Pacific Journal ofMathematics

TWO THEOREMS ON THE PRIME DIVISORS OF ZEROS INCOMPLETIONS OF LOCAL DOMAINS

LOUIS JACKSON RATLIFF, JR.

Vol. 81, No. 2 December 1979

PACIFIC JOURNAL OF MATHEMATICSVol. 81, No. 2, 1979

TWO THEOREMS ON THE PRIME DIVISORS OF ZEROIN COMPLETIONS OF LOCAL DOMAINS

L. J. RATLIFF, JR.

The first theorem concerns the number of minimalprime ideals of a given depth (=dimension) in the comple-tion of a finite integral extension domain of a semi-localdomain. The second theorem characterizes local domainsthat have a depth one prime divisor of zero in their comple-tion as those local domains whose maximal ideal M is aprime divisor (^associated prime) of all nonzero idealscontained in large powers of M.

For arbitrary local (Noetherian) domains R it is of some interestand importance to know as much as possible about the prime divi-sors of zero in the completion R* of R. One reason (among several)for this is that knowledge about these prime ideals is necessary tosolve the catenary chain conjectures. (For example, the Chain Con-jecture, open since 195β, holds if all minimal prime ideals in ί!*have the same depth whenever the integral closure of R is quasi-local.) In this paper we add to the existing knowledge in this areaby proving two new theorems about such prime ideals.

For the first of these, it is well known that if the integralclosure of R has k maximal ideals, then the completion i2* of R hasat least k minimal prime ideals. (For example, see the proof of(33.10) in [3].) It is also known [8, (2.14.1) => (2.14.2)] that if thereexists a maximal chain of prime ideals of length n in some integralextension domain of R, then some minimal prime ideal in i?* hasdepth n. Our first theorem, (4), shows that these two resultscombine in a somewhat unexpected manner for finite integral exten-sion domains of R. The second theorem, (9), characterizes localdomains R such that there exists a depth one prime divisor of zeroin R*. (11) gives a similar characterization of a subclass of thelocal domains whose completions have depth one minimal primeideals.

To help simplify the statement and proof of the first theorem,a few preliminaries are needed. We thus begin by fixing two nota-tional conventions.

(1) NOTATION. We shall denote by A! the integral closure of aring A in its total quotient ring, and by R* the completion (in thenatural topology) of a semi-local ring R.

The following three definitions are also needed.

537

538 L. J. RATLIFF, JR.

(2) DEFINITIONS. For an integral domain A, c(A) = {m; thereexists a maximal chain of prime ideals of length m in A), anduc(A) = {n; there exists a maximal chain of prime ideals of lengthn in some integral extension domain of A}. If PeSpecA, then welet c(P) and uc(P) denote the sets c(AP) and uc(AP). Also, the classof quasi-local domains Q such that uc(Q) = c(Q) will be denoted byif.

It is clear that c(A) £ uc(A), and [3, Example 2, 203-205] inthe case m = 0 shows that this containment may be proper evenwhen A is a local domain, so there are local domains that are notin ^ . It is an open conjecture (the Upper Conjecture) that the onlylocal domains that are not in ^ are all similar to the cited example;that is, those whose altitude ( = Krull dimension) is >1 and whoseintegral closure has a height one maximal ideal. (See [9, (4.10.3)].)

In (3) we list the known results concerning uc(A) and ^ thatwill be needed below.

(3) REMARKS. (3.1) [8, (2.14.1) <=> (2.14.2) ~ (2.14.6)]. If (R, M)is a local domain and X is an indeterminate, then uc(R) = {depth z;z is a minimal prime ideal in R*} = {n — 1; n e uc(R[X]{M>x))}.

(3.2) [9, (4.1)] Every local domain of the form R[X]{M>X) is in<g% where (R, M) is an arbitrary local domain.

(3.3) [9, (4.8.2)] If iZeίT and R' is quasi-local, then, for eachintegral extension domain S of R and for each maximal ideal N inS, SNe%? and uc(N) = ue(R).

The following theorem is the first of the main results in thispaper.

