In this paper we construct a non complete excellent local ring A such that the natural map Spec Â → Spec A is bijective

Lee, Dan

Leer, Leanne

Pilch, Shara A.

Yasufuku, Yu

English

Furman University

Furman UniversityElectronic Journal of Undergraduate MathematicsVolume 5, 1 – 9, 1999EXCELLENT RINGS WITH SINGLETON FORMAL FIBERSDAN LEE, LEANNE LEER, SHARA PILCH, YU YASUFUKUAbstract. In this paper we construct a non-complete excellent local ring Asuch that the natural map Spec bA −→ SpecA is bijective.1. IntroductionAlthough this paper focuses on a commutative algebra result, we shall begin bytalking about algebraic geometry and its relationship with commutative algebra.An algebraic variety is a set of common zeroes of a polynomial with coefficients insome field k. Varieties are very important in algebraic geometry, and to study themwe analyze the set of regular functions from the variety to k. This set turns out tobe an affine ring, that is, a ring of the form k[x1, . . . , xn]/I where I is an ideal.Affine rings have very nice properties, and so a question naturally arises . . .which of these nice properties are the ones that make affine rings geometricallysignificant? Algebraists have tried to capture these nice properties by classifyingnice rings through a definition. The first attempt at doing this was the definition ofNoetherian. A Noetherian ring is a ring in which every ideal is finitely generated.But it turns out that this classification is much too weak and does not capturethe properties that were hoped for. So a new definition was needed. In the 1950’sGrothendieck presented the definition of excellent, and this is the latest attempt atanswering the above question.Received by the editors September 24, 1999.This research was supported by NSF Grant DMS-9820570 and conducted as part of theWilliams College math REU under the guidance of advisor S. Loepp. The authors would liketo thank S. Loepp for many helpful conversations and suggestions, and for wonderful guidanceand enthusiasm.12 DAN LEE, LEANNE LEER, SHARA PILCH, YU YASUFUKUA formal definition of excellent, as well as many other technical definitions, isincluded in section 3 of this paper, but here we shall attempt to describe the ideabehind excellence.Let (A,M) be a local ring.1 By Cohen’s Structure Theorem we know the struc-ture of Â, the M -adic completion of A. In particular, we know that if Â contains afield k, then Â is of the form k[[x1, . . . , xn]]/I, which is quite geometric as it relatesto affine rings. Furthermore, there is a nice relationship between A and Â, namelythat any prime of A can be written as p ∩ A, for some prime ideal p in Â. Thisproperty is called faithful flatness, and it is saying that every prime ideal of A can,in some sense, be described by prime ideals of the completion. So the idea behindexcellence is this—since we know Â is geometric and that the prime ideals of A canbe described by the primes of Â, we hope to carry the geometric properties of Âdown to A by controlling which primes of Â map to which primes of A. Since thewhole goal is to capture nice geometric properties of rings, to say that A is excellentis to say that it satisfies some “nice” conditions about which primes of Â map towhich primes of A, thereby transferring the geometric properties of Â down to A.The definiton of excellent has proved to be successful in some ways. It ensuresthat reducedness and normality are preserved by the completion map, that is tosay that if A is an excellent reduced (or normal) ring, then Â will be reduced (ornormal). This is not always the case when A is not excellent. But excellent rings doseem to fall short of their purpose; they do not appear to capture all the propertiesof affine rings. Matsumura has shown that if R is an affine ring, then there existinfinitely many maximal elements in the generic formal fiber of A, that is, we canfind a countably infinite list of prime ideals p1, p2, . . . of Â such that pi ∩ A = (0)for each i, and for any prime ideal q strictly containing any pi, q∩A 6= (0) [5]. Forexcellent rings this is not necessarily the case. The following theorem, which wasproved in [3] by Loepp, demonstrates how extremely different the situation may befor an excellent ring.1In this paper all rings are commutative with unity. When we write (R,M) is a quasi-localring, we mean that R is a quasi-local ring with maximal ideal M .EXCELLENT RINGS WITH SINGLETON FORMAL FIBERS 3Theorem 1.1. Let (T,m) be a complete regular local ring of dimension at leasttwo containing the rationals, and let |T/m| ≥ |R|. Let p 6= m be a prime ideal of T .Then there exists an excellent regular local ring A with the following properties:• Â = T .