Completions of Countable Excellent Domains and Countable Noncatenary Domains
aa r X i v : . [ m a t h . A C ] M a y COMPLETIONS OF COUNTABLE EXCELLENT DOMAINS ANDCOUNTABLE NONCATENARY DOMAINS
S. LOEPP AND TERESA YU
Abstract.
We find necessary and sufficient conditions for a complete local ring contain-ing the rationals to be the completion of a countable excellent local (Noetherian) domain.Furthermore, we find necessary and sufficient conditions for a complete local ring to bethe completion of a countable noncatenary local domain, as well as necessary and sufficientconditions for it to be the completion of a countable noncatenary local unique factorizationdomain. Introduction
While the structure of complete local (Noetherian) rings is well-understood via Cohen’sStructure Theorem, the structure of local rings that are not complete is much more mys-terious. This asymmetry in understanding has spurred results focusing on the relationshipbetween a local ring and its completion. Past results have characterized complete local ringsthat are completions of local rings with specific ring properties, such as being an integraldomain (see [7]), being an excellent domain in the characteristic zero case (see [8]), andbeing a noncatenary domain or a noncatenary unique factorization domain (see [1]). Theauthors of [2] focus on characterizing completions of local domains with certain cardinalities.In particular, they characterize complete local rings that are the completion of a countablelocal domain. In this paper, we first extend this result to countable excellent local domainsin the case where the complete local ring contains the rationals. In particular, in Section 3,we prove the following result.
Theorem 3.10.
Let T be a complete local ring with maximal ideal M and suppose that T contains the rationals. Then T is the completion of a countable excellent local domain ifand only if the following conditions hold:(1) T is equidimensional,(2) T is reduced, and(3) T /M is countable.Using similar techniques that we use to prove this result, we characterize complete localrings that are the completion of a countable noncatenary local domain in Section 4. Inparticular, we prove the following theorem.
Theorem 4.5.
Let T be a complete local ring with maximal ideal M . Then T is thecompletion of a countable noncatenary local domain if and only if the following conditionshold:(1) no integer of T is a zero divisor,(2) M / ∈ Ass( T ),(3) there exists P ∈ Min( T ) such that 1 < dim( T /P ) < dim T , and(4) T /M is countable.
Finally, in Section 4, we prove the following result characterizing completions of countablenoncatenary local unique factorization domains.
Theorem 4.10.
Let T be a complete local ring with maximal ideal M . Then T is thecompletion of a countable noncatenary local unique factorization domain if and only if thethe following conditions hold:(1) no integer of T is a zero divisor,(2) depth T > P ∈ Min( T ) such that 2 < dim( T /P ) < dim T , and(4) T /M is countable.Interestingly, one can use these results to characterize complete local rings containing therationals that are the completion of an uncountable excellent local domain with a countablespectrum, complete local rings that are the completion of an uncountable noncatenary localdomain with a countable spectrum, and complete local rings that are the completion ofan uncountable noncatenary local unique factorization domain with a countable spectrum.These results appear in the forthcoming paper, “Completions of Uncountable Local Ringswith Countable Spectra,” which will be posted on the arXiv soon.The outline of this paper is as follows. In Section 2, we provide background on completelocal rings and excellent rings. In Section 3, we identify necessary and sufficient conditionson a complete local ring containing the rationals to be the completion of a countable ex-cellent local domain. Using the techniques developed in this section, we then characterizecompletions of countable noncatenary local domains as well as completions of countablenoncatenary local unique factorization domains in Section 4.2.
Background
Throughout this paper, all rings are commutative with unity. We say a ring is quasi-local if it has a unique maximal ideal, and we say that it is local if it is both quasi-local andNoetherian. We denote a quasi-local ring R with unique maximal ideal M by ( R, M ), andwe use b R to denote the completion of R with respect to its maximal ideal when R is local.Finally, we use the standard abbreviation UFD to denote a unique factorization domain andthe standard abbreviations Z to denote the integers, Q to denote the rationals, and R todenote the reals.In [7], Lech characterizes completions of integral domains by proving the following result. Theorem 2.1 ([7], Theorem 1) . A complete local ring (
T, M ) is the completion of a localdomain if and only if(1) no integer of T is a zero divisor, and(2) unless equal to (0), M / ∈ Ass( T ).More recently in [2], the authors characterize completions of countable local domains. Weuse this result to construct a base ring, from which we begin our constructions of countableexcellent local domains and countable noncatenary local domains. Theorem 2.2 ([2], Corollary 2.15) . Let (
T, M ) be a complete local ring. Then T is thecompletion of a countable local domain if and only if(1) no integer is a zero divisor of T ,(2) unless equal to (0), M / ∈ Ass( T ), and OMPLETIONS OF COUNTABLE EXCELLENT DOMAINS AND COUNTABLE NONCATENARY DOMAINS3 (3)
T /M is countable.Throughout this paper, we often need to show that a ring has a certain completion. Inorder to do this, we make use of the following proposition, which gives sufficient conditions.
Proposition 2.3 ([6], Proposition 1) . If (
R, R ∩ M ) is a quasi-local subring of a completelocal ring ( T, M ), the map R → T /M is onto, and IT ∩ R = IR for every finitely generatedideal I of R , then R is Noetherian and the natural homomorphism b R → T is an isomorphism.Note that if R is a local ring and b R = T , then T is a faithfully flat extension of R . Itfollows that R and T satisfy the going-down theorem, implying that if P ∈ Spec( T ), thenht( P ∩ R ) ≤ ht( P ). Furthermore, the fact that T is a faithfully flat extension of R impliesthat for any finitely generated ideal I of R , we have that IT ∩ R = IR . This allows us toshow that the converse of Proposition 2.3 also holds. Proposition 2.4.
If (
T, M ) is a complete local ring and (
R, R ∩ M ) is a local subring of T such that b R = T , then the map R → T /M is onto, and IT ∩ R = IR for every finitelygenerated ideal I of R . Proof.
We first show that since b R = T , the map R → T /M is onto. Let t + M ∈ T /M .Then, since b R = T , we have that t = r + r + r + · · · where r i ∈ ( R ∩ M ) i for all i ≥ t + M = r + r + M , and we have that r + r ∈ R maps to t + M under thenatural map from R to T /M .Since b R = T , the ring T is a faithfully flat extension of R , implying that IT ∩ R = IR forany finitely generated ideal I of R . (cid:3) We then have the following corollary, which provides another way to show that a ring hasa certain completion.
Corollary 2.5.
Suppose (
T, M ) is a complete local ring, and (
R, R ∩ M ) is a local subringof T such that b R = T . Let ( A, A ∩ M ) be a quasi-local subring of T such that R ⊆ A , andsuch that, for every finitely generated ideal I of A , IT ∩ A = IA . Then, A is Noetherianand b A = T . Proof.
