Abhyankar places admit local uniformization in any characteristic

ABHYANKAR PLACES ADMIT LOCAL UNIFORMIZATION IN ANY CHARACTERISTIC

B

H

KNAF

F

-V

KUHLMANN

ABSTRACT. – We prove that every placePof an algebraic function fieldF|Kof arbitrary characteristic admits local uniformization, provided that the sum of the rational rank of its value group and the transcendence degree of its residue fieldF P overK is equal to the transcendence degree ofF|K, and the extensionF P|Kis separable. We generalize this result to the case whereP dominates a regular local Nagata ringR⊆Kof Krull dimensiondimR2, assuming that the valued field(K, vP)is defectless, the factor groupvPF/vPKis torsion-free and the extension of residue fieldsF P|KP is separable. The results also include a form of monomialization.

2005 Published by Elsevier SAS

RÉSUMÉ. – Nous montrons que toute placePd’un corps de fonctions algébriqueF|Ken caractéristique quelconque admet une uniformisation locale, pourvu que la somme du rang rationnel de son groupe de valeurs et du degré de transcendance de son corps résiduelF PsurKsoit égal au degré de transcendance deF|K, et que l’extensionF P|Ksoit séparable. Nous généralisons ce résultat au cas oùP domine un anneau de Nagata local régulierR⊂Kde dimension de Krull au plus2, en supposant que le corps valué (K, vP)soit sans défaut, que le groupe quotientvPF/vPKsoit sans torsion, et que l’extension des corps résiduelsF P|KP soit séparable. Les résultats contiennent aussi une forme de monomialisation.

2005 Published by Elsevier SAS

1. Introduction and main results

In [20], Zariski proved the Local Uniformization Theorem for places of algebraic function fields over base fields of characteristic0. In [22], he uses this theorem to prove resolution of singularities for algebraic surfaces in characteristic0. This result was generalized by Abhyankar to the case of positive characteristic [4] and to the case of arithmetic surfaces over a Dedekind domain [5]. More recently de Jong [8] proved that the singularities of an algebraic or arithmetic variety X can be resolved by successively applying a finite number of morphisms called alterations. An alteration f:Y →X is a composition f =g◦h of a birational morphism h:X →X and a finite morphism g:Y → X. In general it leads to a finite extension K(Y)|K(X)of function fields. Applying de Jong’s results to a proper algebraic variety over a field of positive characteristic or to a proper arithmetic variety over a discrete valuation ring implies Local Uniformization in these cases, provided one allows finite extensions of the variety’s function field or of the discrete valuation ring, respectively. In [13], we replace the application of de Jong’s results by a purely valuation theoretical proof, thereby obtaining a more detailed description of the extensions of the function field.

1The author thanks Hans Schoutens, Peter Roquette, Dale Cutkosky and Olivier Piltant for many inspiring discussions.

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE

As resolution of singularities in the birational sense in positive characteristic for arbitrary dimensions is still an open problem, one is interested in generalizations of the original Local Uniformization Theorem. In the present paper we investigate the class of so-called Abhyankar places of a function fieldF|K. We prove that Abhyankar places that are trivial onKadmit local uniformization on algebraic varieties without any extension ofF, provided they have a separable residue field extension. We also consider the case of Abhyankar places dominating a regular local ringR⊂Kand provide sufficient conditions for local uniformization onR-models ofF|K. The core of our method is purely valuation-theoretical. Before stating the main results precisely, we have to introduce the necessary terminology.

Throughout this paper the termfunction field will always meanalgebraic function field. We consider placesP of a function fieldF|Knot necessarily inducing the identity on the constant fieldK. Places that do induce the identity onK(or an isomorphism) are calledK-trivial. The valuation associated withP will be denoted byvP, thevP-value of an element abyvPaand consequently, the value group ofv_P onF byv_PF. Places are considered to operate on the right:

the residue of an elementa∈F is denoted byaP and consequently,F P stands for the residue field ofF with respect toP. We frequently do not distinguish between a placePon the fieldF and its restrictions to subfieldsE⊆F. Since we are usually working with one fixed place, this does not lead to confusion. The valuation ring ofPon a subfieldE⊆Fis denoted byOE. If we have to distinguish between several places on the same fieldF, then we use the notationOPfor the valuation ring ofPonF. The maximal ideal of the valuation ringsOP andOEis denoted by MPandME, respectively. Finally, it should be mentioned that by abuse of language we refer to a pair(F, P)consisting of a fieldFand a placePofFas a valued field, and to a pair(F|K, P), F|Kan extension of fields, as an extension of valued fields, or as a valued function field ifF|K is a function field.

For every placePof a function fieldF|K, we have the following inequality:

trdegF|KtrdegF P|KP+ dim_Q

(v_PF/v_PK)⊗Q . (1)

Note thatdim_Q(v_PF/v_PK⊗Q)is therational rankof the Abelian groupv_PF/v_PK, i.e., the maximal number of rationally independent elements in v_PF/v_PK. We callP anAbhyankar place ofF|Kif equality holds in (1).

