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008 DIMENSION AND SINGULARITY THEORY FOR LOCAL RINGS OF FINITE

EMBEDDING DIMENSION

HANS SCHOUTENS

ABSTRACT. In this paper, an algebraic theory for local rings of finite embedding dimen-sion is developed. Several extensions of (Krull) dimensionare proposed, which are thenused to generalize singularity notions from commutative algebra. Finally, variants of thehomological theorems are shown to hold in equal characteristic.

This theory is then applied to Noetherian local rings in order to get: (i) over a Cohen-Macaulay local ring, uniform bounds on the Betti numbers of aCohen-Macaulay modulein terms of dimension and multiplicity, and similar bounds for the Bass numbers of afinitely generated module; (ii) a characterization for being respectively analytically unram-ified, analytically irreducible, unmixed, quasi-unmixed,normal, Cohen-Macaulay, pseudo-rational, or weakly F-regular in terms of certain uniform arithmetic behavior; (iii) in mixedcharacteristic, the Improved New Intersection Theorem when the residual characteristic orramification index is large with respect to dimension (and some other numerical invariants).

1. INTRODUCTION

This paper is devoted to the study of local rings of finite embedding dimension, whereby a local ring, we mean a not necessarily Noetherian, commutative ringR with a uniquemaximal idealm,1 and where theembedding dimensionof R, denotedembdim(R), isthe minimal number of elements generatingm. We will see that there are various waysof extending the dimension and singularity theory of Noetherian local rings to this largerclass. The motivation for this study comes from the subclassof ultra-Noetherianlocalrings: these are the ultraproducts of Noetherian local rings of fixed embedding dimension.I had used these ultra-Noetherian rings in my previous work on transfer from positive tozero characteristic ([38, 41]) and on non-standard tight closure ([4, 45, 40, 43, 48]), butthe actual study of their properties was only prompted by thepapers [39, 47], where it wasessential to have a generalized dimension and singularity theory to get asymptotic versionsof various homological theorems in mixed characteristic. It was this realization that led meto develop a systematic ‘local algebra’ for these rings. Consequently, we will be able toderive from this study some improved asymptotic versions inthe final section. For someother recent papers studying ultraproducts of Noetherian rings, see [31, 32, 33].

Closely related to a local ring of finite embedding dimensionare two local rings whichare always Noetherian: its graded ring and its completion. Especially through the latterthe study of local rings of finite embedding dimension is greatly facilitated. Accordingly,I will use the modifiercata- to indicate that a property is inherited by completion. Incontrast, for ultra-Noetherian local rings, the prefixultra- is used to refer to properties that

Date: October 29, 2018.1991Mathematics Subject Classification.13L05,13D22,13H10,03C20.Key words and phrases.smooth equivalence relations, jets, classification, ultraproducts, cataproducts.Partially supported by the National Science Foundation anda PSC-CUNY grant.1In the literature, such rings are also calledquasi-local.

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http://arxiv.org/abs/0809.5267v1

2 HANS SCHOUTENS

are inherited by the ultraproduct. The main goal is now to findconditions under whichboth versions agree, which often requires the introductionof a third, intrinsic (pseudo-)variant. To study these variants, we introduce the notion ofa cataproduct, defined as thecompletion of the ultraproduct. In fact, the cataproduct isobtained from the ultraproductby factoring out theideal of infinitesimals, that is to say, the ideal of elements lying ineach power of the maximal ideal. In [49], both the ultraproduct and the cataproduct arecalledchromatic products, inspired by our musical notationR♮ andR♯ respectively (a thirdchromatic product, not discussed in this paper, is called theproto-productand denotedR♭).

What follows is a brief outline of the present paper. To illustrate the methods and con-cepts, I will here only treat the special case that(R♮,m♮) is an ultra-Noetherian local ring,realized as the ultraproduct of Noetherian local rings(Rn,mn) of the same embeddingdimension. Section2 contains general facts of local rings of finite embedding dimension,by far the most important of which is the already mentioned result that its completion isNoetherian (Theorem2.2). In particular, the cataproductR♯ is Noetherian.2 Our first taskis now to develop a good dimension theory, which is done in§§3–5. Krull dimension inthis context is of minor use, as it is always infinite for example for ultra-Noetherian localrings, except when almost allRn are Artinian of a fixed lengthl, in which caseR♮ is alsozero-dimensional and has lengthl. A first variant, calledgeometric dimension, is inspiredby the geometric intuition that dimension is the least number of hypersurfaces cutting outa finitely supported subscheme. Specifically, thegeometric dimension, gdim(R♮), of R♮

is the least numberd of non-unitsx1, . . . , xd such thatR♮/(x1, . . . , xd)R♮ is Artinian.Other variants are obtained by the general principle discussed above: theultra-dimension,udim(R♮), ofR♮ is the common dimension of almost allRn; and itscata-dimensionis thedimension of its completion, that is to say, ofR♯. It turns out that the cata-dimension isequal to the geometric dimension (Theorem3.4). These dimensions also have a combina-torial nature: whereas Krull dimension is the combinatorial dimension of the full spectrumSpec(R♮), the ultra-dimension ofR♮ is equal to the combinatorial dimension of the subsetof all associated prime ideals of finitely generated ideals;the cata-dimension is equal tothe combinatorial dimension of the subset of allm♮-adically closed prime ideals (Theo-rem 5.18; see also [33] for some related results). The ultra-dimension ofR♮ is at mostits cata-dimension, with equality precisely when almost all Rn have the same parameterdegree (Theorem5.22).

Our next step is to develop a singularity theory for local rings of finite embedding di-mension. Three options present themselves to us: cata-singularities via completions (§6);ultra-singularities via ultraproducts (§8); and pseudo-singularities via homological algebra(§7). For instance,R♮ is calledcata-regularif R♯ is regular;ultra-regular if almost allRn

are regular; andpseudo-regularif its depth equals its embedding dimension. Requiringeach of the quantities

(1) depth(R) ≤ udim(R♮) ≤ gdim(R♮) ≤ embdim(R♮)

in this chain of inequalities to be equal to the last turns outto determine these regularityconditions, in decreasing order of strength: pseudo-regularity, ultra-regularity, and cata-regularity respectively (note that we do not observe such a distinction in the Noetheriancase). In fact, the two first conditions are equal (Theorem8.1). Moreover, Serre’s crite-rion for regularity extends to this larger class (Theorem7.11). In particular, for coherentlocal rings of finite embedding dimension, regular in the sense of Bertin ([7, 18]) implies

2Special cases of this result were already observed and used by various authors [4, 6, 32].

DIMENSION AND SINGULARITY THEORY FOR LOCAL RINGS OF FINITE EMBEDDING DIMENSION 3

pseudo-regular, and the converse holds for uniformly coherent local rings of finite em-bedding dimension (Theorem7.18). Next, variants of the Cohen-Macaulay property areanalyzed—for instance, by equating the first quantity in (1) with respectively the secondand third, we get the notions of ultra-Cohen-Macaulay and pseudo-Cohen-Macaulay localrings. Unfortunately, these variants behave less well. Forinstance, although the class of lo-cal Cohen-Macaulay rings of fixed dimension and multiplicity is closed under cataproducts(Corollary8.7), the converse need not be true, that is to say,R♯ can be Cohen-Macaulaywithout theRn being Cohen-Macaulay. At the source of these discrepancieslies the factthat a sequence can be quasi-regular without being regular in non-Noetherian rings. In5.19, we present an example showing that all of the four quantities in (1) can be different.AlthoughR♮ is rarely coherent, under an additional pseudo-Cohen-Macaulay assumption,it behaves much like one: anym♮-primary ideal, and more generally, any finitely generatedultra-Cohen-Macaulay module is finitely presented. Another generalization of the Cohen-Macaulay condition for local rings of finite embedding dimension, motivated by model-theoretic considerations, was introduced in [37]; we show that up to a Nagata extensionof the ring (which can be taken to be trivial in the ultra-Noetherian case), this conditionis equivalent with being pseudo-Cohen-Macaulay (Theorem7.26). Some further charac-terizations of the various types of Cohen-Macaulay singularities are given in§9 by meansof an analogue of Noether Normalization for the class of local rings of finite embeddingdimension.

Once we have developed a sufficiently well-behaved singularity theory, we analyze thehomological theory of the class of local rings of finite embedding dimension; this is thecontents of§10. We show that most homological theorems, properly restated, hold in anarbitrary equicharacteristic local ring of finite embedding dimension. The main tool isthe existence of an analogue of big Cohen-Macaulay algebrasfor this class of rings. Infact, it suffices to assume that only the completion is equicharacteristic, which is a strictlyweaker condition, as I will explain below. As an application, we provide the followingpartial answer to a question raised by Glaz ([19]) about the extent to which split subringsof coherent regular local rings are Cohen-Macaulay (note that in the Noetherian case, eithercondition would imply thatR is Cohen-Macaulay; for a different answer, see [20, Corollary4.5]).

1.1.Corollary. If (R,m) is a local ring of finite embedding dimension containing a field,and if S a coherent regular local ring locally of finite type overR, such thatR → S iscyclically pure (e.g., split), then there exists a (Noetherian) regular local subring(A, p) ofR such that each maximalA-regular sequence is a maximal quasi-regular sequence inR,and eachR/pnR is a finite, freeA/pn-module.

In the final three sections, we apply the theory to ultra-Noetherian rings to obtain newresults about Noetherian local rings. In§11, we derive uniform bounds on Betti and Bassnumbers. In the literature, one often studies the asymptotic growth of theBetti numbersβn(M) = dimk(Tor

Rn (M,k)), asn goes to infinity, forM a finitely generated module

over a Noetherian local ringR with residue fieldk. In contrast, varying the module andfixing n, we show in Theorem11.1 that over a local Cohen-Macaulay ringR, then-thBetti number of a moduleM of finite length is bounded by a function which only dependson the dimension and multiplicity ofR and the length ofM . In particular, ifPR(t) :=∑

n βn(k)tn denotes thePoincare seriesof R, then we show:

4 HANS SCHOUTENS

1.2.Corollary. For eachd, e ≥ 0, there exists a power seriesPd,e(t) ∈ Z[[t]] such that thePoincare seriesPR(t) of anyd-dimensional local Cohen-Macaulay ringR of multiplicitye is dominated byPd,e(t), meaning thatPd,e(t)− PR(t) has non-negative coefficients.

Recall that a Cohen-Macaulay local ringR is called ofbounded multiplicity typeifthere is a bound on the multiplicity of all of its indecomposable maximal Cohen-Macaulaymodules. According to the Brauer-Thrall conjectures such aring is expected to be offiniterepresentation type, meaning that there exist only finitely many indecomposablemaximalCohen-Macaulay modules (see, for instance, [11, 34, 54] for some known cases of thisconjecture). In support of this, we prove the following ‘universal resolution’ for maximalCohen-Macaulay modules:

1.3.Corollary. SupposeR is a local Cohen-Macaulay ring of bounded multiplicity type.There exists anR-algebraZ, and a complex of finite freeZ-modulesF•, such that for everyindecomposable maximal Cohen-Macaulay moduleM , there exists a sectionZ → R, suchthatF• ⊗Z R is a free resolution ofM .

The theory also gives applications to preservation of properties under infinitesimal de-formations, of which the next result is but an example (recall that aninvertible idealis aprincipal ideal generated by a non zero-divisor):

1.4. Corollary. LetR be a local Cohen-Macaulay ring and letI ⊆ R be an invertibleideal. There exists a positive integera := a(I) with the property that ifJ ⊆ R suchthatR/J is Cohen-Macaulay of multiplicity at most the multiplicityofR/I, and such thatI +ma = J +ma, thenJ is invertible.

It is not clear yet whether similar bounds exist if we drop theCohen-Macaulay assump-tion in these results. In§12, we characterize ring-theoretic properties in terms of uniformarithmetic in the ring. For instance, in Theorem12.1, we reprove, as an illustration of ourmethods, that multiplication is bounded inR if and only if R is analytically irreducible.Whereas the ultraproduct method only gives the existence ofa uniform bound, we knowin this particular case, by the work of Hubl-Swanson [27, 53], that these bounds can betaken to be linear. Nonetheless, our method is far more versatile, allowing us to derivein §12.8many more characterizations of ring-theoretic propertiesin terms of certain uni-form asymptotic behavior of (m-adic)order and (parameter)degree. For instance, one cancharacterize the Cohen-Macaulay property as follows:

Theorem12.14. For each quadruple(d, e, a, b) there exists a boundδ(d, e, a, b) with thefollowing property. Ad-dimensional Noetherian local ring(R,m) of multiplicitye is Coh-en-Macaulay if and only if for each idealI generated byd − 1 elements, and for any twoelementsx, y ∈ R, if R/(I + xR) has length at mosta andy does not belong toI + mb,thenxy does not belong toI +mδ(d,e,a,b).

As already mentioned, our methods only prove the existence of uniform bounds (andpossibly their dependence on other invariants), but say nothing about the nature of thesebounds. It would be interesting to see whether for instance these new bounds also have alinear character.

However, the main application of this paper is discussed in the final section. Herewe derive some asymptotic versions of the homological theorems in mixed characteris-tic. Whereas the papers [39, 47] relied on a deep result from model theory, the so-calledAx-Kochen-Ershov theorem, to carry out transfer from mixedto equal characteristic,3 the

3In fact, although not mentioned explicitly in these papers (but see [49, §14]), these methods make heavilyuse of proto-products, one of the chromatic products not studied in this paper.


present paper departs from the following simple observation: if the (Rn,mn) have mixedcharacteristicpn, then their cataproductR♯ is equicharacteristic in the following two cases:(i) the pn grow unboundedly (in which case the ultraproductR♮ is already equicharacter-istic), or (ii), almost allpn are equal to a fixed prime numberp, but the ramification index,that is to say, themn-adic order ofp, grows unboundedly (in which caseR♮ still has mixedcharacteristicp). Thus we prove:

Theorem13.6(Asymptotic Improved New Intersection Theorem). For each triple of posi-tive integers(m, r, l) there exists a boundκ(m, r, l)with the following property. Let(R,m)be a mixed characteristic Noetherian local ring of embedding dimensionm and letF• bea finite complex of finitely generated freeR-modules of rank at mostr. If eachHi(F•), fori > 0, has length at mostl andH0(F•) has a non-zero minimal generator generating asubmodule of length at mostl, then the length ofF• is at least the dimension ofR, providedeither the residual characteristic or the ramification index ofR is at leastκ(m, r, l).

It should be noted that some Homological Conjectures, such as the Direct SummandConjecture and the Hochster-Roberts theorem on the Cohen-Macaulayness of pure sub-rings of regular local rings, at present elude our methods, and so no asymptotic versionsin the style of this paper are known (but see [47, §9 and§10] for different asymptoticversions).

I conclude the paper with a sketch of an argument that derivesthe full version fromits asymptotic counterpart, provided the bounding function does not grow too fast. Forexample, if for some primep, the boundκ(m, r, l) on the ramification in the above theoremcan be taken to be of the formc(m, r)lα(m,r), for some real valued functionsc(m, r) andα(m, r) with α(m, r) < 1, for all m andr, then the Improved New Intersection Theoremholds in mixed characteristicp.

2. FINITE EMBEDDING DIMENSION

Although we will mainly be interested in the maximal adic topology of a local ring, westart our exposition in a more general setup.

2.1. Filtrations. Recall that afiltration I = (In)n on a ringA is a descending chain ofidealsA = I0 ⊇ I1 ⊇ · · · ⊇ In ⊇ . . . . An important instance of a filtration is obtained bytaking the powers of a fixed idealI ⊆ A, that is to say,In := In; we call this theI-adicfiltration onA. A filtration I defines a topology onA, called theI-adic topologyof A,by taking for basic open subsets all cosets of allIn. If B is anA-algebra, thenIB is theextended filtrationonB given by the idealsInB, and hence the natural homomorphismA→ B is continuous with respect to the respective adic topologies. The intersection of allIn will be denoted byI∞. Hence theI-adic topology is Haussdorf (separated) if and onlyif I∞ = (0). Accordingly, the quotientA/I∞ is called theI-adic separated quotientofA. TheI-adic completionof A is defined as the inverse limit of theA/In and is denotedAI. There is a natural mapA→ AI whose kernel is equal toI∞. In fact,A and itsI-adicseparated quotient have the sameI-adic completion. In general,AI, although complete inthe inverse limit topology, need not be complete in theIAI-adic topology.

Given a filtrationI = (In)n we define itsassociated graded module, where we viewAwith its trivial grading, as the direct sum

grI(A) :=

∞⊕

n=0

In/In+1.

6 HANS SCHOUTENS

The initial form inI(a) ∈ grI(A) and theI-adic orderordI(a) of an elementa ∈ A aredefined as follows. Ifa ∈ In\In+1 for somen, then we setordI(a) := n and we letinI(a)be the image ofa in In/In+1; otherwisea ∈ I∞, in which case we setordI(a) := ∞ andinI(a) := 0. For J an ideal inA, we let inI(J) be the ideal ingrI(A) generated by allinI(a) with a ∈ J . If J = (a1, . . . , an)A, theninI(J) is in general larger than the idealgenerated by theinI(ai) (even ifA is Noetherian!).

Alternatively, we may think of a filtration as given by a functionf : A→ N := N∪{∞}such thatf(a + b) andf(ab) are greater than or equal to respectively the minimum andthe maximum off(a) andf(b); we express this by callingf filtering. Given a filteringfunction f , the idealsIn of all elementsa ∈ A for which f(a) ≥ n form a filtration.Conversely, given a filtrationI, the functionordI(·) is filtering. Supposef is filtering. Iff(ab) ≥ f(a) + f(b), then we callf multiplicative(this then corresponds to the propertythat InIm ⊆ In+m); and if 0 is the only element of infinitef -value (so that the corre-sponding filtration is separated) andf(ab) = f(a) + f(b), thenf is called avaluation. IfI is multiplicative, thengrI(A) admits the structure of a ring and as such is graded. Thisapplies in particular to any ideal adic filtration.

We now specify these notions to the case of interest, whereI is them-adic filtration ofa local ring(R,m). The topology onR is always assumed to be them-adic topology, sothat when we say thatR is separated or complete, we are always referring to this topology.With this in mind, theideal of infinitesimalsof R is the intersection of allmn, and will bedenotedIR. Them-adic order of an elementx ∈ R is denotedordR(x) or justord(x).The (m-adic) separated quotientR/IR is denotedRsep; the graded ring associated tom isdenotedgr(R); and the completion ofR is denotedR. By construction,R is a completelocal ring whose maximal ideal is equal to the inverse limit of them/mn. However, thismaximal ideal may be strictly larger thanmR, so thatR need not be complete in themR-adic topology.

Let (S, n) be a second local ring and letR → S be a ring homomorphism. We callthis homomorphismlocal, or we say thatS is a local R-algebra, ifmS ⊆ n; if we haveequality, then we call the homomorphismunramified. A local homomorphism induceslocal homomorphismsRsep→ SsepandR → S. The natural mapR → R is local. It is flatif R is Noetherian, but no so in general.

Finite embedding dimension. Suppose from now on thatR has moreover finite embed-ding dimension, that is to say, thatm is finitely generated. Sincegr(R) is generated bym/m2 as an algebra over the fieldR/m, it is itself a Noetherian local ring. For eachn, letmn be the kernel of the natural mapR → R/mn. It follows thatmn/mn+1 ∼= mn/mn+1,so thatgr(R) is equal to the graded ringgrM(R) associated to the filtrationM := (mn)non R. By [12, Proposition 7.12], an idealI ⊆ R is generated by elementsa1, . . . , an ifits initial from inM(I) in grM(R) is generated by the initial formsinM(a1), . . . , inM(an).Therefore, sincegrM(R) ∼= gr(R) is Noetherian, so isR. Moreover, sincemnR hasthe same initial form asmn, both ideals are equal. In particular, for eachn, we have anisomorphismR/mn ∼= R/mnR. In conclusion, we have proven:

2.2.Theorem. If (R,m) is a local ring of finite embedding dimension, then its completionR is a complete Noetherian local ring with maximal idealmR. �

2.3. Corollary. If a local ring (R,m) has finite embedding dimension, then eachm-primary ideal is finitely generated.


Proof. Immediate from the fact thatR/mn is Artinian andmn is finitely generated, foreveryn. �

An idealI in a local ring(R,m) is calledclosedif it is closed in them-adic topology,that is to say, ifI is equal to the intersection of allI +mn with n ∈ N.

2.4. Lemma. Let (R,m) be a local ring of finite embedding dimension and letI be anarbitrary ideal inR. The completion ofR/I is R/IR. In particular, IR ∩ R = I if andonly if I is closed.

Proof. Let R := R/I and letS := R/IR = R⊗R R. The isomorphismR/mn ∼= R/mnRinduces by base change an isomorphismR/mnR ∼= S/mnS. HenceR andS have thesame completion. However, sinceR is complete, so isS, showing that it is the completionof R.

Applied withI anm-primary ideal, we get an isomorphismR/I ∼= R/IR showing thatIR ∩ R = I, that is to say, thatI is contracted fromR. Since this property is preservedunder arbitrary intersections, every closed idealI is contracted fromR, as it is the inter-section of them-primary idealsI + mn. Conversely, ifIR ∩ R = I, thenR/I embeds inR/IR, and by the first assertion, this is its completion. In particular,R/I is separated, thatis to say,I is closed. �

The above proof shows that the closure of an idealI is equal toIR ∩ R. In partic-ular, any closed ideal is the closure of a finitely generated ideal, sinceR is Noetherianby Theorem2.2. Moreover, the ascending chain condition holds for closed ideals inR:if I1 ⊆ I2 ⊆ . . . is an increasing chain of closed ideals inR, then, sinceR is Noethe-rian, their extension toR must become stationary, sayInR = In+kR for all k, and hencecontracting back toR givesIn = In+k for all k. This immediately yields:

2.5.Corollary. A local ring is Noetherian if and only if it has finite embedding dimensionand every ideal is closed. �

2.6.Corollary. A closed ideal in a local ringR of finite embedding dimension has finitelymany minimal primes and each of them is closed.

Proof. Let I be a closed ideal and letQ1, . . . ,Qs be the minimal prime ideals ofIR. Letqi := Qi ∩R and letJ be their product. HenceJn ⊆ IR for somen. By Lemma2.4, wehaveJn ⊆ IR ∩ R = I. Hence any prime idealp of R containingI contains one of theqi. This shows that all minimal prime ideals ofI must be among theqi. �

2.7.Corollary. If (R,m) is a local ring of finite embedding dimension, then the image ofthe mapSpec(R) → Spec(R) consists precisely of the closed prime ideals ofR.

Proof. By Lemma2.4, the image of the map consists of closed prime ideals. To prove theconverse, letp be an arbitrary closed prime ideal ofR. By Lemma2.4, we havep = pR∩R.LetN be maximal inR with the property thatp = N ∩R. I claim thatN is a prime ideal,showing thatp lies in the image ofSpec(R) → Spec(R). To prove the claim, supposefg ∈ N, butf, g /∈ N. By maximality, there exista, b ∈ R \ p such thata ∈ N+ fR andb ∈ N + gR. Henceab ∈ N + fgR = N and sinceab ∈ R, we getab ∈ N ∩ R = p,contradicting thatp is prime. �

2.8. Lemma. If the completion of a local ring(R,m) of finite embedding dimension isArtinian, then so isR.

8 HANS SCHOUTENS

Proof. By assumption,mnR = 0, for somen. SinceR/mn+1 ∼= R/mn+1R = R, wegetmn/mn+1 = 0. Sincem is finitely generated, we may apply Nakayama’s Lemma andconclude thatmn = 0, which implies thatR is Artinian. �

2.9. Infinite ramification. We conclude this section with a note on ramification in mixedcharacteristic, which we will use occasionally. Let(R,m) be a local ring with residuefield k. We say thatR is equicharacteristic(or hasequal characteristic) if R andk havethe same characteristic; in the remaining case, that is to say, if R has characteristic0 andkcharacteristicp, we say thatR hasmixed characteristicp. A local ring is equicharacteristicif and only if it contains a field.

For the next definition, assume that the residue field ofR has characteristicp. We callord(p) theramification indexof R. We sayR is unramifiedif its ramification index is one;andinfinitely ramified, if its ramification index is infinite, that is to say, ifp ∈ IR. If R isinfinitely ramified and Noetherian (or just separated), thenin fact it has equal characteristicp (in the literature this is also deemed as an instance of an ‘unramified’ local ring, but forus, it will be more useful to make the distinction). However,in the general case, a localring can have characteristic zero and be infinitely ramified (see Lemma13.5below). Itfollows that the separated quotient and the completion of aninfinitely ramified local ringare both equicharacteristic.

3. GEOMETRIC DIMENSION

Thedimensiondim(A) of a ringA will always mean its Krull dimension, that is to say,the maximal length (possible infinite) of a chain of prime ideals inA. Thedimensionof anidealI ⊆ A is the dimension of its residue ringA/I. If R is local and Noetherian, then itsdimension is always finite, but without the Noetherian assumption, it is generally infinite.In this section, we propose a first substitute for Krull dimension for an arbitrary local ring(R,m); other alternatives will be discussed in§4.

3.1.Definition. We define thegeometric dimensionof R recursively as follows. We saythatR has geometric dimension zero, and we writegdim(R) = 0, if and only if R isArtinian. For arbitraryd, we say thatgdim(R) ≤ d, if there existsx ∈ m such thatgdim(R/xR) ≤ d − 1. Finally, we say thatR has geometric dimension equal tod ifgdim(R) ≤ d, but notgdim(R) ≤ d− 1, and we simply writegdim(R) := d. If there isnod such thatgdim(R) ≤ d, then we setgdim(R) := ∞.

It follows thatgdim(R) ≤ embdim(R). In fact,R has finite geometric dimension ifand only if it has finite embedding dimension. IfR has finite embedding dimension thengdim(R) = 0 if and only if m is nilpotent. The following fact is immediate from thedefinition.

3.2.Lemma. If (R,m) is a local ring anda ∈ m, then

gdim(R)− 1 ≤ gdim(R/aR) ≤ gdim(R).

�

The geometric dimension can be formulated, as in the Noetherian case, in terms of theminimal number of generators of anm-primary ideal (showing that geometric dimensionand Krull dimension agree for Noetherian local rings):

3.3.Lemma. The geometric dimension of a local ring(R,m) of finite embedding dimen-sion is the least possible number of elements generating anm-primary ideal.


Proof. Let d := gdim(R). By Lemma3.2, there exists no sequencey of length less thand such thatR/yR has geometric dimension zero. It follows that anym-primary ideal isgenerated by at leastd elements. So remains to show that there exists a tuple of length dgenerating anm-primary ideal. We induct ond, where the cased = 0 is clear, since then(0)ism-primary. By definition, we can choosex1 ∈ m such thatgdim(R/x1R) = d− 1. Byinduction, there exist elementsx2, . . . , xd whose image inR/x1R generate anm(R/x1R)-primary ideal. Hence(x1, . . . , xd)R ism-primary. �

3.4.Theorem. Let (R,m) be a local ring of finite embedding dimension. The followingnumbers are all equal.