By Proposition 2.4, we have that the map R → T /M is onto. It follows that the map A → T /M , is onto. By assumption, for every finitely generated ideal I of A , IT ∩ A = IA .Thus, by Proposition 2.3, A is Noetherian with completion T . (cid:3) We now provide some background on excellent rings, starting with a few definitions. Recallthat a ring A is catenary if, for any pair of prime ideals P ( Q of A , all saturated chainsof prime ideals between P and Q have the same length. If a ring is not catenary, then it iscalled noncatenary . A Noetherian ring A is universally catenary if A [ x , . . . , x n ] is catenaryfor every n ≥
0; although this is not the classical definition, it is equivalent to the classicaldefinition (see [11], pg. 118). For any P ∈ Spec( A ), define k ( P ) := A P /P A P . Definition 2.6 ([12], Definition 1.4) . A local ring A is excellent if(a) for all P ∈ Spec( A ), b A ⊗ A L is regular for every finite field extension L of k ( P ), and(b) A is universally catenary.A local ring is formally equidimensional if its completion is equidimensional. A conse-quence of Theorem 31.6 from [11] is that a formally equidimensional Noetherian local ringis universally catenary. Thus, we have the following result from [10]. S. LOEPP AND TERESA YU
Theorem 2.7 ([10], Theorem 2.4) . Let A be a local ring such that its completion, b A , isequidimensional. Then A is a universally catenary.It is noted in [12] that, for Definition 2.6, it is enough to only consider the purely insepa-rable finite field extensions L of k ( P ). Because of this, we have the following modification of[10, Lemma 2.5] that gives sufficient criteria for a subring of a complete local ring satisfyingcertain conditions to be excellent. The proof is almost verbatim from the proof given in [10]. Lemma 2.8.
Let (
T, M ) be a complete local ring that is equidimensional and suppose Q ⊆ T . Given a subring ( A, A ∩ M ) of T with b A = T , A is excellent if, for every P ∈ Spec( A )and for every Q ∈ Spec( T ) with Q ∩ A = P , ( T /P T ) Q is a regular local ring. Proof.
We know that A is a local ring, so we must show now that both conditions of Defi-nition 2.6 hold. By Theorem 2.7, we have that A is universally catenary. Thus, it remainsto consider T ⊗ A L for every purely inseparable finite field extension L of k ( P ) for each P ∈ Spec( A ). Since Z ⊆ A and all nonzero integers are units, we have that Q ⊆ k ( P ),so k ( P ) has characteristic 0. Every finite field extension with characteristic 0 is separable.Since it is sufficient to check only purely inseparable field extensions, this leaves only thetrivial field extension, as this is the only field extension that is both separable and purelyinseparable. Thus, we need only show that T ⊗ A k ( P ) is regular for every P ∈ Spec( A ).Note that, for Q ∈ Spec( T ) with Q ∩ A = P , the ring T ⊗ A k ( P ) localized at Q ⊗ k ( P ) isisomorphic to ( T /P T ) Q . Thus, it suffices to show that ( T /P T ) Q is a regular local ring. (cid:3) The following result from [12] concerns the structure of Sing( R ) for excellent rings. Lemma 2.9 ([12], Corollary 1.6) . If R is excellent, then Sing( R ) is closed in the Zariskitopology, i.e., Sing( R ) = V ( I ) for some ideal I of R .We end this section with two results on the cardinalities of local rings and their quotientrings. Proposition 2.10.
Let (
T, M ) be a local ring. If
T /M is finite, then
T /M is finite. If T /M is infinite, then | T /M | = | T /M | . Proof.
The result follows from Lemma 2.12 in [2]. (cid:3)
Let c denote the cardinality of R . Lemma 2.11 ([4], Lemma 2.2) . Let (
T, M ) be a complete local ring with dim T ≥
1. Let P be a nonmaximal prime ideal of T . Then, | T /P | = | T | ≥ c .3. Completions of Countable Excellent Local Domains
In this section, we give necessary and sufficient conditions for a complete local ring con-taining Q to be the completion of a countable excellent local domain.We first cite the following result from [8], which gives necessary and sufficient conditionson a complete local ring containing the integers to be the completion of an excellent localdomain. Theorem 3.1 ([8], Theorem 9) . Let (
T, M ) be a complete local ring containing the in-tegers. Then T is the completion of a local excellent domain if and only if it is reduced,equidimensional, and no integer of T is a zero divisor. OMPLETIONS OF COUNTABLE EXCELLENT DOMAINS AND COUNTABLE NONCATENARY DOMAINS5
To prove that the conditions in Theorem 3.1 are sufficient, the author of [8] assumes thatthe conditions hold on a complete local ring T and then constructs a subring A of T suchthat A is an excellent domain with b A = T . If the dimension of T is at least two, the ring A for that construction has uncountably many prime ideals, and so A is uncountable. As wewant to construct a countable excellent domain, the construction in [8] does not work. Theproof of our main result in this section, then, is fundamentally different than the proof ofTheorem 9 in [8].The conditions in Theorem 3.1, along with conditions (1), (2), and (3) of Theorem 2.2, are,of course, necessary for a complete local ring containing the rationals to be the completion ofa countable excellent local domain. Note that in the case that dim T ≥ T being reducedis a stronger condition than M / ∈ Ass( T ), since if T is a Noetherian reduced ring, thenMin( T ) = Ass( T ). In addition, if a complete local ring contains the rationals, then everynonzero integer is a unit, and thus not a zero divisor. It is therefore enough to considerthe conditions of being equidimensional, reduced, and having a countable residue field. Thebulk of this section is dedicated to showing that these conditions are also sufficient in thecase that dim T ≥ Definition 3.2.
Let (
T, M ) be a complete local ring and let ( R , R ∩ M ) be a countablelocal subring of T such that R is a domain and b R = T . Let ( R, R ∩ M ) be a quasi-localsubring of T with R ⊆ R . Suppose that(a) R is countable, and(b) R ∩ P = (0) for every P ∈ Ass( T ).Then we call R a built-from- R subring of T , or a BR -subring of T for short.In other words, BR -subrings of T are countable quasi-local rings ( R, R ∩ M ) such that R ⊆ R ⊆ T and R contains no zero divisors of T . Note that the countable union of anascending chain of BR -subrings of T is also a BR -subring of T .The general outline of our construction of a countable excellent local domain is as follows.Let ( T, M ) be a complete local ring such that Q ⊆ T and dim T ≥
1, and further supposethat
T /M is countable, T is reduced, and T is equidimensional. Beginning with a countablelocal domain R whose completion is T , we construct an ascending chain of BR -subrings of T , all with completion T . For each BR -subring in our chain, to get the next BR -subringin the chain, we first adjoin generators of certain prime ideals of T , whose properties aredescribed in Lemma 3.4. We adjoin the generators in a way detailed in Lemma 3.6 so thatthe resulting ring is indeed a BR -subring of T . We then use Lemma 3.8 to show thatone can construct a countable local domain from this BR -subring whose completion is T .This is the next ring in our ascending chain. The union of this ascending chain of rings isexcellent, as shown in Theorem 3.9. The union is also countable and has compleition T , sothis is our desired countable excellent local domain. S. LOEPP AND TERESA YU
When constructing our countable excellent domain, we adjoin elements of T to BR -subrings so that the resulting ring is also a BR -subring of T . The next lemma, adaptedfrom [9, Lemma 11], provides sufficient conditions on the elements that we are able to adjoin. Lemma 3.3.