In the context of local uniformization we shall be concerned with the prime factorization in a regular local ring and use the following terminology: LetObe a commutative ring andH⊆ O.

An elementa∈ Ois called anO-monomial inH if a=u

d

i=1

h^µ_iⁱ, u∈ O^×, h_i∈H, µ_i∈N0, i= 1, . . . , d,

holds, whereN0:=N∪ {0}.

We investigate local uniformization over regular base rings: let(F|K, P)be a valued function field, and letR⊆ OP be a subring ofK withFracR=K. Moreover, letZ⊂ OP be a finite set. The pair(P, Z)is calledR-uniformizableif there exists an integral separatedR-schemeX of finite type with field of rational functions F—an R-model of F—such that P is centered in a regular pointx∈X andZ is a subset of the local ringOX,x atx. If this holds, then we also use the phraseR-uniformizable onX. The placeP is calledR-uniformizable if(P, Z)is R-uniformizable for every finite setZ⊂ OP. Note that including the finite setZ in the problem of uniformization allows to prove statements like: given an integralR-schemeX such thatP is centered in the (singular) point x∈X there exist an integral R-scheme Y and a morphism

e ◦

π:U→V, whereU⊆Y andV ⊆Xare open subschemes such thatx∈V andPis centered in a regular pointy∈π⁻¹x.

Throughout the paper we restrict ourselves to the case of aregular local ringRwith maximal idealM =MP∩R, i.e., we assume thatOP dominatesR.

We prove a Local Uniformization Theorem for K-trivial Abhyankar places in arbitrary characteristic:

THEOREM 1.1. – Let P be a K-trivial Abhyankar place of the function field F|K, and assume that F P|K is separable. Take any finite set Z ⊂ OP. Then the pair (P, Z) is K-uniformizable on a variety X such that P is centered in a smooth point x∈X and dimOX,x = dim_Q(vPF ⊗Q). Moreover, X can be chosen such that all ζ∈Z are OX,x- monomials in{a1, . . . , ad}for some regular parameter system(a1, . . . , ad)ofOX,x.

One can also achieve that the OX,x-ideal generated by Z is principal, generated by each element of minimal value. For this, one applies the above theorem to the set Z =Z ∪ {^a_b | a, b∈Z, vavb}in the place ofZ.

Theorem 1.1 is essentially proved by embedding F in the field of fractions of the strict henselization of the valuation ring OK(T)=OP ∩K(T) for a suitable transcendence basis T ⊂F of F|K. The methods used in the proof of Theorem 1.1 are applicable even if P is not trivial onK. They then lead to a uniformization result for integral schemes of finite type over certain base ringsR⊂K. For the definition of the notion “defectless”, see Section 3.

THEOREM 1.2. – Let P be an Abhyankar place of the function field F|K, which is non- trivial onK. Assume that(K, P)is defectless,F P|KP is separable and the groupvPF/vPK is torsion-free.

Let R⊂K∩ OP, FracR =K, be a noetherian, regular local ring with maximal ideal M=MP∩Rand of dimensiondimR2. Assume thatRis a Nagata ring ifdimR= 2.

Then for every finite setZ⊂ OP the pair(P, Z)isR-uniformizable on anR-schemeXsuch that the centerx∈X ofP onXsatisfies:

• dimOX,x= dim_Q(v_PF/v_PK⊗Q) + 1ifdimR= 1ortrdeg(KP|R/M)>0.

• dimOX,x= dim_Q(v_PF/v_PK⊗Q) + 2in the remaining cases.

Moreover,X can be chosen such that allζ∈Z areOX,x-monomials in{a₁, . . . , a_d}for some regular parameter system(a₁, . . . , a_d)ofOX,x.

Some remarks concerning the conditiondimR2may be helpful at this point. Recall that the domainRis calledNagata, if the integral closure of every factor ringR/p,p∈SpecR, in every finite extension ofFracR/pis finite. In the case of dimR= 1, the ringRis a discrete valuation ring and moreover, in the situation of the theorem, we haveR=OK. It is well known that then the property of being Nagata is equivalent to the valued field(K, P)being defectless.

We conclude that all base rings appearing in Theorem1.2are Nagata.

A further important ring-theoretic notion that we will have to use is universal catenarity: the domainR is calleduniversally catenaryif every polynomial ringR[X1, . . . , Xn],n∈N, has the following property: for every pair of prime ideals p, q∈R[X1, . . . , Xn] with p⊂q, all non-refinable chains of primesp=:p0⊂p1⊂ · · · ⊂p:=q have a common finite length (depending onp, q).

Since Cohen–Macaulay rings are universally catenary, all base rings appearing in Theo- rem1.2are universally catenary.

Every universally catenary domainRsatisfies thealtitude formula: for every primeq∈SpecA of every domainAfinitely generated overRthe equation

heightq+ trdeg(A/q|R/p) = heightp+ trdeg(A|R), p:=q∩R, (2)

holds.

The conditiondimR2can be replaced by a more general but rather technical condition that ensures the existence of certain monoidal transforms ofRalong the valuationv_P.