• the geometric dimensiond ofR;• the least possible number of elementsd′ generating anm-primary ideal;• the dimensiond of the completionR ofR;• the dimensiond of the graded ringgr(R) associated toR;• the degreed of theHilbert-Samuel polynomialHSR, whereHSR is the unique

polynomial with rational coefficients for whichHSR(n) equals the length ofR/mn for all large n;

• the geometric dimensiondsepof the separated quotientRsep;

Proof. The equality ofd andd′ is given by Lemma3.3. We already observed thatgr(R)andR are Noetherian and that we have isomorphismsmn/mn+1 ∼= mnR/mn+1R for alln. HenceHSR = HS bR andgr(R) ∼= gr(R). It follows thatd = d, by the Hilbert-Samuel

theory and thatd = d by [29, Theorem 13.9]. This shows already thatd = d = d.Let (y1, . . . , yd) be a tuple inR generating anm-primary ideal. Since(y1, . . . , yd)R is

thenmR-primary,d ≤ d. Finally, let(ξ1, . . . , ξd) be a homogeneous system of parametersof gr(R) and choosexi ∈ R such thatξi = in(xi). Let I := (x1, . . . , xd)R. By [12,Exercise 5.3], we have an isomorphism

gr(R)/ in(I) ∼= gr(R/I).

Since(ξ1, . . . , ξd) gr(R) ⊆ in(I), we see thatgr(R)/ in(I) is Artinian, whence so isgr(R/I). This in turn means thatR/I has a nilpotent maximal ideal, so thatd ≤ d bydefinition of geometric dimension. This proves that the firstfive numbers in the statementare equal. That they are also equal to the last,dsep, follows by applying the result toRsep

together with the fact thatR andRsephave the same completion. �

3.5.Remark.If the leading coefficient of the Hilbert-Samuel polynomialis written ase/d!,with d := gdim(R), then we calle the multiplicity of R and we denote itmult(R). Itfollows thatR has the same multiplicity as its completion and as its separated quotient.

3.6.Corollary. If R is a local ring of geometric dimension one, then there existsN ∈ Nsuch that every closed ideal is the closure of anN -generated ideal.

Proof. By Theorem3.4, the completionR is a one-dimensional Noetherian local ring, andhence by the Akizuki-Cohen theorem ([1, 10]), there is someN such that every ideal inRis generated by at mostN elements. LetI ⊆ R be an arbitrary ideal. SinceIR is generatedby at mostN elements, we may choose by Nakayama’s Lemmaa1, . . . , aN ∈ I such thatIR = (a1, . . . , aN )R. Contracting this equality back toR shows, by Lemma2.4, thatI isthe closure of(a1, . . . , aN )R. �

10 HANS SCHOUTENS

It is well-known that one may takeN to be equal to the multiplicity ofR, in case thelatter is Cohen-Macaulay. In view of Remark3.5and our definition in§6 below, the sameholds true under the assumption thatR is cata-Cohen-Macaulay.

3.7. Generic sequences.A tuple x is calledgeneric, if it generates anm-primary idealand its length is equal to the geometric dimension ofR; it is called part of a genericsequence, if it can be extended to a generic sequence. Ifx is a single element which is partof a generic sequence, then we simply callx a generic element.

3.8.Lemma. Let(R,m) be a local ring of geometric dimensiond. A tuple(x1, . . . , xe) ispart of a generic sequence if and only ifR/(x1, . . . , xe)R has geometric dimensiond− e.

In particular,x is generic if and only ifgdim(R/xR) = gdim(R)− 1.

Proof. Suppose(x1, . . . , xe) is part of a generic sequence and enlarge it to a generic se-quence(x1, . . . , xd). One checks that (the image of)(xe+1, . . . , xd) is a generic sequencein R/(x1, . . . , xe)R. This shows thatgdim(R/(x1, . . . , xe)R) = d − e. Conversely, as-sumegdim(R/(x1, . . . , xe)R) = d−e. Choose a tuple(xe+1, . . . , xd) inR so that its im-age inR/(x1, . . . , xe)R is a generic sequence. Since(x1, . . . , xd) generates anm-primaryideal and has lengthd, it is generic. �

3.9.Proposition. Let(R,m) be a local ring of finite embedding dimension. A sequence inR is generic if and only if its image inR is a system of parameters.

Proof. One direction has already been noted, so letx be a tuple inR whose image inRis a system of parameters. By Theorem3.4, the geometric dimension ofR is equal to thelength of this tuple. LetJ := xR. By Lemma2.4, the completion ofR/J is R/JR. Asthe latter is Artinian, so must the former be by Lemma2.8, showing thatx is generic. �

It follows that (x1, . . . , xd) is generic if and only if so is(xn1

1 , . . . , xnd

d ). However,this does in general not imply that(in(x1), . . . , in(xd)) is a system of parameters ingr(R)(this even fails in the Noetherian case as the example{ξ2, ξζ + ζ3} in k[[ξ, ζ]] shows).Immediately from Proposition3.9and [29, Theorem 14.5] we get:

3.10.Corollary. Any generic sequencex inR is analytically independentin the sense thatif F (ξ) is a homogeneous form overR such thatF (x) = 0, then all coefficients ofF (ξ)lie in the maximal ideal ofR. �

3.11. Threshold primes. By Proposition3.9, x is generic if and only if the image ofx inR is part of a system of parameters. More concretely, letd be the geometric dimension ofRand letp1, . . . ps be thed-dimensional prime ideals ofR. Note thatR itself has dimensiond by Theorem3.4, so that all itsd-dimensional primes are minimal (but there may be otherminimal prime ideals, of lower dimension). We call theqi := pi ∩R the thresholdprimesof R. By Corollary2.7, every threshold primeq is closed and contains no proper closedprime ideals. Moreover,R/q has the same geometric dimension asR by Theorem3.4,sinceR/qR has the same dimension asR. By a threshold primeof an idealI, we mean athreshold prime of its residue ringR/I. Proposition3.9yields the following criterion forgenericity.

3.12.Corollary. An elementx ∈ R is generic if and only if it is not contained in anythreshold prime ofR. In particular, the product of any two generic elements is againgeneric. �


3.13.Corollary. Anym-primary ideal contains a generic sequence. More precisely, if Ris aZ-algebra andI ⊆ Z an ideal such thatIR is m-primary, then there exists a tuplexoverZ with entries inI such that its image inR is a generic sequence.

Proof. We prove the last assertion by induction on the geometric dimensiond of R. Sincethere is nothing to show ifd = 0, we may assumed > 0. Let q1, . . . , qs be the thresholdprimes ofR. Towards a contradiction, supposeI is contained in the union of theqi∩Z. Byprime avoidance, there is somei such thatI ⊆ qi ∩ Z. But thenIR ⊆ qi, forcingqi = m,thus contradicting by Corollary3.12thatd > 0. Hence there existsx ∈ I so that its imagein R lies outside every threshold prime ofR, and therefore is generic by Corollary3.12.By Lemma3.8, the geometric dimension ofR/xR is d − 1. Therefore, by induction, wecan find a tupley of lengthd − 1 with entries inI so that its image inR/xR is generic.The desired sequence is now given by addingx to this tupley. �

In [20], the authors introduce the notion of astrong parameter sequence. It should benoted that this is different from our present notion of generic sequence. For example, ifV is an ultra-discrete valuation ring (see Example6.3 for more details), andx a non-zeroinfinitesimal inV , thenx is V -regular, whence a strong parameter by [20, Proposition3.3(f)], butx is clearly not generic (in fact, the unique threshold prime of V is the ideal ofinfinitesimalsIV ).

3.14. Geometric codimension.Given an idealI in a local ring(R,m) of finite embed-ding dimension, we call itsgeometric codimensionthe maximal length of a tuple inI thatis part of a generic sequence and we denote itgcodim(I). In particular, an ideal ism-primary if and only if its geometric codimension equals the geometric dimension ofR.Our terminology is justified by the next result.

3.15.Proposition. Let (R,m) be a local ring of finite embedding dimension. For everyidealI ⊆ R, we have an equalitygcodim(I) = gdim(R)− gdim(R/I).

Proof. Let d be the geometric dimension ofR and leth be the geometric codimensionof I. Choose a tupley in I of lengthh which is part of a generic sequence ofR. PutS := R/yR, so thatgdim(S) = d − h by Lemma3.8. SinceIS contains no genericelement, it must be contained in some threshold primeq of S by Corollary3.12. From theinclusionsIS ⊆ q we getgdim(S) ≥ gdim(S/IS) ≥ gdim(S/q) = gdim(S), and henceall these geometric dimensions are equal tod− h. SinceS/IS = R/I, we are done. �

3.16. Parameter degree and degree.We conclude this section with another genericitycriterion, in terms of an invariant which was introduced forNoetherian rings in [46, 47]and which will play a crucial role in what follows. Theparameter degreeof a local ringRof finite embedding dimension is by definition the minimal length of a residue ringR/xR,wherex runs over all possible generic sequences ofR. We denote the parameter degree ofR bypardeg(R). We will show in Lemma6.10below that the multiplicity ofR is boundedby its parameter degree and indicate when they are equal.

Closely related to this is an invariant, which for want of a better name, we calldegreeand which is defined as follows. LetR be a local ring of geometric dimensiond ≥ 1.We define thedegreedegR(x) of an elementx to be the least possible length of a residueringR/(xR+ yR), wherey runs over all tuples of lengthd− 1 inside the maximal ideal.Hence, ifx is a unit, its degree is zero; ifx is generic, its degree is the parameter degree ofR/xR; and in the remaining case, its degree is infinite. In particular, we showed:

12 HANS SCHOUTENS

3.17.Corollary. An non-unit in a non-Artinian local ringR of finite embedding dimensionis generic if and only if its degree is finite. Moreover, the parameter degree ofR is theminimum of the degrees of all non-units inR. �

4. EXTENDED DIMENSIONS

In this section, we introduce several other dimension notions for a local ring(R,m).With anextended dimension, we mean an invariant on the class of local rings taking valuesin N := N ∪ {∞} which agrees with Krull dimension on the subclass of all Noetherianlocal rings. Clearly, Krull dimension itself is an extendeddimension, and so is geomet-ric dimension by the results from the previous section. Note, however, that embeddingdimension isnot an extended dimension.

Recall that a partially ordered setΓ hascombinatorial dimension(or, height) d if anyproper (ascending) chain inΓ has length at mostd (meaning that it contains at mostd+ 1elements). Hence, the dimension of a ringA is the combinatorial dimension ofSpec(A)(the set of all prime ideals ordered by inclusion). Given idealsJ ⊆ p in A with p prime,we say thatp is anassociatedprime ofJ if p is of the form(J : a); a minimalprime ofJif no prime ideal is properly contained betweenJ andp; and aminimal associatedprimeof J if it is associated and no associated prime ofJ is properly contained betweenJ andp.

4.1. Cl-dimension. Let CL-Spec(R) be the subset ofSpec(R) consisting of all closedprime ideals ofR. Note that the maximal ideal as well as the threshold primes (see§3.11)belong toCL-Spec(R). In fact, we showed in Corollary2.7thatCL-Spec(R) is the imageof the canonical mapSpec(R) → Spec(R). We call the combinatorial dimension ofCL-Spec(R) thecl-dimensionof R and denote itcldim(R). It is clear thatcldim(R) =dim(R) whenR is Noetherian, showing that cl-dimension is an extended dimension.

4.2. Fr-dimension. We say that an idealI ⊆ R is n-generated, if there exists a tuplexof lengthn such thatxR = I. We say that an ideala ⊆ R is n-related if it is of theform a = (I : a) with I ann-generated ideal. An ideala is calledfinitely relatedif it isn-related for somen < ∞. Let FR-Spec(R) be the subset ofSpec(R) consisting of allfinitely related prime ideals, that is to say, all associatedprime ideals of finitely generatedideals ofR. We call the combinatorial dimension ofFR-Spec(R) the fr-dimensionofR and denote itfrdim(R). WhenR is Noetherian, every ideal is finitely related whencefrdim(R) = dim(R), showing that fr-dimension is an extended dimension. We definethe related notion of astrongly finitely relatedprime ideal as a prime idealp of the form(I : a) with I finitely generated anda /∈ p. A priori, not every finitely related prime idealis strong, but see Corollaries5.3and5.26.

4.3. Pi-dimension. We say thatR haspi-dimensionat mostd, if m is a minimal associatedprime of ad-generated ideal. The pi-dimension,pidim(R), of R is then the leastd suchthatR has pi-dimension at mostd. That pi-dimension is an extended dimension followsfrom Krull’s Principal Ideal theorem (from which it borrowsits name; see for instance [29,Theorem 8.10]).

4.4. Theorem. For an arbitrary local ring (R,m), we have the following inequalitiesbetween extended dimensions:

(4.4.1) frdim(R), cldim(R) ≤ dim(R);(4.4.2) pidim(R) ≤ gdim(R);(4.4.3) cldim(R) ≤ gdim(R), with equality ifgdim(R) is finite.


Moreover, each of these inequalities can be strict.

Proof. Inequalities (4.4.1) are immediate from the definition. In order to show inequal-ity (4.4.2), we may assume thatgdim(R) = d < ∞. By definition,R/I is an Artinianlocal ring for somed-generated idealI. It follows thatm is a minimal associated prime ofI, whence the pi-dimension ofR is at mostd.

So remains to prove (4.4.3). There is nothing to show ifR has infinite geometric dimen-sion, so assumeR has finite geometric dimension, say,d (whence also finite embeddingdimension). By Corollary2.7, there is a surjective mapSpec(R) → CL-Spec(R). Inparticular, the combinatorial dimension ofCL-Spec(R) is at most the dimension ofR,that is to say, in view of Theorem3.4, at mostd. So remains to prove the other inequalityby induction ond. There is nothing to show ifd = 0, so we may assumed > 0. ByCorollary2.7, the minimal elements inCL-Spec(R) are the contractions of the minimalprimes ofR. Hence there are only finitely many of them, all different from the maxi-mal idealm. By prime avoidance, we may choosex ∈ m outside all these finitely manyprime ideals. In particular, since the threshold primes areamong these,x is generic andhenceR/xR has geometric dimensiond − 1. By induction, the combinatorial dimen-sion ofCL-Spec(R/xR) is d − 1. By Lemma2.4, the completion ofR/xR is R/xR.The homomorphismR → R/xR induces an injectionSpec(R/xR) → Spec(R), whoseimage is the subset of all prime ideals ofR containingx. It follows that the canonicalinjectionSpec(R/xR) → SpecR mapsCL-Spec(R/xR) into the subset ofCL-Spec(R)consisting of all closed prime ideals containingx. Using this and the fact that the com-binatorial dimension ofCL-Spec(R/xR) is d − 1, we can find a proper chain of closedprimes idealsq1 q2 · · · qd = m in R with x ∈ q1. Let q0 be a minimal elementof CL-Spec(R) lying insideq1. Since by constructionx /∈ q0, theqi form a proper chainof lengthd, showing that the combinatorial dimension ofCL-Spec(R) is at leastd. Thisproves (4.4.3).

Finally, the local ring in Example4.5(respectively, in Example4.6) shows that in gen-eral, the inequalities (4.4.1) and (4.4.2) (respectively, inequality (4.4.3)) are strict. �

4.5.Example. Let R♮ be the ultraproduct (see§5 for more details) of theA/pn for n =1, 2 . . . , where(A, p) is ad-dimensional Noetherian local ring, ford > 0. Its pi-dimensionand fr-dimension are equal to zero, its geometric dimensionand cl-dimension are equal tod, and its Krull dimension is infinite.

4.6. Example. Let (R♮,m♮) be the ultraproduct of theAn/m2n for n = 1, 2 . . . , where

(An,mn) is the power series ring over a fieldk in n indeterminates. Sincem2♮ = 0 in R♮,

the local ringR♮ has cl-dimension and Krull dimension equal to zero, but its embeddingdimension, whence its geometric dimension, is infinite.

There is a more instructive way to see (4.4.2): the geometric dimension of a local ring(R,m) of finite embedding dimension is at mostd if and only ifm is a minimal prime of ad-generated ideal (that is to say, the same definition as for pi-dimension, but omitting theterm ‘associated’).

Let ‘e-dim’ be some extended dimension. We call e-dimfirst-order if the propertye-dim(·) = d is first-order in the sense of§5.5below, for everyd ∈ N. Moreover, to provethis, it suffices to show that the propertye-dim(·) ≤ d is first-order.

4.7. Lemma. Fr-dimension and pi-dimension are first-order; geometric dimension, cl-dimension and Krull dimension are not.

14 HANS SCHOUTENS

Proof. The assertion is obvious for pi-dimension, since we can express in a first-order waythat the maximal idealm of a local ring is of the form(I : a) for somed-generated idealIsuch that no prime ideal of the form(I : b) is properly contained inm (note thatm admitsa first-order definition as the collection of all non-units).As for fr-dimension, for eachn,let τn,d be the statement expressing that there does not exist a proper chain of lengthd+ 1consisting ofn-related prime ideals. Hence a local ring has fr-dimension at mostd if andonly if τn,d holds in it, for alln.

The local ring in Example4.5shows that Krull dimension, cl-dimension and geometricdimension are not first-order. �

5. ULTRA-NOETHERIAN RINGS

Before we further develop the ‘local algebra’ of local ringsof finite embedding dimen-sion, we introduce an important subclass, arising as ultraproducts of Noetherian local rings.Fix an infinite index setW and a non-principal ultrafilter onW . We will moreover assumethat the ultrafilter is countably incomplete. This is equivalent with the existence of a func-tion f : W → N such that for eachk, the set of allw ∈W for whichf(w) ≥ k belongs tothe ultrafilter. IfW is countable, then any non-principal ultrafilter is countably incomplete,and this is the situation we will find ourselves in all applications.4 For eachw ∈ W , letRw be a local ring and letR♮ be theultraproductof theRw (for a quick review on ultra-products, see [40, §1]; for more details see for instance, [14, 26, 35, 49]). It is importantto note thatRw are not uniquely defined byR♮ (not even almost all; see the example in§5.5). By Łos’ Theorem,R♮ is a local ring with maximal idealm♮ equal to the ultraproductof the maximal idealsmw. If for somem, almost allRw have embedding dimension atmostm, then we say that theRw havebounded embedding dimension; a similar usagewill be applied to other numerical invariants. Hence if theRw have bounded embeddingdimension, thenR♮ has finite embedding dimension, whence finite geometric dimension.In case allRw are equal to a single local ringR, we refer toR♮ as theultrapowerof R.

When dealing with ultraproducts, Łos’ Theorem is an extremely useful tool for trans-ferring properties between almost allRw andR♮. However, this only applies to first-orderproperties (see§5.5 below for more details). In view of this, we introduce the followingmore general set-up for discussing transfer through ultraproducts. LetP be a property oflocal rings of finite embedding dimension and letR be a local ring. We callR cata-Pif it has finite embedding dimension and its completion has propertyP. In particular, byTheorem2.2, any such ring is, in our newly devised terminology, cata-Noetherian. Wecall a local ringultra-P if it is equal to an ultraproductR♮ of local ringsRw of boundedembedding dimension almost all of which satisfy propertyP. In particular,R♮ has finiteembedding dimension too. In fact, according to this terminology, anultra-ring is any ul-traproduct of local rings of bounded embedding dimension; and anultra-Noetherianringis any ring isomorphic to an ultraproduct of Noetherian local rings of bounded embeddingdimension. It is important to notice that the well-known duality between rings and affineschemes breaks down under ultraproducts:

5.1. Proposition. Let Rw be Noetherian local rings of bounded embedding dimensionand letR♮ be their ultraproduct. Then the ultraproduct of theSpec(Rw) is equal toFR-Spec(R♮).

4In fact, it is consistent with ZF to assume that every non-principal ultrafilter on any infinite set is countablyincomplete. Moreover, for most of what we say, we will not need to assume that the ultrafilter is countablyincomplete; it is only used explicitly in Lemma5.6below.


Proof. Recall thatFR-Spec(R♮) consists of all finitely related prime ideals ofR♮ (see§4.2). If I♮ is a finitely generated ideal inR♮, say of the form(x1♮, . . . , xn♮)R♮, and ifxiw ∈ Rw are such that their ultraproduct is equal toxi♮, then the ultraproduct of theidealsIw := (x1w, . . . , xnw)Rw is equal toI♮. Moreover, ify♮ ∈ R♮ is the ultraproductof elementsyw ∈ Rw, then(I♮ : y♮) is equal to the ultraproduct of the(Iw : yw). Since(I♮ : y♮) is prime, so are almost all(Iw : yw) by Łos’ Theorem. Hence any finitely relatedprime ideal inR♮ lies in the ultraproduct of theSpec(Rw).

Conversely, for eachw, let pw be a prime ideal inRw, and letp♮ be their ultraproduct.By Łos’ Theorem,p♮ is prime. Since theRw have bounded embedding dimension, theyalso have bounded dimension. Therefore, there is ad such that almost eachRw has di-mensiond (in the terminology of§5.17below,d is the ultra-dimension ofR♮). By Krull’sPrincipal Ideal theorem, almost eachpw isd-related, whence so isp♮ by Łos’ Theorem. �

In particular, the ultraproduct of theSpec(Rw) does not depend on the choice of theRw having as ultraproductR♮. The local algebra of rings of finite embedding dimensionis hampered by the fact that very few localizations have finite embedding dimension. Wewill discuss one case here (see Corollary8.3 for another one). We first prove a bound forNoetherian rings.5 For a Noetherian ringA, let γ(A) ∈ N ∪ {∞} be the supremum of allembdim(Ap), wherep runs through all prime ideals ofA.

5.2.Proposition. If A is a d-dimensional, excellent ring, thenγ(A) < ∞. In fact, ifA isequicharacteristic and local, thenγ(A) ≤ d+ ρ, whereρ is the parameter degree ofA.

Proof. We prove the first statement by induction ond. Letp1, . . . , ps be the minimal primeideals ofA, and letN be a bound on their number of generators. Since any prime ideal pcontains one of thepi, we see thatγ(A) is bounded by the maximum of allγ(A/pi) +N .Hence we may assume without loss of generality thatA is an excellent domain. Therefore,its regular locus is non-empty and open. LetU = SpecAf be a non-empty affine opencontained in the regular locus ofA. By regularity,embdim(Ap) ≤ d, for anyp ∈ U ,and so we only need to show a bound for those prime ideals containing f . Put A :=A/fA. Note thatA has Krull dimensiond− 1 and is again excellent, so that by inductionγ(A) < ∞. Therefore, for any prime idealp of A containingf , we have an estimateembdim(Ap) ≤ γ(A) + 1, finishing the proof of the first assertion.

Assume next thatA is moreover equicharacteristic and local, with parameter degreeρ.I claim thatγ(A) ≤ γ(A), whereA is the completion ofA. Assuming the claim, we maytakeA to be complete, since parameter degree does not change undercompletion. By theCohen structure theorem,A contains ad-dimensional regular local subringR over whichit is finite. Moreover, by [46, Proposition 3.5], we may chooseR so thatA is generatedby ρ elements as anR-module. Letp be a prime ideal inA and putg := p ∩ R. Bybase change, the fiber ringAg/gAg has dimensionρ over the residue field ofg. Moreover,Ap/gAp is a direct summand ofAg/gAg by the structure theorem of Artinian local rings([12, Corollary 2.16]), whence has length at mostρ. In particular,embdim(Ap/gAp) ≤ ρ.SinceR is regular,gRg is generated by at mostd elements, whence so isgAp. It followsthatembdim(Ap) ≤ ρ+ d, as we wanted to show.

To prove the claim, letq be a minimal prime ideal ofpA. SinceA/p is excellent, itscompletionA/pA is reduced. Therefore, the localization ofA/pA at q is a field, showingthat pAq = qAq, an ideal generated by at mostγ(A) elements. SinceAp → Aq is

5In §§11 and12, we adopt the reverse strategy, by developing bounds from our local algebra results.

16 HANS SCHOUTENS

faithfully flat, pAp is therefore also generated by at mostγ(A) elements, showing thatγ(A) ≤ γ(A). �

5.3.Corollary. If R is an excellent local ring, then any localization of its ultrapowerR♮

at a finitely related prime ideal has finite embedding dimension. Moreover, every finitelyrelated prime ideal ofR♮ is strong.

Proof. Let p be a finitely related prime ideal ofR♮. By Proposition5.1, we can find primeidealspw inRwith ultraproduct equal top. Letγ(R) be the bound given by Proposition5.2on the embedding dimension of allRpw

. Since(R♮)p is the ultraproduct of theRpw, its

embedding dimension is at mostγ(R) as well. In fact, we can find idealsIw ⊆ pwgenerated by at mostγ(R) elements, so thatIwRpw

= pwRpw. Hence, there existsaw /∈

pw, such that(Iw : aw) = pw. Taking ultraproducts, we see thatp is strongly finitelyrelated (see§4.2for the definition). �

In fact, we have the following more general version of the second assertion.

5.4.Proposition. A finitely related prime idealp in an ultra-Noetherian local ringR♮ isstrongly finitely related if and only if(R♮)p has finite geometric dimension.

Proof. Note that a local ring has finite geometric dimension if and only if it has finiteembedding dimension. One direction is true in any ringA: if p is strongly finitely related,say, of the form(I : s) with I ⊆ A finitely generated ands /∈ p, thenpAp = IAp, showingthatAp has finite embedding dimension.

Conversely, suppose(R♮)p has finite geometric dimension, whence finite embeddingdimension. In particular, there exists a finitely generatedidealI ⊆ p such thatI(R♮)p =p(R♮)p. By Łos’ Theorem and Proposition5.1, we can find idealsIw ⊆ pw so thattheir respective ultraproducts areI and p. In particular, almost allpw are prime andIw(Rw)pw

= pw(Rw)pw, for almost allw. Hence, we can findsw /∈ pw such that

pw = (Iw : sw). Letting s♮ be the ultraproduct of thesw, we getp = (I : s♮) ands♮ /∈ p, showing thatp is strong. �

5.5. First-order properties. A propertyP of rings is calledfirst-order if there exists afirst-order theoryΠ, in the language of rings, such thatR is a model ofΠ if and only ifRsatisfiesP. Łos’ Theorem states that ifP is first-order, then ultra-P impliesP. Althoughwe will not use this here, the converse is also true, due to a theorem of Keisler-Shelah(see for instance [26, Theorem 9.5.7]). It follows that ifP is not first-order, then thereexists an ultra-ringS♮ which is at the same time ultra-P and ultra-non-P. Indeed, by whatwe just said, there existRw of bounded embedding dimension satisfyingP so that thereultraproductR♮ does not satisfyP. LetS♮ be any ultrapower ofR♮. SinceS♮ is then alsoan ultraproduct of theRw, but for a larger underlying index set,S♮ is both ultra-P andultra-non-P.

For an ultra-Noetherian example, consider the propertyC0: ‘being a Noetherian localring of characteristic zero’. The ultraproductV ♮ of all the rings ofp-adic integersZp (withrespect to some non-principal ultrafilter on the set of primenumbers) is ultra-C0, but bythe Ax-Kochen-Ershov theorem, this ring can also be realized as the ultraproduct of non-C0 local rings, to wit, theFp[[t]], wheret is a single indeterminate andFp is thep elementfield (see also Example9.7below).

Cataproducts. Let Rw be Noetherian local rings of bounded embedding dimension andlet R♮ be their ultraproduct. The separated quotient ofR♮, that is to say, the factor ring


R♯ := R♮/IR♮, is called thecataproductof theRw. If all Rw are equal to a single ringR,

then we callR♯ thecatapowerof R. This terminology is justified by:

5.6.Lemma. The cataproduct of local rings of bounded embedding dimension is equal tothe completion of their ultraproduct, whence in particularis Noetherian.