Suppose (
T, M ) is a complete local ring and ( R , R ∩ M ) is a countablelocal domain with R ⊆ T and b R = T . Let C be the maximal elements of Ass( T ). Let( R, R ∩ M ) be a BR -subring of T , and let x ∈ T such that, for all P ∈ C , we have that x + P is transcendental over R/ ( R ∩ P ) ∼ = R as an element of T /P . Then, R ′ = R [ x ] R [ x ] ∩ M is a BR -subring of T . Proof.
First notice that since R is countable and R ⊆ R , we have that R ′ is countable and R ⊆ R ′ . We now show that if P ′ ∈ Ass( T ), then R ′ ∩ P ′ = (0). It suffices to show that R [ x ] ∩ P ′ = (0). Let P ∈ C such that P ′ ⊆ P . Suppose u ∈ R [ x ] ∩ P ′ . Then u ∈ R [ x ] ∩ P ,and u is of the form u = a n x n + · · · + a x + a with a i ∈ R . Notice that R/ ( R ∩ P ) ∼ = R injects into T /P , so we can view R as a subring of T /P . Thus, since u ∈ P and x + P istranscendental over R as an element of T /P , a i ∈ P for all i . But a i ∈ R ∩ P = (0). Thus, u = 0, so R ′ ∩ P ′ = (0), and R ′ is a BR -subring of T . (cid:3) In the next lemma, which is adapted from [10], we describe a certain set of prime ideals.Our goal is to adjoin generators of such prime ideals to the ring we are constructing, as thiswill allow us to show that our final ring is excellent.
Lemma 3.4.
Let (
T, M ) be a complete local reduced ring and let (
R, R ∩ M ) be a countablelocal domain with R ⊆ T and b R = T . Then, [ P ∈ Spec( R ) { Q ∈ Spec( T ) | Q ∈ min I for I where Sing( T /P T ) = V ( I/P T ) } is a countable set. Furthermore, for any prime ideal Q in this set, Q * p for all p ∈ Ass( T ). Proof.
Since R is countable and Noetherian, Spec( R ) is countable. Thus, it suffices to showthat the set is countable with respect to any fixed P ∈ Spec( R ). Let P ∈ Spec( R ). Since T is a complete local ring, T /P T is excellent. By Lemma 2.9, Sing(
T /P T ) = V ( I/P T ) forsome ideal I of T containing P T . Consider the set of minimal prime ideals Q of I . Since T is Noetherian, this set is finite.We now show that for any Q in this set, we have that Q * p for all p ∈ Ass( T ). Fix P ∈ Spec( R ), and consider C = { Q ∈ Spec( T ) | Q ∈ min I for I where Sing( T /P T ) = V ( I/P T ) } . First suppose ht( P ) ≥ R . Then, since R is a domain and b R = T , thereexists a ∈ P such that a is not a zero divisor of T , i.e., a / ∈ p for all p ∈ Ass( T ). Since P T ⊆ I ⊆ Q , we have that a ∈ Q ; thus, Q * p for all associated prime ideals p of T .Now suppose ht( P ) = 0. Since R is a domain, this means that P = (0). Then, weare considering I such that Sing( T ) = V ( I ). Since T is reduced, it satisfies Serre’s ( R )condition, meaning for all q ∈ Spec( T ) with ht( q ) = 0, we have that T q is regular. Thus,it cannot be that I is contained in a prime ideal of height 0; otherwise, such a prime idealwould be in the singular locus of T , which is impossible. Notice that Min( T ) = Ass( T )since T is a reduced Noetherian ring; this implies that ht( p ) = 0 for all p ∈ Ass( T ). Thus,it must be that I * p for all p ∈ Ass( T ). Since I ⊆ Q , we also have that Q * p for all p ∈ Ass( T ). (cid:3) Next, we adjoin generating sets for the prime ideals described in the previous lemma. Inorder to accomplish this, we make use of the following result, which is a stronger version of
OMPLETIONS OF COUNTABLE EXCELLENT DOMAINS AND COUNTABLE NONCATENARY DOMAINS7 the Prime Avoidance Theorem. In particular, it enables us to avoid cosets of certain primeideals.
Lemma 3.5 ([3], Lemma 2.4) . Let (
T, M ) be a complete local ring such that dim T ≥
1, let C be a finite set of nonmaximal prime ideals of T such that no ideal in C is contained inanother ideal of C , and let D be a subset of T such that | D | < | T | . Let I be an ideal of T (notnecessarily a proper ideal) such that I * P for all P ∈ C . Then I * S { r + P | P ∈ C, r ∈ D } .The following lemma is inspired by a procedure from [1, Lemma 3.6]. It describes amethod for adjoining a generating set for a prime ideal to a BR -subring to obtain a larger BR -subring. Lemma 3.6.
Suppose (
T, M ) is a complete local ring with dim T ≥
1, ( R , R ∩ M ) is acountable local domain with R ⊆ T and b R = T , and ( R, R ∩ M ) is a BR -subring of T .Let Q ∈ Spec( T ) such that Q * P for all P ∈ Ass( T ). Then there exists a BR -subring of T , ( R ′ , R ′ ∩ M ), such that R ⊆ R ′ and R ′ contains a generating set for Q . Proof.