(4) THEOREM. Let (R, M) he a local domain such that R' isquasi-local, let S be a finite integral extension domain of R, andlet N be a maximal ideal in S. Then the following statements hold:

(4.1) There exists a depth n minimal prime ideal in (SN)* ifand only if there exists a depth n minimal prime ideal in iϋ*.

(4.2) If ne uc(R) and if there are k maximal ideals in S' thatlie over N, then there are at least k minimal prime ideals of depthn in (SN)*.

(4.3) If ne uc(R) and if there are exactly m maximal ideals inS', then there are at least m minimal prime idealsζof depth n in S*.

Proof. (4.1). By (3.1), there exists a depth n minimal primeideal in (SN)* if and only if neuc(N), and there exists a depth nminimal prime ideal in iϋ* if and only if n e uc(R). Also, if n e uc(N),then clearly neuc(S), and so neuc(R), since S is integral over R.

TWO THEOREMS ON THE PRIME DIVISORS OF ZERO 539

Thus it remains to show that if neuc(R), then neuc(N).For this, let D = R[X][M,X) and C = S[X]{BU]_iM,x)). Then ΰ e ^ ,

by (3.2), and C is integral over D. Also, D' is quasi-local, by thehypothesis on R, so by (3.3) uc(P) = uc(D) for each maximal idealP in C. Now (N, X)C is a maximal ideal in C, since N is maximalin S, and Cw,x ) ί 7 = S^[JL] ( ^^, X ) . Therefore, by (3.1), if neuc(R),then w + leuc(D) = w((i\Γ, X)C), so neuc(N).

(4.2): Since there are only finitely many maximal ideals in S',there exists a finite integral extension domain BQS' of S such thatB and S' have the same number of maximal ideals. Therefore thereexist k maximal ideals in B that lie over N. Also, if n e uc(R),then for each maximal ideal Q in B there exists a minimal primeideal of depth n in (J5ρ)*, by (3.1) and (4.1). Now (£<>)* ^ E§5*, sothere exists a minimal prime ideal w in 5* such that w £ Q23* andheight QB*/w — n. Then B*/w is a complete domain, so B*/w islocal, and so QB* is the only maximal ideal in B* that contains w.Therefore depth w = height QB*/w = w.

Now 5* and S* have the same total quotient ring T (since 5is a finite S-algebra and S Q B £ S'), so there exists a one-to-onecorrespondence between the minimal prime ideals z in S* and theminimal prime ideals w in i?* given by w = zT Π 5* and 2 = w Π S*,and then depth 2 = depth w (since I?*/w is integral over S*/z).Therefore, since there exist k maximal ideals in B* that lie overNS*9 it follows from the preceding paragraph that there are at leastk minimal prime ideals of depth n in S* that are contained in NS*.Then, as in the preceding paragraph, if z is such a minimal primeideal, then height NS*/z — depth z = n. Hence, since (SN)* = S%s*,it follows that there are at least k minimal prime ideals of depth nin (SN)\

(4.3) follows from (4.2), since S* = φ (SNi)*9 where the JV, arethe maximal ideals in S.

Concerning (4), it is an open problem if there exists n e uc(R)such that n < altitude R. If such n exist, then, as noted in theintroduction, the Chain Conjecture does not hold.

(5) REMARKS. With the notation of (4), the following state-ments hold:

(5.1) R is quasi-unmixed if and only if some SN is quasi-unmixed if and only if all SN are quasi-unmixed.

(5.2) If there exists a minimal prime ideal of depth d in (SN)*9

then there exist at least k minimal prime ideals of depth d in (SN)*and there exist at least m minimal prime ideals of depth d in S*.

540 L. J. RATLIFF, JR.

(5.3) There exists a local integral extension domain So £ S ofR such that (SJ = S' and (So)* has at least m minimal prime idealsof depth n, for each neuc(R).

Proof. (5.1) is clear by (4.1).(5.2). If there exists a minimal prime ideal of depth d in (SN)*9

then deuc(R), by (4.1) and (3.1), so the conclusion follows from (4.2)and (4.3).