Proof. Let (R♮,m♮) be the ultraproduct of Noetherian local rings(Rw,mw) of embeddingdimension at moste, and letR♯ be their cataproduct, that is to say,R♮/IR♮

. We start withshowing that any Cauchy sequencea♮ : N→ R♮ has a limit. After taking a subsequence ifnecessary, we may assume thata♮(n) ≡ a♮(n+1) mod mn

♮ , for all n. For eachn, chooseaw(n) ∈ Rw such that their ultraproduct is equal toa♮(n). By Łos’ Theorem, we have fora fixedn that

(2) aw(n) ≡ aw(n+ 1) mod mnw

for almost allw, say, for allw in Dn. I claim that we can modify theaw(n) in suchway that (2) holds for alln and allw. More precisely, for eachn there existaw(n) withultraproduct equal toa♮(n), such that

(3) aw(n) ≡ aw(n+ 1) mod mnw

for all n andw. We will construct theaw(n) recursively from theaw(n). Whenn = 0,no modification is required (since by assumptionm0

w = Rw), and hence we setaw(0) :=aw(0) andaw(1) := aw(1). So assume we have defined already theaw(j) for j ≤ n suchthat (3) holds for allw. Now, for thosew for which (2) fails for somej ≤ n, that is to say,for w /∈ (D0 ∪ · · · ∪ Dn), let aw(n + 1) be equal toaw(n); for the remainingw, that isto say, for almost allw, we make no changes:aw(n + 1) := aw(n + 1). It is now easilyseen that (3) holds for allw. Since, for everyn, almost eachaw(n) is equal toaw(n), theirultraproduct isa♮(n), thus establishing our claim.

So we may assume (2) holds for alln andw. Letf : W → N be a function on the indexsetW such that for eachn, almost allf(w) ≥ n (this is where we use that the ultrafilteris countably incomplete; ifW = N, we can of course simply take the identity map). Letb♮ be the ultraproduct of theaw(f(w)). Sinceaw(f(w)) ≡ aw(n) mod mn

w for almostall w by (3), Łos’ Theorem yieldsb♮ ≡ a♮(n) mod mn

♮ , for eachn, showing thatb♮ is alimit of a♮. Although this limit might not be unique, it will be in the separated quotientR♯,

showing that the latter is a complete local ring, equal therefore toR♮. Noetherianity nowfollows from Theorem2.2. �

5.7.Corollary. The closure of an idealI in an ultra-Noetherian ringR♮ is equal toI+IR♮.

In particular, if R♯ is the cataproduct of theRw, andI♮ the ultraproduct of idealsIw ⊆Rw, thenR♯/I♮R♯ is the cataproduct of theRw/Iw.

Proof. SinceR♯ := R♮/IR♮is Noetherian by Lemma5.6, the idealIR♯ is closed by

Krull’s intersection theorem. All assertions now follow from Lemma2.4. �

5.8.Corollary. The cataproductR♯ of Noetherian local ringsRw of bounded embeddingdimension is equal to the cataproductS♯ of their completions.

Proof. Let (R♮,m♮) and(S♮, n♮) be the ultraproduct of respectively theRw and theRw.By Łos’ Theorem,m♮S♮ = n♮ andR♮ is dense inS♮. Hence both rings have the samecompletion, which by Lemma5.6 is respectively the cataproduct of theRw and of theRw. �

18 HANS SCHOUTENS

However, this is not the only case in which different rings can have the same cataprod-uct. Let(R,m) be a local ring of finite embedding dimension. A filtrationI = (In)n onRis calledanalytic if its extensionIR induces a Haussdorf topology onR, or, equivalently,if the intersection of allInR is zero. In particular, them-adic filtration is analytic by Theo-rem2.2. Given two filtrationsI = (In)n andJ = (Jn)n, we say thatI is boundedby J, iftheI-adic topology is stronger than or equal to theJ-adic topology, that is to say, for eachfixedN , we haveIn ⊆ JN for all sufficiently bign.

5.9.Lemma (Chevalley). A filtration on a Noetherian local ring(R,m) is analytic if andonly if it is bounded by them-adic filtration.

Proof. If I = (In)n is analytic, then the intersection of allInR is zero. By Chevalley’stheorem (see for instance [29, Exercise 8.7]) we have for fixedN an inclusionInR ⊆ mN Rfor n sufficiently big. By faithful flatness,In ⊆ mN for n≫ 0. The converse is immediatefrom Krull’s intersection theorem (see for instance [29, Theorem 8.10]). �

5.10.Corollary. If (In)n is an analytic filtration on a Noetherian local ringR, then thecatapowerR♯ ofR is isomorphic to the cataproductS♯ of theR/In.

Proof. Without loss of generality, we may assumeR is complete. The natural surjectionsR → R/In induce a mapR♯ → S♯, which is again surjective by Łos’ Theorem. Letx♮ be an element in the ultrapowerR♮ of R so that its image inR♯ is in the kernel ofR♯ → S♯. Choosexn ∈ R with ultraproduct equal tox♮ and fixN . Sincex♮ ∈ IS♮

,almost eachxn ∈ mN(R/In). By Lemma5.9, almost eachIn ⊆ mN and hence almosteachxn ∈ mN . By Łos’ Theorem,x♮ ∈ mNR♮. SinceN was arbitrary,x♮ lies inIR♮

andhence its image is zero inR♯, showing thatR♯ → S♯ is also injective. �

It should be noted that the corresponding ultraproductsR♮ andS♮, however, are far fromequal, as, for instance,FR-Spec(S♮) is always a singleton by Proposition5.1. Contraryto the Noetherian case, the natural mapR → R does not need to be flat ifR has finiteembedding dimension. We nevertheless expect some vestige of (faithful) flatness to hold.One example of this is given by Lemma2.4, namelyI = IR∩R for any closed idealI. Itis well-known (see for instance [44, Theorem 2.2]) that the latter property already followsfrom the vanishing ofTorR1 (R, k), wherek is the residue field ofR. For ultra-Noetherianlocal rings, where completion and separated quotient coincide by Lemma5.6, this latterproperty does indeed hold:

5.11.Proposition. For every ultra-Noetherian local ringR♮ with residue fieldk♮, we have

TorR♮

1 (R♯, k♮) = 0.

Proof. From the exact sequence

0 → IR♮→ R♮ → R♯ → 0

we get after tensoring overk♮ an exact sequence

0 → TorR♮

1 (R♯, k♮) → IR♮/m♮IR♮

→ k♮ → k♮ → 0,

wherem♮ is the maximal ideal ofR♮. In particular, the first Betti number ofR♯ vanishesif and only if m♮IR♮

= IR♮. To prove the latter equality, let(Rw,mw) be Noetherian

local rings with ultraproductR♮. Leta♮ be a non-zero element inIR♮and choose non-zero

aw ∈ Rw so that their ultraproduct is equal toa♮. Let m♮ be generated byx1♮, . . . , xe♮and, for eachi, choosexiw ∈ Rw whose ultraproduct equalsxi♮. By Łos’ Theorem,


mw = (x1w, . . . , xew)Rw. If aw has ordernw, then we can findbiw ∈ Rw of ordernw− 1 such thataw = x1wb1w + · · ·+xewbew. Let bi♮ be the ultraproduct of thebiw. FixsomeN . Sincea♮ ∈ IR♮

, its order is strictly bigger thanN and hence so is almost eachnw. Therefore, almost eachbiw has order at leastN and hencebi♮ ∈ mN

♮ . Since this holdsfor all N , we getbi♮ ∈ IR♮

. Sincea♮ = x1♮b1♮ + · · · + xe♮be♮ by Łos’ Theorem, we aredone. �

5.12.Corollary. Let (R♮,m♮) be an ultra-Noetherian local ring andI an ideal inR♮. If Iis closed, then so isImn

♮ for everyn.

Proof. By Corollary5.7, we haveIR♮⊆ I. SinceIR♮

= mn♮ IR♮

by the proof of Propo-sition 5.11, we getIR♮

⊆ Imn♮ , showing thatImn

♮ is closed by another application ofCorollary5.7. �

We may extend the notion of cataproduct to modules as well: for eachw, let Mw beanRw-module, and letM ♮ be their ultraproduct. It follows thatM ♮ is anR♮-module. Wedefine thecataproductof theMw as theR♯-moduleM ♯ := M ♮ ⊗R♮

R♯ = M ♮/IR♮M ♮

given by base change. IfNw ⊆ Mw are submodules, thenN ♮ ⊆ M ♮. However, theinduced homomorphismN ♯ → M ♯ may fail to be injective. The following result is anexercise on Łos’ Theorem (see for instance [36]), and the proof is left to the reader.

5.13.Proposition. LetM ♮ andM ♯ be the respective ultraproduct and cataproduct of theMw. Almost eachMw is minimally generated bys elements (respectively, has lengths),if and only ifM ♮ is minimally generated bys elements (respectively, has lengths), if andonly if so doesM ♮. �

Flatness of catapowers.A key result about catapowers, one which will be used frequentlyin our characterizations through uniform behavior in§12, is the following theorem and itscorollary:

5.14.Theorem. LetR be a Noetherian local ring andR♯ its catapower. There is a canon-ical homomorphismR → R♯ which is faithfully flat and unramified.

Proof. Let R♮ be the ultrapower ofR andR → R♮ the diagonal embedding. Composedwith the canonical surjectionR♮ → R♯ = R♮/IR♮

, we get the mapR → R♯. By Corol-lary 5.8 and the fact that completion is faithfully flat, we may already assume thatR iscomplete. SincemR♯ is the maximal ideal ofR♯, the mapR → R♯ is unramified. So re-mains to show that this map is flat. Let us first prove this underthe additional assumptionthatR is regular. We induct on its dimension. Letx be a regular parameter ofR, that is tosay, an element of order one. I claim thatx is R♯-regular. This follows for instance fromthe results in§8 (proving among other things thatR♯ is then regular), but we can give adirect argument here. Indeed, supposes♮ ∈ R♮ is such thatxs♮ ∈ IR♮

. If sw ∈ R haveultraproduct equal tos♮, then for a fixedN , almost eachxsw ∈ mN . SinceR is regular andx has order one,sw ∈ mN−1 and hence by Łos’ Theorem,s♮ ∈ mN−1R♮. Since this holdsfor all N , we gets♮ ∈ IR♮

, showing thatx isR♯-regular. It is not hard to see thatR♯/xR♯

is the catapower of the regular local ringR/xR, so that by induction,R/xR→ R♯/xR♯ isfaithfully flat. Since anyR/xR-regular sequence is thenR♯/xR♯-regular,R♯ is a balancedbig Cohen-Macaulay algebra overR. SinceR is regular,R → R♯ is therefore faithfullyflat (see for instance [42, Theorem IV.1] or [25, Lemma 2.1(d)]).

For the general case, we may writeR as a homomorphic imageS/I of a completeregular local ringS by Cohen’s theorem. By what we just proved,S → S♯ is faithfully

20 HANS SCHOUTENS

flat, whereS♯ is the catapower ofS. Hence the base changeR = S/I → R♯ = S♯/IS♯ isalso flat. �

5.15.Corollary. LetR be an excellent local ring (e.g., a complete Noetherian local ring)with catapowerR♯. The natural mapR → R♯ is regular. In particular,R is regular(respectively, normal, reduced, Cohen-Macaulay or Gorenstein), if and only if, so isR♯.

Proof. The second assertion is a well-known consequence of the first(see for instance [29,Theorem 32.2]). As for the first, let us first show this in the special case thatR = k isa field. Note that in this case, the catapower is equal to the ultrapowerk♮ of k. Hence,we need to show thatk → k♮ is separable, and so we may assume thatk has positivecharacteristicp. We will establish separatedness by verifying MacLane’s criterion (seefor instance [29, Theorem 26.4]). Letb1, . . . , bn be elements ink1/p which are linearlyindependent overk. Supposex1♮b1+ · · ·+xn♮bn = 0 for somexi♮ ∈ k♮. Choosexiw ∈ kwith ultraproduct equal toxi♮ ∈ k♮. Takingp-th powers, using Łos’ Theorem and thentakingp-th roots, we getx1wb1 + · · · + xnwbn = 0 for almost allw. Since thebi arelinearly independent overk, almost allxiw are zero. By Łos’ Theorem, eachxi♮ is zero,

showing that thebi, viewed as elements ink1/p♮ , remain linearly independent overk♮, aswe wanted to show.

ForR arbitrary, Theorem5.14yields thatR → R♯ is faithfully flat and unramified. Bywhat we just proved, the induced residue field extension is separable. ThereforeR → R♯ isformally smooth by [29, Theorem 28.10]. Regularity then follows from a result by Andrein [2] (see also [29, p. 260]). �

5.16.Proposition. LetR ⊆ S be an injective, local homomorphism between Noetherianlocal rings and letR♯ → S♯ be the induced map of catapowers.

(5.16.1) If R ⊆ S is finite, thenR♯ → S♯ is finite and injective.(5.16.2) If R ⊆ S is cata-injective, that is to say, ifR → S is injective, thenR♯ → S♯

is injective too.

Proof. Let m andn be the maximal ideals of respectivelyR andS. AssumeR ⊆ S isfinite, so thatna ⊆ mS for somea. By the Artin-Rees Lemma,mnS ∩ R ⊆ mn−c forsomec and alln ≥ c. Hencenna ∩R ⊆ mn−c for all n ≥ c and hence by Łos’ Theorem,the same inclusions hold in the extensionR♮ ⊆ S♮ of ultrapowers. Using this, it is not hardto show thatIS♮

∩R♮ = IR♮, showing thatR♯ ⊆ S♯ is injective (and clearly also finite).

If R ⊆ S is cata-injective, then the filtrationnk ∩ R, for k = 0, 1, . . . , is easily seento be analytic, whence bounded by them-adic filtration by Lemma5.9. Again one derivesfrom this thatIS♮

∩R♮ = IR♮, whence thatR♯ ⊆ S♯ is injective. �

5.17. Extended dimensions in ultra-Noetherian local rings.We extend the nomencla-ture introduced in the beginning of this section to include invariants. In particular, wedefine thecata-dimensionof R, denotedcdim(R), as the (Krull) dimension of its com-pletion R. For an ultra-Noetherian local ringR♮ given as the ultraproduct of Noetherianlocal ringsRw of embedding dimension at mostm, we define itsultra-dimension, denotedudim(R♮), as the dimension of almost allRw. Since almost allRw have dimension atmostm, the ultra-dimension ofR♮ is finite.

5.18.Theorem. For an ultra-Noetherian local ringR♮, we have inequalities

(4) depth(R♮) ≤ pidim(R♮) = frdim(R♮) = udim(R♮)

≤ cldim(R♮) = gdim(R♮) = cdim(R♮) ≤ embdim(R♮).


Proof. By Theorems3.4 and4.4, the cata-dimension ofR♮ is equal to its geometric di-mension and to its cl-dimension. On the other hand, Łos’ Theorem and Lemma4.7yieldthat the ultra-dimension ofR♮ coincides with its pi-dimension and its fr-dimension. Depthis also first-order, as it is cast in terms of the vanishing of the Koszul homology of a gen-erating set ofm (see§7.1below for more details). Since in a Noetherian local ring depthnever exceeds dimension, the first inequality is then also clear. �

There are no further constraints on the above invariants of an ultra-Noetherian ring, asthe following examples show (in the discussion of these examples, we will also use someterminology from later sections).6

5.19.Example. Let e ≤ h ≤ d ≤ m. We will construct an ultra-Noetherian local ringR♮

with depthe, ultra-dimensionh, cata-dimensiond, and embedding dimensionm. First weintroduce some notation. LetR♮ be the ultraproduct of theRw and letn♮ be a non-standardpositive integer, that is to say, an ultraproduct of an unbounded sequence of positive inte-gersnw. For an elementa♮ ∈ R♮, realized as an ultraproduct of elementsaw ∈ Rw, we

write an♮

♮ to denote the ultraproduct of the elementsanww ; one verifies that this is inde-

pendent of the choice ofaw or nw. Let S♮ be the ultrapower ofS := k[[ξ]], for someindeterminatesξ := (ξ1, . . . , ξm) and some fieldk, let

I := (ξn♮

e+1ξm, . . . , ξn♮

h ξm, ξn♮

h+1, . . . , ξn♮

d , ξ2d+1, . . . , ξ2m)S♮

and putR♮ := S♮/I. By Łos’ Theorem,(ξ1, . . . , ξe) isR♮-regular and since the maximal

ideal ofR♮/(ξ1, . . . , ξe)R♮ is annihilated by the elementξn♮−1e+1 · · · ξ

n♮−1d · ξd+1 · · · ξm, we

see thatR♮ has depthe. Sinceξd+1, . . . , ξm are nilpotent, we get from Proposition5.21below that the ultra-dimension ofR♮ is the same as the ultra-dimension of

R♮/(ξd+1, . . . , ξm)R♮ = S♮/(ξn♮

h+1, . . . , ξn♮

d , ξd+1, . . . , ξm)S♮,

that is to say, equal toh. On the other hand,IS♯ = (ξ2d+1, . . . , ξ2m)S♯, whereS♯ is the

catapower ofS (note thatS♯∼= k♮[[ξ]], wherek♮ is the ultrapower ofk; see for instance

[4, Proposition 3.1]). Hence the catapowerR♯ of R has dimensiond. By Lemma5.6, thecata-dimension ofR♮ is therefored. Finally, it follows from Łos’ Theorem thatR♮ hasembedding dimensionm. Note that sinceR♯ is Cohen-Macaulay,R♮ is cata-Cohen-Mac-aulay.

More generally, letq be any number betweene andd and letR′♮ := S♮/I

′, whereI ′ isthe sum of the idealI above and the ideal(ξq+1ξm, . . . , ξdξm)S♮. ThenR′

♮ has still thesame depth, ultra-dimension, cata-dimension and embedding dimension asR♮, but nowthe depth ofR♯, that is to say, thecata-depthof R♮, is q, since(ξ1, . . . , ξq) is a regularsequence.

5.20.Example. The previous example might one lead to think that the depth ofR is alwaysat most its cata-depth. However, this is not the case as the following example shows. LetS♮ be as in the previous example withm = 3, and letR♮ := S♮/(ξ

21 , ξ1ξ2, ξ1ξ3 − ξ

n♮

2 )S♮,with n♮ a non-standard positive integer. Sinceξ3 is R♮-regular and sinceR♮ has ultra-dimension one, the depth ofR♮ is one by Theorem5.18. On the other hand,R♯ is equalto S♯/(ξ

21 , ξ1ξ2, ξ1ξ3)S♯, whence has depth zero. Note thatR♯ has dimension two, so that

6One should note that for Noetherian rings, other than the obvious restriction that pi-dimension and dimensionagree, we also have the remarkable fact that when dimension and embedding dimension agree, that is to say, whenthe ring is regular, then this common value must also be equalto its depth.

22 HANS SCHOUTENS

R♮ itself has cata-dimension two. HenceR♮ is ultra-Cohen-Macaulay, but not cata-Coh-en-Macaulay.

Isodimensionality. We call a local ringR of finite embedding dimensionisodimensionalif (4.4.2) is an equality, that is to say, if the geometric dimension ofR is equal to its pi-dimension. In view of Theorem5.18, an ultra-Noetherian local ring is isodimensional ifand only if its ultra-dimension is equal to its cata-dimension.

5.21.Proposition. LetR♮ be an ultra-Noetherian local ring. Ifa is a finitely related idealcontained innil(R♮), thenR♮ andR♮/a have the same ultra-dimension. In particular,R♮

is isodimensional if and only ifR♮/a is.

Proof. Let h be the ultra-dimension ofR♮, so thatR♮ is the ultraproduct ofh-dimensionalNoetherian local ringsRw of bounded embedding dimension. Sincea is finitely related, itcan be realized as the ultraproduct of finitely related idealsaw by the argument in the proofof Proposition5.1. By Łos’ Theorem, almost eachaw is nilpotent, and thereforeRw/awhas again dimensionh. HenceR♮/a has ultra-dimensionh as well.

The final assertion follows from the fact thatR♮ andR♮/a have the same geometricdimension (this is true in general, sincea is contained in every threshold prime ofR♮). �

For ultra-Noetherian local rings, we have the following important criterion for isodi-mensionality:

5.22.Theorem. LetR♮ andR♯ be the respective ultraproduct and cataproduct of Noethe-rian local ringsRw of bounded embedding dimension. The following are equivalent:

(5.22.1)R♮ is isodimensional;(5.22.2) almost allRw have dimension equal togdim(R♮);(5.22.3) almost allRw have the same dimension asR♯;(5.22.4) almost allRw have the same parameter degree (which is then also the param-

eter degree ofR♮ and ofR♯).

Proof. The equivalence of (5.22.2) and (5.22.3) follows from Lemma5.6and Theorem3.4.Let d ≤ m be the respective geometric dimension and embedding dimension of R♮. ByTheorem5.18, the cata-dimension ofR♮ is d. Sincedim(Rw) ≤ m, almost allRw havea common dimensionh ≤ m, which is then the ultra-dimension ofR♮ by definition, fromwhich we get the equivalence of (5.22.1) and (5.22.2).

So remains to show that equivalence of (5.22.2) and (5.22.4). Supposepardeg(Rw) = efor almost allw. In eachRw, choose anh-tuplexw so that almost allRw/xwRw havelengthe. Letx♮ be the ultraproduct of thexw. By Proposition5.13, the length ofR♮/x♮R♮,being the ultraproduct of theRw/xwRw, is alsoe. It follows thatR♮ has geometric dimen-sion at mosth. We already argued that its geometric dimension is at leasth, so that wegeth = d. In particular, the parameter degree ofR is at moste, and by reversing thisargument, one can also show that it cannot be less thane, whence must be equal toe.

Conversely, assumeh = d. Let x♮ be a generic sequence inR♮ and choosed-tuplesxw whose ultraproduct isx♮. By Łos’ Theorem, almost eachxw generates anmw-primaryideal, and therefore must be a system of parameters inRw, since almost eachRw hasdimensionh = d. Let l be the length ofR♮/x♮R♮. By Proposition5.13, almost eachRw/xwRw has lengthl, showing thatpardeg(Rw) ≤ l, for almostw. �

5.23. Example. We cannot replace parameter degree with multiplicity in thepreviousresult as the following example shows. Fix somee > 0 and putRw := S/(ξw, ξeζw−e)Sfor eachw ≥ e, whereS := k[[ξ, ζ]] andk is a field. LetR♮ be the ultraproduct of the


Rw, let k♮ be the ultrapower ofk and letS♯∼= k♮[[ξ, ζ]] be the catapower ofS. Since

the ultraproduct of theξw and theξeζw−e are infinitesimals, the cataproduct of theRw isR♯ = S♯, showing thatR♮ is not isodimensional (since theRw are one-dimensional andR♯ is two-dimensional). Therefore, by the theorem, the parameter degree of theRw isunbounded (in fact, equal tow). On the other hand,ζ is a parameter in eachRw so that wecan calculate the multiplicity ofRw by Lech’s lemma ([29, Theorem 14.12]) as the limitof ewn/n asn tends to infinity, whereewn is the length ofRw/ζ

nRw. One calculates thatewn = w(w − 1) + e(n − w + 2) and hencemult(Rw) = e. This shows, in view ofRemark3.5, that multiplicity is in general not first-order.

5.24.Remark.In view of Theorem5.22, we will often require that a collection of Noethe-rian local ringsRw have (almost all) the same embedding dimension and the same param-eter degree, to ensure that their cataproduct is again Noetherian of the same dimension. Infact, we can replace this requirement with the more natural requirement that (almost all)Rw have the same dimension and parameter degree. Indeed, if a Noetherian local ringRhas dimensiond and parameter degreee, then its embedding dimension is at mostd+e−1.

Note that by Lemma6.10below, if almost allRw are Cohen-Macaulay we may furthersimplify this to the requirement that almost allRw have the same dimension and multiplic-ity. The previous example shows that this is no longer true without the Cohen-Macaulayassumption.

5.25.Corollary. If R♮ is an isodimensional ultra-Noetherian local ring andx♮ the ultra-product of elementsxw , thenx♮ is generic if and only ifdeg(xw) is bounded.

Proof. LetRw be Noetherian local rings with ultraproductR♮. By Theorem5.22, almosteachRw has dimensiond := gdim(R♮). Supposex♮ is generic. Hence,R♮/x♮R♮ hasgeometric dimensiond − 1, whence ultra-dimension at mostd − 1. In particular, almosteachRw/xwRw must have dimensiond − 1. Hencexw is generic inRw andR♮/x♮R♮

is again isodimensional. By Theorem5.22, this means that theRw/xwRw must havebounded parameter degree, proving the direct implication.

Conversely, suppose thedeg(xw) are bounded, that is to say, almost allxw are genericand the parameter degrees of theRw/xwRw are bounded. By Theorem5.22once more,R♮/x♮R♮ has geometric dimensiond− 1, showing thatx♮ is generic. �

Without the isodimensional assumption, the result is false: for instance ifR♮ has ultra-dimension zero (e.g., the ultraproduct of theR/mn), then no element inR♮ is realized asan ultraproduct of elements of finite degree.

Conform with our previous nomenclature, we call a local ringultra-excellent, if it is theultraproduct of excellent local rings of bounded embeddingdimension. We can now givethe following improvement of Corollary5.3.

5.26.Corollary. LetR♮ be an ultra-Noetherian local ring, realized as the ultraproduct ofequicharacteristic excellent local ringsRw. If R♮ is isodimensional, then any localizationat a finitely related prime ideal has finite embedding dimension, and any finitely relatedprime ideal is strong.

Proof. Let p ∈ FR-Spec(R♮). By Proposition5.1, there exist prime idealspw ⊆ Rw

with ultraproduct equal top. By Theorem5.22, there is someρ, such that almost eachRw

has parameter degreeρ. Hence, by Proposition5.2, almost each(Rw)pwhas embedding

dimension at mostd+ρ, whered is the common dimension of almost allRw (that is to say,the ultra-dimension, whence geometric dimension, ofR♮). Since(R♮)p is the ultraproduct

24 HANS SCHOUTENS

of the(Rw)pw, its embedding dimension is at mostd+ρ. Proposition5.4then implies that

p is strong. �

We actually showed that each stalk ofSpec(R♮) at a point belonging toFR-Spec(R♮)has embedding dimension at mostd + ρ, whered is the geometric dimension ofR♮ andρ its parameter degree. Inspecting the proof of Proposition5.2, we see that almost each(Rw)pw

has parameter degree at mostρ, showing that each stalk is also isodimensional, ofultra-dimension, whence geometric dimension, at mostd.

6. CATA -SINGULARITIES

According to the definitions in§5, a local ring of finite embedding dimension iscata-regular if its completion is a regular (Noetherian) local ring.

6.1.Theorem. Let(R,m) be a local ring of geometric dimensiond and letk be its residuefield. The following are equivalent:

(6.1.1) R is cata-regular;(6.1.2) Rsep is cata-regular;(6.1.3) gdim(R) = embdim(R);(6.1.4) m is generated by a generic sequence;(6.1.5) m is generated by a quasi-regular sequence;(6.1.6) gr(R) is isomorphic tok[ξ], with ξ a d-tuple of indeterminates.