Let ( x , . . . , x n ) be a generating set for Q . We inductively define a chain of BR -subrings of T , R = R ⊆ R ⊆ · · · ⊆ R n +1 such that R n +1 contains a generating set for Q .To construct R i +1 from R i , we show that there exists an element ˜ x i of T so that R i +1 := R i [˜ x i ] ( R i [˜ x i ] ∩ M ) is a BR -subring of T and such that we can replace x i in the generating setof Q with ˜ x i . By assumption and the Prime Avoidance Theorem, Q * S P ∈ Ass( T ) P ; thus,there exists y ∈ Q such that y / ∈ P for all P ∈ Ass( T ).We first construct R by considering R = R . We find ˜ x = x + α y with α ∈ M so that˜ x + P is transcendental over R / ( R ∩ P ) = R/ ( R ∩ P ) ∼ = R as an element of T /P for all P maximal in Ass( T ). To do this, first fix P , a maximal element of Ass( T ), and consider x + ty + P for some t ∈ T . Notice that since R / ( R ∩ P ) is countable, its algebraic closurein T /P is also countable. In addition, each choice of t + P gives a different x + ty + P , since y / ∈ P . Thus, for at most countably many choices of t + P , the element x + ty + P of T /P is algebraic over R / ( R ∩ P ).Let D ( P ) be a full set of coset representatives of elements t + P of T /P that make x + ty + P algebraic over R / ( R ∩ P ). Let C be the maximal elements of Ass( T ) and let D = S P ∈ C D ( P ) . Then, D is countable since there are finitely many associated prime ideals P of T and D ( P ) is countable for each P . Since dim T ≥ T is uncountable by Lemma 2.11. Itfollows that | D | < | T | . In addition, T is the completion of a local domain and dim T ≥ M / ∈ Ass( T ) by Theorem 2.1; then, M * P for all P ∈ C . Applying Lemma 3.5 with I = M , C the maximal elements of Ass( T ), and D = D , we have that M * [ { r + P | r ∈ D , P ∈ C } , so there exists α ∈ M such that x + α y + P is transcendental over R / ( R ∩ P ) forevery P ∈ C . Let ˜ x := x + α y . By Lemma 3.3, we have that R := R [˜ x ] ( R [˜ x ] ∩ M ) is a BR -subring of T .We now show that Q = (˜ x , x , . . . , x n ). Write y ∈ Q as y = β , x + · · · + β ,n x n , for some β ,i ∈ T . Notice that ˜ x ∈ Q , since x , y ∈ Q ; then,˜ x = x + α y = (1 + α β , ) x + α β , x + · · · + α β ,n x n . S. LOEPP AND TERESA YU
Rearranging, we have that x = (1 + α β , ) − (˜ x − α β , x − · · · − α β ,n x n ) ∈ (˜ x , x , . . . , x n ) , where (1 + α β , ) is a unit because α ∈ M . Thus, we are able to replace x with ˜ x in ourgenerating set for Q . Notice that this argument works even if n = 1.To construct R , let D = S P ∈ C D ( P ) , where D ( P ) is a full set of coset representatives ofthe elements t + P of T /P that make x + ty + P ∈ T /P algebraic over R / ( R ∩ P ). Note that D is countable. Again using Lemma 3.5, there exists α ∈ M such that x + α y + P ∈ T /P is transcendental over R / ( R ∩ P ) for every P ∈ C . Let ˜ x := x + α y . Then, R := R [˜ x ] R [˜ x ] ∩ M is a BR -subring of T by Lemma 3.3. We have that Q = (˜ x , ˜ x , x , . . . , x n )by a similar argument as above by writing y = β , ˜ x + β , x + · · · + β ,n x n to show that x ∈ (˜ x , ˜ x , x , . . . , x n ).Repeat the above process for each i = 4 , . . . , n + 1 to obtain a chain of BR -subrings of T , R ⊆ · · · ⊆ R n +1 and have Q = (˜ x , ˜ x , . . . , ˜ x n ). By construction, each ˜ x i ∈ R i +1 , so R n +1 contains a generating set for Q . Thus, R ′ = R n +1 is our desired BR -subring of T . (cid:3) After adjoining generating sets for prime ideals via Lemma 3.6, we obtain a BR -subringof T whose completion is not necessarily T . If R is a BR -subring of T , then our goal is tobuild a BR -subring R ′ from R such that R ⊆ R ′ , and b R ′ = T .The following lemma, which is a modification of [3, Lemma 2.6], is used to accomplish thisgoal. In particular, we use Lemma 3.7 to show in Lemma 3.8 that one can close up ideals ofa BR -subring of T . We then use Corollary 2.5 to show that the resulting ring is Noetherianand its completion is T . Lemma 3.7.
Suppose (
T, M ) is a complete local ring with dim T ≥
1, ( R , R ∩ M ) is acountable local domain with R ⊆ T and b R = T , and ( R, R ∩ M ) is a BR -subring of T .Let I be a finitely generated ideal of R and let c ∈ IT ∩ R . Then there exists a BR -subring( R ′ , R ′ ∩ M ) of T such that R ⊆ R ′ and c ∈ IR ′ . Proof.
We induct on the number of generators of I . For the base case, suppose I = aR . If a = 0, then c = 0, so R is the desired BR -subring. Now consider the case where a = 0.Then, c = au for some u ∈ T . We show that R ′ := R [ u ] ( R [ u ] ∩ M ) is the desired BR -subring.First, note that R [ u ] ( R [ u ] ∩ M ) is countable, since R is countable. Now suppose f ∈ R [ u ] ∩ P for P ∈ Ass( T ). Then, f = r n u n + · · · + r u + r ∈ P for r i ∈ R . Multiplying through by a n , we get a n f = r n ( au ) n + · · · + r a n − ( au ) + r a n = r n c n + · · · + r a n − c + r a n . Notice that r i , c, a ∈ R for all i so a n f is an element of P ∩ R = (0). Since a ∈ R with a = 0, itcannot be that a is a zero divisor in T . Thus, it must be that f = 0, showing that R [ u ] ( R [ u ] ∩ M ) is indeed a BR -subring of T . Since c = au , we have that c ∈ IR [ u ] ( R [ u ] ∩ M ) = IR ′ .Now suppose I is generated by m > R generated by m − I = ( y , . . . , y m ) R where y i = 0 for all i = 1 , , . . . , m . Then, c = y t + · · · + y m t m for some t i ∈ T . Note that c = y t + ( y y t − y y t ) + y t + · · · + y m t m = y ( t + y t ) + y ( t − y t ) + y t + · · · + y m t m for any t ∈ T . Let x = t + y t and x = t − y t , where we will choose the element t later.Now, let P be a a maximal element of Ass( T ). If ( t + y t ) + P = ( t + y t ′ ) + P for some t, t ′ ∈ T , then it must be the case that y ( t − t ′ ) ∈ P . But y ∈ R with y = 0 and R ∩ P = (0),so we must have that t − t ′ ∈ P . Thus, t + P = t ′ + P . The contrapositive of this result OMPLETIONS OF COUNTABLE EXCELLENT DOMAINS AND COUNTABLE NONCATENARY DOMAINS9 indicates that if t + P = t ′ + P , then ( t + y t )+ P = ( t + y t ′ )+ P . Let D ( P ) be a full set of cosetrepresentatives of the cosets t + P that make x + P algebraic over R/ ( R ∩ P ) in T /P . Since R is countable, the algebraic closure of R/ ( R ∩ P ) in T /P is countable, so D ( P ) is countable. Let C be the maximal elements of Ass( T ) and let D = S P ∈ C D ( P ) . Note that D is also countable,and, by Lemma 2.11, T is uncountable. Therefore, | D | < | T | . In addition, since T is thecompletion of a local domain and dim T ≥
1, we have by Theorem 2.1 that
M / ∈ Ass( T ),and so M * P for all P ∈ C . We now use Lemma 3.5 with C the maximal elements ofAss( T ) and I = M to find an element t ∈ M ⊆ T such that x + P ∈ T /P is transcendentalover R/ ( R ∩ P ) for every P ∈ C . By Lemma 3.3, we have that R ′′ := R [ x ] ( R [ x ] ∩ M ) is a BR -subring of T . Now let J = ( y , . . . , y m ) R ′′ and c ∗ = c − y x . Notice that c, y x ∈ R ′′ ,and c ∗ = ( y ( t + y t ) + y ( t − y t ) + y t + · · · + y m t m ) − y x = y ( t − y t ) + y t + · · · + y m t m , because of how we have defined x . Thus, we have that c ∗ ∈ J T ∩ R ′′ .We can now use our induction assumption to draw the conclusion that there exists a BR -subring ( R ′ , R ′ ∩ M ) of T such that R ′′ ⊆ R ′ ⊆ T and c ∗ ∈ J R ′ . Thus, c ∗ = y r + · · · + y m r m for some r i ∈ R ′ . It follows that c = y x + y r + · · · + y m r m ∈ IR ′ , and thus R ′ is thedesired BR -subring. (cid:3) The following lemma ensures that, given a BR -subring R of T , one can construct a BR -subring from R whose completion is T . Lemma 3.8.