(5.3). Let So = R + J, where J is the Jacobson radical of S.Then So is a finite integral extension domain of R (since S is andR Ω SQζZ S), and J is the only maximal ideal in So and is the con-ductor of So in S. Therefore So is a local domain and (So)' = S', sothe conclusion follows from (4.3) (with So in place of S).

(4) can be used to say something about the number of minimalprime ideals in the completion of finite integral extension domainsof arbitrary semi-local domains, as will now be shown.

(6) COROLLARY. Let S be a finite integral extension domain ofa semi-local domain R. Let Mu , Mg be the maximal ideals inR', assume that there are kt maximal ideals in S' that lie over Mt

{for i = 1, , g) and that n e uc{M%) if and only if i = 1, , h ^ g.Then there are at least k^+ + kh minimal prime ideals of depthn in S*.

Proof. There exist finite integral extension domains A of Rand B of S such that A £ R', B £ S', R' and A have the samenumber of maximal ideals, and S' and B have the same number ofmaximal ideals; and it can be assumed that A £ B. Therefore,Lt = AM.nA is a local domain such that (Lt)' is quasi-local and thereexist kt maximal ideals in B that lie over Pi = Mt Π A. Also, bythe Going Up Theorem, n e uc(L%) if and only if i <Lh. Hence, by(4.3), for i = 1, •••, h there are at least kt minimal prime ideals ofdepth n in (BA_Pi)*. Now (BA_Pi)* is isomorphic to a direct summand(with ki maximal ideals) of B*, so the kt minimal prime ideals ofdepth n in (JB^_P.)* correspond to kt minimal prime ideals of depthn in 2?*. Thus it follows that there are at least kx + ••• + kh

minimal prime ideals of depth n in J5*. Therefore the conclusionfollows as in the second paragraph of the proof of (4.2), since S*and JS* have the same total quotient ring.

Before considering the second of the main theorems, we givetwo small applications of (6). The second of these uses a number

TWO THEOREMS ON THE PRIME DIVISORS OF ZERO 541

of new results from the recent paper [10].

(7) With the notation of (6), if Z = RndS*, then (S*/Z)' hasat least kλ + + kh maximal ideals of height n. In particular,if S is analytically unramified, then there exist at least kt + + kh

maximal ideals of height n in S*\ For, (S*/Z)' = 0 {(S*/zY; z isa minimal prime ideal in S*}, so since each S*/z is a complete localdomain and since at least kt + + kh of the z's have depth n, by(6), it follows that (S*/Z)' has at least &!+•••+&* maximal idealsof height n. And if S is analytically unramified, then Z = (0), sothe second statement follows from the first.

(8) With the notation of (6), let Q be an open ideal in S, letr be a large integer, and let I be an ideal in S such that Qr £ / Q(Qr)a — the integral closure of Qr in S. Then there exist at leastkγ + . . . -\- kh minimal prime ideals of depth n in the form ring{ — associated graded ring) J^~(S, I) of S with respect to I. To provethis, note first that it is known [11, Theorem 2.1] that ^~(S, I) =&\t~γ&, where & — &(S, I) = S[tl, t'1] (t is an indeterminate) isthe Rees ring of S with respect to I. (The restriction to local ringsin [11] is not essential.) Therefore it suffices to prove that t~~γ&has at least k^+ + kh minimal prime divisors of depth n. Forthis, it may be assumed that / = (Qr)a, by [10, (7.7)]. Let ^ ° =^ ( S * , IS*), let 2 be a minimal prime ideal in S*, and let z* =zT[t, r 1 ] Π ° ' , where T is the total quotient ring of S*. Then [7,Corollary 2.23] says there exists a height one prime ideal pof in &*such that (z*, r 1 ) ^ 0 ' £ p". Next, since r is large, [10, (7.1)] showsthat z* is the only minimal prime ideal in <^?0' that is contained inp°\ Also, by [10, (7.3)] and since r is large, there exists a one-to-one correspondence between the minimal prime divisors p0' of ί"1^?0',p° of t~γ&\ and p of t~γ&, and then depth p = depth p° = depthp0' = depth z, by [10, (7.3) and (5.2.1)]. Therefore, since S* has atleast kx + + kh minimal prime ideals of depth n, by (6), it followsthat t'1^ has at least kx + + kh minimal prime divisors of depthn9 and so ^~(S, I) has at least kx+ + kh minimal prime idealsof depth n.