Proof. The equivalence of (6.1.1) and (6.1.2) is clear sinceRsep has the same completionasR, and their equivalence with (6.1.6) follows from [29, Theorem 14.4], since we havean isomorphism of graded ringsgr(R) ∼= gr(R). The equivalence of (6.1.3) and (6.1.4)is clear from the definition of geometric dimension. Suppose(6.1.4) holds, so thatm isgenerated by a generic sequence(x1, . . . , xd). There is a natural surjective homomor-phismk[ξ] → gr(R) which mapsξi to in(xi), whereξ = (ξ1, . . . , ξd). Since both ringshave the same dimension by Theorem3.4, the kernel must be zero, proving (6.1.6). Con-versely, assumegr(R) ∼= k[ξ]. Hencem/m2 is generated byd elements, and therefore, byNakayama’s Lemmam is generated byd elements, showing that (6.1.4) holds.

Remains to show the equivalence of the other conditions with(6.1.5). Recall thatxis quasi-regular ifF (x) = 0, for a homogeneous polynomialF ∈ R[ξ], implies thatF has all its coefficients inI := xR. This is equivalent with the natural epimorphism(R/I)[ξ1, . . . , ξd] → grI(R) being injective, whence an isomorphism (see for instance[29, §16]). Hence takingI = m, we see that (6.1.5) is equivalent with (6.1.6). �

6.2.Remark.In the above proof, we actually showed that ifR is cata-regular of geometricdimensiond, then anyd-tuple generatingm is quasi-regular. We will shortly show (The-orem6.8below) that then every generic sequence is quasi-regular. The ringR in the nextexample shows that a generic sequence generating the maximal ideal in a cata-regular localring is not necessarily a regular sequence.

6.3.Example. A local ring of geometric dimension zero is cata-regular if and only if it isa field. A local ring of geometric dimension one is cata-regular if and only if its maximalideal is generated by a non-nilpotent element. For instance, let V ♮ be an ultraproductof discrete valuation rings (anultra-DVR for short), or more generally, a valuation ringof finite embedding dimension (which is then automatically one). If x is an element inthe ideal of infinitesimalsIV ♮

of V ♮, thenR := V ♮/xV ♮ is cata-regular of geometricdimension one. Ifx 6= 0, thenR is not a domain. In fact,R has then depth zero (and so isnot pseudo-regular in the sense of§7.6below).


The following fact, however, is noteworthy: ifR is moreover separated, then any quasi-regular element is regular; see for instance [29, Theorem 16.3]. In fact, we have the fol-lowing result:

6.4.Corollary. If a cata-regular local ring is separated, then it is a domain. More gener-ally, the separated quotient of a cata-regular local ring isa domain.

Proof. Immediate from the fact thatRsepembeds inR and the fact that Noetherian regularlocal rings are always domains. �

6.5.Corollary. If R is cata-regular, then so is any homomorphic imageR/I, for I ⊆ IR.

Proof. SinceR andR/I have the same separated quotient, the result follows from Theo-rem6.1. �

6.6.Corollary. For eachd, the class of cata-regular local rings of geometric dimensiondis first-order definable.

Proof. Observe that a ring is local if and only if any sum of two non-units is again a non-unit. In fact, an element lies in the maximal ideal of a local ring if and only if it is not aunit. Therefore, the maximal ideal of a local ring is definable, as is expressing that someelement lies in the maximal ideal. In particular, the formulaλd,n(x, a) is first order, whereλd,n(x, a) is the formula in the variablesx := (x1, . . . , xd) anda := (aν)ν , for ν runningover alld-tuples inNd whose sum|ν| is n, expressing that

if x generates the maximal ideal and if∑

|ν|=n

aνxν = 0,

then someaν lies in the maximal ideal.(5)

Let Td be the theory consisting of all sentences(∀x, ∀a)λd,n(x, a), for n = 1, 2, . . . ,together with the sentenceσd expressing that the maximal ideal is generated by somed-tuple. I claim thatTd axiomatizes the class of cata-regular local rings of geometricdimensiond. Indeed, suppose that(R,m) satisfiesTd. By σd, there is ad-tuplex such thatm = xR. Sinceλd,n(x, a) holds for all tuplesa inR, we see thatx is quasi-regular. HenceR is cata-regular by Theorem6.1. Conversely, ifR is cata-regular of geometric dimensiond, then it satisfiesTd by Remark6.2. �

This immediately gives a large class of cata-regular local rings. Namely, any ultraprod-uct of regular local rings of dimensiond is cata-regular, of geometric dimensiond. We willaddress this situation further in§8 below.

6.7.Corollary. A local ringR of geometric dimension one is cata-regular if and only ifRsep is a discrete valuation ring.

Proof. AssumeR is cata-regular so thatR is a discrete valuation ring with valuationordbR(·). SinceordRsep(a) = ordbR(a) for all a ∈ Rsep, alsoordRsep(·) is a valuation,showing thatRsep is a discrete valuation ring. Conversely, ifRsep is a discrete valuationring, thenR is cata-regular by Theorem6.1. �

Cata-Cohen-Macaulay local rings.We now turn to the study of cata-Cohen-Macaulaylocal rings of finite embedding dimension, that is to say, local rings whose completion isCohen-Macaulay. Clearly, any cata-regular local ring is cata-Cohen-Macaulay.

26 HANS SCHOUTENS

6.8.Theorem. A local ring of finite embedding dimension is cata-Cohen-Macaulay if andonly if its separated quotient is cata-Cohen-Macaulay if and only if some (equivalently,every) generic sequence is quasi-regular.

Proof. Let (R,m) be a local ring of geometric dimensiond and letx be a generic sequence.Sincegr

xR(R)∼= gr

xbR(R), the sequencex is quasi-regular inR if and only if it is so in

R. SinceR andRsep have the same completion, we only need to show the equivalenceof the first and last condition. Suppose thatx is generic. Sincex is R-quasi-regular, it isR-regular by [29, Theorem 16.3] and the fact thatR is Noetherian. SinceR has dimensiond by Theorem3.4, it is Cohen-Macaulay, showing thatR is cata-Cohen-Macaulay.

Conversely, supposeR is Cohen-Macaulay. Sincex is a system of parameters inR, itis R-regular, whenceR-quasi-regular. By our previous observation,x is then quasi-regularin R. �

6.9.Corollary. A local ring of finite embedding dimension is cata-regular ifand only if itis cata-Cohen-Macaulay and has multiplicity one.

Proof. If a local ringR is cata-regular, its completionR is regular, whence has multiplicityone. SinceR and its completionR have the same multiplicity by Remark3.5, the directimplication is clear. Conversely, ifR is cata-Cohen-Macaulay andmult(R) = 1, thenRis Cohen-Macaulay withmult(R) = 1 by Remark3.5. SinceR is unmixed, it is regularby [30, Theorem 40.6], showing thatR is cata-regular. �

6.10.Lemma. The multiplicity ofR is at most its parameter degree. IfR has infiniteresidue field then we have equality if and only ifR is cata-Cohen-Macaulay.

Proof. Let x be a generic sequence ofR. By Proposition3.9, it is a system of parametersin R andR/xR ∼= R/xR by Lemma2.4. The common length of the latter two quotientsis at least the multiplicity of the idealxR by [29, Theorem 14.10] which in turn is at mostmult(R) by [29, Formula 14.4]. The desired inequality now follows from this, sinceRandR have the same multiplicity by Remark3.5.

The last assertion holds ifR is Noetherian by [46, Lemma 3.3]. The general case followsfrom this sinceR andR have the same multiplicity and the same parameter degree.�

6.11.Theorem. A local ring of finite embedding dimension is cata-Gorenstein (respec-tively, a cata-‘complete intersection’) if and only if so isits separated quotient, if and onlyif it admits a quasi-regular, generic sequencex such thatR/xR is Gorenstein (respectively,a complete intersection).

Proof. Let (R,m) be a local ring of geometric dimensiond. SinceR andRsep havethe same completion, we only need to show the equivalence of the first and last condi-tion. Supposex is a quasi-regular, generic sequence. In particular,R is cata-Cohen-Macaulay by Theorem6.8, whenceR is Cohen-Macaulay andx is R-regular. Moreover,R/xR ∼= R/xR by Lemma2.4. Therefore the former is Gorenstein (respectively, a com-plete intersection) if and only if the latter is, if and only if R is (see [9, Theorem 2.3.4 andProposition 3.1.19]). �

6.12.Proposition. A local ring of finite embedding dimension is cata-Gorenstein if andonly if there exists a quasi-regular, generic sequence generating an irreducible ideal. Whenthis is the case, every generic sequence is quasi-regular and generates an irreducible ideal.


Proof. Let x be a quasi-regular, generic sequence. The result is now immediate from thefact thatxR is irreducible if and only ifR/xR is Gorenstein. �

7. PSEUDO-SINGULARITIES

The cata-singularities from the previous section do not always correspond to their ‘ul-tra’ versions (which will be treated in the next section). Tothis end we will define somestronger versions of these cata-singularities, defined intrinsically, that is to say, withoutreference to the completion. Throughout this section,(R,m) is a local ring of finite em-bedding dimension.

7.1. Grade and depth. Let A be an arbitrary ring andI a finitely generated ideal inA.Choose a tuple of generatorsx = (x1, . . . , xn) of I. Thegradeof I, denotedgrade(I),is by definition equal ton − h, whereh is the largest valuei for which thei-th KoszulhomologyHi(x;A) is non-zero. One shows that the grade ofI does not depend on thechoice of generatorsx. For a local ringR of finite embedding dimension, we define itsdepthas the grade of its maximal ideal; it is non-zero if and only ifits maximal ideal is notan associated prime.

Grade, and hence depth,deforms well, in the sense that the

(6) grade(I(A/xA)) = grade(I)− |x|

for everyA-regular sequencex contained inI. If R has geometric dimensiond, thenits depth is at mostd. Indeed, by definition, the grade of a finitely generated ideal neverexceeds its minimal number of generators, and by [9, Proposition 9.1.3], the depth ofR isequal to the grade of any of itsm-primary ideals.

The relationship between depth and the length of a regular sequence (sometimes calledthe naive depthof R) is less straightforward in the non-Noetherian case and requires anadditional definition. For a local ring(R,m) and a finite tuple of indeterminatesξ :=(ξ1, . . . , ξn), we will denote the localization ofR[ξ] at the idealmR[ξ] by R(ξ) (this issometimes called then-fold Nagata extensionofR). It follows thatR → R(ξ) is faithfullyflat and unramified, with closed fiber equal to the residue fieldextensionk ⊆ k(ξ), wherek is the residue field ofR andk(ξ) the field of fractions ofk[ξ].

7.2.Lemma. Let(R,m) be a local ring of finite embedding dimension and letξ be a tupleof indeterminates. ThenR andR(ξ) have the same geometric dimension and the samedepth.

Proof. Let d be the geometric dimension ofR ande its depth. We will induct ond to showthatgdim(R(ξ)) = d. It is easy to see thatR is Artinian if and only ifR(ξ) is, thus provingthe cased = 0. In the general case, we may choosex ∈ m so thatgdim(R/xR) = d− 1.By induction,(R/xR)(ξ) ∼= R(ξ)/xR(ξ) has geometric dimensiond − 1, showing thatgdim(R(ξ)) ≤ d. On the other hand, induction also shows thatgdim(R(ξ)) > d − 1, sothat we getgdim(R(ξ)) = d, as required.

As for depth, this follows from [9, Proposition 9.1.2] sinceR → R(ξ) is faithfullyflat. �

We can now characterize depth in terms of regular sequences:

7.3.Lemma. For a local ringR of finite embedding dimension, its depth is equal to themaximal length of anR(ξ)-regular sequence, whereξ runs over all finite tuples of indeter-minates. More precisely, ifR has depthe, then we can find a regular sequence(y1, . . . , ye)in R(ξ1, . . . , ξe) which is part of a generic sequence.

28 HANS SCHOUTENS

Proof. In view of Lemma7.2, it suffices to prove the second assertion. To this end, weneed to construct, by Lemma3.8, anR(ξ)-regular sequence(y1, . . . , ye) such that thegeometric dimension ofR(ξ)/(y1, . . . , ye)R(ξ) is d − e, whereξ := (ξ1, . . . , ξe). Weinduct on the depthe of R, where there is nothing to show ife = 0. Let (x1, . . . , xd)be a generic sequence and letn be the ideal generated by this sequence. Sincen is thenm-primary, its grade ise. By [9, Proposition 9.1.3], the element

y1 := x1 + x2ξ1 + · · ·+ xdξd−11

is anR[ξ1]-regular element. SinceR[ξ1] → R(ξ1) is flat, y1 is R(ξ1)-regular. LetS := R(ξ1)/y1R(ξ1). SinceS/(x2, . . . , xd)S ∼= (R/n)(ξ1), it is Artinian. Therefore,the geometric dimension ofS is at mostd − 1. By Lemma7.2, the geometric dimensionof S cannot be less, and hence it is equal tod− 1. In particular, we are done in casee = 1.

Assume thereforee > 1. It follows from Lemma7.2 and (6) thatS has depthe −1. By induction, there exists anS(ξ2, . . . , ξe)-regular sequence(y2, . . . , ye) such thatS(ξ2, . . . , ξe)/(y2, . . . , ye)S(ξ2, . . . , ξe) has geometric dimensiond− e. Hence withξ :=(ξ1, . . . , ξe), the sequence(y1, . . . , ye) isR(ξ)-regular and part of a generic sequence.�

7.4.Remark.The argument even shows that, for a given generic sequence(x1, . . . , xd),we may choose anR(ξ)-regular sequence(y1, . . . , ye) so that

(y1, . . . , ye, xe+1, . . . , xd)R(ξ) = (x1, . . . , xd)R(ξ).

In particular, ifR is moreover cata-regular, then we may take(y1, . . . , ye) equal to a gen-erating set of the maximal ideal ofR(ξ).

For ultra-Noetherian rings, no such extension is necessary, since depth is first-orderdefinable:

7.5.Proposition. The depth of an ultra-Noetherian local ringR is equal to the maximallength of anR-regular sequence. �

7.6. Pseudo-singularities.We now introduce some singularity variants that are based ondepth. LetR be a local ring of finite embedding dimension. If the depth ofR is equal toits embedding dimension, then we callR pseudo-regular, and if it is equal to its geometricdimension, we callR pseudo-Cohen-Macaulay. Immediate from the definitions we get:

7.7.Proposition. A local ring of finite embedding dimension is pseudo-regularif and onlyif it is cata-regular and pseudo-Cohen-Macaulay. �

In order to derive a homological characterization of pseudo-regularity analogous toSerre’s characterization for regularity, we need some additional definitions.

7.8. Finite presentation type. We say that anR-moduleM admits afinite free resolution(of lengthn), if there exists an exact sequence

(7) 0 → Fn → Fn−1 → · · · → F1 → F0 →M → 0

with eachFi a finitely generated freeR-module. The alternating sum of the ranks oftheFi is called theEuler numberEul(M) of M . It follows from Schanuel’s Lemma thatEul(M) does not depend on the choice of finite free resolution, and by[29, Theorem 19.7],it is always non-negative. Also, if

0 → H → Gm → Gm−1 → · · · → G1 → G0 →M → 0

is an arbitrary exact sequence with allGi finitely generated freeR-modules, thenH is alsofinitely generated, andEul(M) is the alternating sum of the ranks of theGi and ofEul(H)(see [29, §19] for more details).


In general, very few modules admit a finite free resolution, and hence we introduce thefollowing weaker version: we say that anR-module isfinitely n-presented, if it admitsfinitely generatedi-th syzygies fori = 0, . . . , n, or equivalently, if there exists an exact se-quence as in (7), but without the initial zero, with allFi finitely generated freeR-modules.HenceM is finitely 0-presented if and only if it is finitely generated, andM is finitely1-presented if and only if it is finitely presented. We will saythat anR-module hasfinitepresentation type, if it is finitely n-presented, for alln. Although these definitions do notrequireR to be local, the next one does: we call aR-module complex(G•, d•) minimal ifthe kernel of each morphismdi lies insidemGi.

7.9.Lemma. Let (R,m) be a local ring with residue fieldk. AnR-moduleM is finitelyn-presented if and only if there exist a minimal exact sequence

(F•) Fn → Fn−1 → · · · → F1 → F0 →M → 0

with eachFi a finitely generated freeR-module. Moreover, if this is the case then thei-thBetti numberβR

i (M) ofM , that is to say, the vector space dimension ofTorRi (M,k), isequal to the rank ofFi, for all i ≤ n, showing thatF• is unique up to isomorphism.

Proof. One direction is immediate and the other can by induction be reduced to the casethatM is finitely 0-presented, that is to say, finitely generated. This case is then simply areformulation of Nakayama’s Lemma. To prove the last assertion, augmentF• by addingon the left a free moduleFn+1, possibly of infinite rank, which maps onto the kernel ofFn → Fn−1. Tensoring this exact sequence withk gives a complex in which all morphismsare zero and hence itsi-th homology isFi ⊗ k, for i = 0, . . . , n. Since this homology isalso equal toTorRi (M,k), we proved the second assertion. �

Since a projective module over a local ring is always free ([29, Theorem 2.5]), a neces-sary and sufficient condition for anR-moduleM to have a finite free resolution is thatMhas projective dimensionn < ∞ and is finitelyn-presented. By the previous result, sucha module then admits a unique minimal finite free resolution.

7.10.Lemma. Any direct summand of anR-module with a finite free resolution has itselfa finite free resolution. Similarly, any direct summand of a finitelyn-presented module isagain finitelyn-presented.

Proof. We prove both results simultaneously. SupposeM ⊕N has a finite free resolutionof lengthn as in (7) (respectively, of the formF•). We will show by induction onn thatM has a finite free resolution (respectively, is finitelyn-presented). Ifn = 0, that is to say,if M ⊕ N is free, thenM is projective whence free (respectively, ifn = 0, that is to say,M ⊕ N is finitely generated, then so isM ). Hence assumen > 0 and choose an exactsequence

(8) 0 → K → Rm →M ⊕N → 0

such thatK admits a finite free resolution of lengthn − 1 (respectively is finitelyn − 1-presented). Clearly,M andN must also be finitely generated, so that we can choose exactsequences

0 → G→ Ra →M → 0

0 → H → Rb → N → 0.

Taking the direct sum of these last two exact sequences and comparing it with (8), weget from Schanuel’s Lemma an isomorphismK ⊕ Ra ⊕ Rb ∼= G ⊕ H ⊕ Rn. Since the

30 HANS SCHOUTENS

module at the left hand side has a finite free resolution of length n − 1 (respectively, isfinitely n− 1-presented), our induction hypothesis yields thatG has a finite free resolution(respectively, is finitelyn − 1-presented), whence so doesM (respectively, whenceM isfinitely n-presented). �

7.11.Theorem. A local ring of finite embedding dimension is pseudo-regularif and onlyif its residue field admits a finite free resolution.

Proof. Suppose first that(R,m) is pseudo-regular of geometric dimensiond. Let x be ageneric sequence generatingm. SinceR has depthd, all Hi(x;R) vanish, showing thatthe Koszul complexK•(x) of x is exact, yielding the desired finite free resolution of theresidue fieldk.

Conversely, assume thatk has a finite free resolution

0 → Ran → Ran−1 → · · · → Ra1 → R → k → 0.

Letm be the embedding dimension ofR (so that we may choosea1 = m). Observe thatboth hypothesis and conclusion are invariant under taking aNagata extension of the formR ⊆ R(ξ) (by faithful flatness), so that at any time we may make such an extension ifneeded. There is nothing to show ifm = 0, so we induct onm > 0. By [29, Theorem19.6], the depth ofR must be positive. By Lemma7.3, we may assume after making aNagata extension, that some minimal generatorx of m is R-regular. PutS := R/xR, sothat its embedding dimension ism− 1. For eachi > 1, we have an isomorphism

TorRi (S, k)∼= TorRi−1(S,m) ∼= TorSi−1(S,m/xm) = 0

sincex isR-regular, whence alsom-regular. This implies that the complex

0 → San → San−1 → · · · → Sa1

is acyclic, that is to say, is a finite free resolution ofm ⊗ S = m/xm. I claim thatk is adirect summand ofm/xm. Assuming the claim, Lemma7.10then yields thatk admits afinite free resolution as anS-module. Therefore, by our induction hypothesis,S is pseudo-regular, whence has depthm − 1. It follows from (6) thatR has depthm, showing that itis pseudo-regular.

To prove the claim, choosex2, . . . , xm ∈ m so that(x, x2, . . . , xm)R = m. LetH betheR-submodule ofm/xm generated by the image ofx. HenceH ∼= k and we want toshow thatH is a direct summand ofm/xm. Let N be the submodule generated by theimages of thex2, . . . , xm in m/xm, so thatm/xm = H +N . Let a ∈ m and suppose itsimage inm/xm lies inH ∩N . It follows that we can writea in two different ways, namelyasa = a1x = a2x2 + · · ·+ amxm + rx with ai ∈ R andr ∈ m. By Nakayama’s lemma,we therefore must havea1 ≡ r ≡ 0 mod m, that it so say,a ∈ xm. In other words, weshowed thatH ∩N = 0 and hence thatm/xm ∼= H ⊕N , as required. �

7.12.Remark.Under the assumptions of the theorem,k has projective dimension equal tothe geometric dimension ofR andEul(k) = 0 (use the Koszul complex to calculate bothnumbers). The Koszul complex is minimal and thereforeTorRi (k, k) has dimension equal

to(ni

)for all i.

7.13.Remark.Using a similar argument, one can show thatR is pseudo-Cohen-Macaulayif and only if there exists a generic sequencex such thatR/xR has a finite free resolution(which then can be chosen to be the Koszul complexK•(x) of x). For a related result, seeProposition8.9below.


To not confuse with our present terminology we deviate from [7] or [18, §5] by calling aring Bertin-Serre regular, if every finitely generated ideal has finite projective dimension.If R is moreover coherent, then it is shown that any finitely generated ideal admits a finitefree resolution. Applied to the maximal ideal, we get immediately from Theorem7.11:

7.14.Corollary. A coherent Bertin-Serre regular local ring of finite embedding dimensionis pseudo-regular. �

For the converse, we have the following:

7.15.Corollary. Let(R,m) be a pseudo-regular local ring of geometric dimensiond, andlet beM an R-module. IfM is finitely d + 1-presented, thenM has finite projectivedimension (at mostd).

Proof. By Lemma7.9, there exists a minimal exact sequenceF• with n = d + 1, andthe i-th Betti number ofM is the rank ofFi. However,k has projective dimensiond byRemark7.12, and henceβd+1(M) = 0, showing thatFd+1 = 0. �

7.16.Corollary. LetR be a pseudo-regular local ring of geometric dimension one. If R iscoherent, then it is Bertin-Serre regular.

Proof. Let I be a finitely generated ideal. SinceR is coherent, it is finitely presented.HenceR/I is finitely 2-presented, and therefore has finite projective dimension by Corol-lary 7.15. �

We cannot expect for this result to also hold if the geometricdimensiond is strictlybigger than one, since a coherent Bertin-Serre regular ringis Cohen-Macaulay in the senseof [20] and therefore admits a regular sequence of lengthd (that is to say, in such a ring,naive depth always equals depth). To obtain a converse, we require a stronger coherencecondition:

7.17.Theorem. A local ring of finite embedding dimension is coherent and Bertin-Serreregular if and only if it is pseudo-regular and every finitelygenerated ideal has finite pre-sentation type.

Proof. If R is coherent and Bertin-Serre regular, then any finitely generated ideal has afinite free resolution by [17], whence has in particular finite presentation type. Moreover,R is pseudo-regular by Corollary7.14. To prove the converse, letI be a finitely generatedideal. By assumption,I, whence alsoR/I, is finitelyn-presented, and therefore has finiteprojective dimension by Corollary7.15applied withn sufficiently large. �

In [52], Soublin calls a ringR uniformly coherent7 if there exists a functionα : N→ N

such that any morphismRn → R has a kernel generated by at mostα(n) elements.

7.18.Theorem. LetR be a uniformly coherent local ring of finite embedding dimension.Then every finitely generated ideal ofR has finite presentation type. In particular,R ispseudo-regular if and only if it is Bertin-Serre regular.

Proof. By [52] or [3, Corollary 2.3], the countable direct productRN is coherent. Sincea finitely generated submodule of a finitely generated freeR-module embeds inRN, it isfinitely presented. Applied to the syzygies of a finitely generated idealI, we see thatI hasfinite presentation type. The second assertion then followsfrom Theorem7.17. �

7This is quite a strong hypothesis, even for Noetherian rings, for which it forces, among other things, that thedimension is at most two.

32 HANS SCHOUTENS

Pseudo-Cohen-Macaulay local rings.Recall that we calledR pseudo-Cohen-Macaulay,if its depth equals its geometric dimension.

7.19.Theorem. A pseudo-Cohen-Macaulay local ring is cata-Cohen-Macaulay.

Proof. LetR be a pseudo-Cohen-Macaulay local ring of geometric dimensiond and letxbe a generic sequence. SinceR has depthd, the grade ofn := xR is d, implying that allHi(x;R) vanish, fori > 0. For i = 1, this yields thatx is quasi-regular by [8, Ch. X,§9,Theoreme 1]. HenceR is cata-Cohen-Macaulay by Theorem6.8. �

The converse is in general false:R can be cata-Cohen-Macaulay without being pseudo-Cohen-Macaulay; an example is provided by the depth zero cata-regular ring in6.3. Onthe other hand, neither is it the case that in a pseudo-Cohen-Macaulay local ringR everyR-regular element isR-regular. For instance,R could be a non-separated domain, in whichcase any non-zero element in the ideal of infinitesimals isR-regular, but zero inR. Thisalso gives an example of anR-regular element which is not part of a generic subset. Fromthe proof of [29, Theorem 16.3], it follows that ifR is separated and cata-Cohen-Macaulay,then every generic element isR-regular. In particular, we showed that ifR has geometricdimension one, thenR is cata-Cohen-Macaulay if and only ifRsep is pseudo-Cohen-Mac-aulay.

7.20.Example. Let Rw := A/(ξ2, ξζw)A whereA := k[[ξ, ζ]]. It follows that allRw

have depth zero and dimension one. Hence their ultraproductR♮ has depth zero and ultra-dimension one. The cataproductR♯ is isomorphic tok♮[[ξ, ζ]]/ξ2k♮[[ξ, ζ]], wherek♮ is theultrapower ofk. This is a one-dimensional Cohen-Macaulay local ring. HenceR♮ is cata-Cohen-Macaulay and has geometric dimension one. In conclusion,R♮ is isodimensionaland cata-Cohen-Macaulay, but not pseudo-Cohen-Macaulay.

7.21.Corollary. A local ring of finite embedding dimension is pseudo-regularif and onlyif it is pseudo-Cohen-Macaulay and has multiplicity one.

Proof. The direct implication follows from Proposition7.7and Corollary6.9. Conversely,if R has multiplicity one and is pseudo-Cohen-Macaulay, then itis cata-Cohen-Macaulayby Theorem7.19, whence cata-regular by Corollary6.9, and the result now follows fromProposition7.7. �

7.22.Corollary. LetR be a pseudo-Cohen-Macaulay local ringR of geometric dimensiontwo. IfR is either a domain or separated, then any generic sequence isR-regular.