Suppose (
T, M ) is a complete local ring with dim T ≥ R , R ∩ M ) isa countable local domain with R ⊆ T and b R = T . Let ( R, R ∩ M ) be a BR -subring of T . Then there exists a BR -subring of T , ( R ′ , R ′ ∩ M ), such that R ⊆ R ′ ⊆ T , and, if I isa finitely generated ideal of R ′ , then IT ∩ R ′ = IR ′ . Thus, R ′ is Noetherian and b R ′ = T . Proof.
Consider the setΩ = { ( I, c ) | I is a finitely generated ideal of R , c ∈ IT ∩ R } . Since R is countable, so is Ω. Enumerate Ω and let 1 denote its first element. Now, werecursively define an ascending chain of BR -subrings, beginning with R = R . Given the k th element ( I, c ) with the BR -subring R k defined, we will construct R k +1 . Let I = ( a , . . . , a ℓ ) R for a i ∈ R . Notice that I ′ = ( a , . . . , a ℓ ) R k is a finitely generated ideal of R k , and if c ∈ IT ∩ R , then c ∈ IT ∩ R ⊆ I ′ T ∩ R k , since R = R ⊆ R k . Now let R k +1 be the BR -subring obtained from Lemma 3.7 so that R k ⊆ R k +1 ⊆ T and c ∈ I ′ R k +1 . Thendefine S = S ∞ i =1 R i . Since this is a countable union of BR -subrings, we have that S is a BR -subring of T . By our construction, if I is a finitely generated ideal of R and c ∈ IT ∩ R ,then for some i ∈ Z + , we have that c ∈ IR i ⊆ IS ; thus, IT ∩ R ⊆ IS .We repeat this process using S in place of R to obtain a BR -subring S such that IT ∩ S ⊆ IS for every finitely generated ideal I of S . Continuing this process results inan ascending chain of BR -subrings R ⊆ S ⊆ S ⊆ · · · such that IT ∩ S i ⊆ IS i +1 for everyfinitely generated ideal I of S i .Let R ′ := S ∞ i =1 S i . Note that R ′ is a BR -subring of T . We show that, for every finitelygenerated ideal I of R ′ , IT ∩ R ′ = I . Consider some finitely generated ideal I of R ′ . Clearly, IR ′ ⊆ IT ∩ R ′ , so suppose that c ∈ IT ∩ R ′ . Let I = ( a , . . . , a m ) R ′ . For some k , we havethat a i ∈ S k for all i and c ∈ S k . Then, c ∈ ( a , . . . , a m ) T ∩ S k ⊆ ( a , . . . , a m ) S k +1 ⊆ IR ′ . Thus we have IT ∩ R ′ = IR ′ .Since R ⊆ R ⊆ R ′ and for every finitely generated ideal I of R ′ , we have that IT ∩ R ′ = IR ′ , by Corollary 2.5, we have that R ′ is Noetherian and b R ′ = T . (cid:3) We are now able to give sufficient conditions for a complete local ring containing therationals and of dimension at least one to be the completion of a countable excellent localdomain. The proof of Theorem 3.9 is adapted from the proof of [10, Theorem 3.5].
Theorem 3.9.
Let (
T, M ) be a complete local ring containing the rationals with dim T ≥ T is equidimensional,(2) T is reduced, and(3) T /M is countable.Then there exists a countable excellent local domain (
S, S ∩ M ) such that S ⊆ T and b S = T . Proof.
Since T contains the rationals, all nonzero integers are units, and so no integer of T is a zero divisor. Because T is reduced and dim T ≥
1, we have that
M / ∈ Ass( T ); thus, T is the completion of a countable local domain, ( R , R ∩ M ), by Theorem 2.2.Let S = R . Notice that S is a local countable domain and has completion T . We definean ascending chain of rings recursively. For each S i , we ensure that it contains R and thatit satisfies the criteria of Lemma 3.4, i.e., that ( S i , S i ∩ M ) is a countable local domain with S i ⊆ T and b S i = T . Suppose that ( S i , S i ∩ M ) is a countable local domain satisfying suchproperties. We construct S i +1 to satisfy these as well.By Lemma 3.4, the set [ P ∈ Spec( S i ) { Q | Q ∈ min I for I where Sing( T /P T ) = V ( I/P T ) } is countable, and for any Q in the set, we have that Q * p for all p ∈ Ass( T ).Enumerate this set, ( Q j ) j ∈ Z + . Since T is Noetherian, each Q j is finitely generated. Werecursively define an ascending chain of BR -subrings of T , S ∗ ⊆ S ∗ ⊆ · · · . Let S ∗ = S i .For each successive S ∗ k , we ensure that it contains a generating set of Q , . . . , Q k .First consider S ∗ = S i and Q . Since S ∗ is a domain and c S ∗ = T , we have that S ∗ ∩ p = (0)for all p ∈ Ass( T ). It follows that S ∗ is a BR -subring of T . By Lemma 3.6, there existsa BR -subring S ∗ of T such that S ∗ ⊆ S ∗ and S ∗ contains a generating set for Q . Nowsuppose that S ∗ k has been defined, and it is a BR -subring of T that contains generating setsfor Q , . . . , Q k . Consider Q k +1 ; we can again apply Lemma 3.6 to find a BR -subring of T , S ∗ k +1 , such that S ∗ k ⊆ S ∗ k +1 and such that S ∗ k +1 contains a generating set for Q k +1 .Let S ′ i := S ∞ k =0 S ∗ k . Notice that this is a countable union of ascending BR -subrings of T ,so we have that S ′ i is a BR -subring of T . Let ( S i +1 , S i +1 ∩ M ) be the BR -subring of T obtained by applying Lemma 3.8 to S ′ i , so ( S i +1 , S i +1 ∩ M ) is countable local domain suchthat S ′ i ⊆ S i +1 ⊆ T and b S i +1 = T . Thus, S i +1 satisfies the conditions of Lemma 3.4 asdesired. This also implies by Proposition 2.4 that IT ∩ S i +1 = I for any finitely generatedideal I of S i +1 . Note that, for any Q ∈ [ P ∈ Spec( S i ) { Q | Q ∈ min( I ) for I where Sing( T /P T ) = V ( I/P T ) } , we have that S i +1 contains a generating set for Q . OMPLETIONS OF COUNTABLE EXCELLENT DOMAINS AND COUNTABLE NONCATENARY DOMAINS11
Define S := S ∞ i =0 S i . We show that ( S, S ∩ M ) is a countable excellent local domain suchthat R ⊆ S ⊆ T and b S = T . First, since each S i a BR -subring of T , we have that S is a BR -subring of T as well so ( S, S ∩ M ) is a countable quasi-local domain with S ⊆ T . Notethat, for every i , since b S i = T , if I is a finitely generated ideal of S i , then IT ∩ S i = IS i . Weshow that this holds for S as well.Suppose that I is a finitely generated ideal of S . We clearly have that IS ⊆ IT ∩ S , sowe will show that IT ∩ S ⊆ IS . Since I is finitely generated, I = ( a , . . . , a m ) S for a i ∈ S .Let c ∈ IT ∩ S . Choose ℓ such that a i , c ∈ S ℓ . Then, c ∈ ( a , . . . , a m ) T ∩ S ℓ = ( a , . . . , a m ) S ℓ ⊆ IS.