• The generic formal fiber of A is local with p its maximal ideal.• For all q ∈ SpecT , if q 6⊆ p, then (q ∩A)T = q.The above A is very strange for two reasons. First, the generic formal fiberof A has exactly one maximal element, whereas there were infinitely many in theaffine case. Second, by the third property of the theorem, there is a one-to-onecorrespondence between the primes of A and the primes of T not contained in thegeneric formal fiber of A. Although this property is nice in some sense, it is farfrom the behavior exhibited by affine rings. Such an odd ring hardly deserves tobe called “excellent,” and yet A is an excellent ring.In this paper, we use the above theorem to help us construct an excellent regularlocal ring B such that there is a one-to-one correspondence between the primes of Band the primes of B̂ while B 6= B̂. Since B is excellent, there are strong conditionson the formal fibers of B, and yet all of the formal fibers turn out to be singletonsets.2. Main TheoremTheorem 2.1. Let T be a complete regular local ring with maximal ideal m suchthat dimT ≥ 2, Q ⊆ T , and |T/m| ≥ |R|. Suppose x ∈ T is a prime element.Then there exists an excellent regular local ring B such that B̂ = T , the naturalmap SpecT −→ SpecB is bijective, and x /∈ B.Proof. Since T is a regular local ring, T is a UFD by (20.3) of [4]. Since dimT ≥ 2,m is a prime ideal generated by at least two elements, so by factoring the genera-tors into prime elements, we see that there are at least two prime elements in T .Therefore let y 6= x be a prime element. Using Theorem 1.1, we construct A sothat its generic formal fiber is {(0), xT}.4 DAN LEE, LEANNE LEER, SHARA PILCH, YU YASUFUKUWe now define B = A[xy]m∩A[xy]. We will prove that this B satisfies the con-clusion of the theorem. Clearly B is quasi-local. Moreover, by the Hilbert BasisTheorem, A[xy] is Noetherian, and since the localization of a Noetherian ring isNoetherian, it follows that B is Noetherian. Thus B is local. Obviously, A ⊆ B,and since elements in A[xy]− (m∩A[xy]) ⊆ T −m are all units in T , it follows thatB ⊆ T . By completeness of T , every Cauchy sequence in T converges in T , so inparticular, every Cauchy sequence in B converges in T . Moreover, Â = T impliesthat every element of T can be written as the limit of a Cauchy sequence in A, andtherefore also as the limit of a Cauchy sequence in B. Hence B̂ = T . Also notethat B is a regular local ring since T is a regular local ring (see (10.15)(iii) and(11.12)(ii) of [1]). Note that this implies that B is a UFD.Claim 2.1. x /∈ BProof. Assume x ∈ B. Then x = a0+a1xy+···+am(xy)ma′0+a′1xy+···+a′n(xy)n , where the de-nominator is not in m. So x(a′0+a′1xy+· · ·+a′n(xy)n) = a0+a1xy+· · ·+am(xy)m. So a0 ∈ xT ∩A = (0). Then since T is a domain, we can can-cel x on both sides to conclude that a′0 +a′1xy+ · · ·+a′n(xy)n ∈ yT ⊆ m,which is a contradiction.Since T is the completion of B, we know that the map SpecT −→ SpecB issurjective by faithful flatness. We also want to show that it is injective.Claim 2.2. For all p ∈ SpecT such that p 6⊆ xT , we have (p∩B)T = p.Proof. By our construction of A and by Theorem 1.1, we know that(p ∩A)T = p. So p = (p ∩A)T ⊆ (p ∩B)T ⊆ p.The only primes contained in xT are (0) and xT , so in order to get injectivity weneed to check the following four cases: for all distinct p, q ∈ SpecT not containedin xT :(i) p ∩B 6= (0) ∩B(ii) p ∩B 6= xT ∩B(iii) xT ∩B 6= (0) ∩B(iv) p ∩B 6= q ∩BEXCELLENT RINGS WITH SINGLETON FORMAL FIBERS 5We know p ∩ A 6= (0) by the construction of A, so (i) follows. For (ii), supposep ∩ B = xT ∩ B. Then by Claim 2.2, p = (p ∩ B)T = (xT ∩ B)T ⊆ xT . This is acontradiction, so we have (ii). Clearly xy is nonzero and xy ∈ xT ∩B, thus we get(iii). Case (iv) follows directly from Claim 2.2. Therefore SpecT −→ SpecB is abijection.We also want B to be excellent. Because T is an integral domain and contains Q,it suffices to show that all of its formal fiber rings are regular rings (see “excellent”in section 3). In fact, we will show that