We now consider the second theorem on prime divisors of zeroin i?*. Concerning (9), it is shown in [1] that every power of themaximal ideal in an analytically irreducible local domain R containsa prime ideal of height = altitude R — 1. (9) characterizes localdomains that fail about as completely as possible to have this latterproperty.

(9) THEOREM. The following statements are equivalent for a

542 L. J. RATLIFF, JR.

local domain (22, M) such that altitude R ^ 1:(9.1) There exists a depth one prime divisor of zero in 22*.(9.2) There exist M-primary ideals Q in R such that every

nonzero ideal contained in Q has M as a prime divisor.

Proof. (9.1) => (9.2). Let zlf , zg be the prime divisors of zeroin 22* with depth zx = 1 and let (0) = Πί Q% be a primary decomposi-tion of zero in R*. Let x be a nonzero element in Πf& Then,for all P* e Spec 22* - fo, ikf*}, x e Ker (22* -> 22|*), so a; is in everyP* primary ideal. Hence if / is a nonzero ideal in R such thatI: M — I, then a primary decomposition of IR* shows that xeIR*,since IR*: M* = 122* and IR* g zlβ Since a ^ 0, there exist M-primary ideals Q in R such that xgQR*, so xglR*, for all idealsIQ Q. Thus it follows that M is a prime divisor of all nonzeroideals IQQ.

(9.2) => (9.1). Assume (9.1) does not hold and let z19 - *,zk bethe maximal prime divisors of zero in 22*, so depth zt > l(i = 1, , k).Assume it is known that there exist P19 , Pk e Spec 22* — {ikf*}such that ^ c P t and such that there are no containment relationsamong the pt = Pt Π 22 (so there are no containment relations amongthe PJ . Let S = 22 - Uί &, let S* = 22* - Uί Ή, and f or n ^ 1 letL = (Πί PΓ)Λ5 Π 22 and let I* = (Πf PΓ)22|* n 22*. Then /» C /* n 22and /Λ (resp., /*) is the intersection of the k primary ideals pln) =pΓΛP< Π 22 (resp., P f > - Pf22*, Π 22*). Also, /* 2 J*+1 and f| /ί = (0),since Γ\n (flf P?)-B** = (0) and Ker (22* -> 22**) = (0). Therefore foreach m ^ 1 there exists w(ra) such that /*(m) £ Λί*w, by [3, (30.1)].Hence /w ( m>£/»* ( w )ni2C M*mn22 = Mm and 2lf is not a prime divisorof /Λ ί m ), so (9.2) does not hold. Therefore it remains to show theexistence of the P,.

For this, let 0 Φ a e M and for i — 1, , k let P ΐ ? α be a minimalprime divisor of (zi9 α)22*. Let ^ = {Piia n R; 0 Φ ae M}. It willnow be shown that for each i: (i) ^ is an infinite set such thatU . = M"; and, (ii) the intersection of each infinite subset of ^ iszero. Namely, depth P i>αn22 ^ depth P ί i β ^ l , so each ^ is an infiniteset such that U ^ = M (since α is an arbitrary nonzero element inM), so (i) holds. Also, (ii) holds since the intersection of eachinfinite subset of {Piya; 0 Φ aeM} is z€.

Now assume 1 <; h < k and p1 e &*19 , ph e &*h have beenchosen such that are no containment relations among pί9 , ph.Then by (i) (and since M is not the union of finitely many primeideals) there exist infinitely many p e ^h+1 such that p gΞ U? Vu a n dby (ϋ) Πf Pi is n ° t contained in at least one of these p, so thereexists ph+16 &*h+1 such that there are no containment relations among

TWO THEOREMS ON THE PRIME DIVISORS OF ZERO 543

Pi> , PΛ+I From this the existence of Plf , Pk e Spec R* — {M*}such that Zi c P* and such that there are no containment relationsamong the Pi Π R readily follows.