Proof. Let (x, y) be a generic sequence inR. If R is a domain, thenx is R-regular. Letus show that the same holds ifR is separated. SinceR has depth two by assumption,H2(x, y;R) = 0. This means that wheneverax+ by = 0 for somea, b ∈ R then(a, b) =r(y,−x) for somer ∈ R. In particular, ifa ∈ AnnR(x), then(a, 0) = r(y,−x) for somer ∈ R, showing thata ∈ yAnnR(x). In other words,AnnR(x) = yAnnR(x) so thatby inductionAnnR(x) = ynAnnR(x) whenceAnnR(x) ⊆ IR = 0. This concludes theproof thatx isR-regular. Using once more the above characterization ofH2 = 0, we seethat in either case,y isR/xR-regular, whence(x, y) isR-regular. �

We can generalize Proposition5.11substantially under an additional Cohen-Macaulayassumption.

7.23.Proposition. LetR be a local ring of finite embedding dimension and letM be anR-module of finite length. IfR is pseudo-Cohen-Macaulay, thenTorRi (R,M) vanishes forall i > 0.


Proof. SinceM has finite length, its annihilator ism-primary, and hence contains a genericsequence by Corollary3.13. SinceR → R(ξ) is faithfully flat, the vanishing of the Tor’sis unaffected by such an extension. Hence, after some Nagataextension, we may assume,using Remark7.4, thatR admits anR-regular, generic sequencex contained in the an-nihilator of M . SinceR is Cohen-Macaulay by Theorem7.19, the sequencex is alsoR-regular. By a well-known deformation property of Tor modules, we get

TorRi (R,M) ∼= TorR/xRi (R/xR,M)

for all i > 0. From this the vanishing then follows sinceR/xR ∼= R/xR by Lemma2.8.�

Given a moduleM over a local ringR of finite embedding dimension, we define itsgeometric dimensionto be the geometric dimension ofR/AnnR(M), and we denote itgdim(M). Since the notions of grade and depth also extend to modules,we may calla finitely generatedR-moduleM pseudo-Cohen-Macaulay, if its geometric dimensionequals its depth.

7.24.Corollary. LetR be a local ring of finite embedding dimension and letM be a finitelygeneratedR-module. If bothR andM are pseudo-Cohen-Macaulay, thenTorRi (R,M) =0, for all i > 0.

Proof. We induct on the geometric dimensione of M . If e = 0, thenM is a finitelygenerated module over the Artinian local ringR/AnnR(M), whence has finite length,and the result follows from Proposition7.23. So assumee > 0. As far as proving thevanishing is concerned, we may always, by faithfully flat descent, take a Nagata extensionof R. Hence, by the module analogue of Lemma7.3 (the proof of which is left to thereader), we may assume, after possibly taking a Nagata extension, thatx is anM -regularelement. From the exact sequence

0 →Mx

−−→M →M/xM → 0

we get, by tensoring withR, part of a long exact sequence

0 = TorRi+1(R,M/xM) → TorRi (R,M)x

−−→TorRi (R,M) → TorRi+1(R,M/xM) = 0

where the two outer modules are zero by induction. Fixi and putT := TorRi (R,M). SinceT = xT , we haveT = IRT . As R is Noetherian,IRR vanishes, whence so doesIRT ,sinceT is the homology of a complex of modules overR. This showsT = 0, completingour proof. �

7.25.Example. In [37], a class of local rings was introduced which extends the class ofCohen-Macaulay local rings. More precisely, for eachd, e ≥ 0, letCMd,e be the class ofall local ringsR such that there exists anR-regular sequence of lengthd and such that theminimal length of a homomorphic imageR/xR is e, wherex is an arbitrary tuple inR oflengthd. The latter condition implies thatR has geometric dimension at mostd, and theformer that its depth is at leastd. It follows thatR is pseudo-Cohen-Macaulay of geometricdimensiond. Letx be an arbitrary tuple of lengthd. SupposeR/xR is Artinian of lengthl (by assumptionl ≥ e). HenceR/xR ∼= R/xR andx is generic inR. Moreover,x isR-regular, sinceR is Cohen-Macaulay. It follows that the idealxR has multiplicityl. Fora general choice of system of parametersy in R, the idealyR is a reduction ofmR ([29,Theorem 14.14]), so that the multiplicity ofyR is equal tomult(R). By assumption, theminimal value of the multiplicity of an ideal generated by ad-tuple fromR is e. Since these

34 HANS SCHOUTENS

form a general subset of alld-tuples inR, we showed thatR has multiplicitye, whence sodoesR by Remark3.5. In fact, we have the following characterization of these classes:

7.26.Theorem. A local ringR is pseudo-Cohen-Macaulay of geometric dimensiond andmultiplicity e if and only ifR(ξ) belongs to the classCMd,e for some (d-)tuple of indeter-minatesξ.

Proof. SinceR andR(ξ) are easily seen to have the same multiplicity (by comparingtheir completions), one direction follows from the previous discussion. On the other hand,supposeR is pseudo-Cohen-Macaulay of geometric dimensiond and multiplicity e. Bythe same argument as above, we may choose a generic sequencex in R such thatxR isa reduction ofmR, whence has multiplicitye. It follows from Lemma2.8 thatR/xRhas lengthe and by a similar argument that this is the least possible length. In order toconstruct anR-regular sequence, we have to go to an extensionR(ξ) by Lemma7.3 andthis extension is then in the classCMd,e. �

In particular, by Corollary7.21, a local ringR is pseudo-regular if and only ifR(ξ)belongs toCMd,1 for somed and somed-tuple of indeterminatesξ. Moreover, by Propo-sition7.5, an ultra-Noetherian local ring belongs toCMd,e if and only if it is pseudo-Coh-en-Macaulay of geometric dimensiond and multiplicitye.

Let R be a local ring of finite embedding dimension. We say thatR is pseudo-Goren-stein, if it is pseudo-Cohen-Macaulay and there exists a generic sequencex such thatR/xR is an Artinian Gorenstein ring.

7.27.Proposition. A pseudo-Cohen-Macaulay local ring is pseudo-Gorenstein if and onlyif it is cata-Gorenstein.

In fact, let (R,m) be a pseudo-Cohen-Macaulay local ring of geometric dimension dand letk be its residue field. IfR is pseudo-Gorenstein, thenExtiR(k,R) = 0, for alli 6= d andExtdR(k,R)

∼= k. Conversely, ifExtiR(k,R) vanishes for somei > d or ifExtdR(k,R)

∼= k, thenR is pseudo-Gorenstein.

Proof. Let x be a generic sequence inR. By Lemma7.2, the extensionR(ξ) is againpseudo-Cohen-Macaulay andx is generic inR(ξ). Since

R/xR→ (R/xR)(ξ) ∼= R(ξ)/xR(ξ)

is faithfully flat and unramified, the former is Gorenstein ifand only if the latter is. Sincethe Ext-functors commute with faithfully flat base change, we may replaceR by R(ξ)everywhere and assume by Lemma7.3thatx is a regular sequence.

In particular,R is pseudo-Gorenstein if and only ifR/xR ∼= R/xR is Gorenstein ifand only if R is Gorenstein, sincex is R-regular. This already proves the first assertion.Sincex isR-regular, we have

(9) ExtiR(k,R)∼= Exti−d

R/xR(k,R/xR)

where we letExtjR(·, ·) be the zero functor for negativej (see for instance [9, Lemma3.1.16] and the proof of (3)⇔ (1) of [29, Theorem 16.6]). The final assertion now followsfrom [29, Theorem 18.1] applied to the Artinian local ringR/xR. �

It follows that if R is pseudo-Gorenstein, thenR/xR is Gorenstein for every genericsequencex.


8. ULTRA-SINGULARITIES

We now compare the ‘cata’ and ‘pseudo’ versions from the previous two sections withtheir ‘ultra’ counterparts. Throughout this section, unless mentioned explicitly,R♮ is anultra-Noetherian local ring with maximal idealm♮ and residue fieldk♮, realized as theultraproduct of Noetherian local rings(Rw,mw) of bounded embedding dimension andresidue fieldkw. Recall (Lemma5.6) that the cataproductR♯ of theRw is the separatedquotient as well as the completion ofR♮, and it is in particular Noetherian.

8.1.Theorem. For an ultra-Noetherian local ringR♮, the following are equivalent

(8.1.1) R♮ is pseudo-regular;(8.1.2) R♮ is ultra-regular;(8.1.3) R♮ is cata-regular and isodimensional.

Proof. Let R♮ be the ultraproduct of Noetherian local ringsRw of bounded embeddingdimension. IfR♮ is pseudo-regular, then it is isodimensional by Theorem4.4and thereforecata-regular by Theorem6.1. Moreover, by Łos’ Theorem, almost allRw are regular sinceembedding dimension and depth are first-order definable. This shows thatR♮ is ultra-regular, and the converse follows along the same lines. Finally, if R♮ is cata-regular andisodimensional, then it is pseudo-regular, again by Theorem 4.4. �

The same proof also shows thatR♮ is ultra-regular if and only if it is not ultra-singular.In view of Lemma5.6, we may rephrase the theorem as follows:

8.2.Corollary. LetRw be Noetherian local rings of the same dimension and parameterdegree and letR♯ be their cataproduct. Then almost allRw are regular if and only ifR♯

is. �

8.3.Corollary. Any localization of an ultra-regular local ring at a finitelyrelated primeideal is ultra-regular.

Proof. LetR♮ be an ultra-regular local ring, obtained as the ultraproduct of d-dimensionalregular local ringsRw, and letp ∈ FR-Spec(R♮). By Proposition5.1, there exist primeidealspw ⊆ Rw whose ultraproduct is equal top. Since almost each(Rw)pw

is regular ofdimension at mostd, their ultraproduct(R♮)p is ultra-regular (of geometric dimension atmostd). �

We conclude our discussion of ultra-regular rings with an ultraproduct version of Corol-lary 5.15.

8.4.Corollary. The canonical embeddingR → R♮ of an excellent local ring in its ultra-power has ultra-regular fibers at finitely related prime ideals: for everyp ∈ FR-Spec(R♮),the fiber ring(R♮/gR♮)p is ultra-regular, whereg = p ∩R.

Proof. To show that(R♮/gR♮)p is ultra-regular, we may replaceR byR/g, sinceR♮/gR♮

is the ultrapower ofR/g, and assume without loss of generality thatR is a domain andp∩R = (0). By Corollary5.3, the localization(R♮)p has finite embedding dimension, andp is the ultraproduct of prime idealspw ∈ Spec(R). SinceR is an excellent domain, itssingular locus is a proper, closed subset, say, defined by a non-zero idealI ⊆ R. If almostall pw would belong to this singular locus, then they would almost all containI, whenceso wouldp, contradicting thatp ∩ R = (0). Hence almost allpw are in the regular locus,and the result now follows from the proof of Corollary8.3. �

36 HANS SCHOUTENS

Ultra-Cohen-Macaulay local rings. Recall thatR♮ is calledultra-Cohen-Macaulayifalmost allRw are Cohen-Macaulay. We can characterize this property in terms of thefundamental inequalities (1).

8.5.Theorem. For an ultra-Noetherian local ringR♮, the following are equivalent

(8.5.1) R♮ is ultra-Cohen-Macaulay;(8.5.2) the depth ofR♮ equals its ultra-dimension.

In particular,R♮ is pseudo-Cohen-Macaulay if and only if it is ultra-Cohen-Macaulay andisodimensional.

Proof. The first assertion follows immediately from the fact that depth is first-order. Thesecond assertion is now also clear, since a pseudo-Cohen-Macaulay must be isodimen-sional by Theorem5.18. �

8.6.Remark.Note that unlike in the regular case, isodimensionality together with beingcata-Cohen-Macaulay is not sufficient for being pseudo-Cohen-Macaulay, as example7.20shows.

Also note that ultra-Cohen-Macaulay does not imply pseudo-Cohen-Macaulay nor evencata-Cohen-Macaulay. Namely, let(R,m) be a non-Cohen-Macaulay local ring and letR♮

andR♯ be the respective ultraproduct and cataproduct of theR/mn. Corollaries5.10and5.15together imply thatR♯ is not Cohen-Macaulay. HenceR♮ is not cata-Cohen-Macau-lay, although it is clearly ultra-Cohen-Macaulay (there isno contradiction with the abovetheorem, sinceR♮ is not isodimensional).

8.7.Corollary. The cataproduct of Cohen-Macaulay local rings having the same dimen-sion and the same multiplicity, is again Cohen-Macaulay.

Proof. Let R♮ andR♯ be the respective ultraproduct and cataproduct of Noetherian localringsRw of the same multiplicity and the same dimension. If almost all Rw are Cohen-Macaulay, thenR♮ is isodimensional by Remark5.24. Therefore,R♮ is pseudo-Cohen-Macaulay by Theorem8.5, and henceR♯ is Cohen-Macaulay by Theorem7.19. �

Let us call an ultra-moduleM ♮, that is to say, an ultraproduct ofRw-modulesMw,ultra-Cohen-Macaulay, if almost allMw are Cohen-Macaulay. Although such a moduleneed not be finitely generated, we have:

8.8.Lemma. For eachw, letMw be a finitely generated module overRw, and letM ♮ betheir ultraproduct. If almost allRw are Cohen-Macaulay, of the same dimension and mul-tiplicity, thenM ♮ is finitely generated and pseudo-Cohen-Macaulay if and onlyif almostall Mw are Cohen-Macaulay of the same multiplicity.

Proof. If almost allMw are Cohen-Macaulay of multiplicityl, then there exists, by [9,Theorem 4.6.10], anRw-regular andMw-regular sequencexw such thatMw/xwMw haslengthl. Since each sequence can have length at mostd, almost all have the same lengths ≤ d. The ultraproductM ♮/x♮M ♮, too, has lengthl by Proposition5.13, wherex♮ is theultraproduct of thexw. In particular,M ♮ is finitely generated. Moreover,x♮ isM ♮-regular,showing thatM ♮ has depth at leasts. On the other hand, sinceM ♮/x♮M ♮ has finite length,the geometric dimension ofM ♮ is at mosts. This proves thatM ♮ is pseudo-Cohen-Mac-aulay.


Conversely, assumeM ♮ is pseudo-Cohen-Macaulay and finitely generated. As depthis first-order, by the (module version of) Proposition7.5, there exists anM ♮-regular se-quencex♮ such thatM ♮/x♮M ♮ has geometric dimension zero. AsM ♮ is finitely gener-ated,M ♮/x♮M ♮ has finite length, say,l. Lettingxw be tuples inRw having as ultraprod-uct x♮, the ultraproduct of theMw/xwMw is equal toM ♮/x♮M ♮, and hence almost allMw/xwMw have lengthl by Proposition5.13. Moreover, almost eachxw isMw-regular,showing thatMw is Cohen-Macaulay, of multiplicity at mostl, by another application of[9, Theorem 4.6.10]. �

The next result, which is some type of coherence property forultra-Cohen-Macaulaylocal rings, will be used in§11 to deduce some uniform bounds on Betti numbers. Recallthat thei-th Betti numberβi(M) of a module over a local ringR with residue fieldk is the(possibly infinite) dimension ofTorRi (M,k); for the notion of finite presentation type, see§7.8.

8.9.Proposition. If R♮ is an isodimensional, ultra-Cohen-Macaulay local ring, then ev-ery finitely generated pseudo-Cohen-MacaulayR♮-module (e.g., everyR♮-module of finitelength) has finite presentation type. More precisely, for any givene, if almost eachMw isa Cohen-MacaulayRw-module of multiplicitye, then, for eachn, almost allMw have thesamen-th Betti number as their ultraproductM ♮ and as their cataproductM ♯.

Proof. In view of Lemma8.8, it suffices to prove the second assertion. We will show, byinduction onn, that

βn(M ♮) = βn(M ♯) = βn(Mw)

for almost allw. The casen = 0 follows from Proposition5.13, sinceM ♮ is finitelygenerated by Lemma8.8. So assumen ≥ 1.

LetFn,w → Fn−1,w → · · · → F1,w →Mw → 0

be a minimal finite free resolution ofMw, with eachFi,w a finite freeRw-module of rankri,w := βi(Mw) (see§7.8). Taking ultraproducts, we get by Łos’ Theorem a minimalresolution

(10) Fn,♮ → Fn−1,♮ → · · · → F1,♮ → M ♮ → 0

By induction and Lemma7.9, we getFi,♮∼= Rri

♮ , for i < n, whereri is the common valueof almost allβi(Mw). Theorem8.5implies thatR♮ is pseudo-Cohen-Macaulay, and hence

by Corollary7.24, all TorR♮

i (R♯,M ♮) vanish fori > 0. Therefore, if we tensor (10) withR♯, we get again a minimal resolution

Fn,♯ → Rrn−1

♯ → · · · → Rr1♯ → M ♯ → 0

SinceR♯ is Noetherian and the resolution is minimal,ri = βi(M ♯) for i < n, and the lastmodule in this resolution,Fn,♯, is generated byrn := βn(M ♯) elements. Tensoring withthe common residue fieldk♮ of R♮ andR♯, we get

krn♮∼= Fn,♯/m♮Fn,♯

∼= Fn,♮/m♮Fn,♮.

Since the latter module is the ultraproduct of theFn,w/mwFn,w∼= k

rn,ww , wherekw is the

residue field ofRw, we getrn,w = rn for almost allw, as we wanted to show. �

8.10.Theorem. A pseudo-Cohen-Macaulayultra-Noetherian local ringR♮ is cata-Goren-stein if and only if it is ultra-Gorenstein; and it is a cata-‘complete intersection’ if and onlyif it is an ultra-‘complete intersection’.

38 HANS SCHOUTENS

In particular, if Rw are Cohen-Macaulay local rings having the same dimension andmultiplicity, then their cataproductR♯ is respectively Gorenstein or a complete intersectionif and only if so are almost allRw.

Proof. The second assertion follows from the first in view of Theorem5.22 and Theo-rem 8.5. We already observed thatR♮ is isodimensional, so thatR♯ and almost allRw

have the same dimension,d, say. Hence ifx♮ is a generic sequence inR♮, realizedas an ultraproduct of tuplesxw in Rw, then almost eachxw is a system of parametersin Rw, whenceRw-regular. Therefore, almost allRw are Gorenstein if and only if soare almost allRw/xwRw. This in turn is equivalent withR♮/x♮R♮ being Gorenstein byŁos’ Theorem (using that these are Artinian local rings; see[36] for more details). SinceR♮/x♮R♮

∼= R♯/x♮R♯, the latter is then equivalent withR♯ being Gorenstein.By Proposition8.9, we have a minimal free resolution ofR♯-modules

Rr♯ → Rm

♯ → R♯ → k♮ → 0

wherer = β2(k♮) = β2(kw) andm = β1(k♮) = β1(kw), for almost allw. Moreover,R♯ has the same dimensiond as almost allRw by Theorem8.5. By [9, Theorem 2.3.3],therefore,R♯ is a complete intersection if and only ifr = m(m+ 2)/2− d, if and only ifalmost allRw are complete intersections. �

Lefschetz Hulls. In [4], we showed that every Noetherian local ringR of equal charac-teristic zero (that is to say, containing the rationals) admits an ultra-Noetherian faithfullyflat extensionD(R) which isLefschetz, meaning thatD(R) is the ultraproduct of Noethe-rian local ringsRw of prime characteristic. In fact, theRw may be chosen to be completewith algebraically closed residue field. We callD(R) a Lefschetz hullof R. Although theconstruction can be made more functorial, it still depends on a choice of a cardinal numberlarger than the cardinality ofR. However, in caseR is of finite type over an uncount-able8 algebraically closed field of characteristic zero, there isa canonical choice forD(R),called thenon-standard hullof R and denotedR∞; see [40] for details. In view of ourcharacterizations of pseudo-singularities in this section, the following result is immediatefrom [4, Theorem 5.2]:

8.11.Theorem. A Noetherian local ringR of equal characteristic zero with Lefschetzhull D(R) is Cohen-Macaulay (respectively, Gorenstein or regular) if and only ifD(R) ispseudo-Cohen-Macaulay (respectively, pseudo-Gorenstein or pseudo-regular).

9. CATA -NORMALIZATIONS

An extremely useful fact in commutative algebra is the existence of Noether normaliza-tions: any finitely generated algebra over a field or any complete Noetherian local domainadmits a regular subring over which it is module-finite. Thisresult is not hard to showin equal characteristic, so that we will adopt this additional assumption in this section toformulate an analogue for local rings of finite embedding dimension. In the sequel, let(R,m) be an equicharacteristic local ring with residue fieldk and letπ : R → k denote theinduced surjection.

8Strictly speaking, of cardinality equal to2λ, for some infinite cardinalλ.


Weak coefficient fields. A subfieldκ of R is called aweak coefficient fieldof R if therestriction ofπ to κ induces an algebraic extensionπ(κ) ⊆ k. If this extension is anisomorphism, then we callκ a coefficient fieldof R (in the literature one also encountersthe notion of aquasi-coefficientdefined as a weak coefficient fieldκ for which the inducedextensionπ(κ) ⊆ k is also separable). The next result is well-known, but its proof isincluded for convenience.

9.1.Lemma. Let (R,m) be an equicharacteristic local ring. For any subfieldκ0 ofR, wecan find a weak coefficient fieldκ ofR containingκ0.

If, moreover,R has characteristic zero and is Henselian, then we can chooseκ to be acoefficient field.

Proof. Let κ be maximal among all subfields ofR containingκ0 (such a field exists byZorn’s lemma). We need to show that the extensionπ(κ) ⊆ k is algebraic, wherek is theresidue field ofR andπ : R → k the residue map. To this end, take an arbitrary elementu ∈ k \ π(κ). Leta ∈ R be such thatπ(a) = u. It follows thata /∈ κ. By maximality, thesubringκ[a] ofR generated byamust contain a non-zero non-invertible element (lestκ(a)be a larger subfield ofR). This means thatP (a) ∈ m, for some non-zeroP ∈ κ[ξ]. Hencetaking reductions, we getP π(u) = 0 in k, whereP π is the polynomial obtained fromPby applyingπ to each of its coefficients. SinceP π is not identical zero,u is algebraic overπ(κ).

To prove the last assertion, assume by way of contradiction thatR has characteristiczero and is Henselian, but thatπ(κ) is strictly contained ink. Takeu ∈ k \ π(κ). Let pbe a minimal equation ofu overπ(κ) and letP ∈ κ[ξ] be such that its imageP π is equalto p. Sinceu is a single root ofp, Hensel’s Lemma yields the existence of a roota ∈ Rof P with π(a) = u. However, this implies that the fieldκ[ξ]/Pκ[ξ] embeds inR via theassignmentξ 7→ a, contradicting the maximality ofκ. �

A local homomorphismA → R is calledcata-integral(respectively,cata-finite, ca-ta-injective, cata-surjective, cata-flat) if its completionA → R is integral (respectively,finite, injective, surjective, flat). Let(R,m) be a local ring of finite embedding dimension.

Cata-normalization. A cata-normalizationof R is a cata-integral local homomorphismθ : (A, p) → (R,m) such thatA is a (Noetherian) regular local ring andpR is m-primary.We say that a cata-normalizationθ is Cohen, if pR = m, andNoetherif θ is injective.

9.2. Theorem. An equicharacteristic local ring of finite embedding dimension admits acata-normalization, which can be chosen to be either Cohen or Noether.

Proof. Let (R,m) be an equicharacteristic local ring of finite embedding dimension. ByLemma9.1, there exists a weak coefficient fieldκ of R. Choose a tuplex := (x1, . . . , xn)generating somem-primary ideal. LetA be the localization of the polynomial ringκ[ξ]at the ideal generated by the indeterminatesξ = (ξ1, . . . , ξn). Let θ : A → R be the(unique)κ-algebra homomorphism which sendsξi to xi, for eachi. To show thatθ is acata-normalization, we only need to show that its completion is integral, since the otherconditions are immediate. Therefore, without loss of generality, we may already assumethatA andR are complete, so that both rings are now Noetherian. Letπ : R → k be theresidue map and letl be a finite extension ofπ(κ) contained ink. PutBl := π−1(l). Sinceκ + m ⊆ Bl, one checks easily thatBl is a local ring with maximal idealm. The localhomomorphismA → Bl induces a finite extension of residue fields. Therefore, sinceAis complete andBl is separated,Bl is finitely generated as anA-module by [29, Theorem

40 HANS SCHOUTENS

8.4]. Sincek is the union of all its finite extensionsl containingπ(κ), so isR the union ofall theBl, showing thatR is integral overA.

It is clear that if we choosex so that it generatesm, thenθ is Cohen. Assume next thatx is a generic sequence. In particular,R has dimensionn by Theorem3.4. SinceA is ann-dimensional domain andA → R is integral, this map must be injective. But then somustA→ R be, that is to say,θ is Noether. �

9.3. Remark. If I is a finitely generated ideal ofR, then we can always choose a cata-normalizationA → R with the additional property that there is some idealJ ⊆ A withJR = I. Simply choosex so that it contains a set of generators ofI.

9.4.Remark.From the above proof it is also clear that ifR admits a coefficient field, thenwe can choose the cata-normalizationA→ R to be cata-finite.

9.5.Theorem. An equicharacteristic local ringR of finite embedding dimension is cata-Cohen-Macaulay if and only if there exists a cata-flat, cata-normalizationA→ R.

Proof. If A → R is flat withA regular, thenR is Cohen-Macaulay by [29, Corollary toTheorem 23.3], since the closed fiber has dimension zero. This proves one direction. Toprove the converse implication, assume thatR is Cohen-Macaulay. LetA → R be anyNoether cata-normalization. SinceA → R is a local homomorphism of Noetherian localrings of the same dimension, with closed fiber having dimension zero, it is flat by [29,Theorem 23.1], becauseA is regular andR is Cohen-Macaulay. �

From the proof it follows that any Noether cata-normalization of a cata-Cohen-Mac-aulay local ring is cata-flat. We conclude this section with an instance of true NoetherNormalization:

9.6.Theorem. If R♮ is an ultraproduct of equicharacteristic completed-dimensional Noe-therian local rings, thenR♮ is isodimensional if and only if there exists an ultra-regularlocal subringS♮ ⊆ R♮ such thatR♮ is module-finite over it.

Proof. Let us show that the if-direction holds for any ultra-Noetherian local ring of finiteembedding dimension. LetS♮ ⊆ R♮ be a finite extension withS♮ ultra-regular, realizedas the ultraproduct of regular local ringsSw. By Proposition5.13, if R♮ is generated by atmostN elements overS♮, then almost eachRw is generated by at mostN elements overSw. If yw is a regular system of parameters inSw, then its image is a system of parametersin Rw. SinceRw/ywRw has vector space dimension at mostN over the residue field ofSw, its length is at mostN , showing that theRw have bounded parameter degree. Hence,R♮ is isodimensional by Theorem5.22.

Conversely, assumeR♮ is as in the statement, so that in particular its geometric dimen-sion isd. By Theorem5.22, almost allRw have parameter degreeρ, for someρ <∞. By[46, Corollary 3.8], almost eachRw is a module-finite extension of a regular subringSw,generated as anSw-module by at mostρ elements. LetS♮ be the ultra-regular local ringgiven as the ultraproduct of theSw. Another application of Proposition5.13yields thatR♮

is generated by at mostρ elements overS♮. �

9.7. Example. The equicharacteristic condition is necessary as the following exampleshows. Fix a prime numberp and an indeterminateξ, and letZp denote the ring ofp-adic integers. PutRw := Zp[ξ]/(ξ

2w+1 − p2)Zp[ξ] and letR♮ be the ultraproduct of theRw. EachRw is a one-dimensional complete local Cohen-Macaulay domainwith mul-tiplicity (=parameter degree) two. HenceR♮ is isodimensional (indeed, the cataproductR♯

∼= (Zp♯/p2Zp♯)[[ξ]] is also one-dimensional, whereZp♯ is the catapower ofZp).


