Strong linkage for function fields of surfaces
SSTRONG LINKAGE FOR FUNCTION FIELDS OF SURFACES
KARIM JOHANNES BECHER AND PARUL GUPTA
Abstract.
Over a global field any finite number of central simple algebras ofexponent dividing m is split by a common cyclic field extension of degree m .We show that the same property holds for function fields of two-dimensionalexcellent schemes over a henselian local domain of dimension one or two withalgebraically closed residue field. Classification (MSC 2010): 13J15, 16K20, 16S35, 19C30, 19D45Keywords: cyclic algebra, splitting field, period-index problem, Brauer group,Milnor K -theory, symbol, common slot, linkage, henselian ring, arithmetic sur-face, pseudo-algebraically closed field, quasi-finite field Introduction
Let F be a field and m a positive integer. We say that F is strongly linked indegree m if any finite number of central simple F -algebras of exponent dividing m is split by a cyclic field extension of degree dividing m . It follows from classfield theory that global fields have this property. In [20], H. Lenstra showed asimilar statement for elements of the second K -group of a global field.In this article we show that F satisfies strong linkage in degree m in the fol-lowing three cases:(1) F is the fraction field of a two-dimensional excellent henselian local do-main with algebraically closed residue field of characteristic not dividing m (e.g. a finite extension of C pp X, Y qq ).(2) F is the function field of a curve over the fraction field of an excellenthenselian discrete valuation ring with algebraically closed residue field ofcharacteristic not dividing m (e.g. a finite extension of C pp X qqp Y q ).(3) F is the function field of a curve over a perfect pseudo-algebraically closedfield of characteristic not dividing m .Let Br m p F q denote the m -torsion part of the Brauer group of F . Just as forglobal fields, in the cases p q´p q the period-index problem has a positive answer,that is, any element of Br m p F q is given by a central simple F -algebra of degree m . Date : 08.01.2020.This work was supported by the FWO Odysseus Programme (project
Explicit Methods inQuadratic Form Theory ), funded by the Fonds Wetenschappelijk Onderzoek – Vlaanderen, andby the Bijzonder Onderzoeksfonds (BOF), Universiteit Antwerpen, (project BOF-DOCPRO4,2865,
Nieuwe methoden in de arithmetiek van lichamen en de theorie van kwadratische vormen ). a r X i v : . [ m a t h . K T ] J a n KARIM JOHANNES BECHER AND PARUL GUPTA
This was shown in [15] and [14] for the cases p q and p q , respectively, and for thecase p q this follows from [12, Theorem 3.4]. The method in all these cases is tostudy the ramification of elements of Br m p F q with respect to discrete valuationson F and to construct a cyclic field extension that splits the ramification. Thismethod relies on the fact that the unramified part of the Brauer group is trivialin these cases. We apply the same technique to show strong linkage in degree m .If F is strongly linked in degree m then it follows that any central F -divisionalgebra of exponent m is cyclic of index m . In particular, strong linkage in alldegrees implies a positive answer to the period-index problem. The field of it-erated Laurent series C pp X qqpp Y qqpp Z qq gives an example of a field where stronglinkage fails in all degrees m ě while the period-index problem has a positiveanswer. This strange example is disposed off by restricting to fields of cohomo-logical dimension , which still covers global fields as well as the fields in thecases p q ´ p q .It is more difficult to show that there exist also fields of cohomological dimen-sion where the period-index problem has a positive answer but where stronglinkage fails in some degree m . A. Chapman and J.-P. Tignol showed very re-cently that F “ C p X, Y q , the rational function field in two variables over thecomplex numbers, does not satisfy strong linkage for m “ . More specifically,it is shown in [6] that no quadratic field extension of F splits all the four F -quaternion algebras p X, Y q F , p X, Y ` q F , p X ` , Y q F , p X, XY ` q F . In view of this result it seems reasonable to expect that strong linkage fails inany degree m ě over this field, and more generally over the function field ofany algebraic surface over C .The result of [6] is interesting in view of the fact that the field C p X, Y q hasthe C -property (in terms of Tsen-Lang theory, see [28, Chapter 5]) and that theperiod-index problem has a positive answer for this field, by [9].The structure of this article is as follows. In Section 2, we will recall the setupof Milnor K -theory and formulate the problem of strong linkage in these terms.This seems the most natural context for this problem and for some of the toolsthat we need, and it further gives an opening to studying the analogous problemfor higher K -groups. In this article, however, we focus on the second K -groupof a field and its quotient modulo m , which in characteristic not dividing m isnaturally isomorphic to Br m p F q , by the Merkurjev-Suslin Theorem. We recall thisrelation in Section 3, where we further collect the relevant results on the vanishingof the unramified part of the Brauer group of certain function fields. In Section 4,a general strategy is