(10) REMARKS. (10.1) It is clear by (9) that if there exists anideal Q in R as in (9.2), then for all finite local integral extensiondomains (L, P) Q Rf of R there exists a P-primary ideal Q' suchthat P is a prime divisor of every nonzero ideal in L that is con-tained in Q'.

(10.2) It is interesting to note that the equivalent conditionsin (9) imply that if P e Spec R — {0, M} and i is a large integer,then P£ is not P-primary (since P* Q Mi Q Q). It would be interest-ing to know if this is, in fact, equivalent to the equivalent state-ments in (9).

(10.3) It was shown in [6, (4.8) (5)] that if R is as in [3,Example 2, 203-205] in the case m = 0, then for all i ^ 2 and forall PeSpeci? - {0, M}, Pι is not P-primary.

(10.4) If R is as in (10.3), then ikP is a suitable Q for (9.2).

Proof. (10.4). Rr is a regular domain with exactly twomaximal ideals, say P, Q, such that height P = 1 < height Q = r + 1,so there exist exactly two minimal prime ideals in R'*f say z*, w*,with z* Π w* = (0) and depth z* = 1 < depth w* = r + 1. Thenw = w* Π R* and z = z* Π i?* are the prime divisors of zero in iϋ*and zΠw = (0). Also, Λf = P n Q, so ikί* = P* n Q*. Thereforew — w* Π M* = ^ * Π (P* Π Q*) = w* Π P* is a minimal prime idealin i2* and w g M*2 (since i2p** - (w* n P*)i2'P** = P*ΛJί g P*2i2'Pΐ =M*2i2p*). So a; in the proof of (9.1) => (9.2) can be chosen in w, ί ilί*2.

(10.4) and (10.2) give a different proof of the result noted in(10.3).

It follows from the next result (see (12.2)) that if Rr is a finitei?-algebra (for example, if R is analytically unramified), then (9.2)characterizes when there exists a depth one minimal prime divisorof zero in R*.

(11) PROPOSITION. Let (R, M) be a local domain such thataltitude R ^ 1 and every M-primary ideal contains a nonzerointegrally closed ideal. Then the following statements are equivalent:

(11.1) There exists a depth one minimal prime ideal in i2*.(11.2) There exist M-primary ideals Q in R such that every

nonzero ideal contained in Q has M as a prime divisor.

Proof. (11.1) =- (11.2) by (9).

544 L. J. RATLIFF, JR.

For the converse, let Q be such an ideal and let I be a nonzerointegrally closed ideal contained in Q. Let b e I, 6 ^ 0 . Then(bR)a C I Q Q, so M is a prime divisor of (bR)a. Now (6i2)β = bR' n #,so there exists a (height one) prime divisor p' of 6i2' that lies overM. Therefore p' is a height one maximal ideal, and so there existsa depth one minimal prime ideal in i?* by [4, Proposition 3.3].

This paper will be closed with the following three remarksrelated to (11).

(1.2) REMARKS. (12.1) There exists a local domain (R, M) suchthat altitude R > 1 and there exist Af-primary ideals Q in R suchthat there does not exist a nonzero integrally closed ideal containedin Q. Therefore if 0 Φ b e Q, then (b*R)a gΞ Q, for all n ^ 1.

(12.2) If (R, Λf) is a local domain and if there exists b in Msuch that Rb Π i2' is a finite i?-algebra (in particular, if R! is afinite iϋ-algebra), then every M-primary ideal contains an integrallyclosed ideal.

(12.3) If (jβ, M) is a local domain, if there exists b in M suchthat RbC]R{1) is a finite i2-algebra (where R{1) = n{5P; peSpeci?and height p = 1}), and if there exists an ideal Q in iϋ as in (11.2),then altitude R = 1.

Proof. (12.1). By (9) and (11) it suffices to show the existenceof a local domain R such that iϋ* has a depth one prime divisor ofzero but does not have a depth one minimal prime ideal. Such anexample is given in [2, Proposition 3.3].

(12.2). Let Q be an M"-primary ideal and let n such thatMn C Q, so bn e Q. Now {r/bk; r e bkR' n R} = Rb Π R' is a finite R-algebra, so there exists k ^ 1 such that bk(Rb ΓΊ i?') £ R. Thereforeif r e (bk+ίR)a = bMRf Π R, then r/bk+i e Rf n Λ6, so r e 6*β, hence

£ ft'Λ for all i ^ 1. Thus (6fe+wi2)α C bnR Q Q.

