V Σ. Since R is a Prüfer domain, there exists a prime ideal P of R such that V = R P. Let M be a maximal ideal of R containing P. Then, as we have established, R M Σ. Since each member of Σ is hidden above D and centered on a distinct maximal ideal of D, it follows that the members of Σ are incomparable with respect to inclusion. Therefore, since R M R P = V, this forces R M = V, which proves the lemma. Recall that the rational rank of a totally ordered abelian group G is the Q-dimension of Q Z G. The rank of G is the order type of the set of proper convex subgroups of G. The group G is discrete if it is order-isomorphic to the Hahn product of copies of Z. In case G has finite rank, then G is discrete if and only if G is order-isomorphic to a direct sum of copies of Z with the lexicographic ordering. Recently, F. V. Kuhlmann has proved very general existence theorems for valuation overrings of polynomial rings. We quote only a very special case that is needed here: Let K be a field, and let w be a valuation on K with value group G w and residue field k. Let G be a countably generated ordered abelian group extension of G w such that G/G w is an infinite torsion group. Then there exists an extension v of w from K to K(X) such that the value group of v is G and the residue field of v is k [31, Proposition 3.17]. Lemma 4.3. Let D = K[X 1,..., X n ], where K is a field and X 1,..., X n are indeterminates for K. If m is a maximal ideal of D having residue field K and G is a totally ordered abelian group of rational rank < n or a discrete group of rank n, then there exists a hidden valuation overring V of D centered on m such that V has value group G. Proof. Since m has residue field K, there exist k 1,..., k n K such that m = (X 1 k 1,..., X n k n )D. Thus after a change of variables we may assume without loss of generality that m = (X 1,..., X n )D. Let g 1,..., g k be a maximal linearly independent set of positive elements of G (so that k = rational rank of G), and such that g 1 is not in any proper convex subgroup of G. Define H = Zg Zg k. We may in addition assume that if G is a finitely generated abelian group, then g 1,..., g k are chosen so that G = H. Moreover, by replacing {g 1, g 2,..., g k } with the set {g 1, g 2 + g 1,..., g k + g 1 }, which also is a maximal linearly independent subset of G that generates H, we may assume that g 1 < g i for all i > 1. We will use Kuhlmann s theorem to show there exists a valuation v on K(X 1,..., X n ) with value group G such that v is trivial on K, v(x i ) = g i for all 1 i k and v(x i ) = g 1 for all k < i n. For given such a valuation v with corresponding valuation ring V, X i V X 1 V for all 1 i n, and since g 1 is not in any proper convex subgroup of G and v(x 1 ) = g 1, X 1, hence each X i, is an element of every nonzero prime ideal of V [15, Lemma 2.3.1]. Thus V is a valuation ring hidden over the maximal ideal m of D and having value group G. To construct such a valuation v, we first observe that since the elements g 1,..., g k are linearly independent in G, there exists a valuation w : K(X 1,..., X k ) G { }

14 14 Bruce Olberding such that w(x i ) = g i for all 1 i k; cf. [8, VI.10.3, Theorem 1] or [31, Lemma 2.6]. Thus if G is a finitely generated abelian group, then by assumption H = G, and we may extend w to a valuation v : K(X 1,..., X n ) G { } such that v(x i ) = g i for all 1 i k and v(x j ) = g 1 for all k < j n [15, Theorem 2.2.1]. Therefore, the claim holds when G is a finitely generated abelian group. On the other hand, suppose that G is not a finitely generated abelian group. Then by assumption the rational rank k of G is less than n. Let w be a valuation extension of w to K(X 1,..., X n 1 ) such that w (X j ) = g 1 for all k < j n 1 [15, Theorem 2.2.1]. Then w has the same value group H as w. Now, since G is a torsion-free group and H is a finitely generated torsion-free group but G is not, it must be that G/H is an infinite group. Moreover, since H is generated by a maximal linearly independent subset of G, G/H is a torsion group. Thus by the theorem of Kuhlmann discussed above, there exists an extension v of w to K(X 1,..., X n ) such that v has value group G. Thus we have verified the claim in all cases, and the proof is complete. We use the Kronecker function ring construction in the proof of Theorem 4.4. Let R be an integrally closed domain with quotient field Q, and let T be an indeterminate for Q. If V is a valuation overring of R, then V b denotes the trivial extension of V to Q[T ] (i.e., V b is the valuation domain of Q(T ) determined by the valuation that assigns to a polynomial in Q[T ] the minimum of the values of its coefficients). We define R b = V V b, where V ranges over the valuation overrings of R. Then R b is a Bézout domain [21, Theorem 32.7]. If R is an h-local Prüfer