provided for finding a common slot for a finite set of symbolsin K -theory modulo m (or of the corresponding cyclic algebras). In Section 5,we treat case p q . For treating the cases p q and p q we need some results fromalgebraic geometry. In these cases we need to achieve that the ramification of theset of symbols is contained in a normal crossing divisor on some regular model TRONG LINKAGE FOR FUNCTION FIELDS OF SURFACES 3 of the field F . Sections 6 and 7 revisit some results related to divisors on regularsurfaces over henselian local domains. In Section 8 we show strong linkage in thecases p q and p q . In this situation one can obtain a candidate for this slot as afunction passing along all components of the support of the ramification divisorwith multiplicity . Finally, in Section 9 we make the observation that a discretevaluation on a quasi-finite field is henselian and the residue field is algebraicallyclosed of characteristic zero. As a consequence we obtain that algebraic functionfields over a discretely valued quasi-finite field fall under case p q , and thereforethey satisfy strong linkage in all degrees.The techniques that we use to obtain our results are not new. They have beendeveloped and applied to show structure properties for central simple algebrasand quadratic forms over function fields of certain surfaces, by Ford and Saltman([15], [14], [31], [32]) and further by Colliot-Thélène, Ojanguren, Parimala andSuresh ([26], [7], [27]). Here, we are focussing on function fields of cohomologicaldimension and hence do not cover the case of function fields of p -adic curves,which were in the focus of several of the articles just mentioned. We shift thefocus from the study of a single object like a central simple algebra, an elementof a Milnor K -group or of a Galois cohomology group to the problem of havinga simultaneous representation of an arbitrary finite number of such objects. Itis probably clear to experts that the methods carry over to settle that problem,leading to most of our results. However, to achieve a good presentation, we in-clude an exposition of these tools, and we further adapt them to the set-up ofMilnor K -theory as far as possible. For the crucial fact that splitting ramifica-tion splits the elements in question (which also holds in some other situationsthan those considered here) we rely entirely the existing vanishing results for theunramified part of certain cohomology groups. On the other hand, the compu-tations on ramification become straightforward in the setup of generators andrelations, and we underline this by including these computations.This article is based on Parul Gupta’s PhD-thesis prepared under the supervi-sion of Karim Johannes Becher ( Universiteit Antwerpen ) and Arno Fehm (
Tech-nische Universität Dresden ) in the framework of a joint PhD at
UniversiteitAntwerpen and
Universität Konstanz .2. K -groups and ramification We recall some basic terminology and facts from Milnor K -theory. Our mainreferences are Milnor’s seminal article [25] and [17, Chapter 7].Let F be a field. Let F ˆ denote the multiplicative group of F . For n, m P N ,let K p m q n F be the n th Milnor K -group of F modulo m , that is, the additive abeliangroup defined by generators and relations as follows: K p m q n F is generated by so-called symbols t a , . . . , a n u with parameters a , . . . , a n P F ˆ , and the defining KARIM JOHANNES BECHER AND PARUL GUPTA relations are that the map p F ˆ q n Ñ K p m q n F, p a , . . . , a n q ÞÑ t a , . . . , a n u is multi-linear, that t a , . . . , a n u “ holds for any a , . . . , a n P F ˆ with a i ` a i ` “ forsome index i ă n , and that m ¨ K p m q n F “ . Note that K p q n F is the full Milnor K -group, usually denoted by K n F , and that K p m q n F “ K n F { m K n F . For a fieldextension F { F there is a natural homomorphism K p m q n F Ñ K p m q n F which mapsthe symbol t a , . . . , a n u given by parameters a , . . . , a n P F ˆ to the symbol withthe same notation in K p m q n F ; this map is generally not injective, because symbolsfrom K p m q n F are subject to additional relations in K p m q n F . Given a field exten-sion F { F and α P K p m q n F , we denote by α F the image of α under the naturalhomomorphism K p m q n F Ñ K p m q n F .We now fix a positive integer m . The direct sum of Z -modules À n P N K p m q n F hasa natural multiplication which makes it into a graded Z -algebra; the multiplica-tion is determined by the rule that, for n, n P N and a , . . . , a n , b , . . . , b n P F ˆ ,we have t a , . . . , a n u ¨ t b , . . . , b n u “ t a , . . . , a n , b , . . . , b n u . Let α P K p m q n F . A symbol σ P K p m q r F is called a factor of α if α “ σ ¨ γ for someelement γ P K p m q n ´ r F . For the case where r “ , we also say that a P F ˆ is a slotof α if the symbol t a u is a factor of α .Given a symbol σ P K p m q r F , we denote by σ ¨ K p m q n ´ r F the subgroup of K p m q n F consisting of the elements having σ as a factor, that is, the image of the map K p m q n ´ r F Ñ K p m q n F given by multiplication from the left by σ . We say that K p m q n F is strongly linked if for every finite subset S of K p m q n F there exists a symbol σ P K p m q n ´ F such that S Ď σ ¨ K p m q F . Trivially, if K p