in this case each formal fiber ring is a field.This will complete the proof of the theorem.Fix a prime ideal P of B. Let k(P ) = BP /PBP be the residue field of BP . Weneed to show that the formal fiber ring at P , T ⊗B k(P ), is a field.As noted on page 56 of [2], T ⊗B k(P ) ∼= S−1T/((PT )(S−1T )), where S =B − P and the bar denotes saturation (see section 3 for the definition of saturation).Moreover, the prime ideals of this formal fiber ring correspond to the prime idealsof T which contract down to P , so by the (1-1) correspondence proved above, thisring has a unique prime ideal. Thus it suffices to show that for each P ∈ SpecB,(PT )(S−1T ) is a prime ideal of S−1T . This would show that S−1T/((PT )(S−1T ))is an integral domain, and therefore the unique prime ideal must be the zero ideal,proving that the formal fiber ring at P is a field. Now we divide into cases dependingon p, where p is the unique prime ideal of T such that p ∩B = P .(a) Suppose p = xT .P = xT ∩B is a nonzero prime ideal as it contains xy. Since the completionmap is flat, the Going Down Theorem holds (see exercise 5.11 in [1]). Thisimplies that xT ∩ B has height at most 1, since xT has height 1. BecausexT ∩B is nonzero, it follows that xT ∩B is a height 1 prime, and since B isa UFD, it follows that xT ∩B = bB for some prime element b ∈ B.We know that b = xt for some t ∈ T and we also know that xy = bb′ forsome b′ ∈ B, since xy ∈ xT ∩ B = bB. Then we have xy = bb′ = xtb′, soy = tb′ since T is an integral domain. Further, since y is a prime elementin T , it follows that either t or b′ must be a unit in T . We claim that t can6 DAN LEE, LEANNE LEER, SHARA PILCH, YU YASUFUKUnever be a unit. Assume on the contrary that t is a unit. Then xt = b ∈ B =A[xy]m∩A[xy], so we can writext =a0 + a1xy + · · ·+ am(xy)ma′0 + a′1xy + · · ·+ a′n(xy)n,where the denominator is not in m. Clearing the denominator, we getxt(a′0 + a′1xy + · · ·+ a′n(xy)n) = a0 + a1xy + · · ·+ am(xy)m.Thus, a0 ∈ xT ∩A = (0), so a0 = 0. Now, observe that the RHS is divisible byy, y is a prime element in T , and y - xt. So a′0+a′1xy+· · ·+a′n(xy)n ∈ yT ⊆ m.But this is a contradiction since a′0 +a′1xy+ · · ·+a′n(xy)n was assumed not tobe in m. So we have proved that t cannot be a unit. Thus b′ must be a unit.Now, PT = (xT ∩B)T = bT = xy(b′)−1T = xyT . Note that yT 6⊆ xT . So bythe construction of A, it follows that yT ∩A 6= (0). So let 0 6= yt′ ∈ A. If yt′were in P , then yt′ ∈ PT ∩ A ⊆ xT ∩ A = (0), yielding a contradiction. Soyt′ ∈ B − P , and thus by definition of saturation, y ∈ S. Thus, in S−1T , y isa unit, so (PT )(S−1T ) = (xyT )(S−1T ) = (xT )(S−1T ). Since xT is a primeideal of T , this is a prime ideal of S−1T if we can show that xT ∩ S = ∅.But this is clear, since if xu ∈ S, then xut′′ ∈ B − P for some t′′ ∈ T ,so xut′′ ∈ xT ∩ B = P , giving a contradiction. Thus, we have shown that(PT )(S−1T ) is a prime ideal, so we are done with this case.(b) Suppose p = (0).Since p = (0), P = (0), and so PT = (0). So (PT )(S−1T ) = (0) and istherefore prime, since S−1T is an integral domain.(c) Suppose p 6⊆ xT .Recall that P = p ∩B. Then PT = (p ∩B)T = p by Claim 2.2. ThereforePT is prime in T . Thus, to show that (PT )(S−1T ) is prime, we only need toshow that PT ∩ S = ∅. Assume t ∈ PT ∩ S. Then since t ∈ S, tt′ ∈ B − Pfor some t′ ∈ T , which implies tt′ ∈ PT ∩ B = P . This is a contradiction.Therefore (PT )(S−1T ) is prime.This concludes the proof of Theorem 2.1.EXCELLENT RINGS WITH SINGLETON FORMAL FIBERS 73. DefinitionsIn this section we define some necessary terms. See [7], [1], or [4] for moreexplanation.• Spec R: In a ring R, SpecR denotes the set of all prime ideals in R.•Quasi-local: A ring R is called quasi-local when it has exactly one maximalideal.•Local: A ring R is called local when it is quasi-local and also Noetherian.•Localization: Let S be a multiplicatively closed subset of a commutative ringR. Then the ring of fractions S−1R is the ring we obtain by inverting all elementsof S. When S = R− P for some P ∈ SpecR, we write RP for S−1R. In this case,RP is a quasi-local ring, called the localization of R at P , with maximal ideal PRP .• Saturation: Let S be a multiplicatively closed subset of R. Define the satura-tion of S to be S = {r ∈ R|rr′ ∈ S for some r′ ∈ R}. Note that S−1R = (S)−1R.