Suppose there is an ultra-regular subringS♮ ⊆ R♮ such thatR♮ is generated as anS♮-module byN elements. Hence by Łos’ Theorem, there is a regular subringSw ⊆ Rw, suchthatRw is generated as anSw-module byN elements, for almost allw. This, however,contradicts [46, Proposition 3.5 and Example 3.2], where it is shown that theleast numberof generators for any regular subring ofRw must be equal to the length ofRw/pRw (theso-called equi-parameter degree ofRw), that is to say, must be at least2w + 1.

By varying the primep as well (say, by lettingpw be an enumeration of all primenumbers and replacingp by pw in the definition ofRw), we can construct a similar coun-terexampleR♮ which itself is equicharacteristic zero. This latter ring also shows the extentto which the Ax-Kochen-Ershov theorem ([5, 15, 16]) holds. Namely, letV ♮ be the ultra-product of theZpw

, so thatV ♮ is in particular ultra-regular. By Ax-Kochen-Ershov,V ♮ isalso the ultraproduct of theFpw

[[t]] whereFpwis thepw-element field andt a single inde-

terminate. PutR′w := Fpw

[[t, ξ]]/(ξ2w+1 − t2)Fpw[[t, ξ]] and letR′

♮ be their ultraproduct(so thatR′

♮ andR′w are the equicharacteristic analogues ofR♮ andRw). BothR♮ andR′

♮

containV ♮ as a subring in a natural way, but neither extension is finite.However, there isa second embedding ofV ♮ intoR′

♮ making the latter a finite extension. Namely,V ♮ is alsoisomorphic with the subring given as the ultraproduct of theFpw

[[ξ]]. Under this identifi-cation,R′

♮ is isomorphic toV ♮[t]/(t2 − α)V ♮[t], whereα is the ultraproduct of theξ2w+1.

In conclusion,R♮ andR′♮ cannot be isomorphic (note, however, that their cataproducts are

isomorphic, toF ♮[[t, ξ]]/t2F ♮[[t, ξ]], whereF ♮ is the ultraproduct of theFpw

).

9.8.Remark.Using [46, Proposition 3.5], we can use the same argument to show that if R♮

is an ultraproduct of completed-dimensional Noetherian local rings of mixed characteristicand of bounded equi-parameter degree, thenR♮ admits an ultra-regular local subringS♮

over which it is module-finite. Recall that theequi-parameterdegree of a Noetherian localringA of mixed characteristicp is the least possible length of a homomorphic imageA/Imodulo a parameter idealI ⊆ A containingp.

9.9. Corollary. If S♮ ⊆ R♮ is a local module-finite extension of ultra-Noetherian localrings withS♮ ultra-regular, thenR♮ is pseudo-Cohen-Macaulay if and only if it is flat overS♮.

Proof. Let (Sw, nw) andRw be Noetherian local rings with ultraproduct equal to(S♮, n♮)andR♮ respectively. By Łos’ Theorem, almost allSw ⊆ Rw are finite extensions withSw

regular. Suppose first thatR♮ is pseudo-Cohen-Macaulay, whence ultra-Cohen-Macaulayby Theorem8.5. Hence almost allRw are Cohen-Macaulay, whence flat overSw. To show

thatR♮ is flat overS♮, it suffices by [29, Theorem 7.8(3)] to show thatTorS♮

1 (R♮, S♮/I♮)vanishes for all finitely generated idealsI♮ of S♮. ChooseIw ⊆ Sw whose ultraproduct

equalsI♮. Since ultraproducts commute with homology,TorS♮

1 (R♮, S♮/I♮) is the ultra-

product of theTorSw

1 (Rw, Sw/Iw). Since the latter are zero by flatness, so is the former.Conversely, supposeS♮ ⊆ R♮ is flat. In particular,R♮ isodimensional by (the proof

of) Theorem9.6. By the same argument as above, the vanishing ofTorS♮

1 (R♮, S♮/n♮)

implies the vanishing of almost allTorSw

1 (Rw, Sw/nw). By the local flatness criterion,this implies that almost allRw are flat overSw, whence are Cohen-Macaulay. HenceR♮

is ultra-Cohen-Macaulay, and therefore pseudo-Cohen-Macaulay by Theorem8.5. �

42 HANS SCHOUTENS

10. HOMOLOGICAL THEOREMS

In this section, we prove for local rings of finite embedding dimension the counterpartsof the homological theorems from commutative algebra, under the the assumption that thecompletion is equicharacteristic. We start with an immediate corollary of the definitions:

10.1.Corollary (Monomial Theorem). LetR be a local ring of geometric dimensiond andlet x be a generic sequence inR. SupposeR has either equal characteristic or otherwiseis infinitely ramified (see§2.9). If ν0, . . . , νs ∈ Nd are multi-indices such thatν0 does notbelong to the semigroup generated byν1, . . . , νs, thenxν0 does not lie in the ideal inRgenerated byxν1 , . . .xνs .

Proof. If the contrary were true, then the same ideal membership holds in the completionR. However, by Proposition3.9, the image ofx in R is a system of parameters, thus violat-ing the usual Monomial Theorem (see for instance [21]), sinceR is equicharacteristic.�

A special instance of the assertion (which is often already referred to as the MonomialTheorem) is the fact that for any generic sequence(x1, . . . , xd) in R, we have

(11) (x1 · · ·xd)t /∈ (xt+1

1 , . . . , xt+1d )R

for all t. In the Noetherian setup, the latter result suffices to show the so-called DirectSummand Theorem (see for instance [9, Lemma 9.2.2]). However, it is not clear how toderive this in the present setup (presently, I can only get a weaker version, which I omithere).

Next we have a look at the Hochster-Roberts theorem. Although one can formulatea more general version, we will only give the result for localhomomorphismsR → Swhich arelocally of finite type, meaning thatS is a localization of some finitely generatedR-algebra. Note that the class of local rings of finite embedding dimension is closed undersuch maps: if(R,m) → (S, n) is locally of finite type, then so isR/m → S/mS. Inparticular,S/mS is Noetherian, andn(S/mS) is finitely generated. Hence ifm is finitelygenerated, then so isn.

10.2.Theorem (Hochster-Roberts). LetR → S be a local homomorphism between localrings of finite embedding dimension. SupposeR has equal characteristic or is infinitelyramified. IfR → S is cyclically pure and locally of finite type, and ifS is cata-regular,thenR is cata-Cohen-Macaulay.

Proof. It suffices to show thatR → S is cyclically pure, for then the classical Hochster-Roberts theorem shows thatR is Cohen-Macaulay by [25, Theorem 2.3], sinceS is regularand equicharacteristic. To prove cyclical purity, we need to show thatI = IS ∩ R for eachidealI in R. Since any ideal is an intersection ofmR-primary ideals, it suffices to showthis for I anmR-primary ideal, wherem is the maximal ideal ofR. By Lemma2.8, anysuch ideal is extended fromR, that is to say of the formI = aR with a anm-primary idealofR. SinceS/aS is locally of finite type over the Artinian local ringR/a, it is Noetherian.Therefore,aS is closed, so thatIS ∩ S = aS ∩ S = aS by Lemma2.4. Hence, in thecomposition

R/I ∼= R/a → S/aS → S/IS

all maps are injective, as the first is an isomorphism by Lemma2.4 and the second isinjective by assumption. This proves thatIS ∩ R = I, as required. �


10.3.Remark.Combining this result with Theorems6.8, 8.1 and9.5, and Corollary7.14yields Corollary1.1 from the introduction. In the theorem, less than cyclical purity isrequired; it suffices thatR → S is pure-closed, in the sense thatIS ∩ R = I for everyclosed (equivalently, everym-primary) idealI ⊆ R. Furthermore, we may weaken thecondition thatR → S is locally of finite type to requiring that its closed fiberS/mS isNoetherian. In order to apply the techniques from§13 and deduce an asymptotic versionof the Hochster-Roberts theorem in mixed characteristic, we would like to prove a strongerresult: namely, under an additional isodimensionality assumption, may we conclude thatR is pseudo-Cohen-Macaulay?

To obtain other homological properties, we follow Hochster’s treatment [21], by gener-alizing the notion of big Cohen-Macaulay modules. In fact, as in the Noetherian case, wecan even put a ring structure on the latter:

Big Cohen-Macaulay algebras.We call anR-algebraB a big Cohen-Macaulay algebraif some generic sequence ofR is B-regular; we callB a balanced big Cohen-Macaulayalgebraif every generic sequence isB-regular.

10.4.Theorem. Let R be a local ring of finite embedding dimension. IfR has equalcharacteristic or is infinitely unramified, then it admits a balanced big Cohen-Macaulayalgebra.

Proof. By the work of Hochster and Huneke ([23, 25]) or the more canonical construc-tion of [4, §7] (note that the algebras in the latter paper are local), anyequicharacteristicNoetherian local ring admits a balanced big Cohen-Macaulayalgebra. This applies in par-ticular to the completionR as it is always equicharacteristic by the discussion in§2.9. Soremains to show that any balanced big Cohen-MacaulayR-algebraB is a balanced bigCohen-MacaulayR-algebra. However, this is clear for ifx is a generic sequence, then it isa system of parameters inR by Proposition3.9, whenceB-regular. �

10.5.Remark.We may drop the requirement on the characteristic whenR has geometricdimension at most three, since in that case, regardless of characteristic,R admits a balancedbig Cohen-Macaulay algebra by [22]. In particular, all the homological theorems belowalso hold under this assumption.

10.6.Remark.In fact, we may choose balanced big Cohen-Macaulay algebrasin a weaklyfunctorial way in the following sense. We will call a local homomorphismR → S ca-ta-permissible, if R → S is permissible in the sense of [28, §9] or [4, §7.9]. In thatcase, we may choose a balanced big Cohen-MacaulayR-algebraB (whence a balancedbig Cohen-MacaulayR-algebra), a balanced big Cohen-MacaulayS-algebraB′ (whencea balanced big Cohen-MacaulayS-algebra) and a homomorphismB → B′ extendingR → S, whence also extendingR → S. Recall from the cited sources that any localalgebra is permissible over an equidimensional and universally catenary Noetherian localring (e.g., a complete local domain).

10.7.Proposition. If R is pseudo-regular with residue fieldk and ifB is a balanced bigCohen-MacaulayR-algebra, then allTorRi (B, k) vanish fori > 0, andIB ∩ R is equalto the closure ofI for each idealI ⊆ R.

Proof. It is not hard to verify thatB ⊗R S is a balanced big Cohen-MacaulayS-algebra,for S := R(ξ) andξ a tuple of indeterminates. SinceR → S is faithfully flat, we may

44 HANS SCHOUTENS

pass fromR to S and therefore assume in view of Remark7.4that the maximal ideal ofRis generated by a regular sequencex. Sincex is alsoB-regular andk = R/xR,

TorRi (B, k)∼= Tor

R/xRi (B/xB, k) = 0

for all i > 0. Therefore, for anym-primary idealn, we getTorR/n1 (B/nB, k) = 0 by

[44, Lemma 2.1]. SinceR/n is Artinian,B/nB is faithfully flat by the Local FlatnessCriterion, and hence in particularn = nB ∩ R. The last assertion then follows since anyclosed ideal is the intersection of allm-primary ideals containing it. �

Using an argument similar to the one in the proof of Corollary7.24, one can show thatunder the above assumptions, eachTorRi (B,M) vanishes, fori > 0 andM a finitelygenerated pseudo-Cohen-Macaulay module: for the Artiniancase, induct on the length ofM , and for the general case, on the depth ofM , using thatIRB = 0 by construction;details are left for the reader. Before stating the next result, we need to introduce someterminology. We will follow the treatment in [9, §9.4] and refer to this source for moredetails. LetF• be a complex

0 → Fsϕs

−−−→Fs−1ϕs−1

−−−−→ . . .ϕ2

−−−→F1ϕ1

−−−→F0 → 0

with eachFi a finitely generated freeR-module. We calls the length ofF• and we call thecokernel ofϕ1 thecokernelof the complex. For each1 ≤ n ≤ s, we will define then-thFitting idealIn(F•) of F• as follows. Fix1 ≤ n ≤ s and put

r :=s∑

i=n

(−1)i−n rankFi.

LetΓ be a matrix representing the morphismϕn (by choosing bases forFn andFn−1) andlet In(F•) be the ideal inR generated by allr × r-minors ofΓ. One shows that this isindependent from the choices made.

We say thatF• is acyclic if all Hi(F•) vanish, fori > 0; if alsoH0(F•) vanishes (thatis to say, if the cokernel ofF• is zero), then we say thatF• is exact. In particular, ifF• isacyclic, then it is a finite free resolution of its cokernel.

10.8.Theorem. Let (R,m) be an equicharacteristic or an infinitely ramified local ringoffinite embedding dimension. LetF• be a finite complex of finitely generated freeR-modulesof lengths and letM be its cokernel. If the geometric codimension ofIn(F•) is at leastnfor eachn = 1, . . . , s, then the geometric codimension ofAnnR(µ) is at mosts, for anynon-zero minimal generatorµ ofM .

Proof. Let d ande be the geometric dimension ofR andR/AnnR(µ) respectively. Inview of Proposition3.15, we need to show thatd − e ≤ s, and we do this by inductionon e. Assume first thate = 0, so thatAnnR(µ) is m-primary. By Theorem10.4, thereexists a balanced big Cohen-MacaulayR-algebraB. By Proposition3.15, we can findpart of a generic sequence of lengthn in In(F•), which is thereforeB-regular. HenceeachIn(F•)B has grade at leastn, and the Buchsbaum-Eisenbud Acyclicity criterion ([9,Theorem 9.1.6]) then yields that the complexF• ⊗R B is acyclic. SinceB, whence eachmodule inF•⊗RB, has depthd, and sinceM ⊗RB is the cokernel ofF•⊗RB, the depthofM ⊗RB is at leastd−s by [9, Proposition 9.1.2(e)]. By Nakayama’s lemma, the imageof µ in M/mM is non-zero, which implies thatµ ⊗ 1 is non-zero inM ⊗R B. Since theannihilator ofµ⊗ 1 containsAnnR(µ), it is m-primary. It follows thatM ⊗R B has depthzero, and hence thatd− s ≤ 0.


Assume now thate > 0. The threshold primes ofR andAnnR(µ) are all differentfrom m, and so are the threshold primes of thoseIn(F•) that are notm-primary. Byprime avoidance, we may therefore choosex ∈ m outside all these finitely many thresholdprimes. PutRn := R/In(F•) andS := R/xR. We want to apply the induction hypothesisto the complexF• ⊗R S and the image ofµ in M ⊗R S. By Corollary3.12, the geometricdimension ofS andR0⊗RS ared−1 ande−1 respectively, and the geometric dimensionof S/In(F• ⊗R S) ∼= Rn ⊗R S is at mostd − n − 1 (this is trivially true if In(F•) ism-primary and follows from Lemma3.8 in the remaining case). SinceS/AnnS(µ) is aresidue ring ofR0 ⊗R S, its geometric dimension is at moste − 1, so that our inductionhypothesis applies, yieldingd− 1− (e − 1) ≤ gcodim(AnnS(µ)) ≤ s. �

We can now generalize the new intersection theorems due to Evans-Griffith and Peskine-Szpiro-Roberts.

10.9.Corollary. Let (R,m) be an equicharacteristic or an infinitely ramified local ringoffinite embedding dimension. LetF• be a finite complex of finitely generated freeR-modulesof lengths and letM be its cokernel.

(10.9.1) If F• is acyclic when localized at any closed prime ideal ofR different fromm and there exists a non-zero minimal generator ofM whose annihilator ism-primary, thengdim(R) ≤ s.

(10.9.2) If F• is exact when localized at any closed prime ideal ofR different fromm

ands < gdim(R), thenF• is exact.

Proof. To prove (10.9.1), assumes < d := gdim(R). We reach the desired contradictionfrom Theorem10.8, if we can show thatR/In(F•) has geometric dimension at mostd−n,for all n = 1, . . . , s. Fix n and letI := In(F•). There is nothing to show ifI ism-primary,so that we may exclude this case. By Remark9.3, we can choose a cata-normalizationA0 → R and an idealJ ⊆ A0 such thatJR = I (note thatI is finitely generated byconstruction). LetA be the image ofA0 in R, so thatA ⊆ R is also cata-integral and cata-injective (although Noetherian,A will, in general, no longer be regular). SinceA → R isintegral and injective,A, whence alsoA, has dimensiond. SupposeJA has heighth andlet q be a minimal prime ofJA of heighth. By [29, Theorem 9.3], we can find a primeidealP in R lying aboveq. Let p := P ∩ R (which is therefore closed by Corollary2.7).Note that sinceI is notm-primary,h < d, and thereforep 6= m. By assumption,(F•)pis acyclic, so that the grade ofIRp is at leastn by the Buchsbaum-Eisenbud Acyclicitycriterion ([9, Theorem 9.1.6]). By [9, Proposition 9.1.2(g)], the grade ofJAq is thereforealso at leastn. In particular,Aq has depth at leastn, showing thatn ≤ h. This in turnimplies thatA/JA has dimension at mostd − n. SinceA/JA → R/IR is integral, thedimension of the first ring is at least that of the second ring.Hence we showed thatR/IRhas dimension at mostd − n. By Lemma2.4 and Theorem3.4, this in turn implies thatR/I has geometric dimension at mostd− n, as required.

The second assertion follows from the first by a standard argument (see for instancethe proof of [9, Corollary 9.4.3]). Namely, it implies that the cokernelM of F• has finitelength. The only way that this does not contradict (10.9.1) is thatM = 0 (by Nakayama’sLemma). This in turn implies that we can split of the last termin F• and then an inductiveargument ons finishes the proof. �

We can translate these results to more familiar versions of the homological theorems.

10.10.Theorem (Superheight). Let R → S be a local homomorphism of equicharac-teristic or infinitely ramified local rings of finite embedding dimension and letM be an

46 HANS SCHOUTENS

R-module admitting a finite free resolutionF• of lengths. If M ⊗R S has finite length,thenS has geometric dimension at mosts.

Proof. Let m andn be the respective maximal ideals ofR andS. Let q be an ideal inSdifferent fromn and putp := q∩R. Since the localization ofM ⊗R S atq is zero, we get

Mp/pMp ⊗k(p) Sq/pSq = 0,

wherek(p) is the residue field ofp. SinceSq/pSq is non-zero,Mp/pMp must be zero,and thereforeMp = 0, by Nakayama’s Lemma. Hence(F•)p is exact whence split ex-act. Therefore, this remains so after tensoring withSq. In other words, the conditionsof (10.9.2) are met for the complexF• ⊗R S over the ringS, showing thatS must havegeometric dimension at mosts. �

10.11.Theorem (Intersection Theorem). LetR be an equicharacteristic or an infinitelyramified local ring of finite embedding dimension and letM,N beR-modules. IfM has afinite free resolution of lengths, thengdim(N) ≤ s+ gdim(M ⊗R N).

Proof. Assume first thatM ⊗R N has finite length and letS := R/AnnR(N). It followsthatM ⊗R S has finite length, so that the geometric dimension ofS is at mosts byTheorem10.10. For the general case, we induct on the geometric dimension of M ⊗ N .Using Proposition3.9, one can findx ∈ R such that it is part of a generic subset of bothR/AnnR(N) andR/AnnR(M ⊗N). It follows that the geometric dimensions ofN/xNandM ⊗N/xN both have dropped by one, so that we are done by induction. �

10.12.Theorem(Canonical Element Theorem). Let(R,m) be an equicharacteristic or aninfinitely ramified local ring of finite embedding dimension.LetF• be a free resolution ofthe residue fieldk ofR and letx be a generic sequence inR. If γ is a complex morphismfrom the Koszul complexK•(x) to F•, extending the natural mapγ0 : K0(x) = R/xR→k, then the morphismγd : Kd(x) → Fd is non-zero, whered is the geometric dimension ofR.

Proof. Supposeγd is zero. LetB be a local balanced big Cohen-Macaulay algebra forRand lety ∈ B be such that its image inB/xB is a non-zero socle element. Defineψ0 : R→B by sending1 to y. Sincex isB-regular, the Koszul complexK•(x;B) := K•(x) ⊗ Bis acyclic. It follows thatψ0 extends to a morphism of complexesψ : F• → K•(x;B).Let α := ψ ◦ γ be the compositionK•(x) → K•(x;B). In particularα0(1) = y andαd = 0. On the other hand,α0 induces by tensoring a morphism of complexesβ :=1 ⊗ α0 : K•(x) → K•(x) ⊗ B = K•(x;B). SinceK•(x;B) is acyclic,α andβ differby a homotopyσ. In particular,βd = βd − αd = σd−1 ◦ δd, whereδd : Kd(x) = R →Kd−1(x) = Rd is the left most map in the Koszul complex. Since the image ofδd lies inxRd, we gety = βd(1) = σd−1 ◦ δd(1) ∈ xB, contradicting our choice ofy. �

To formulate the next result, which extends a result of Eisenbud and Evans in [13],recall that for anR-moduleM and an elementz ∈ M , theorder idealof z is the idealOM (z) consisting of all imagesα(z) for α ∈ HomR(M,R). Moreover, ifR is a domainwith field of fractionsK, then therank of M is defined as the dimension of the vectorspaceM ⊗R K.

10.13.Theorem (Generalized Principal Ideal Theorem). Let (R,m) be an equicharacter-istic or an infinitely ramified local domain of finite embedding dimension, and letM be afinitely generatedR-module. Ifz ∈ mM , then the geometric codimension ofOM (z) is atmost the rank ofM .


Proof. Leth be the geometric codimension ofOM (z), let r be the rank ofM , and letd bethe geometric dimension ofR. By definition, there exists a generic sequence(x1, . . . , xd)with xi ∈ OM (z), for i = 1, . . . , h. ReplacingM by M ⊕ Rd−h andz by the element(z, xh+1, . . . , xd), so that both the rank ofM and the geometric codimension ofOM (z)increase byd − h, we may assume thatOM (z) contains a generic sequencex. Let y bea finite tuple generatingm. As explained in the proof of [9, Theorem 9.3.2], the canonicalhomomorphismR/xR → R/yR induces a morphism of Koszul complexesα : K•(x) →K•(y). LetF• be a free resolution of the residue fieldR/yR ofR andβ : K•(y) → F• bean induced morphism of complexes. By Theorem10.12, applied to the compositionβ ◦α,we get in degreed a non-zero morphismβd ◦αd, showing in particular thatαd is non-zeroas well. Sinceαd is just thed-th exterior power ofα1, the rank ofα1 is at leastd. On theother hand,α1 factors by construction throughHomR(M,R), whence has rank at mostr,yielding the desired inequalityd ≤ r (see [9, Theorem 9.3.2] for more details). �

11. UNIFORM BOUNDS ONBETTI NUMBERS

In the next two sections, we apply the previous theory to derive uniformity results forNoetherian local rings. In this section, we study Betti numbers. Recall that given a moduleM over a local ringR with residue fieldk, its n-th Betti numberβn(M) is defined asthe vector space dimension ofTorRn (M,k) ∼= ExtnR(M,k). It is equal to the rank of then-th module in a minimal free resolution ofM (provided such a resolution exists), andby Nakayama’s Lemma, it is then also equal to the least numberof generators of ann-thsyzygy ofM . One usually studies the behavior of these Betti numbers fora fixed moduleasn goes to infinity. In contrast, we will study their behavior for fixed n as we vary themodule.

11.1.Theorem. For each quadruple(d, e, l, n) of non-negative integers, there exists abound∆(d, e, l, n) with the following property. IfR is ad-dimensional local Cohen-Mac-aulay ring of multiplicitye, andM is a Cohen-MacaulayR-module of multiplicity at mostl, then

βn(M) ≤ ∆(d, e, l, n).

Proof. Suppose not, so that for some quadruple(d, e, l, n), we cannot define such an upperbound. This means that for everyw, we can find ad-dimensional Cohen-Macaulay localringRw of multiplicity e, and a Cohen-MacaulayRw-moduleMw of multiplicity at mostl, such thatβn(Mw) ≥ w. By Theorem5.22, the ultraproductR♮ is isodimensional, and byLemma8.8, the ultraproductM ♮ is finitely generated and pseudo-Cohen-Macaulay. Sincethe cataproductM ♯ is therefore finitely generated over the (Noetherian) cataproductR♯, itsn-th Betti numberβn(M ♯) is finite, and by Proposition8.9, equal to almost allβn(Mw),contradiction. �

Theorem11.1applied to the residue field ofR yields Corollary1.2 from the introduc-tion. We can also reformulate the previous theorem in terms of universal resolutions:

11.2.Corollary. For each triple(d, e, l), there exists a countably generatedZ-algebraZ and a complexF• of finite freeZ-modules, with the following property. IfR is a d-dimensional local Cohen-Macaulay ring of multiplicitye, andM a finitely generated Coh-en-Macaulay module of multiplicity at mostl, then there exists a homomorphismZ → R,such that for anyn and anyR-moduleN , we have

TorRn (M,N) ∼= Hn(F• ⊗Z N) and ExtnR(M,N) ∼= Hn(HomZ(F•, N)).

48 HANS SCHOUTENS

If we impose furthermore thatR is regular (whencee = 1) or, more generally, thatMhas finite projective dimension, then we may takeZ to be a finitely generatedZ-algebraandF• a complex of lengthd.

Proof. For eachn, let δn := ∆(d, e, l, n) be the bound given by Theorem11.1, and letΞn

be a tuple of indeterminates viewed as aδn−1 × δn-matrix. LetZ be the polynomial ringoverZ generated by all indeterminatesΞn modulo the relationsΞn ·Ξn+1 = 0, expressingthat the product of two consecutive matrices is zero. We thendefine the complexF• byletting itsn-th term beZδn , and itsn-th differential the matrixΞn. By construction,F• isa free complex. Now, givenR andM as in the statement, Theorem11.1implies that wemay assign to each entry inΞn, a value inR so that under the induced mapZ → R, thecomplexF• ⊗Z R becomes a free resolution ofM . The statement now follows from thedefinition of Tor and Ext. �

Then-th Bass numberµn(M) of a finitely generatedR-moduleM is the vector spacedimension ofExtnR(k,M), wherek is the residue field ofR. Theq-th Bass number, withq equal to the depth ofM , is also called thetypeof M .

11.3.Corollary. The type (respectively, for eachn, then-th Bass number) of a finitelygenerated moduleM over a local Cohen-Macaulay ringR is bounded above by a function(in n) depending only on the dimension and multiplicity ofR, and on the minimal numberof generators ofM .

Proof. Since the depth ofM is at most the dimension ofR, it suffices to prove theclaim for a fixedn. By Corollary 11.2, there is a resolutionF• of k by finite freeR-modulesFn whose ranksβn(k) are bounded by the dimension and multiplicity ofR. SinceExtnR(k,M) is then-th cohomology ofHomR(F•,M), its lengthµn(M) is at most thenumber of generators ofHomR(Fn,M) ∼=Mβn(k), and the claim follows. �

Let us extend some definitions from [50]. We will call a homomorphismR → S ofNoetherian local ringsformally etale(or a scalar extension), if it is faithfully flat andunramified (=the maximal ideal ofR extends to the maximal ideal ofS). Let (R,m) and(S, n) be Noetherian local rings, and letM be a finitely generatedR-module andN afinitely generatedS-module. We define theinfinitesimal neighborhood distancebetweenM andN as the real number

d(M,N) := e−α

whereα is the (possibly infinite) supremum of allj such that there exists an Artinian localring T , together with formally etale extensionsR/mj → T andS/nj → T , yieldingM ⊗R T ∼= N ⊗S T . As shown in [50] (where the distance is only defined between rings),the infinitesimal neighborhood distance is a (quasi-)metric, and, roughly speaking, up toa formally etale base change, limits in this metric space canbe calculated by means ofcataproducts.