In [16] it is shown that theK-trivial Abhyankar places lie dense in the Zariski space of all K-trivial places ofF|K, with respect to a “Zariski patch topology”. This topology is finer than the Zariski topology (but still compact); its basic open sets are the sets of the form

{P|P a place ofF|Ksuch thata1P = 0, . . . , akP = 0; b1P= 0, . . . , bP= 0}

witha₁, . . . , a_k, b₁, . . . , b∈F\ {0}. Theorem 1.1 thus yields:

COROLLARY 1.3. –The K-uniformizable places of F|K lie dense in the Zariski space ofF|K, with respect to the Zariski patch topology, provided thatKis perfect.

2. Valuation independence

The following theorem, together with Theorem 3.1 below, gives the motivation for the definition of the distinguished class of Abhyankar places. For its proof see [7, Chapter VI, §10.3, Theorem 1].

THEOREM 2.1. – Let(F|K, P)be an extension of valued fields. Take elementsxi, yj∈F, i∈I,j∈J, such that the values vPxi,i∈I, are rationally independent overvPK, and the residuesyjP,j∈J, are algebraically independent overKP. Then the elementsxi, yj,i∈I, j∈J, are algebraically independent overK.

Moreover, if we write

f=

k

c_k

i∈I

x^µ_i^k,i

j∈J

y_j^νk,j∈K[x_i, y_j|i∈I, j∈J]

in such a way that for everyk=there is someis.t.µ_k,i=µ_,ior somejs.t.ν_k,j=ν_,j, then vPf= min

k

vP

ck

i∈I

x^µ_i^k,i

j∈J

y_j^νk,j

= min

k

vPck+

i∈I

µk,ivPxi

.

That is, the value of the polynomialf is equal to the least of the values of its monomials. In particular, this implies:

vPK(xi, yj|i∈I, j∈J) =vPK⊕

i∈I

ZvPxi, K(xi, yj|i∈I, j∈J)P=KP(yjP|j∈J).

It also implies that the valuationvP and the placeP onK(xi, yj|i∈I, j∈J)are uniquely determined by their restrictions toK, the valuesvPxiand the residuesyjP.

Every finite extension L of the valued field (K, P) satisfies the fundamental inequality (cf. [10]):

[L:K]

g

i=1

e_if_i (3)

e ◦

whereP₁, . . . , P_g are the distinct extensions of P from K toL,e_i= (v_PiL:v_PK) are the respective ramification indices andf_i= [LP_i:KP]are the respective inertia degrees. Note that g = 1if(K, P)is Henselian.

COROLLARY 2.2. – Let (F|K, P) be an extension of valued fields of finite transcendence degree. Then the following inequality holds:

trdegF|KtrdegF P|KP + dim_Q

(v_PF/v_PK)⊗Q . (4)

If in additionF|Kis a function field, and if equality holds in(4), then the extensionsvPF|vPK andF P|KP are finitely generated. In particular, ifP is trivial onK, thenvPF is a product of finitely many copies ofZ, andF P is again a function field overK.

Proof. –Choose elementsx₁, . . . , x_ρ, y₁, . . . , y_τ∈F such that the valuesv_Px₁, . . . , v_Px_ρare rationally independent overv_PK and the residuesy₁P, . . . , y_τP are algebraically independent overKP. Then by the foregoing theorem,ρ+τtrdegF|K. This proves thattrdegF P|KP and the rational rank ofv_PF/v_PKare finite. Therefore, we may choose the elementsx_i, y_jsuch thatτ= trdegF P|KP andρ= dim_Q((v_PF/v_PK)⊗Q)to obtain inequality (4).

Assume that this is an equality. This means that for F₀:=K(x₁, . . . , x_ρ, y₁, . . . , y_τ), the extensionF|F₀is algebraic. SinceF|Kis finitely generated, it follows thatF|F₀is finite. By the fundamental inequality (3), this yields thatvPF|vPF0andF P|F0Pare finite extensions. Since alreadyvPF0|vPKandF0P|KP are finitely generated by the foregoing theorem, it follows that alsovPF|vPKandF P|KP are finitely generated. 2

If equality holds in (4) we will either say that(F|K, P)is without transcendence defector as already defined earlier thatP is an Abhyankar place ofF|K.

3. Inertially generated function fields

In this section we provide the valuation-theoretic core of the present paper. It is a generalization of the “Grauert–Remmert Stability Theorem” and is proved in [17]. To state it, we introduce a fundamental notion: a valued field(K, P) is calleddefectless (or stable) if equality holds in the fundamental inequality (3) for every finite extensionL|K. IfcharKP = 0, then(K, P)is defectless (this is a consequence of the “Lemma of Ostrowski”, cf. [10,18]).

THEOREM 3.1 (Generalized Stability Theorem). – Let(F|K, P)be a valued function field without transcendence defect. If(K, P)is a defectless field, then also(F, P)is a defectless field.