Thus, IT ∩ S ⊆ IS . Since R ⊆ S , by Corollary 2.5, S is Noetherian and has completion T .Finally, since T is equidimensional, we can show that S is excellent using Lemma 2.8, i.e.,by showing that, for P ∈ Spec( S ) and Q ∈ Spec( T ) such that Q ∩ S = P , ( T /P T ) Q is aregular local ring. Let P ∈ Spec( S ) and Q ∈ Spec( T ) such that Q ∩ S = P . Suppose forcontradiction that Q/P T ∈ Sing(
T /P T ) = V ( I/P T ). Then, Q ⊇ p ⊇ I for some minimalprime ideal p ∈ Spec( T ) of I . Since p ⊇ I ⊇ P T , p /P T ∈ Sing(
T /P T ). Furthermore, P = P T ∩ S ⊆ p ∩ S ⊆ Q ∩ S = P. Thus, p ∩ S = P .Note that P is finitely generated, so let P = ( p , . . . , p m ) S . Choose i so that p j ∈ S i forall j = 1 , . . . , m . Then define P ′ := P ∩ S i , and note that P ′ ∈ Spec( S i ). We first show that P ′ T = P T . Observe that P ′ = P ∩ S i = ( P T ∩ S ) ∩ S i = P T ∩ S i = ( p , . . . , p m ) T ∩ S i = ( p , . . . , p m ) S i , so then P ′ T = (( p , . . . , p m ) S i ) T = ( p , . . . , p m ) T = P T.
Thus,
T /P T = T /P ′ T , so since p /P T ∈ Sing(
T /P T ), we have p /P ′ T ∈ Sing(
T /P ′ T ). Next,we show that p = P T . Since p ∩ S = P , we have p ∩ S i = P ∩ S i = P ′ . So then, p ∈ [ P ′ ∈ Spec( S i ) { q | q ∈ min I for I where V ( I/P ′ T ) = Sing( T /P ′ T ) } . This means that, for some generating set for p , { q , . . . , q ℓ } ⊆ T , we have { q , . . . , q ℓ } ⊆ S i +1 .Thus, p ∩ S i +1 = ( q , . . . , q ℓ ) T ∩ S i +1 = ( q , . . . , q ℓ ) S i +1 . Since S ⊇ S i +1 , p ∩ S ⊇ p ∩ S i +1 = ( q , . . . , q ℓ ) S i +1 . Recall that p ∩ S = P , so P T = ( p ∩ S ) T ⊇ (( q , . . . , q ℓ ) S i +1 ) T = ( q , . . . , q ℓ ) T = p . We already know that p ⊇ P T ; thus p = P T . Then, (
T /P T ) p = ( T /P T ) P T is a field, andtherefore a regular local ring. However, p /P T ∈ Sing(
T /P T ), so (
T /P T ) p is not a regularlocal ring, which is a contradiction. It follows that Q/P T / ∈ Sing(
T /P T ), so (
T /P T ) Q is aregular local ring. Thus, by Lemma 2.8, we have that S is excellent. (cid:3) We are now ready to prove our main result.
Theorem 3.10.
Let (
T, M ) be a complete local ring with Q ⊆ T . Then T is the completionof a countable excellent local domain if and only if the following conditions hold:(1) T is equidimensional,(2) T is reduced, and (3) T /M is countable.
Proof.
First suppose (
T, M ) is a complete local ring with dim T = 0 and Q ⊆ T . Fur-ther suppose that T is reduced, equidimensional, and T /M is countable. Notice that T P ∈ Spec( T ) P = p (0) = (0), since the nilradical of a reduced ring is the (0) ideal. Butthere is only one prime ideal in a local ring of dimension 0, namely the maximal ideal. Thus, M = (0) and T is a field; then, T /M = T / (0) = T is countable. In addition, T itself isexcellent, as it is a complete local ring. Thus, we have that T is a countable excellent localdomain, with completion itself.Now suppose ( T, M ) is a complete local ring with dim T ≥ Q ⊆ T . In addition,suppose that T is reduced, equidimensional, and T /M is countable. By Theorem 3.9, thereexists a countable excellent local domain S such that b S = T .Suppose now that T is the completion of a countable excellent local domain A and dim T ≥
0. By Theorem 2.2,
T /M is countable. Notice that A is universally catenary, since it isexcellent. By [11, Theorem 31.7], this means that A is formally catenary, i.e., A/P is formallyequidimensional for every P ∈ Spec( A ). Since A is a domain, we can consider A/ (0) ∼ = A . Note that A is formally equidimensional, so we have that b A = T is equidimensional.Finally, since A is a domain, it is reduced. By [11, Theorem 32.2], its completion, T , is alsoreduced. (cid:3) As a corollary, we characterize complete local UFDs containing the rationals that are thecompletion of a countable excellent local UFD.
Corollary 3.11.
Let (
T, M ) be a complete local UFD with Q ⊆ T . Then T is the completionof a countable excellent local UFD if and only if T /M is countable.
Proof.
Suppose (
T, M ) is a complete local UFD with Q ⊆ T and suppose that T /M iscountable. Since T is a domain, it is reduced and equidimensional. Then, by Theorem 3.10, T is the completion of a countable excellent local domain, ( A, A ∩ M ). Since T is a UFD, A must be as well. Thus, T is the completion of a countable excellent local UFD.Now suppose ( T, M ) is the completion of a countable excellent local UFD (
A, A ∩ M ).Then, T is the completion of a countable local domain, so by Theorem 2.2, T /M is countable. (cid:3)
Example 3.12.