11.4.Theorem. For each quadruple of positive integers(d, e, l, n), there exists a boundε := ε(d, e, l, n) > 0 such that ifR and S are d-dimensional local Cohen-Macaulayrings of multiplicity e, andM andN are finitely generated Cohen-Macaulay modulesof multiplicity at mostl overR andS respectively, withd(M,N) ≤ ε, thenβn(N) =βn(M).

Proof. Suppose no such bound exists for the pair(d, e, l, n), resulting in a counterex-ample for eachw, given byd-dimensional Cohen-Macaulay local rings(Rw,mw) and(Sw, nw) of multiplicity e, and finitely generated Cohen-Macaulay modulesMw andNw


of multiplicity at moste overRw andSw respectively, such thatd(Mw, Nw) ≤ e−w, butβn(Mw) 6= βn(Nw). Since Betti numbers are preserved under formally etale extensions,the techniques in [50] allows us to reduce to the case that the distance condition means that

(12) Rw/mww∼= Sw/n

ww and Mw/m

wwMw

∼= Nw/nwwNw

LetM ♮ andN ♮ be the respective ultraproducts of theRw, Sw, Mw, andNw. By Corol-lary 8.7, the respective ultraproductsR♮ andS♮ are pseudo-Cohen-Macaulay local rings,and by Lemma8.8, the respective ultraproductsM ♮ andN ♮ are finitely generated pseudo-Cohen-Macaulay modules overR♮ andS♮ respectively. Moreover, by Łos’ Theorem andmodding out infinitesimals, we get from (12) that the respective cataproductsR♯ andS♯

are isomorphic, as are the respective cataproductsM ♯ andN ♯. By Proposition8.9, wetherefore get for almost allw, the following contradictory equalities

βn(Mw) = βn(M ♯) = βn(N ♯) = βn(Nw).

�

11.5.Corollary. Given a local Cohen-Macaulay ringR, there exists, for eachn ∈ N, aboundδ := δ(n) > 0 such that ifM andN are maximal Cohen-Macaulay modules withd(M,N) ≤ ε, thenβn(N) = βn(M).

Proof. If d(M,N) < 1, thenM andN have the same minimal number of generatorsm. In view of Theorem11.4, it suffices to show that the multiplicity ofM andN areuniformly bounded in terms ofm. Let e andq be respectively the maximum multiplicityof R/p and the maximal length ofRp, wherep runs over the finitely manyd-dimensionalprime ideals ofR. Since we have a surjective mapRm → M , tensoring with one of thesed-dimensional prime idealsp shows that the length ofMp is at mostmq. The bound onthe multiplicity now follows from [9, Corollary 4.6.8]. �

Proofs of Corollaries1.3and 1.4. The first corollary follows immediately from the def-initions and Theorem11.1. To prove the second, lete be the multiplicity ofR/I. SinceI = xR for some regular elementx ∈ R, the residue ringR/I is Cohen-Macaulay and hasprojective dimension one. Henceβ1(R/I) = 1 andβ2(R/I) = 0. Choose someε > 0 asgiven by Theorem11.4such thatd(R/I,M) ≤ ε implies thatR/I andM have the samezero-th, first and second Betti number, forM a Cohen-Macaulay module of multiplicity atmoste. Note that fromβ0(M) = β0(R/I) = 1 it follows thatM is of the formR/J , sothat in the statement, we did not even need to assume thatM was cyclic. Choosea suchthate−a ≤ ε. In particular,d(R/I,R/J) ≤ ε, and thereforeβ1(R/J) = 1, yielding thatJ is cyclic, andβ2(R/J) = 0, yielding that it is invertible. �

In terms of thePoincare seriesof a moduleM , defined asPR(M ; t) :=∑

n βn(M)tn,our results yield:

11.6.Corollary. Over a fixed local Cohen-Macaulay ring, the Poincare series is a con-tinuous map from the metric space of Cohen-Macaulay modulesof multiplicity at moste(respectively, from the space of all maximal Cohen-Macaulay modules), toZ[[t]] with itst-adic topology.

Proof. For eachn, we can choose by Theorem11.4(respectively, by Corollary11.5), anε > 0 such thatd(M,N) ≤ ε implies that the firstn Betti numbers ofM andN arethe same, forM andN Cohen-Macaulay modules of multiplicity at moste (respectively,maximal Cohen-Macaulay modules). HencePR(M ; t) ≡ PR(N ; t) mod tnZ[[t]]. �

50 HANS SCHOUTENS

Although we did not formulate it here, we may even extend thisresult by also varyingthe base ring over all local Cohen-Macaulay rings of a fixed dimension and multiplicity;see [50, §8]. Applied to a regular local ring, we immediately get:

11.7.Corollary. LetR be a regular local ring. For eache, there existsδ := δ(e) > 0 suchthat ifM andN are Cohen-Macaulay modules of multiplicity at moste andd(M,N) ≤ δ,thenPR(M ; t) = PR(N ; t). �

12. UNIFORM ARITHMETIC

In this section, we prove several uniform bounds, and show that the existence of suchbounds is often equivalent with a certain ring theoretic property. We start with examiningthe domain property. It is not true in general that the catapower of a domain is again adomain: letR be the local ring at the origin of the plane curve over a fieldk given byf := ξ2 − ζ2 − ζ3. The catapower ofR is k♮[[ξ, ζ]]/fk♮[[ξ, ζ]], wherek♮ is the ultrapowerof k, and this is not a domain (since1 + ζ has a square root ink♮[[ξ, ζ]]). Clearly, theproblem is thatR is notanalytically irreducible, that is to say, not a cata-domain.

Before we give a necessary and sufficient condition for a catapower to be a domain, letus introduce some terminology which makes for a smoother presentation of our results.PutN := N ∪ {∞}. By ann-ary numericalfunction, we mean a map fromf : Nn → N,with the property thatf(s1, . . . , sn) = ∞ if and only if one of the entriessi is equal to∞.Moreover, we will always assume that a numerical functionf is non-decreasing in any ofits arguments, that is to say, ifsi ≤ ti for i = 1, . . . , n, thenf(s1, . . . , sn) ≤ f(t1, . . . , tn).To indicate that a numerical function depends on a ringR, we will write the ring as asubscript.

Recall thatR hasbounded multiplicationif there exists a binary numerical functionµR

(called auniformity function) such that

ord(xy) ≤ µR(ord(x), ord(y))

for all x, y ∈ R (see§2.1for the definition of order). In view of our definition of numericalfunction, the ideal of infinitesimals in a local ring with bounded multiplication is a primeideal, and hence the separated quotient is a domain.

12.1.Theorem. Let (R,m) be a Noetherian local ring. The following are equivalent:

(12.1.1)R is analytically irreducible;(12.1.2)R has bounded multiplication;(12.1.3) some (equivalently, any) catapowerR♯ ofR is a domain.

Proof. The implication (12.1.2) ⇒ (12.1.1) is clear from the above discussion, since hav-ing bounded multiplication is easily seen to be preserved under completions. In order toprove (12.1.1) ⇒ (12.1.3), assumeR is analytically irreducible and letR♯ be its catapower.SinceR has the same catapower by Corollary5.8, we may moreover assume thatR is acomplete Noetherian local ring. IfR is normal, then so isR♯ by Corollary5.15, and henceagain a domain. For the general case, letS be the normalization ofR, so thatR ⊆ S isa finite extension. SinceR is complete,S is again local. By Proposition5.16, we get anextensionR♯ ⊆ S♯. Since we argued thatS♯ is a domain, the same therefore is true forR♯.

Remains to show (12.1.3) ⇒ (12.1.2). By way of contradiction, suppose no boundexists for the pair(a, b). In other words, we can findxn, yn ∈ R such thatord(xn) = a,ord(yn) = b andxnyn ∈ mn. Lettingx♮ andy♮ be their respective ultraproducts, we getord(x♮) = a, ord(y♮) = b andx♮y♮ ∈ IR♮

. SinceIR♮is by assumption prime,x♮ or y♮

lies inIR♮, neither of which is possible. �


12.2. Remark. The equivalence of (12.1.1) and (12.1.2) is well-known and is usuallyproven by a valuation argument. By [53, Theorem 3.4] and [27, Proposition 2.2] these con-ditions are also equivalent with the existence of alinear uniformity function:µR(a, b) :=kR max{a, b}, for somek := kR ∈ N, in which case we say thatR hask-bounded multi-plication. For a further result along these lines, see [32, Proposition 5.6].

By the same argument proving implication (12.1.3) ⇒ (12.1.2), we get:

12.3.Corollary. LetRn be Noetherian local rings of bounded embedding dimension. If(almost) allRn have bounded multiplication with respect to the same uniformity functionµ = µRn

, then so do their ultraproductR♮ and cataproductR♯. In particular, R♯ is adomain. �

Note that the converse is not true. For instance, ifR is a complete Noetherian localdomain, then the cataproduct of theR/mn is a domain by Corollaries5.10and12.3. Ifinstead of order, we use degree (see§3.16for the definition), we get the following analogueof bounded multiplication, this time in terms of a bound whose dependence on the ring isonly through its embedding dimension.

12.4.Theorem. There exists a ternary numerical functionω with the following property.For every Noetherian local ringR and any two elementsx, y ∈ R, we have an inequality

deg(xy) ≤ ω(embdim(R), deg(x), deg(y)).

Proof. Towards a contradiction, suppose such a function cannot be defined on the triple(m, a, b). This means that for eachn, we can find a Noetherian local ringRn of embed-ding dimensionm and elementsxn, yn ∈ Rn such thatdeg(xn) = a, deg(yn) = b anddeg(xnyn) ≥ n. LetR♮, x♮ andy♮ be the respective ultraproduct of theRn, thexn andtheyn. Let d be the ultra-dimension ofR♮, so that almost allRn have dimensiond. ByCorollary3.17, almost eachRn has parameter degree at mosta and henceR♮ is isodimen-sional by Theorem5.22. Hencex♮ andy♮ are both generic by Corollary5.25, and henceso is their productx♮y♮ by Corollary3.12. However, this contradicts Corollary5.25as thexnyn have unbounded degree. �

From the exact sequence

R/xRy

−−→R/xyR→ R/yR→ 0

where the first map is induced by multiplication byy, we see thatdeg(xy) ≤ deg(x) +deg(y) for all x, y in a one-dimensional Noetherian local ring (in fact, ifR is Cohen-Macaulay, then the first map is injective and we even have equality). I do not know whathappens in higher dimensions.

12.5. Order versus degree.We next investigate the relationship between order and de-gree. IfR is Cohen-Macaulay andx isR-regular, then the degree ofR is just the multiplic-ity of R/xR. By [29, Theorem 14.9], we getord(x) ≤ deg(x)/mult(R). In particular,ord(x) ≤ deg(x), and this latter inequality could very well always be true (see also§12.19below). At any rate, we have:

12.6.Corollary. There exists a binary numerical functionπ with the following property.For every Noetherian local ringR and every elementx ∈ R, we have an inequality

ord(x) ≤ π(embdim(R), deg(x)).

52 HANS SCHOUTENS

Proof. Suppose for some pair(m, a), we have for eachn, a counterexamplexn ∈ mnn of

degreea in the Noetherian local ring(Rn,mn) of embedding dimensionm. Letx♮ ∈ R♮ bethe ultraproduct so that by Theorem5.22, the degree ofx♮ isa, yetx♮ ∈ IR♮

, contradictingCorollary3.17. �

Applying Corollary12.6to a product and then using Theorem12.4, we get the existenceof a ternary numerical functionη such that for any Noetherian local ringR and elementsx, y ∈ R, we have

(13) ord(xy) ≤ η(embdim(R), deg(x), deg(y))

For analytically irreducible Noetherian local rings, order and degree are mutually bounded,and in fact, we have the following more precise result:

12.7.Theorem. There exists a quaternary numerical functionζ with the following prop-erty. For everyd-dimensional Noetherian local domain(R,m) of parameter degree atmoste andk-bounded multiplication, and for everyx ∈ R, we have an inequality

deg(x) ≤ ζ(d, e, k, ord(x)).

Proof. It suffices to show that there exists a functionβ such that iford(x) < a for somex ∈ R and somea ∈ N, thendeg(x) < β(d, e, k, a). Suppose no such bound existsfor the quadruple(d, e, k, a). Hence, for eachn, we can find ad-dimensional Noetherianlocal domain(Rn,mn) of parameter degree at moste andk-bounded multiplication, andan elementxn /∈ ma

n whose degree is at leastn. Let (R♮,m♮) andx♮ be the respectiveultraproduct of the(Rn,mn) and thexn. By Theorem5.22, the geometric dimension ofR♮ is d. Since theRw/xwRw have dimensiond− 1, but unbounded parameter degree, thesame theorem shows that the geometric dimension ofR♮/x♮R♮ is strictly bigger than itsultra-dimensiond − 1, whence is also equal tod. In particular,x♮ is not generic. Sincethe cataproductR♯ is a domain by Corollary12.3, we getx♮ ∈ IR♮

by Corollary3.12.However, by Łos’ Theorem,x♮ /∈ ma

♮ , a contradiction. �

Whereas order is a filtering function (see§2), inducing them-adic filtration onR, thisis no longer true for degree. For instance, letR be the local ring at the origin of the curvewith equationξζ + ξ3 + ζ3 = 0. Then bothξ andζ have degree three, but their sumξ + ζhas degree two. As we will see below in§12.19, on regular local rings, degree is filtering.Can one characterize in general rings for which degree is filtering? Is, for everyn, the setof elements having degree at leastn always a finite union of ideals? In other words, asfar as its properties are concerned, degree is still a mysterious function. However, its mainuse in this paper is to characterize properties via its asymptotic behavior, as we will nowdiscuss.

12.8. Characterizations through uniform behavior. Recall that a Noetherian local ringis analytically unramifiedif its completion is reduced. Any reduced excellent local ring isanalytically unramified ([29, Theorem 32.2]).

12.9.Corollary. A Noetherian local ringR is analytically unramified if and only if thereexists a numerical functionνR, such that for everyx ∈ R, we have an inequality

ord(x2) ≤ νR(ord(x)).

Proof. Since order remains unaffected by completion, we may assumethatR is moreovercomplete. SupposeR is reduced. It suffices to show that there exists a functionνR such thatx2 ∈ mνR(b) impliesx ∈ mb. By way of contradiction, suppose this is false forb. Hence,we can findxn ∈ R such thatx2n ∈ mn, butxn /∈ mb. LetR♮ be the ultrapower ofR and let


x♮ be the ultraproduct of thexn. By Łos’ Theorem,x2♮ ∈ IR♮andx♮ /∈ mbR♮. However,

IR♮is radical by Corollary5.15. Hencex2♮ ∈ IR♮

impliesx♮ ∈ IR♮, contradiction.

Conversely, let the functionνR be as asserted. Ifx2 is zero, then its order is infinite.The only way that this can be bounded byνR(ord(x)), is for x to have infinite order too,meaning thatx = 0. This shows thatR is reduced. �

By a similar argument, one easily shows that if allRn have bounded squares (in thesense of the corollary) with respect to the same functionν = νRn

, then their cataproductis reduced. IfR is analytically irreducible, then the results of Hubl-Swanson (see Re-mark12.2) imply that we may takeνR(b) of the formkRb for somekR and allb. I do notknow whether this is still true in general. Similarly, for the bounds we are about to prove,is their still some vestige of linearity?

12.10.Corollary. A Noetherian local ringR is analytically irreducible if and only if thereexists a numerical functionξR such that for everyx ∈ R, we have an inequality

deg(x) ≤ ξR(ord(x)).

Proof. In view of Remark12.2, the direct implication follows from an application of The-orem12.7. As for the converse, suppose degree is bounded in terms of order. Since bothorder and degree remain the same after passing to the completion, we may moreover as-sumeR is complete. Since a non-zero element has finite order, it hasfinite degree whenceis generic. This shows that there are no non-zero prime ideals of maximal dimension,which in turn forces the zero ideal to be a prime ideal. �

Tweaking (13) slightly (for a fixed ringR), we can characterize the following property.Recall that a Noetherian local ringR is calledunmixed, if each associated primep of itscompletionR has the same dimension asR; if the above is only true for minimal primesof R, then we say thatR is quasi-unmixed(also calledformally equidimensional).

12.11.Lemma. If a Noetherian local ring is (quasi-)unmixed, then so is itscatapower.

Proof. By Corollary 5.8, we may assumeR is a complete (quasi-)unmixed Noetherianlocal ring. Let us first show that the catapowerR♯ is quasi-unmixed. In any case,R andR♯ have the same dimension, sayd. SinceR♯ is complete by Lemma5.6, we need to showthat every minimal primeq ⊆ R♯ has dimensiond. SinceR is complete, it is of the formS/I for some complete regular local ringS and some idealI ⊆ S. By Corollary5.15,the catapowerS♯ of S is regular, whence a domain. Letp := q ∩ R andP := p ∩ S. Byflatness,p is a minimal prime ofR by [29, Theorem 15.1], whence has dimensiond, asRis equidimensional.

SinceS → S♯ is flat, S♯/PS♯ is equidimensional by [29, Theorem 31.5]. SinceS♯/IS♯

∼= R♯, we getS♯/PS♯∼= R♯/pR♯. Sinceq is necessarily a minimal prime ofpR♯,

equidimensionality yields thatR♯/q andR♯/pR♯ have the same dimension. SinceR♯/pR♯

is the catapower ofR/p, this dimension is justd, showing thatq is ad-dimensional prime.Assume next thatR is unmixed. SinceR has no embedded primes, it satisfies Serre’s

condition(S1), whence so doesR♯ by Corollary5.15and [29, Theorem 23.9]. Since wealready know thatR♯ is quasi-unmixed, it is in fact unmixed. �

12.12.Theorem. A Noetherian local ringR is unmixed if and only if there is a binarynumerical functionχR such that for everyx, y ∈ R, we have an inequality

(14) ord(xy) ≤ χR(deg(x), ord(y)).

54 HANS SCHOUTENS

Proof. Assume first thatR is unmixed. Since degree and order remain the same when wepass to the completion, we may assumeR is complete. By Lemma12.11, the catapowerR♯ is then also unmixed. By way of contradiction, assume that for some pair(a, b), wecan find elementsxn, yn ∈ R with deg(xn) ≤ a andord(yn) ≤ b, such thatxnyn ∈ mn.Hence, in the ultrapowerR♮ ofR, the ultraproductx♮ of thexn has degree at mosta and theultraproducty♮ of theyn has order at mostb, butx♮y♮ ∈ IR♮

. Sincex♮ has finite degree, itis generic and hence its image inR♯ lies outside any prime of maximal dimension. SinceR♯ is unmixed,x♮ is thereforeR♮-regular and hencey♮ = 0 in R♯, contradicting that itsorder is at mostb.

Conversely, assume a functionχ with the proscribed properties exists and letx be ageneric element, say, of degreea. We have to show thatx isR-regular. If not, thenxy = 0for some non-zeroy, say, of orderb. However, the order ofxy is bounded byχ(a, b),contradiction. �

By the same argument, one easily proves that the cataproductof Noetherian local ringsRn of bounded embedding dimension is unmixed, provided almosteachRn satisfies thehypothesis of the statement with respect to the same uniformity function χ = χRn

. Inorder to characterize quasi-unmixedness, we have to introduce one more invariant. Givena Noetherian local ringR, we define itsnilpotency degreeto be the leastt such thatnt = 0,wheren is the nilradical ofR. HenceR is reduced if and only if its nilpotency degree isone.

12.13.Proposition. A Noetherian local ringR of nilpotency degree at mostt is quasi-unmixed if and only if there exists a binary numerical function θR such that for everyx, y ∈ R, we have an inequality

ord((xy)t) ≤ θR(deg(x), ord(yt)).

Proof. Again, we may pass to the completion ofR, since all invariants remain unchangedunder completion, and assume from the start thatR is complete. Suppose thatθR has theabove property. To show thatR is quasi-unmixed, which in the complete case is just beingequidimensional, we need to show that any generic elementx lies outside any minimalprime ofR. A moment’s reflection shows that this is equivalent with showing thatx isRred-regular. Hence, towards a contradiction, assumey ∈ R is a non-nilpotent element inR such thatxy is nilpotent. By definition oft, this meansyt 6= 0, but(xy)t = 0. However,the order of(xy)t is bounded by the finite numberθR(deg(x), ord(yt)), contradiction.

Conversely, assumeR is equidimensional, but no functionθR can be defined for somepair (a, b). Hence we can find counterexamplesxn ∈ R of degreea andyn ∈ R such thatytn /∈ mb+1, but (xnyn)t ∈ mn. Let x♮, y♮ andR♮ be the respective ultraproducts, so thatx♮ is generic by Corollary5.25, andyt♮ /∈ mb+1R♮, but (x♮y♮)t ∈ IR♮

by Łos’ Theorem.However, by Lemma12.11, the cataproductR♯ is again equidimensional (note thatR♯ iscomplete), and therefore,x♮, being generic inR♯, is (R♯)red-regular. Hence(x♮y♮)t = 0in R♯ yields thaty♮ is nilpotent inR♯. Let n be the nilradical ofR. SinceR♯/nR♯ is thecatapower ofRred = R/n by Corollary5.7, it is reduced by Corollary5.15. This provesthat the nilradical ofR♯ is justnR♯ and hence in particular,R♯ has nilpotency degreet too.Therefore,yt♮ = 0, contradicting thatyt♮ /∈ mb+1R♯. �

12.14.Theorem. Ad-dimensional Noetherian local ringR is Cohen-Macaulay if and onlyif there exists a binary numerical functionδR such that for alld-tuplesx := (x1, . . . , xd)and(y1, . . . , yd) with x a system of parameters, we have an inequality

(15) ordR(x1y1 + · · ·+ xdyd) ≤ δR(ℓ(R/xR), ordR/(x1,...,xd−1)R(yd)).


Moreover, the functionδR only depends on the dimension and the multiplicity ofR.

Proof. Assume first a functionδR with the asserted properties exists. In order to provethatR is Cohen-Macaulay, we take a system of parameters(z1, . . . , zd) and show thatit is R-regular. Fix somei and supposea1z1 + · · · + aizi = 0. We need to show thatai ∈ I := (z1, . . . , zi−1)R. Fix somek and definexj andyj as follows. Ifj = 1, . . . , d−i,thenxj := zki+j andyj := 0; if j = d− i+1, . . . , d, thenxj := zi+j−d andyj := ai+j−d.In other words, we have

(16)

x := (x1, . . . , xd) = (zki+1, . . . , zkd , z1, . . . , zi)

y := (y1, . . . , yd) = (0, . . . , 0, a1, . . . , ai)

x1y1 + · · ·+ xdyd = a1z1 + · · ·+ aizi = 0.

Apply (15) to these two tuplesx andy. Sincex is again a system of parameters,ℓ(R/xR)is finite. Hence, by the last equation in (16), the order ofyd = ai in R/(x1, . . . , xd−1)Rmust be infinite, that is to say,

ai ∈ (x1, . . . , xd−1)R = I + (zki+1, . . . , zkd)R.

Since this holds for allk, Krull’s intersection theorem yieldsai ∈ I.To prove the converse, supposeR is Cohen-Macaulay, but we cannot defineδR(a, b)

for some pair(a, b). This means that there exists for eachn, a system of parametersxn := (x1n, . . . , xdn) such thatR/xnR has lengtha, and ad-tupleyn := (y1n, . . . , ydn),such that

ordR/(x1n,...,xd−1,n)R(ydn) = b

andx1ny1n + · · · + xdnydn has order at leastn. Let x♮ := (x1♮, . . . , xd♮) andyi♮ bethe respective ultraproducts of thexn andyin inside the ultrapowerR♮ of R. By Łos’Theorem, the order ofyd♮ in R♮/(x1♮, . . . , xd−1♮)R♮ is b, the length ofR♮/x♮R♮ is a, andthe sumx1♮y1♮ + · · · + xd♮yd♮ is an infinitesimal. In particular, the image ofx♮ in thecatapowerR♯ is a system of parameters, whenceR♯-regular, sinceR♯ is Cohen-Macaulayby Corollary5.15. Sincex1♮y1♮+ · · ·+xd♮yd♮ = 0 inR♯, regularity forcesyd♮ to be in theideal(x1♮, . . . , xd−1♮)R♯, contradicting that its order inR♯/(x1♮, . . . , xd−1♮)R♯ is finite.

To prove the final statement, observe that for fixed dimensiond and multiplicity e,we may modify the above proof by taking each counterexamplexn andyn in somed-dimensional local Cohen-Macaulay ringRn of multiplicity e. Indeed, by Corollary8.7,the cataproductR♯ of theRn is again Cohen-Macaulay so that we can copy the aboveargument. �

One can view the previous result as a quantitative version ofthe unmixedness theorem.Namely, we can rewrite condition (15) as follows: for anyd− 1-tuplez and anyx, y ∈ R,if z is part of a system of parameters, then

(17) ordR/zR(xy) ≤ δR(degR/zR(x), ordR/zR(y)).

Comparing this with (14), we can now rephrase Theorem12.14using the following ter-minology: by acurve, we mean a one-dimensional subschemeC of X := Spec(R); wecall a curveC a complete intersection inX if it is of the formSpec(R/I) with I an idealgenerated bydimR − 1 elements; we callC unmixed, if its coordinate ring is (note thatthis is equivalent withC being Cohen-Macaulay).

12.15.Corollary. A Noetherian local ringR is Cohen-Macaulay if and only if every com-plete intersection curveC in Spec(R) is unmixed with respect to a uniformity functionχ = χC (as given by Theorem12.12) independent fromC. �

56 HANS SCHOUTENS

We can depart from other criteria for Cohen-Macaulayness toget some more uniformitycharacterizations. For instance, we could use the criterion proven in [49, Corollary 5.2.11]thatR is Cohen-Macaulay if and only if every system of parametersx := (x1, . . . , xd) isindependent, in the sense that a relationx1y1 + · · · + xdyd = 0 implies that allyi lie inxR. Thus, we get the following modified form of (15): a d-dimensional Noetherian localring R is Cohen-Macaulay if and only if there exists a binary numerical functionδ′R suchthat for every twod-tuplesx := (x1, . . . , xd) and(y1, . . . , yd), we have an inequality

ordR(x1y1 + · · ·+ xdyd) ≤ δ′R(ℓ(R/xR), ordR/xR(yd)).

Next, we characterize normality:

12.16.Theorem. A Noetherian local ringR is normal if and only if there exists a binarynumerical functionεR such that for allx, y, z ∈ R, we have an inequality

(18) mink

{ordR/zkR(xyk)} ≤ εR(ord(x), ordR/zR(y)).