In what follows we consider concepts like the henselization of a valued field(K, P), where one has to fix an extension ofP andv_P to the algebraic closure ofK. Thus, whenever we talk of a valued field(K, P), we will implicitly assume the valuationv_Pand the placePto be extended to the algebraic closure ofK, the extensions denoted byv_P andP again. Therefore, we will talk ofthehenselizationK^h, and of the absolute inertia fieldKⁱ ofK, which we define to be the inertia field of the normal separable extensionK^sep|Kwith respect to the given valuationv_P; here,K^sepdenotes the separable-algebraic closure ofK.

The following lemma is proved in [17] (and partially also in [10]):

LEMMA 3.2. – A valued field(K, P)is defectless if and only if its henselization(K^h, P)is.

An extension (L|K, P) of valued fields is called immediate if the canonical embeddings KP→LP andvPK→vPLare onto.

COROLLARY 3.3. – If(K, P)is defectless, then(K^h, P)does not admit proper immediate algebraic extensions.

Proof. –If(K, P)is defectless, then so is (K^h, P), by the foregoing lemma. Suppose that (L|K^h, P)is a finite immediate algebraic extension. Hence,(v_PL:v_PK^h) = 1 = [LP:K^hP].

Since (K^h, P) is a Henselian field, there is a unique extension of v_P from K^h to L. Since (K^h, P)is defectless, we have that [L:K^h] = (v_PL:v_PK^h)[LP :K^hP] = 1, showing that L=K^h. As every proper immediate algebraic extension would contain a proper immediate finite extension, it follows that(K^h, P)does not admit any proper immediate algebraic extension. 2

From these facts, we deduce:

THEOREM 3.4. – Assume that (F|K, P) is a valued function field without transcendence defect such thatF P|KP is a separable extension,(K, P)is a defectless field andvPF/vPK is torsion-free. Then (F|K, P) is inertially generated, by which we mean that there is a transcendence basisT={x1, . . . , xρ, y1, . . . , yτ}such that

(a) vPF=vPK⊕ZvPx1⊕ · · · ⊕ZvPxρ,

(b) y1P, . . . , yτP is a separating transcendence basis ofF P|KP, (c) (F, P)lies in the absolute inertia field of(K(T), P).

Assertion(c)holds for each transcendence basisT which satisfies assertions(a)and(b).

Proof. –By Corollary 2.2, the factor group vPF/vPK and the residue field extension F P|KP are finitely generated. We choosex1, . . . , xρ∈F such thatvPF =vPK⊕ZvPx1⊕

· · · ⊕ZvPxρ, where ρ= dim_Q((vPF/vPK)⊗Q). Since F P|KP is a finitely generated separable extension, it is separably generated. Therefore, we can choosey1, . . . , yτ∈F such that F P|KP(y1P, . . . , yτP) is separable-algebraic, where τ = trdegF P|KP. We set T :=

{x1, . . . , xρ, y1, . . . , yτ}andF0:=K(T).

Now we can choose some a∈F P such that F P =KP(y1P, . . . , yτP, a). Since a is separable-algebraic overKP(y1P, . . . , yτP), by Hensel’s Lemma there exists an elementη in the henselization of(F, P)such thatηP=aand that the reduction of the minimal polynomial of ηoverF0 is the minimal polynomial ofaoverKP(y1P, . . . , yτP). Thenη lies in the absolute inertia field ofF0. Now the fieldF0(η)has the same value group and residue field asF, and it is contained in the henselizationF^h ofF. As henselizations are immediate extensions and the henselizationF0(η)^hof F0(η)can be chosen inside ofF^h, we obtain an immediate algebraic extension(F^h|F0(η)^h, P). On the other hand,(K, P)is assumed to be a defectless field. By construction,(F0|K, P)is without transcendence defect, and the same is true for(F0(η)|K, P) since this property is preserved by algebraic extensions. Hence we know from Theorem 3.1 that (F0(η), P)is a defectless field. Now Corollary 3.3 shows that the extensionF^h|F0(η)^hmust be trivial. Therefore,F is contained in F0(η)^h, which in turn is a subfield of the absolute inertia field ofF0. This proves our theorem. 2

Theorem 3.4 is central in the proof of the uniformization results presented in this paper. Its importance is based on the fact that the valuation ring OK(T)ⁱ =OP ∩K(T)ⁱ is the strict henselization of the valuation ring OK(T) and therefore the ring extensionOK(T)ⁱ|OK(T) is local-ind-étale; see [19, Chapter X]. Using this property, one can construct an extensionB|Aof finitely generatedR-algebras,R=K orRthe local base ring appearing in Theorem 1.2, such thatB⊂ OP,FracB=F,FracA=K(T), andB_q|Aqis an étale extension forq:=MP∩A.