domain, then R b is an h-local Bézout domain; for example, apply Corollary 3.4 of [4]. Moreover, if R is a Prüfer domain, then the mapping γ : I(R) I(R b ) defined by γ(a) = AR b is an order-isomorphism since every finitely generated ideal of R b can be generated by elements of R, and AR b R = A for all finitely generated ideals A of R [21, Theorem 32.7]. Theorem 4.4. If R is an h-local Prüfer overring of a Noetherian domain D, then I(R) is a cardinal sum i I G i, where I D and {G i : i I} is a collection of totally ordered abelian groups such that each G i has finite rational rank. If in addition D has finite Krull dimension n, then each G i has rational rank < n or is a discrete group of rank n. Conversely, let {G i : i I} be a collection of totally ordered abelian groups, each of finite rational rank. Then there exists a Noetherian domain D with D = max{ℵ 0, I } and an h-local Bézout overring R of D such that I(R) is order-isomorphic to the cardinal sum i I G i. Proof. Suppose that R is an h-local Prüfer overring of a Noetherian domain D. Since each ideal of D is finitely generated, it follows there are at most D many ideals of D. Let Σ = {R M : M is a maximal ideal of R}. Then since R is a finite character domain, Σ is a finite character collection of valuation overrings of R. Hence there are at most finitely many members of Σ centered on a given prime ideal of R, and so there are at most D many members of Σ. Let V Σ, and let M be the maximal ideal of V. Then V is a valuation overring of the Noetherian domain D M D. Let n denote the Krull dimension of D M D. Then since D M D is a Noetherian domain, the value group G of

15 Characterizations and constructions of h-local domains 15 V is either a discrete group of rank n or has rational rank < n [1, Theorem 1]. Thus the claim is a consequence of Theorem 2.3(5). To prove the converse, let K be a countable field, and let {X ij : i I, j N} be a collection of indeterminates for K. Define D = K[X ij : i I, j N]. For each i I, let n i = 1 + rank G i, and define p i = (X i1, X i2,..., X ini )D and F i = K(X i j : i i or j N \ {1, 2,..., n i }). Then D pi = F i [X i1,..., X ini ] pi, so by Lemma 4.3, there exists a hidden valuation overring V i of D pi with value group G i. Define S = D \ ( i I p i). As discussed in Example 3.3, D S is a finite character Noetherian domain whose maximal ideals are of the form p i D S, i I. Since {V i : i I} is a collection of hidden valuation overrings of D S, each centered on a distinct maximal ideal, we have by Lemma 4.2 that R := i I V i is an h-local Prüfer domain with {V i : i I} = {R M : M Max R}. Thus by Theorem 2.3(5), I(R) is order-isomorphic to the cardinal sum i I G i. Also, since K is countable, D S = K(X ij : i I, j N) = I. Finally, as discussed above, the ring R b is an h-local Bézout domain, and I(R) is order isomorphic to I(R b ). Thus the claim is proved, since D S [T ] = D S = I and R b is an overring of the Noetherian domain D S [T ]. From the theorem, we deduce precisely which lattice-ordered groups are realizable as the group divisibility of an h-local Bézout overring of a Noetherian domain; compare this to Example 3.4. Corollary 4.5. Let G be a lattice-ordered group. Then G is order-isomorphic to the group of divisibility of an h-local Bézout overring of a Noetherian domain if and only if G is a cardinal sum of totally ordered abelian groups, each having finite rational rank. 5 H-local Prüfer overrings of polynomial rings We next consider the problem of realizing lattice-ordered groups as the group of invertible fractional ideals of an h-local Prüfer overring of a polynomial ring in finitely many variables. Thus, in comparison to the last section, we seek a realization theorem for a very specific sort of Noetherian ring. As we see in this section, the obstacle here is finding infinite finite character collections of maximal ideals. Proposition 3.9 provides some simple examples of domains possessing an infinite finite character set of maximal ideals, but the approach in the proposition does not allow one to control the residue fields of the maximal ideals, something that we need to be able to do in Theorem 5.8. Thus we give in Proposition 5.1 an argument based on a technique of R. Heitmann in [27], where it is shown that if D is a countable PID of characteristic 0, then D[X] has, in our terminology, a finite character set of maximal ideals with specified residue fields. For a related application, see the construction of Dedekind domains given by Goldman in [23]. Proposition 5.1. If D is a countable domain with a finite character set {m i : i N} of distinct maximal ideals, and X 1,..., X n are indeterminates for D, then there exists a finite character set {n i } of distinct maximal ideals of R := D[X 1,..., X n ] such that for each i, m i = n i D and R/n i = D/mi.