Proof. SupposeR is normal, butεR cannot be defined for a pair(a, b). Hence, for eachn,there exist elementsxn, yn, zn ∈ R such thatxn has ordera andyn has orderb moduloznR, butordR/zk

nR(xny

kn) ≥ n for all k. Let x♮, y♮, z♮ ∈ R♮ be the respective ultraprod-

ucts ofxn, yn, zn ∈ R. In particular,x♮ is non-zero in the catapowerR♯ andy♮ /∈ z♮R♯.On the other hand, sincex♮yk♮ ∈ zk♮R♯ for all k, a well-known criterion shows thaty♮ liesin the integral closure ofz♮R♯. SinceR♯ is normal by Corollary5.15, any principal idealis integrally closed, so thaty♮ ∈ z♮R♯, contradiction.

Conversely, assume a numerical functionεR exists with the proscribed properties. Tak-ing z = 0 in (18), we see thatR is a domain by Theorem12.1. Supposey/z is an elementin the field of fractions ofR which is integral overR. We want to show thaty/z ∈ R.Sincey is then in the integral closure ofzR, there exists a non-zerox such thatxyk ∈ zkRfor all k. The left hand side in (18) is therefore infinite, whence so must the right hand sidebe, forcingy ∈ zR. �

In our last two examples, we show how also tight closure conditions fit in our presentprogram of characterizing properties by certain uniform behavior. We will adopt the usualtight closure notation of writingI [q] as an abbreviation for the ideal(wq

1 , . . . , wqn)R, where

I := (w1, . . . , wn)R is some ideal andq is some power of the prime characteristicp of R.An elementy ∈ R lies in thetight closureI∗ of I, if there existsc ∈ R outside all minimalprime ideals, such thatcyq ∈ I [q] for all powersq of p. We say thatR is F-rationalif some parameter ideal is tightly closed, in which case every parameter ideal is tightlyclosed (recall that aparameter idealis an ideal generated by a system of parameters). Onthe other hand, if every ideal is tightly closed, then we callR weakly F-regular.

12.17.Theorem. An excellent local ringR of characteristicp is pseudo-rational if andonly if there exists a ternary numerical functionϕR such that for all elementsx, y ∈ Rand every (equivalently, some) parameter idealI, we have an inequality

(19) minq

{ordR/I[q](xyq)} ≤ ϕR(deg(x), ℓ(R/I), ordR/I(y))

whereq runs over all powers ofp.

Proof. We will use Smith’s tight closure characterization [51] thatR is pseudo-rational ifand only if it is F-rational. Assume first thatR is pseudo-rational whence F-rational, but anumerical functionϕR cannot be defined on the triple(a, b, c). Hence there exist for eachn, elementsxn, yn ∈ R and a parameter idealIn inR such thatxn has degreea andR/In


is an Artinian local ring of lengthb in which yn has orderc, butordR/I

[q]n(xyq) ≥ n for

all powersq of p. Letx♮, y♮, I♮ be the respective ultraproducts of thexn, yn, In and letR♯

be the catapower ofR. LetJ be a parameter ideal inR. HenceJR♯ is a parameter ideal inR♯. SinceR → R♯ is regular by Corollary5.15and sinceJ is tightly closed, so isJR♯ by[28, Theorem 131.2] or [24], showing thatR♯ is F-rational.

Since a pseudo-rational local ring is a domain,x♮ is generic inR♯ andI♮R♯ is a pa-

rameter ideal inR♯. Moreover,y♮ /∈ I♮R♯, butx♮yq♮ ∈ I

[q]♮ R♯ for all q. By definition of

tight closure,y♮ ∈ (I♮R♯)∗. In particular, every parameter ideal, includingI♮R♯, is tightly

closed and hencey♮ ∈ I♮R♯, contradiction.Conversely, assumeϕR satisfies (19) for some parameter idealI. To verify thatR is

F-rational, lety ∈ I∗. Hence, for somex ∈ R not in any minimal prime,xyq ∈ I [q] forall q. The left hand side of (19) is therefore infinite whence so is the right hand side. Sincex is generic, whence has finite degree, the third argument mustbe infinite, that is to say,y ∈ I. �

12.18.Theorem. A Noetherian local ring(R,m) of characteristicp is weakly F-regular ifand only if there exists a ternary numerical functionψR such that for all elementsx, y ∈ Rand allm-primary idealsI, we have an inequality

(20) minq

{ordR/I[q](xyq)} ≤ ψR(deg(x), ℓ(R/I), ordR/I(y))

whereq runs over all powers ofp.

Proof. Note that forR to be weakly F-regular, it suffices that everym-primary ideal istightly closed, since by Krull’s Intersection Theorem, anyideal is an intersection ofm-primary ideals. Moreover, ifR is weakly F-regular, then so is its catapowerR♯ by [24,Theorem 7.3] in conjunction with Corollary5.15. In view of these facts, the proof is nowalmost identical to the one for Theorem12.17; details are left to the reader. �

12.19. Epilogue: characterization of regularity. Let me make a few further observa-tions, although they do no longer relate to our proof method.If R is regular, then in factord(xy) = ord(x)+ord(y). However, the latter condition does not characterize regularity,but only the strictly weaker condition that the associated graded ringgr(R) is a domain.The following condition, however, does characterize regularity: a Noetherian local ringR is regular if and only iford(x) = deg(x) for all x ∈ R. Indeed, ifR is regular andord(x) = a, then by judiciously choosing a regular system of parameters (x1, . . . , xd),we can ensure thatx still has ordera in V := R/(x1, . . . , xd−1)R. SinceV is a dis-crete valuation ring with uniformizing parameterxd, one checks thatℓ(V/xV ) = a. Sincedeg(x) ≤ ℓ(R/(x, x1, . . . , xd−1)R) = a, we getdeg(x) ≤ ord(x). The other inequalityfollows from our discussion in§12.5.

Conversely, if order and degree agree, then in particular there exists an element of degreeone, and hence a system of parametersx such thatR/xR has length one, whence is a field,showing thatx is a regular system of parameters. �

13. ASYMPTOTIC HOMOLOGICAL CONJECTURES IN MIXED CHARACTERISTIC

In [39, 47], we derived asymptotic versions of the homological conjectures for localrings of mixed characteristicp, where byasymptotic, we mean that the residual charac-teristic p must be large with respect to the complexity of the data. In the above papers,complexity was primarily given in terms of the degrees of thepolynomials defining the

58 HANS SCHOUTENS

data. In this paper, we phrase complexity in terms of (natural) invariants of the ring andthe data only.

Improved New Intersection Theorem. To not have to repeat each time the conditionsfrom this theorem, we make the following definition: given a finite complexF• of finitelygenerated freeR-modules, afinite free complex, for short, we say that itsrank is at mostr,if all Fi have rank at mostr; and we say that itsINIT-degreeis at mostl, if eachHi(F•), fori > 0, has length at mostl, andH0(F•) has a minimal generator generating a submoduleof length at mostl. Recall that thelengthof F• is the largestn such thatFn 6= 0.

13.1.Theorem(Asymptotic Improved New Intersection Theorem). For each triple of non-negative integers(m, r, l), there exists a boundκ(m, r, l) with the following property. LetR be a Noetherian local ring of mixed characteristicp and of embedding dimension atmostm. If F• is a finite free complex of rank at mostr and INIT-degree at mostl, then itslength is at least the dimension ofR, providedp ≥ κ(m, r, l).

Proof. Since the dimension ofR is at mostm, there is nothing to show for complexes oflengthm or higher. Suppose the result is false for some triple(m, r, l). This means thatfor infinitely many distinct prime numberspw, we can find adw-dimensional Noetherianlocal ring (Rw,mw) of mixed characteristicpw and embedding dimension at mostm,and we can find a finite free complexF•w of rank at mostr, of lengthsw ≤ m, andof INIT-degree at mostl, such thatsw < dw. Choose a non-principal ultrafilter and let(R♮,m♮) be the ultraproduct of the(Rw,mw). Sincesw < dw ≤ m, their respectiveultraproducts satisfys < d ≤ m. By Theorem5.18, the geometric dimension ofR♮ is atleastd. LetF•♮ be the ultraproduct of the complexesF•w. Since the ranks are at mostr,each module inF•♮ is a freeR♮-module of rank at mostr. Since ultraproducts commutewith homology, and preserve uniformly bounded length by Proposition5.13, the higherhomology groupsHi(F•♮) have finite length (at mostl). Furthermore, by assumption, wecan find a minimal generatorµw of H0(F•w) generating a submodule of length at mostl.Hence the ultraproductµ♮ of theµw is by Łos’ Theorem a minimal generator ofH0(F•♮),generating a submodule of length at mostl. In conclusion,F•♮ has INIT-degree at mostl. In particular,F•♮ is acyclic when localized at a non-maximal prime ideal, and hence(10.9.1) from Corollary10.9applies, yielding thats ≥ gdim(R♮) ≥ d, contradiction. �

We can even give an asymptotic version of Theorem10.8, albeit in terms of some lessnatural bounds.

13.2.Theorem. For each triple of non-negative integers(m, r, l), there exists a boundσ(m, r, l) with the following property. Let(R,m) be a Noetherian local ring of mixedcharacteristicp and of embedding dimension at mostm, and letF• be a finite free complexof rank at mostr. LetM be the cokernel ofF•, and letµ be a non-zero minimal generatorofM . Assume eachR/Ik(F•) has dimension at mostdimR− k and parameter degree atmostl, for k ≥ 1, andR/AnnR(µ) has parameter degree at mostl.

If p ≥ σ(m, r, l), then the length of the complexF• is at least the codimension ofAnnR(µ).

Proof. Suppose the result is false for some triple(m, r, l). This means that for infinitelymany distinct prime numberspw, we can find adw-dimensional mixed characteristic Noe-therian local ring(Rw,mw) whose residue field has characteristicpw and whose embed-ding dimension is at mostm, and we can find a finite free complexF•w of length swand of rank at mostr, and a non-zero minimal generatorµw of its cokernelMw suchthatRw/Ik(F•w) has dimension at mostdw − k and parameter degree at mostl, for all


k = 1, . . . , sw, and such thatRw/AnnRw(µw) has parameter degree at mostl, but dimen-

sion strictly less thandw − sw. Choose a non-principal ultrafilter and let(R♮,m♮) be theultraproduct of the(Rw,mw). Sincesw ≤ dw ≤ m, their respective ultraproducts satisfys ≤ d ≤ m. By Theorem5.18, the geometric dimension ofR♮ is at leastd. LetF•♮ andµ be the ultraproduct of the complexesF•w and the minimal generatorsµw respectively.Since the ranks are at mostr, each module inF•♮ will be a freeR♮-module of rank at mostr. By Theorem5.22, the geometric dimension ofR♮/Ik(F•♮) is at mostd − k, for allk = 1, . . . , s. Also by Łos’ Theorem,µ is a minimal generator of the cokernel ofF•♮ andR♮/AnnR♮

(µ), being the ultraproduct of theRw/AnnRw(µw), has geometric dimension

strictly less thand − s by Theorem5.22. However, this is in contradiction with Theo-rem10.8, which yields thatR♮/AnnR♮

(µ) has geometric dimension at leastd− s. �

Using the same techniques, we can deduce from Theorem10.12the following asymp-totic version (details are left to the reader).

13.3.Theorem(Asymptotic Canonical Element Theorem). For each triple of non-negativeintegers(m, r, l), there exists a boundρ(m, r, l) with the following property. LetR be ad-dimensional Noetherian local ring of mixed characteristic p and embedding dimensionat mostm, and letF• be a free resolution of the residue fieldk ofR, of rank at mostr.

If x is a system of parameters inR such thatR/xR has length at mostl and if themorphism of complexesγ : K•(x) → F• extends the natural homomorphismR/xR → k,thenγd 6= 0, providedp ≥ ρ(m, r, l). �

13.4.Remark.Perhaps it is not entirely justified to call this theorem a ‘canonical elementtheorem’, since it does not necessarily produce a canonicalelement in local cohomologylike it does in the equicharacteristic case. This is due to the fact that we can not applythe theorem to the various ‘powers’ of a system of parametersas in the discussion in [9,p. 346-347] without having to raise the boundρ(n, r, l). In particular, the above result doesnot imply an asymptotic version of the Direct Summand conjecture.

Ramification. Instead of requiring that the residual characteristic is large in the aboveasymptotic results, we can also require the ramification to be large, as we will now explain.For the proofs, we only need to apply the corresponding versions in §10 for infinitelyramified local rings of finite embedding dimension. The main observation is the followingimmediate corollary of Łos’ Theorem:

13.5.Lemma. LetRw be Noetherian local rings of mixed characteristicp and embeddingdimensionm. If for eachn, almost allRw have ramification index at leastn, then theirultraproductR♮ is infinitely ramified and hence their cataproductR♯ has equal character-istic p. �

13.6. Theorem. For each triple of non-negative integers(m, r, l), there exists a non-negative integerκ(m, r, l) with the following properties. Let(R,m) be ad-dimensionalmixed characteristic Noetherian local ring of embedding dimension at mostm, and letF• be a finite free complex of rank at mostr. If the ramification index ofR is at leastκ(m, r, l), then the following are true:

(13.6.1) If F• has INIT-degree at mostl, then the length ofF• is at leastd.(13.6.2) If eachR/Ik(F•) has dimension at mostd− k and parameter degree at most

l, for k ≥ 1, and ifµ is a non-zero minimal generator of the cokernel ofF•

such thatR/AnnR(µ) has parameter degree at mostl, then the length ofF•

is at least the codimension ofAnnR(µ).

60 HANS SCHOUTENS

(13.6.3) If F• is a free resolution ofR/m, if x is a system of parameters inR such thatR/xR has length at mostl and if the morphism of complexesγ : K•(x) → F•

extends the natural homomorphismR/xR → R/m, thenγd 6= 0.

Proof. Suppose first that such a bound for a triple(m, r, l) cannot be found in a fixedresidual characteristicp. In other words, we can find mixed characteristicp Noetherianlocal ringsRw, whose embedding dimension is at mostm, and whose ramification indexis at leastw, satisfying the negation of one of the above properties. By Lemma13.5,their cataproduct is equicharacteristic and the proof follows by the previous discussion;details are left to the reader. To make this bound independent from p as well, we use thecorresponding bounds from the previous theorems. �

Monomial Theorem. By the same process as above, we can derive some asymptotic ver-sion of the Monomial Theorem from Corollary10.1. Unfortunately, the bounds will alsodepend on the monomials involved, and hence does not lead to an asymptotic version of theDirect Summand conjecture. More precisely, givenν0, . . . , νs ∈ N

d with ν0 not a positivelinear combination of theνi and givenl,m, there is a boundN depending on these data,such that for every mixed characteristicp Noetherian local ringR of embedding dimensionat mostm and dimensiond, and for every system of parametersx := (x1, . . . , xd) in Rsuch thatR/xR has length at mostl, if eitherp or the ramification index ofR is at leastN , thenxν0 does not belong to the ideal inR generated by thexνi .

In particular, for fixedm and l, we get a boundNt, for eacht ≥ 1, such that (11)holds, wheneverx andR satisfy the assumptions from the previous paragraph. To derivefrom this an asymptotic version of the Direct Summand conjecture, we need to show thattheNt can be chosen independently fromt. To derive this conclusion, we would like toestablish the following result. Let(R♮,m♮) be an isodimensional ultra-Noetherian localring, say the ultraproduct ofd-dimensional Noetherian local rings(Rw,mw) of boundedembedding dimension and parameter degree. LetHd

∞(R♮) be the ultraproduct of the localcohomology groupsHd

mw(Rw). There is a natural mapHd

m♮(R♮) → Hd

∞(R♮).

13.7.Conjecture. The canonical mapHdm♮

(R♮) → Hd∞(R♮) is injective.

Without proof, I state that the conjecture is true whenR♮ is ultra-Cohen-Macaulay. Letus show how this conjecture implies that theNt can be chosen to be independent fromt, thus yielding a true asymptotic version of the Monomial Theorem (whence also of theDirect Summand Theorem) in mixed characteristic. Indeed, assume the conjecture and let(x1♮, . . . , xd♮) be a generic sequence inR♮ and choosexiw ∈ Rw so that their ultraproductis xi♮. Since the (image of the) element1/(x1♮ · · ·xd♮) in the top local cohomology mod-uleHd

m♮(R♮) is non-zero by Corollary10.1—here we realizeH•

m♮(R♮) as the cohomology

of theCech complex associated to(x1♮, . . . , xd♮)—its image inHd∞(R♮) is therefore also

non-zero, whence almost each1/(x1w · · ·xdw) is non-zero inHdmw

(Rw). Hence (11) isvalid for almost each(x1w, . . . , xdw) and allt.

Towards a proof of the full Improved New Intersection Theorem. Although our meth-ods can in principle only prove asymptotic versions, a better understanding of the boundscan in certain cases lead to a complete solution of the conjecture. To formulate sucha result, let us say that a numerical functionf grows sub-linearlyif there exists some0 ≤ α < 1 such thatf(n)/nα remains bounded whenn goes to infinity.


13.8. Theorem. Suppose that for each pair(m, r) the numerical functionfm,r(l) :=κ(m, r, l) grows sub-linearly, whereκ is the numerical function given in(13.6.1), then theImproved New Intersection Theorem holds.

Proof. Let Im,r,l be the collection of counterexamples with invariants(m, r, l), that is tosay, all mixed characteristic Noetherian local ringsR of embedding dimension at mostm,admitting a finite free complexF• of rank at mostr and INIT-degree at mostl, such thatthe length ofF• is strictly less than the dimension ofR. We have to show thatIm,r,l isempty for all(m, r, l), so by way of contradiction, assume it is not for the triple(m, r, l).For eachn, let f(n) be the supremum of the ramification indexes of counterexamples inIm,r,n (and equal to0 if there is no counterexample). By Theorem13.6, this supremum isalways finite. By assumption,f grows sub-linearly, so that for some positive real numbersc andα < 1, we havef(n) ≤ cnα, for all n. In particular, forn larger than the(1− α)-throot of clα

f(l) , we have

(21) f(ln) < nf(l).

Let (R,m) be a counterexample inIm,r,l of ramification indexf(l), witnessed by the finitefree complexF• of length strictly less than the dimension ofR. Since the completion ofRwill be again a counterexample inIm,r,l of the same ramification index, we may assumeR is complete, whence by Cohen’s structure theorem of the formR = V [[ξ]]/I for somediscrete valuation ringV , some tuple of indeterminatesξ, and some idealI ⊆ V [[ξ]]. Letn ≫ 0 so that (21) holds, and letW := V [t]/(tn − π)V [t], whereπ is a uniformizingparameter ofV . Let S := W [[ξ]]/IW [[ξ]], so thatR → S is faithfully flat andS has thesame dimension and embedding dimension asR. By construction, its ramification index isequal tonf(l). By faithful flatness,F• ⊗R S is a finite free complex of length strictly lessthan the dimension ofS, with homology equal toH•(F•) ⊗R S. I claim that ifH is anR-module of lengtha, thenH ⊗R S has lengthna. Assuming this claim, it follows thatSbelongs toIm,r,nl, and hence its ramification is by definition at mostf(ln), contradicting(21).

The claim is easily reduced by induction to the casea = 1, that is to say, whenH isequal to the residue fieldR/m = V/πV = k. In that case,H ⊗R S = S/mS = W/πW ,and this is isomorphic tok[t]/tnk[t], a module of lengthn. �

REFERENCES

[1] Y. Akizuki, Zur ldealtheorie der einartigen Ringbereiche mit dem Teilerkettensatz, Proc. Imp. Acad.13(1937), 53–55.9

[2] M. Andre, Localisation de la lissite formelle, Manuscripta Math.13 (1974), 297–307.20[3] M. Aschenbrenner,Bounds and definability in polynomial rings, Quart. J. Math.56 (2005), no. 3, 263–300.

31[4] M. Aschenbrenner and H. Schoutens,Lefschetz extensions, tight closure and big Cohen-Macaulay algebras,

Israel J. Math.161(2007), 221–310.1, 2, 21, 38, 43[5] J. Ax and S. Kochen,Diophantine problems over local fields I, II, Amer. J. Math.87 (1965), 605–630,

631–648.41[6] J. Becker, J. Denef, L. van den Dries, and L. Lipshitz,Ultraproducts and approximation in local rings I,

Invent. Math.51 (1979), 189–203.2[7] J. Bertin,Anneaux coherents reguliers, C. R. Acad. Sci. Paris273(1971), 1–2.2, 31[8] N. Bourbaki,Algebre, vol. 1-10, Masson, New York, 1970.32[9] W. Bruns and J. Herzog,Cohen-Macaulay rings, Cambridge University Press, Cambridge, 1993.26, 27, 28,

34, 36, 37, 38, 42, 44, 45, 47, 49, 59[10] I. S. Cohen,Commutative rings with restricted minimum condition, Duke Math. J.17 (1950), 27–42.9[11] E. Dieterich,Reduction of isolated singularities, Comment. Math. Helv.62 (1987), no. 4, 654–676.4

62 HANS SCHOUTENS

[12] D. Eisenbud,Commutative algebra with a view toward algebraic geometry, Graduate Texts in Mathematics,vol. 150, Springer-Verlag, New York, 1995.6, 9, 15

[13] D. Eisenbud and E. G. Evans,A generalized principal ideal theorem, Nagoya Math. J.62 (1976), 41–53.46[14] P. Eklof,Ultraproducts for algebraists, Handbook of Mathematical Logic, North-Holland Publishing, 1977,

pp. 105–137.14[15] Y. Ershov,On the elementary theory of maximal normed fields I, Algebra i Logica4 (1965), 31–69.41[16] , On the elementary theory of maximal normed fields II, Algebra i Logica5 (1966), 8–40.41[17] S. Glaz,Commutative coherent rings, Lect. Notes in Math., vol. 1371, Springer-Verlag, Berlin,1989.31[18] , Commutative coherent rings: Historical perspective and current developments, Nieuw Arch. Wisk.

10 (1992), 37–56.2, 31[19] , Coherence, regularity and homological dimensions of commutative fixed rings, Commutative al-

gebra (Trieste, 1992), World Sci. Publ., River Edge, NJ, 1994, pp. 89–106.3[20] T. Hamilton and T. Marley,Non-Noetherian Cohen-Macaulay rings, ArXiv: math.AC/0601567, 2006.3,

11, 31[21] M. Hochster,Topics in the homological theory of modules over commutative rings, CBMS Regional Conf.

Ser. in Math, vol. 24, Amer. Math. Soc., Providence, RI, 1975. 42, 43[22] , Big Cohen-Macaulay algebras in dimension three via Heitmann’s theorem, J. Algebra254(2002),

395–408.43[23] M. Hochster and C. Huneke,Infinite integral extensions and big Cohen-Macaulay algebras, Ann. of Math.

135(1992), 53–89.43[24] , F -regularity, test elements, and smooth base change, Trans. Amer. Math. Soc.346 (1994), 1–62.

57[25] , Applications of the existence of big Cohen-Macaulay algebras, Adv. in Math.113(1995), 45–117.

19, 42, 43[26] W. Hodges,Model theory, Cambridge University Press, Cambridge, 1993.14, 16[27] R. Hubl and I. Swanson,Discrete valuations centered on local domains, J. Pure Appl. Algebra161(2001),

no. 1-2, 145–166.4, 51[28] C. Huneke,Tight closure and its applications, CBMS Regional Conf. Ser. in Math, vol. 88, Amer. Math.

Soc., 1996.43, 57[29] H. Matsumura,Commutative ring theory, Cambridge University Press, Cambridge, 1986.9, 10, 12, 18, 20,

23, 24, 25, 26, 28, 29, 30, 32, 33, 34, 40, 41, 45, 51, 52, 53[30] M. Nagata,Local rings, Interscience Tracts in Pure and Applied Mathematics, Interscience Publishers,

1962.26[31] B. Olberding and S. Saydam,Ultraproducts of commutative rings, Commutative ring theory and applica-

tions (Fez, 2001), Lecture Notes in Pure and Appl. Math., vol. 231, Dekker, New York, 2003, pp. 369–386.1

[32] B. Olberding, S. Saydam, and J. Shapiro,Completions, valuations and ultrapowers of Noetherian domains,J. Pure Appl. Algebra197(2005), no. 1-3, 213–237.1, 2, 51

[33] B. Olberding and J. Shapiro,Prime ideals in ultraproducts of commutative rings, J. Algebra285 (2005),no. 2, 768–794.1, 2

[34] D. Popescu,Indecomposable Cohen-Macaulay modules and their multiplicities, Trans. Amer. Math. Soc.323(1991), no. 1, 369–387.4

[35] P. Rothmaler,Introduction to model theory, Algebra, Logic and Applications, vol. 15, Gordon and BreachScience Publishers, Amsterdam, 2000.14

[36] H. Schoutens,Existentially closed models of the theory of Artinian localrings, J. Symbolic Logic64 (1999),825–845.19, 38

[37] , Artin approximation via the model theory of Cohen-Macaulayrings, Logic Colloquium ’98: pro-ceedings of the 1998 ASL European Summer Meeting held in Prague, Czech Republic (S. Buss, P. Hajek,and P. Pudlak, eds.), Lect. Notes in Logic, vol. 13, Association for Symbolic Logic, 2000, pp. 409–425.3,33

[38] , Lefschetz principle applied to symbolic powers, J. Algebra Appl.2 (2003), 177–187.1[39] , Mixed characteristic homological theorems in low degrees, C. R. Acad. Sci. Paris336 (2003),

463–466.1, 4, 57[40] , Non-standard tight closure for affineC-algebras, Manuscripta Math.111(2003), 379–412.1, 14,

38[41] , A non-standard proof of the Briancatacon-Skoda theorem, Proc. Amer. Math. Soc.131 (2003),

103–112.1[42] , Projective dimension and the singular locus, Comm. Algebra31 (2003), 217–239.19


[43] , Canonical big Cohen-Macaulay algebras and rational singularities, Illinois J. Math.48 (2004),131–150.1

[44] , On the vanishing of Tor of the absolute integral closure, J. Algebra275(2004), 567–574.18, 44[45] , Log-terminal singularities and vanishing theorems via non-standard tight closure, J. Alg. Geom.

14 (2005), 357–390.1[46] , Absolute bounds on the number of generators of Cohen-Macaulay ideals of height at most two,

Bull. Soc. Math. Belg.13 (2006), 719–732.11, 15, 26, 40, 41[47] , Asymptotic homological conjectures in mixed characteristic, Pacific J. Math.230(2007), 427–468.

1, 4, 5, 11, 57[48] , Pure subrings of regular rings are pseudo-rational, Trans. Amer. Math. Soc.360(2008), 609–627.

1[49] , Use of ultraproducts in commutative algebra, in preparation, 2008.2, 4, 14, 56[50] , Classifying singularities up to analytic extensions of scalars, arXiv: AC/0508236, in preparation.

48, 49, 50[51] K. Smith,F-rational rings have rational singularities, Amer. J. Math.119(1997), 159–180.56[52] J.-P. Soublin,Anneaux et modules coherents, J. Algebra15 (1970), 455–472.31[53] I. Swanson,Linear equivalence of ideal topologies, Math. Z.234(2000), no. 4, 755–775.4, 51[54] Y. Yoshino, Brauer-Thrall type theorem for maximal Cohen-Macaulay modules, J. Math. Soc. Japan39

(1987), no. 4, 719–739.4

DEPARTMENT OFMATHEMATICS, CITY UNIVERSITY OF NEW YORK, 365 FIFTH AVENUE, NEW YORK,NY 10016 (USA)

E-mail address: [email protected]

arxiv:0809.5267v1 [math.ac] 30 sep 2008 · 2018. 11. 2. · arxiv:0809.5267v1 [math.ac] 30 sep 2008...

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