• Regular Local Ring: Let (R,M) be a local ring. Then (R,M) is called aregular local ring if M can be generated by dimR elements.• Regular Ring: A ring R is called a regular ring when the localization atevery prime ideal is a regular local ring.• Complete: We can define a metric on a local ring (R,M) as follows. For allx, y ∈ R, let n be the largest integer such that x− y ∈Mn, where M0 = R. Thenwe define the metric d(x, y) = 12n when n exists and 0 otherwise.A local ring R is called complete when it is complete with respect to the abovemetric, that is, when all its Cauchy sequences converge.• Completion: The completion of a local ring A is the ring of all Cauchysequences modded out by the following equivalence relation: two Cauchy sequencesare equivalent if their difference converges to zero. The completion of A is denotedÂ; this is always complete.• Formal Fiber: Let (A,m) be a local ring, Â its completion, and P a primeideal in A. We define the formal fiber of A at P to be the inverse image of P underthe map Spec Â −→ SpecA. That is, prime ideals in the formal fiber at a primeideal P are the prime ideals of Â that lie over P .8 DAN LEE, LEANNE LEER, SHARA PILCH, YU YASUFUKU• Generic Formal Fiber: The generic formal fiber of a local integral domainis the formal fiber of (0).• Formal Fiber Ring: We call the ring Â ⊗A (AP /PAP ) the formal fiberring of A at P . The prime spectrum of the formal fiber ring of A at P naturallycorresponds to the formal fiber of A at P .• Flat: Let A be a ring and let M , N , and N ′ be A-modules. We say that M isflat over A if for all injections f : N ′ −→ N , f ⊗ 1 : N ′ ⊗AM −→ N ⊗AM is alsoinjective.• Faithfully Flat: A ring B is said to be faithfully flat over A if B is flat overA and SpecB → SpecA is surjective. For a local ring A, Â is faithfully flat over A.• Excellent: Let (A,M) be a local ring. For each P ∈ SpecA, let k(P ) denotethe field AP /PAP . Then A is said to be excellent if it satisfies the following twoconditions: (a) For each P ∈ SpecA, the formal fiber ring Â⊗Ak(P ) is geometricallyregular over k(P ) , that is,(Â⊗A k(P ))⊗k(P ) L ∼= Â⊗A Lis regular for every finite field extension L over k(P ), and (b) A is universallycatenary.In this paper, it is easier to check excellence because of additional conditions.Specifically, assume that (T,m) is a complete local domain containing Q and let Abe a local ring such that T = Â. Then by Theorem 31.6 in [4], condition (b) isautomatically satisfied for A. Moreover, since every integer in T is invertible, nointeger belongs to m. For P ∈ SpecA, P ⊆ m ∩ A ⊆ m, so it follows that k(P ) =AP /PAP must be a field of characteristic zero. So the only purely inseparableextension of this field is itself. Thus, by Remark 1.3 in [6], for (a), we only need tocheck that each formal fiber ring Â⊗A (AP /PAP ) is a regular ring.EXCELLENT RINGS WITH SINGLETON FORMAL FIBERS 9References[1] M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Perseus Books,1969.[2] S. Greco and P. Salmon, Topics in m-adic Topologies, Springer-Verlag, 1971.[3] S. Loepp, Excellent Rings with Local Generic Formal Fibers, J. Algebra 201 (1998), 573-583.[4] H. Matsumura, Commutative Ring Theory, Cambridge University Press, 1986.[5] H. Matsumura, On the Dimension of Formal Fibres of a Local Ring, Algebraic Geometry andCommutative Algebra in Honor of Masayoshi Nagata, Kinokuniya, 1987, 261-266.[6] C. Rotthaus, Excellent Rings, Henselian Rings, and the Approximation Property, RockyMountain J. Math 27 (1997), 317-334.[7] R. Y. Sharp, Steps in Commutative Algebra, Cambridge University Press, 1990.Dan Lee: Harvard UniversityE-mail address: dalee@fas.harvard.eduLeanne Leer: Bucknell UniversityE-mail address: leer@bucknell.eduShara A. Pilch: Williams CollegeE-mail address: Shara.A.Pilch@williams.eduYu Yasufuku: Harvard UniversityE-mail address: yasufuku@fas.harvard.eduSponsored by S. Loepp of Williams CollegeE-mail address: sloepp@williams.edu

Excellent Rings with Singleton Formal Fibers

Abstract. In this paper we construct a non-complete excellent local ring A such that the natural map Spec bA − → SpecA is bijective. 1

Dan Lee

Leanne Leer

Shara Pilch

Yu Yasufuku

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EXCELLENT RINGS WITH SINGLETON FORMAL FIBERS

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Excellent Rings with Singleton Formal Fibers

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