In order to prove Theorems 1.1 and 1.2, and ignoring the requirement for the elementsζ∈Zfor the moment, by the permanence of smoothness and regularity under étale extension it therefore suffices to construct A in such a way that Aq is smooth over K, or is a regular local ring, respectively. This is done in the next section.

e ◦

4. Abhyankar places on rational function fields

In this section, uniformization of Abhyankar places on rational function fields of the type appearing in Theorem 3.4 is investigated. Throughout this section, let(F|K, P) be a rational function field equipped with a placeP subject to the following conditions:

(T) There exists a transcendence basisT= (x1, . . . , xρ, y1, . . . , yτ)ofF|Ksuch that:

• F=K(T),

• F P=KP(y1P, . . . , yτP),withy1P, . . . , yτPalgebraically independent overKP,

• vPF=vPK⊕ZvPx1⊕ · · · ⊕ZvPxρ. In particular,P is an Abhyankar place ofF|K.

We fix a finite set Z ⊂ OP and a regular local ring R⊆K such that FracR=K and MP∩R=M, whereM is the maximal ideal ofR. The case ofR=Kis included.

In the case ofR =K, we cannot prove uniformization over the base ringRfor an arbitrary pair (P, Z)of given data. Instead, we have to impose rather technical conditions on the pair(R, Z).

These conditions involve the notion of a monoidal transform ofR: assume for the moment that dimR >0holds, and letvbe a valuation of the fieldKsuch thatR⊆ Ov andM=R∩ Mv. Letp∈SpecRbe a prime ofheight(p)1. The monoidal transform ofRalongvwith center pis the local ring

R₁:=R

x⁻¹p_M

v∩R[x⁻¹p],

wherex∈psatisfiesvx= min{va|a∈p}. Ifp=M, then the monoidal transform is also called quadratic transform. It is well known thatR1 does not depend on the choice ofxand that it is a regular local ring of dimensiondimR1dimR provided thatpis generated by a part of a regular system of parameters ofR. Moreover,R1⊆ Ov andM1=R1∩ Mv for the maximal idealM1ofR1. Every member of a finite chainR=:R0R1· · ·Rtof local rings, where R_i+1 is a monoidal transform of R_i along v with center p_i∈SpecR_i, is called an iterated monoidal transform ofRalongv.

The properties of(R, Z)we are interested in can now be formulated as follows:

(NC) There exists an iterated monoidal transform R ofRalong the valuation vP|K such that the regular local ring R admits a regular parameter system(t1, . . . , td)with the following property:everyζ∈Zadmits a representation

ζ= N

i=1r_ix^µiy^νi _N

i=1s_ix^κiy^λi, r_i, s_i∈R, (5)

with x:= (x1, . . . , xρ), y:= (y1, . . . , yτ),µ

i, κ_i∈N^ρ₀, ν_i, λ_i∈N^τ₀, where the prime factorizations inRof the coefficientsri, sihave the form

u d

i=1

t^ε_iⁱ, u∈(R)^×, ε_i∈N0, (6)

i.e., the coefficientsri, siareR-monomials in{t1, . . . , td}.

(V) The values v_Pt₁, . . . , v_Pt_δ of those parameters t₁, . . . , t_δ actually occurring in at least one of the prime factorizations (6), are rationally independent. Without loss of generality, we assume here that these parameters are the firstδof the complete set.

The main result of this subsection now reads as follows:

THEOREM 4.1. – Let (F|K, P) be a valued rational function field satisfying the require- ment(T). Let R⊆K be a regular local ring dominated byOP and let Z⊂ OP be a finite

set such that the pair(R, Z)fulfills the requirements(NC)and(V). Then there exist an iterated monoidal transformRofRalongv_P|Kand elementsx₁, . . . , x_ρ+δ∈ OP,δdimR, such that the localization of theR-algebraA:=R[x₁, . . . , x_ρ+δ, y₁, . . . , y_τ]at the primeq:=MP ∩A is a regular ring having the properties:dimA_q= dimR+ρ, the elementsx₁, . . . , x_ρ+δ are a part of a regular parameter system ofAq,Z⊂Aq, and every element ofZ is anAq-monomial in{x1, . . . , x_ρ+δ}.

In particular, there exists anR-modelXofFsuch thatPis centered in a regular pointx∈X with the propertiesZ⊂ OX,xanddimOX,xdimR+ρ. Moreover, ifR=Kthen the model Xcan be chosen such thatX∼=A^ρ+τ_K anddimOX,x=ρ.

The following lemma is applied in the proof of Theorem 4.1; it was proved (but not explicitly stated) by Zariski in [20] for subgroups ofR, using the algorithm of Perron. We leave it as an easy exercise to the reader to prove the general case by induction on the rank of the ordered Abelian group. However, an instant proof of the lemma can also be found in [9, Theorem 2.2].

LEMMA 4.2. – LetΓbe a finitely generated ordered Abelian group. Take any non-negative elements α1, . . . , α∈ Γ. Then there exist positive elements γ1, . . . , γρ ∈Γ such that Γ = Zγ1⊕ · · · ⊕Zγρand everyαican be written as a sum

jnijγjwith non-negative integersnij. We now turn to the

Proof of Theorem 4.1. –Since iterated monoidal transforms of a local ringRare essentially of finite type overR, the existence of the schemeX is a consequence of the first part of the theorem.

To simplify notation, we replaceR with an iterated monoidal transformR ofR along the valuation vP|K having the properties (NC) and (V) for a specific regular parameter system t1, . . . , td∈M,M the maximal ideal ofR. Recall thatRandR are universally catenary and thatdimRdimR.