(12.3). [5, Lemma 5.15(10)] shows that there exists k such thatbk+\Rb Π R{1))^bR, if Rb n J?(1) is a finite ^-algebra, so bk+n(Rb Π (1)) £bnR. Also, bk+n(Rb Π -B(1)) is a finite intersection of height oneprimary ideals, by [5, Lemma 5.15(4)], and height one prime idealsin Rb Π R{1) lie over height one prime ideals in R, by [5, Lemma5.15(6)]. Therefore with n such that b% e Q, we have bk+n(Rb Π Ra)) £ Qand is a finite intersection of height one primary ideals, so altitudeR = 1 by the property of Q.

TWO THEOREMS ON THE PRIME DIVISORS OF ZERO 545

REFERENCES

1. R. Berger, Zur Idealtheorie analytisch normaler Stellenringe, J. Reine Angew.Math., 201 (1959), 172-177.2. D. Ferrnad and M. Raynaud, Fibres formelles d'un anneau local Noetherien, Ann.Sci. Ecole Norm. Sup., 3 (1970), 295-311.3. M. Nagata, Local Rings, Interscience Tracts 13, Interscience, New York, 1962.4. L. J. Ratliff, Jr., On quasi-unmixed local domains, the altitude formula, and thechain condition for prime ideals (I), Amer. J. Math., 91 (1969), 508-528.5. , On quasi-unmixed local domains, the altitude formula, and the chaincondition for prime ideals (II), Amer. J. Math., 92 (1970), 99-144.6. 1 Chain conjectures and H-domains, 222-238, Lecture Notes in Math. No.311, Conference on Commutative Algebra, Springer-Verlag, Berlin-Heidelberg-New York,1973.7. , Locally quasi-unmixed Noetherian rings and ideals of the principal class,Pacific J. Math., 52 (1974), 185-205.8. L. J. Ratliff, Jr. and S. McAdam, Maximal chains of prime ideals in integral ex-tension domains, I, Trans. Amer. Math. Soc, 224 (1976), 103-116.9. L. J. Ratliff, Jr., Maximal chains of prime ideals in integral extension domains,II, Trans. Amer. Math. Soc, 224 (1976), 117-141.10. , On the prime divisors of zero in form rings, Pacific J. Math., 70 (1977),489-517.11. D. Rees, A note on form rings and ideals, Mathematika, 4 (1957), 51-60.

Received July 12, 1978. Research on this paper was supported in part by theNational Science Foundation, Grant MCS77-00951 A01.

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of divisors of an integer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411Alan Hopenwasser and David Royal Larson, The carrier space of a reflexive

operator algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417Kyung Bai Lee, Spaces in which compacta are uniformly regular Gδ . . . . . . . 435Claude Levesque, A class of fundamental units and some classes of

Jacobi-Perron algorithms in pure cubic fields . . . . . . . . . . . . . . . . . . . . . . . 447Teck Cheong Lim, A constructive proof of the infinite version of the

Belluce-Kirk theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467Dorothy Maharam and A. H. Stone, Borel boxes . . . . . . . . . . . . . . . . . . . . . . . . . 471Roger McCann, Asymptotically stable dynamical systems are linear . . . . . . . 475Peter A. McCoy, Approximation and harmonic continuation of axially

symmetric potentials in E3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481Takahiko Nakazi, Extended weak-∗Dirichlet algebras . . . . . . . . . . . . . . . . . . . . 493Carl L. Prather, On the zeros of derivatives of balanced trigonometric

polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515Iain Raeburn, An implicit function theorem in Banach spaces . . . . . . . . . . . . . . 525Louis Jackson Ratliff, Jr., Two theorems on the prime divisors of zeros in

completions of local domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537Gloria Jean Tashjian, Cartesian-closed coreflective subcategories of

Tychonoff spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547Stephen Edwin Wilson, Operators over regular maps . . . . . . . . . . . . . . . . . . . . . 559

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