For some complete local rings, it is not difficult to find an example of acountable local excellent domain whose completion is the given complete local ring. Forexample, the complete local ring Q [[ x , . . . , x n ]] is the completion of Q [ x , . . . , x n ] ( x ...,x n ) .For other complete local rings, however, Theorem 3.10 is more useful. For example, perhapssurprisingly, we have by Theorem 3.10 that the complete local ring Q [[ x, y, z ]] / ( xy ) is thecompletion of a countable excellent local domain.4. Completions of Countable Noncatenary Local Domains and UFDs
In this section, we give necessary and sufficient conditions for a complete local ring tobe the completion of a countable noncatenary local domain, and neceesary and sufficientconditions for a complete local ring to be the completion of a countable noncatenary localUFD. To do this, we use some of the techniques developed in the previous section, especiallythose concerning BR -subrings.We first focus on characterizing completions of countable noncatenary local domains. Thefollowing result from [1] characterizes completions of noncatenary local domains. OMPLETIONS OF COUNTABLE EXCELLENT DOMAINS AND COUNTABLE NONCATENARY DOMAINS13
Theorem 4.1 ([1], Theorem 2.10) . Let (
T, M ) be a complete local ring. Then T is thecompletion of a noncatenary local domain if and only if the following conditions hold:(1) no integer of T is a zero divisor,(2) M / ∈ Ass( T ), and(3) there exists P ∈ Min( T ) such that 1 < dim( T /P ) < dim T .Using this result, along with Theorem 2.2, we show that a complete local ring is thecompletion of a countable noncatenary local domain if and only if it is the completion ofa noncatenary local domain and the completion of a countable local domain. In order toachieve this result, we first identify sufficient conditions needed by constructing a countablenoncatenary local domain. We first provide some lemmas that will help us show that therings we construct are noncatenary. Lemma 4.2 ([1], Lemma 2.8) . Let (
T, M ) be a local ring with
M / ∈ Ass( T ) and let P ∈ Min( T ) with dim( T /P ) = n . Then there exists a saturated chain of prime ideals of T , P ( Q ( · · · ( Q n − ( M , such that, for each i = 1 , . . . , n −
1, we have that Q i / ∈ Ass( T )and P is the only minimal prime ideal contained in Q i .We now use this result to prove the following lemma, which shows that under certainconditions, there exists a prime ideal with desirable properties. Lemma 4.3.
Let (
T, M ) be a catenary local ring with
M / ∈ Ass( T ). If there exists P ∈ Min( T ) such that 1 < dim( T /P ) < dim T , then there exists Q ∈ Spec( T ) such thatdim( T /Q ) = 1, ht( Q ) + dim( T /Q ) < dim T , and Q * p for all p ∈ Ass( T ). Proof.
Suppose P ∈ Min( T ) and 1 < dim( T /P ) = n < dim T . By Lemma 4.2, thereexists a saturated chain of prime ideals of T , P ( Q ( · · · ( Q n − ( M , such that,for each i = 1 , . . . , n −
1, we have that Q i / ∈ Ass( T ) and P is the only minimal primeideal of T contained in Q i . Since T is catenary and this chain is saturated, we have thatht( Q i ) + dim( T /Q i ) = n < dim( T ) for all i = 1 , . . . , n −
1. In particular, if we consider Q n − ,we have that dim( T /Q n − ) = 1 and ht( Q n − ) + dim( T /Q n − ) < dim T . In addition, we havethat M, Q n − / ∈ Ass( T ), so Q n − * p for all p ∈ Ass( T ) as well. (cid:3) We now identify sufficient conditions for a complete local ring to be the completion of acountable noncatenary local domain.
Proposition 4.4.
Let (
T, M ) be a complete local ring satisfying the following properties:(1) no integer of T is a zero divisor,(2) M / ∈ Ass( T ),(3) there exists P ∈ Min( T ) such that 1 < dim( T /P ) < dim T , and(4) T /M is countable.Then T is the completion of a countable noncatenary local domain. Proof.
Conditions (1), (2), and (4) imply that T is the completion of a countable localdomain, ( R , R ∩ M ) , by Theorem 2.2. Notice that R is itself a BR -subring of T , and thatdim T ≥ T is catenary since it is a complete local ring, and T satisfies conditions (2) and (3); thus, there exists Q ∈ Spec( T ) such that dim( T /Q ) = 1,ht( Q ) + dim( T /Q ) < dim T , and Q * p for all p ∈ Ass( T ) by Lemma 4.3. Then, byLemma 3.6, there exists a BR -subring of T , ( R, R ∩ M ), such that R contains a generatingset for Q . Since dim T ≥
1, by Lemma 3.8, there exists a BR -subring of T , ( A, A ∩ M ), such that R ⊆ A ⊆ T , A is Noetherian, and b A = T . Since A is a BR -subring of T , we havethat A is countable and a domain. It remains to be shown that A is noncatenary.Recall that Q ∈ Spec( T ) such that dim( T /Q ) = 1, ht( Q ) + dim( T /Q ) < dim T , and A contains a generating set for Q . Since A contains a generating set for Q , we have that( Q ∩ A ) T = Q . We show that dim( A/ ( Q ∩ A )) = 1 in order to show that A is noncatenary.Suppose P ′ is a prime ideal of A such that Q ∩ A ( P ′ . Then, ( Q ∩ A ) T = Q ( P ′ T ; otherwise, Q = P ′ T would imply that Q ∩ A = P ′ T ∩ A = P ′ . Since dim( T /Q ) = 1, we have thatdim(
T /P ′ T ) = 0, which in turn implies that dim( A/P ′ ) = 0, since [ A/P ′ = T /P ′ T . Then, P ′ is actually the maximal ideal of A , i.e., P ′ = A ∩ M , and dim( A/ ( Q ∩ A )) = 1. A localring and its completion satisfy the going-down theorem, so we have that ht( Q ∩ A ) ≤ ht( Q );then, ht( Q ∩ A ) + 1 ≤ ht( Q ) + 1 < dim T = dim( A ) . Thus, we have shown that A is noncatenary. (cid:3) We now characterize completions of countable noncatenary local domains.
Theorem 4.5.
Let (
T, M ) be a complete local ring. Then T is the completion of a countablenoncatenary local domain if and only if the following conditions hold:(1) no integer of T is a zero divisor,(2) M / ∈ Ass( T ),(3) there exists P ∈ Min( T ) such that 1 < dim( T /P ) < dim T , and(4) T /M is countable.
Proof. If T satisfies conditions (1), (2), (3), and (4), then by Proposition 4.4, T is thecompletion of a countable noncatenary local domain. If T is the completion of a countablenoncatenary local domain, then by Theorem 4.1, T must satisfy conditions (1), (2), and (3).By Theorem 2.2, T must satisfy condition (4) as well. (cid:3) Next, we characterize completions of countable noncatenary local UFDs. The following isa result from [1] that characterizes completions of noncatenary local UFDs.