The coefficients appearing in the representations (5) can now be replaced by their prime factorizations inR:

ζ= N

i=1uit^εix^µiy^νi _N

i=1v_it^δix^κiy^λi, u_i, v_i∈R^×, (7)

wheret= (t1, . . . , tδ)are those regular parameters among thet1, . . . , tdthat actually occur in at least one of the prime factorizations (6) of the coefficients. Note that in our notation in (7) the dependence of the coefficients and exponents onζdoes not explicitly appear, in order to avoid overloading our notation. We use this simplification throughout the present proof.

Next, one divides numerator and denominator by a monomial intandxwith leastv_P-value among the monomials appearing in the denominator. One can assume that this monomial is the first one in the denominator, thus obtaining the expression:

ζ= _N

i=1uit^εi−δ1x^µi−κ1y^νi N

i=1vit^δi−δ1x^κi−κ1y^λi, ui, vi∈R^×. (8)

By construction, the monomial expressionst^δi−δ1x^κi−κ1all have non-negativevP-value. Since vP(ζ)0 and since P is an Abhyankar place the same holds for the monomial expressions t^εi−δ1x^µi−κ1 appearing in the numerator; see Theorem 2.1.

Among the monomials intandxappearing in the numerator of (8) we choose the unique one having leastvP-value,hζ:=t^εm−δ1x^µm−κ1say, and define the finite sets

e ◦

H_ζ:={h_ζ} ∪

t^δi−δ1x^κi−κ1|i= 1, . . . , N

∪

t^εi−εmx^µi−µm=t^εi−δ1x^µi−κ1

h_ζ |i= 1, . . . , N

. The finite set

H:={t1, . . . , tδ, x1, . . . , xρ} ∪

ζ∈Z

Hζ

then consists of elements with non-negativevP-value only.

Let G⊂K(x1, . . . , xρ)^× be the group (freely) generated by the elements t1, . . . , tδ and x1, . . . , xρ. By assumption, the valuationvP induces an isomorphism

v_P:G→ δ

i=1

Zv_Pt_i⊕ ρ

i=1

Zv_Px_i⊆v_PF.

Applying Lemma 4.2 yields a basis(v_Px₁, . . . , v_Px_ρ+δ)of the groupv_PGconsisting of positive elements such that everyv_Ph∈v_PH⊂v_PG can be expressed as a linear combination with non-negative coefficients. It follows that everyh∈Hcan be expressed in the form

h=

ρ+δ

i=1

x_i^µi, µ_i∈N0. (9)

By definition of the setHζthe monomials intandxappearing in the numerator of (8) have the formhhζfor someh∈Hζ. Therefore, each of these monomials has the form

t^εi−δ1x^µi−κ1= _ρ+δ

i=1

x_i^βi _ρ+δ

i=1

x_i^αi

, β_i∈N0, (10)

where

h_ζ=

ρ+δ

i=1

x_i^αi, α_i∈N0. (11)

Substituting (9) and (10) in (8) yields

ζ=

i=1u_ix^βiy^νi

i=1v_ix^κiy^λi _ρ+δ

i=1

x_i^αi

, ui, vi∈R^×, (12)

with non-negative exponentsβ_i, κ_i∈N^ρ+r₀ .

We set A:=R[x₁, . . . , x_ρ+δ, y1, . . . , yτ]⊆ OP and claim: AqA, where qA:=MP ∩A, is a regular ring of dimension ρ+ dimR, Z⊂AqA and all ζ∈Z are AqA-monomials in {x1, . . . , x_ρ+δ}.

LetB:=R[x₁, . . . , x_ρ+δ]⊆Aand consider the ideals

J:=

ρ+δ

i=1

Bx_i+ d

i=δ+1

Bti⊆qB:=MP∩B.

Due to Eqs. (9) for the elementst_i,i= 1, . . . , δ, one getsJ∩R=M. Hence,B/J=R/Mand therefore,q_B=J is a maximal ideal ofBgenerated byρ+delements,d= dimR.

According to the altitude formula in finitely generated integral domains over universally catenary rings, one has:

heightqB= heightM+ trdeg(B|R)−trdeg(B/qB|R/M)

=d+ρ,

where we use that due to Eqs. (9) for the elementsxithe relationFracB=K(x1, . . . , xρ)holds.

Consequently,BqBis a regular local ring of Krull dimensiond+ρ.

Considering that AqA =BqB[y1, . . . , yτ]p, where p= MP ∩ BqB[y1, . . . , yτ], and that BqB[y1, . . . , yτ] is a polynomial ring overBqB, we get the desired regularity ofAqA because a polynomial ring over a regular ring is regular.

It remains to calculate the dimension ofAqA. By the choice of the transcendence basisT we have that BqB[y1, . . . , yτ]/p=R/M[y1+p, . . . , yτ+p] is a polynomial ring in the variables yi+p, i= 1, . . . , τ. Thus p=qB[y1, . . . , yτ] and therefore, heightp= heightqB, where the latter equation holds in any polynomial ring over a noetherian ring. Consequently,AqAis a local ring of Krull dimensionρ+d.