m q n F “ then K p m q n F isstrongly linked. It is also obvious that, if K p m q n F is strongly linked, then everyelement of K p m q n F is a symbol.In this article we focus on the case n “ , hence on the problem of stronglinkage for K p m q F . When F is a finite field, then K p q F “ (see [17, Example7.1.3]). In [20], H. Lenstra showed that K p q F is strongly linked in the case where F is a global field. Apart from these fields and their algebraic extensions, we donot expect strong linkage to hold for K p q F in any other cases, and we thereforeconsider the problem of strong linkage of K p m q F in the sequel for m ą .A crucial tool in the context of Milnor K -theory is given by residue maps withrespect to discrete valuations. Our main reference for valuation theory is [13].By a Z -valuation we mean a valuation with value group Z .Let v be a Z -valuation on F . We denote by O v , m v and κ v the correspondingvaluation ring, its maximal ideal and its residue field, respectively. An element π P F ˆ is called a uniformizer of v if v p π q “ . Given a P O v we write ¯ a for theimage of a under the residue map O v Ñ κ v , a ÞÑ a ` m v . We recall the definitionof the ramification homorphism with respect to v , which is denoted by B v . TRONG LINKAGE FOR FUNCTION FIELDS OF SURFACES 5
Proposition.
For each n ě there exists a unique homomorphism B v : K p m q n F Ñ K p m q n ´ κ v satisfying B v pt x, u , . . . u n uq “ v p x q ¨ t ¯ u , . . . , ¯ u n u in K p m q n ´ κ v for all x P F ˆ and for all u , . . . u n P O v ˆ .Proof: See [17, Proposition 7.1.4] for the case m “ ; this implies the statementfor any m P N . l For n “ the homomorphism B v : K p m q F Ñ K p m q κ v from Proposition 2.1 isgiven on symbols by the rule t a, b u ÞÑ p´ q v p a q v p b q t a ´ v p b q b v p a q u for a, b P F ˆ . Recall that the valuation v on F is henselian if it extends uniquely to everyfinite field extension of F , or equivalently, if for any polynomial f P O v r X s anysimple root of f in κ v is the residue of a root of f in O v (see [13, Theorem 4.1.3]).2.2. Proposition.
Assume that v is henselian, char p κ v q does not divide m and K p m q n κ v “ . Then B v : K p m q n F Ñ K p m q n ´ κ v is an isomorphism. Furthermore, anyuniformizer of v is a slot of every element of K p m q n F .Proof: This follows from the statement and the proof given in [17, Corollary7.1.10] with the assumption that v is complete; the proof given there only usesthat v is henselian. l Let m and n be positive integers with m ě and let α P K p m q n F . We call B v p α q the ramification of α at v . We call α unramified at v if B v p α q “ in K p m q n ´ κ v ,otherwise we say that α is ramified at v . We say that a finite field extension L { F splits the ramification of α at v if for every Z -valuation w on L with O v Ď O w we have B w p α L q “ .2.3. Lemma.
Let f P F ˆ be such that v p f q is coprime to m and let L { F be afinite field extension L { F with f P L ˆ m . Then L { F splits the ramification at v of all elements of K p m q n F .Proof: Consider a Z -valuation w of L such that O w X F “ O v . We have that r w p L ˆ q : w p F ˆ qs ď r L : F s and set e “ r w p L ˆ q : w p F ˆ qs . By [17, Remarks 7.1.6(2)] we have the commutative diagram K p m q n L K p m q n ´ κ w K p m q n F K p m q n ´ κ v B w B v ¨ e KARIM JOHANNES BECHER AND PARUL GUPTA where the vertical arrows refer to the natural homomorphisms multiplied by and e , respectively. But it follows from the hypothesis on v p f q that m divides e .Hence B w p α L q “ for every α P K p m q n F . l Symbol algebras and linkage
Let F be a field and m a positive integer. In this section, we connect stronglinkage of K p m q F to strong linkage of F in degree m as we defined it in theintroduction, namely in terms of central simple algebras. This is done by usingthe Merkurjev-Suslin Theorem, which identifies the Brauer group with the secondMilnor K -group of a field. Our standard references for central simple algebrasand the Brauer group of a field are [17], [33, Chapter 8] and [11].The Brauer group of F and its m -torsion part are denoted by Br p F q and by Br m p F q , respectively. For a central simple F -algebra A , we denote by r A s itsclass in Br p F q . Let A be a central simple F -algebra. Then dim F p A q “ d forsome positive integer d , called the degree of A and denoted by deg p A q , and wesay that A is split if A » M d p F q .Let L { F be a field extension. Then A b F L is a central simple L -algebra ofthe same degree as A , which we denote by A L . This gives rise to a naturalhomomorphism Br p F q Ñ Br p L q defined by r A s ÞÑ r A b F L s , which restricts to ahomomorphism Br m p F q Ñ Br m p L q . We say that A splits over L or that L splits A if A L is a split L -algebra.We assume in the sequel that char p F q does not divide m and we denote by ω a primitive m th root of unity over F , contained in an algebraic extension of F .Assume for now that ω P F . Then, for a, b P F ˆ , we denote by p a, b q F,ω the F -algebra generated by two elements x, y subject to the relations x m “ a, y m “ b and yx “ ωxy . Such algebras are called symbol algebras . By [11, Chapter 11, Theorem 1], for a, b P F ˆ the F -algebra p a, b q F,ω is central simple of degree m .3.1. Proposition.