Theorem 4.6 ([1], Theorem 3.7) . Let (
T, M ) be a complete local ring. Then T is thecompletion of a noncatenary local UFD if and only if the following conditions hold:(1) no integer of T is a zero divisor,(2) depth ( T ) >
1, and(3) there exists P ∈ Min( T ) such that 2 < dim( T /P ) < dim T .We first identify sufficient conditions for a complete local ring to be the completion of acountable noncatenary local UFD. We do this by studying the construction of noncatenarylocal UFDs given in [1]. The following result in [1] identifies sufficient conditions on acomplete local ring for it to be the completion of a noncatenary local UFD. Lemma 4.7 ([1], Lemma 3.6) . Let (
T, M ) be a complete local ring such that no integerof T is a zero divisor. Suppose depth T > P ∈ Min( T ) such that2 < dim( T /P ) < dim T . Then T is the completion of a noncatenary local UFD.In the proof of this characterization, the authors of [1] use the proof of the following resultof Heitmann’s from [5]. OMPLETIONS OF COUNTABLE EXCELLENT DOMAINS AND COUNTABLE NONCATENARY DOMAINS15
Theorem 4.8 ([5], Theorem 8) . Let (
T, M ) be a complete local ring such that no integeris a zero divisor in T and depth T ≥
2. Then there exists a local UFD A such that b A ∼ = T and | A | = sup( ℵ , | T /M | ). If p ∈ M where p is a nonzero prime integer, then pA is a primeideal.In particular, the proof of Lemma 4.7 is as follows. Given a complete local ring ( T, M )satisfying the conditions of the lemma’s hypothesis, adjoin generators of a particular primeideal of T to the prime subring of T . Then use this ring as the base ring in the proof ofTheorem 4.8, in which elements from a set of cardinality | T /M | are adjoined to the basering, so that the resulting ring A has cardinality sup( ℵ , | T /M | ). The ring A can then beshown to be a noncatenary local UFD with completion T .Using this outline, we identify sufficient conditions on a complete local ring to be thecompletion of a countable noncatenary local UFD. Proposition 4.9.
Let (
T, M ) be a complete local ring such that the following conditionsare satisfied:(1) no integer of T is a zero divisor,(2) depth T > P ∈ Min( T ) such that 2 < dim( T /P ) < dim T , and(4) T /M is countable.Then T is the completion of a countable noncatenary local UFD. Proof.
We show that given the added assumption that
T /M is countable, the ring constructedin the proof of Lemma 4.7 is countable. The ring in the proof of this lemma is initialized bytaking the prime subring of T and adjoining generators of a prime ideal of T . The resultingring, which we call R , is countable, since the prime subring of T is countable, and adjoiningfinitely many elements to the prime subring yields a countable ring. Then, to prove Lemma4.7, the authors of [1] use R as the base ring in Heitmann’s construction in the proof ofTheorem 4.8. The resulting ring A is countable, since | T /M | = | T /M | by Proposition 2.10so A has cardinality sup( ℵ , | T /M | ) = ℵ . Finally, it is shown, in the proof of [1, Lemma3.6], that A is a noncatenary local UFD with completion T . (cid:3) We are now able to characterize completions of countable noncatenary local UFDs.
Theorem 4.10.
Let (
T, M ) be a complete local ring. Then T is the completion of a countablenoncatenary local UFD if and only if the following conditions hold:(1) no integer of T is a zero divisor,(2) depth T > P ∈ Min( T ) such that 2 < dim( T /P ) < dim T , and(4) T /M is countable.
Proof. If T satisfies conditions (1), (2), (3), and (4), then by Proposition 4.9, T is thecompletion of a countable noncatenary local UFD. If T is the completion of a countablenoncatenary local UFD, then by Theorem 4.6, T must satisfy conditions (1), (2) and (3).By Theorem 2.2, T must satisfy condition (4) as well. (cid:3) Example 4.11.
By Theorem 4.5, an example of a complete local ring that is the completionof a countable noncatenary local domain is Q [[ x, y, z, w ]] / ( x ) ∩ ( y, z ). By Theorem 4.10,the complete local ring Q [ x, y , y , z , z ]] / ( x ) ∩ ( y , y ) is the completion of a countablenoncatenary local UFD. Acknowledgments
We thank the Clare Boothe Luce Scholarship Program for supporting the research of thesecond author.
References [1] Chloe I. Avery, Caitlyn Booms, Timothy M. Kostolansky, S. Loepp, and Alex Semendinger,
Charac-terization of completions of noncatenary local domains and noncatenary local UFDs , J. Algebra (2019), 1–18, DOI 10.1016/j.jalgebra.2018.12.016. MR3902351 ↑
1, 7, 12, 13, 14, 15[2] Erica Barrett, Emil Graf, S. Loepp, Kimball Strong, and Sharon Zhang,
Cardinalities of Prime Spectraof Precompletions (2019), available at arXiv:1911.06648 . ↑
1, 2, 4, 5[3] P. Charters and S. Loepp,
Semilocal generic formal fibers , J. Algebra (2004), no. 1, 370–382, DOI10.1016/j.jalgebra.2004.01.011. MR2068083 ↑
7, 8[4] A. Dundon, D. Jensen, S. Loepp, J. Provine, and J. Rodu,
Controlling formal fibers of principalprime ideals , Rocky Mountain J. Math. (2007), no. 6, 1871–1891, DOI 10.1216/rmjm/1199649827.MR2382631 ↑ Characterization of completions of unique factorization domains , Trans. Amer.Math. Soc. (1993), no. 1, 379–387, DOI 10.2307/2154327. MR1102888 ↑
14, 15[6] ,
Completions of local rings with an isolated singularity , J. Algebra (1994), no. 2, 538–567,DOI 10.1006/jabr.1994.1031. MR1262718 ↑ A method for constructing bad Noetherian local rings , Algebra, algebraic topology andtheir interactions (Stockholm, 1983), 1986, pp. 241–247. ↑
1, 2[8] S. Loepp,
Characterization of completions of excellent domains of characteristic zero , J. Algebra (2003), no. 1, 221–228, DOI 10.1016/S0021-8693(03)00239-4. MR1984908 ↑
1, 4, 5[9] ,
Constructing local generic formal fibers , J. Algebra (1997), no. 1, 16–38, DOI10.1006/jabr.1997.6768. MR1425557 ↑ Uncountable n -dimensional excellent regular local rings with countablespectra , Trans. Amer. Math. Soc. (2020), no. 1, 479–490, DOI 10.1090/tran/7921. MR4042882 ↑ Commutative ring theory , 2nd ed., Cambridge Studies in Advanced Mathemat-ics, vol. 8, Cambridge University Press, Cambridge, 1989. Translated from the Japanese by M. Reid.MR1011461 ↑
3, 12[12] Christel Rotthaus,
Excellent rings, Henselian rings, and the approximation property , Rocky MountainJ. Math. (1997), no. 1, 317–334, DOI 10.1216/rmjm/1181071964. MR1453106 ↑↑