IfR=Kis a field, then theK-algebraBis a polynomial ring in the variablesx₁, . . . , x_ρand therefore,AqAis a localization of the polynomial ringK[x₁, . . . , x_ρ, y1, . . . , yτ].

Turning to the claim aboutZwe first observe that the denominator of the expression (12) has vP-value0, which implies the inclusionZ⊂AqA. The numerator of the first factor in (12) has vP-value0too, because itsm-th summand hasvP-value0. Thus this first factor is a unit inAqA, which verifies the claim. 2

We finish this section with a discussion of the requirements (NC) and (V) in various cases to give examples in which Theorem 4.1 applies.

(1) dimR1: In this case,Ris either the fieldKitself or a discrete valuation ring. In the first case, the coefficients appearing in the representations (5) are units, hence (NC) and (V) are trivially satisfied. In the second case, every prime elementt∈Rsatisfies the requirements (NC) and (V), soR=R.

(2) OK is a discrete valuation ring: It is known that in this case,OK=_∞

k=0Rk, whereRk

is the quadratic transform ofRk−1alongvP|K—see for example [12]. Given a finite set Z⊂ OK, one chooses prime factorizationsz=uzt^εz,uz∈ O^×_K,ta prime element, for allz∈Z. It is now easy to verify that these prime factorizations remain valid in someR, thus showing that (NC) and (V) are satisfied for arbitrary finite setsZ⊂ OP.

(3) dimR= 2: One applies Abhyankar’s results [6]. If trdeg(KP|R/M)>0, then the valuationvP|K is discrete andOK is an iterated quadratic transform ofR, thus we can takeR=R. In the case oftrdeg(KP|R/M) = 0, due to [6, §4, Theorem 2], for every finite setZ⊂ OK there exists a2-dimensional monoidal transformR with a regular parameter system(t1, t2)such that the elementsz∈Z have prime factorizations of the type (6) with property (V), whereδ∈ {1,2}depending on the rational rank ofvPK. In the case ofdimOK= 2one has to assume in addition thatRis a Nagata ring. It follows that Theorem 4.1 applies for2-dimensional regular local Nagata ringsR.

(4) dimR= 3: In this case, the following two results are relevant in the current context:

If R is excellent and charR = charR/M holds, then every finite set Z ⊂ OK is contained in an iterated monoidal transformR ofRalongvP|K such that the elements z∈Zhave prime factorizations of the form (6)—[3, (5.2.3)]. Thus (NC) is satisfied for every finite setZ⊂ OP.

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D. Fu proved [11, Proposition 3.5] that (NC) and (V) are satisfied for every finite setZ⊂ OP, provided thatRis essentially of finite type over an algebraically closed fieldkandKP =k.

5. The general case

In this section we provide the proofs for Theorems 1.1 and 1.2. The ingredients are already given in Sections 3 and 4. We need one more fact— an ascend property forR-uniformizability—

to bring them together.

Consider a finite extension(F|E, P)of valued fields such thatF is contained in the absolute inertia field of (E, P|E). It is then well known that the extension OF|OE is local-étale [19, Chapter X, Theorem 1]:OF=Aq for an étaleOE-algebraAandq=A∩ MF. According to [19, Chapter V, Theorem 1] we can assume thatAis standard-étale, i.e.,

A=OE[x]_g(x), (13)

where OE[x] =OE[X]/fOE[X] with a monic polynomial f ∈ OE[X]. Furthermore, g∈ OE[X] is chosen such that the image of the derivative f under the natural morphism φ:OE[X]→Ais a unit.

Claim. – In the definition ofAwe can assumef to be prime.

The Lemma of Gauß allows to factorize f as_r

i=1p^ν_iⁱ with pairwise distinct, monic prime polynomials pi ∈ OE[X]. Among them there is a unique pj with φ(pj) =pj(x) = 0. We consider the natural surjectionψ:A→(OE[X]/pjOE[X])_g(x). The equationψ(^h+fO_g(x)^Es^[X]) = 0 implies h∈pjOE[X]. Since in A we have pj(x) = 0, there exists t∈N such that g^tpj∈ fOE[X], thusg^th∈fOE[X]and hence^h+fO_g(x)^Es^[X] = 0. Finally, for somep∈ OE[X]we have φ(f) =φ((ppj)) =φ(pp_j)∈A^×, which implies thatφ(p_j)∈A^×. This proves thatψis an isomorphism and hence the claim.

Next, we fix a set of structural constants determiningAuniquely, which we shall use to define an étale algebra over a subring ofOE. Leth∈ OE[X]be chosen such thatf(x)⁻¹= _g(x)^h(x)s, s∈N, and letC(f, g, h)⊂ OEbe the set of coefficients of the polynomialsf, gandh.

LetZ⊂ OF be a finite set and split it asZ= (Z∩ O^×_F)∪(Z∩ MF). SincevP is unramified in the extensionF|Ewe can write everyζ∈Z∩ MF in the formζ=uζζwithuζ∈ O^×_F and ζ∈ ME; letZ:={ζ|ζ∈Z∩ MF}.