16 16 Bruce Olberding Proof. It is enough to prove the theorem for the case n = 1, since an inductive argument then finishes the proof. We first show there exist only finitely many positive integers i such that D/m i < n. Suppose there exists N > 0 such that D/m i < N for infinitely many i. Let d be a nonunit in D. By assumption, D/m i < N for infinitely many i, so d N! + m i = 1 + m i for infinitely many i. Thus d N! 1 m i for infinitely many i, and since {m i } has finite character, d N! = 1, implying d is a unit. This contradiction means that for each n > 0, there exist only finitely many positive integers i such that D/m i < n. In particular, for each n > 0, there exists i > 0 such that D/m j > n for all j > i. Now we show the theorem holds for the ring D[X], where X is an indeterminate, by constructing a finite character set of maximal ideals of D[X] of the form (m i, X d i ), d i D. (Here we use Heitmann s technique from Lemma 2.5 of [27].) Observe that for f(x) D[X], then f(x) (m i, X d i ) if and only if f(d i ) m i. Thus we seek elements d 1, d 2, d 3,... of D such that for all f(x) D[X], f(d i ) m i for only finitely many i. Since D[X] is countable, we may enumerate its elements: f 1 (X), f 2 (X), f 3 (X),.... We define an ascending sequence of positive integers t 1, t 2, t 3,... in the following manner. As we have shown, we may select t 1 such that D/m t > (deg f 1 (X)) + 1 for all t > t 1. More generally, we may select t n > t n 1 such that D/m t > ( n degf j (X)) + 1 j=1 for all t > t n. In this way we obtain a sequence {t n } of positive integers having the property that if t > t n, then the image of g n (X) := f 1 (X) f n (X) in (D/m t )[X] is not identically zero for all elements of the field D/m t. Indeed, the degree of g n (X) is less than D/m t1 1 for all t > t n, yet g(x) has at most deg g(x) roots modulo m t. In particular, for each n and t > t n, there exists d D such that g n (d) m t. Arbitrarily choose d 1,..., d t1. If m > 0, select d tm+1,..., d tm+1 such that g m (d t ) m t for all t such that t m +1 t t m+1. Then for all n > 0, f n (d t ) m t for all t > t n. Hence f n (m t, X d t ) for all t > t n. For each t, set n t = (m t, X d t ). Then it follows that {n t } t>0 is a finite character set of maximal ideals of D[X]. Moreover, for each t, m t = n t D and D[X]/n t = D/mt. This proves the proposition. In Corollary 5.3, we extend the proposition to affine domains over countable domains. This is done via: Lemma 5.2. Let D R be an integral extension of domains, and let {p i } i I be a finite character collection of distinct prime ideals of D. If {q i } i I is a collection of prime ideals of R such that for each i I, q i lies over p i, then {q i } i I has finite character. Proof. Suppose r is a nonzero element of R. Since R is integral over D, there exists a monic polynomial g(x) D[X] such that g(r) = 0. Write g(x) = X n +a n 1 X n a 0. Since R is integral domain, we may assume without loss of generality that a 0 0. Let f(x) = X n 1 + a n 1 X n a 1. Now 0 = g(r) = rf(r) + a 0, and we have rf(r) = a 0. If r is an element of infinitely many of the q i, then so is