For eachξ∈Z^×:= (Z∩ O^×_F)∪ {u_ζ|ζ∈Z∩ MF}we choose a representation ξ=a(x)

b(x)g(x)^k, a, b∈ OE[X], b(x)∈/q, k∈Z. (14)

Finally we define the finite set C(Z^×)⊂ OE as the collection of the coefficients of the polynomialsa, bappearing in the representations (14) for the elementsξ.

LEMMA 5.1. – Let(F|E, P)be a finite extension of valued fields such thatF is contained in the absolute inertia fieldEⁱ of(E, P|E). LetZ⊂ OF be a finite set and letR⊆ OE be a regular local ring with maximal idealM=ME∩R.

Let A⊆ OF be the étale OE-algebra (13) and assume that there exists a set of re- presentations(14)for the elementsξ∈Z^×, such that the pair(P|E, C(f, g, h)∪C(Z^×)∪Z) isR-uniformizable. Then the pair(P, Z)isR-uniformizable, too.

Moreover, one can findR-modelsX ofEandY ofFand a morphismπ:U→V, whereUis an affine open neighborhood of the centeryofP onY, andV is an affine open neighborhood

of the centerxofP|EonX such that:x, yare regular points on the respective model,πy=x, and the extensionOY,y|OX,xis local-étale.

In particular,dimOY,y = dimOX,x holds. Furthermore, if R=K is a field and OX,x is smooth overK, thenOY,yis smooth overK.

If for some regular system of parameters(a₁, . . . , a_d)of OX,x every ζ ∈Z is an OX,x- monomial in {a₁, . . . , a_d}, then every ζ∈Z is an OY,y-monomial in the regular system of parameters(a1, . . . , ad)ofOY,y.

Proof. –By assumption there exists a finitely generatedR-algebraB⊆ OE such that BqB, qB := ME ∩B, is regular and contains the finite set C(f, g, h)∪C(Z^×)∪ Z. Define C:=B[x]_g(x)⊆A, where x is the element appearing in the definition (13) of A. We have FracC=F, and theBqB-algebra CqB =BqB[x]_g(x) is standard-étale: this is a consequence of the construction ofCqB once we have verified thatBqB[x]∼=BqB[X]/f BqB[X]. So assume h(x) = 0for someh∈BqB[X]; sincef is the minimal polynomial ofxoverEwe geth=f h, h∈E[X]. NowBqB is integrally closed inEandf is monic, thus the Lemma of Gauß yields h∈BqB[X].

Since regularity ascends in étale extensions, the domain CqB and hence also CqC, where qC:=MF ∩C, are regular. Moreover, we have Z ⊂CqC by construction. Étale extension preserves the Krull dimension and smoothness. Moreover, if(a1, . . . , ad)is a regular parameter system ofBqB, then it is a regular parameter system forCqC, too. These facts yield the remaining assertions. 2

We are now prepared to prove our main results.

Proof of Theorems 1.1 and 1.2. –One starts by choosing a transcendence basisT⊂F with the properties described in Theorem 3.4; in particular, the valued field(F, P)lies in the absolute inertia field of(K(T), P|K(T)).

According to Lemma 5.1, the pair(P, Z)isR-uniformizable for a given finite set Z⊂ OP

once the pair (P|K(T), Z) is R-uniformizable for a certain finite setZ⊂ OK(T) derived fromZand the extensionOP|OK(T)—see the discussion preceding Lemma 5.1. Moreover, the elementsζ∈Zpossess the required factorization property once the elements of a certain subset Z⊆Zpossess this factorization property.

The valued rational function field(K(T), P|K(T))satisfies the requirements (T) of Section 4.

Points (1) and (3) of the discussion at the end of Section 4 show that the pair(R, Z)fulfills the requirements (NC) and (V) foreveryfinite setZ⊂ OK(T). An application of Theorem 4.1 thus yieldsR-uniformizability of(P|K(T), Z)and the factorization property for the elements ζ∈Zfor an arbitrary finite setZ.

In the case ofR=K, the pair (P|K(T), Z)is K-uniformizable on the affine space X= A^ρ+τ_K ,ρ= dim_Q(v_PF⊗Q), due to Theorem 4.1. Moreover, the centerx∈XofP|K(T)satisfies dimOX,x=ρ. Lemma 5.1 thus gives the remaining assertions of Theorem 1.1.

In the case ofR=K, it remains to verify the dimension statements of Theorem 1.2. They are direct consequences of the remarks (1) and (3) made at the end of Section 4 combined with Lemma 5.1. 2

As an immediate corollary of Theorem 1.1 we get:

COROLLARY 5.2. – Let the situation be as in Theorem 1.1, except for the separability of F P|K. Then there exists a finite purely inseparable extensionL|K such that(P , Z), where P is the unique extension ofP to the constant extensionF.LofF, is L-uniformizable. All other assertions of Theorem1.1remain valid overL.

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