Assume that ω P F . Let A be a central simple F -algebra ofdegree m and let x P A be such that x m P F ˆ and F r x s is a field which is maximalas a commutative subring of A . Then A » p a, b q F,ω for a “ x m and some b P F ˆ .Proof: This follows from the proof of [33, Theorem 8.12.2]. l Lemma.
Assume that ω P F . Let a, b P F ˆ . For any positive integer s , thesymbol algebras p a, b q F,ω s and p a, b s q F,ω are Brauer equivalent.Proof:
See [11, Chapter 11, Lemma 5 and 6]. l For a P F ˆ we denote by F p m ? a q the splitting field of X m ´ a over F . Notethat ω P F p m ? a q for any a P F ˆ . Recall that, if ω P F , then every cyclic fieldextension of F of degree m is of the form F p m ? a q for some a P F ˆ .For a finite field extension F { F we denote by N F { F : F Ñ F the norm map. TRONG LINKAGE FOR FUNCTION FIELDS OF SURFACES 7
Proposition.
Assume that ω P F . Let A be a central simple F -algebra ofdegree m and let a P F ˆ . Then A F p m ? a q is split if and only if A » p a, b q F,ω forsome b P F ˆ .Proof: Set L “ F p m ? a q . For b P L ˆ we have that p a, b q L,ω » p , b q L,ω » M m p L q .Hence, if A » p a, b q F,ω for some b P F ˆ , then A L is split.Assume now that A L is split. Then there exists a central simple F -algebra B containing L as a maximal commutative subring and such that r A s “ r B s in Br m p F q . Set m “ r L : F s . Then deg p B q “ m . Note further that m divides m . Fix an element z P L with z m “ a . Note that N L { F p z q “ ω r ¨ z m for some r P N and hence z m P F ˆ . We set a “ z m and s “ mm and obtain that a s “ a and L “ F p z q “ F p m ? a q . By Proposition 3.1, there exists b P F ˆ such that B » p a , b q F,ω s . By Lemma 3.2, p a , b q F,ω s is Brauer equivalent to p a, b q F,ω . Thus A is Brauer equivalent to p a, b q F,ω . Since both central simple F -algebras are ofdegree m , we conclude that A » p a, b q F,ω . l The link between the Brauer group and Milnor K -theory is given by the fol-lowing deep theorem.3.4. Theorem (Merkurjev-Suslin) . If ω P F , then the rule t a, b u ÞÑ rp a, b q F,ω s for a, b P F ˆ determines an isomorphism Φ ω : K p m q F Ñ Br m p F q . Proof:
See [17, Theorem 2.5.7]. l Theorem.
Assume that r F p ω q : F s is coprime to m and let a P F ˆ . Thekernel of the natural homomorphism K p m q F Ñ K p m q F p m ? a q consists of the symbols t a, b u with b P F ˆ .Proof: For b P F ˆ we clearly have that t a, b u P ker p K p m q F Ñ K p m q F p m ? a qq .Conversely, consider α P ker p K p m q F Ñ K p m q F p m ? a qq . Set F “ F p ω q . Since α F p ω q “ and F p m ? a q Ď F p m ? a q , it follows that Φ ω p α F p m ? a q q “ . By Propo-sition 3.3, we obtain that Φ ω p α F q “ rp a, c q F ,ω s for some c P F . Since Φ ω isinjective, we have that α F “ t a, c u in K p m q F . By [17, Chapter 7, Remark 7.3.1],it follows that r F : F s ¨ α “ t a, N F { F p c qu . Since r F : F s is coprime to m , thereexists r P Z with r ¨ r F : F s ” m Z . Then α “ t a, N F { F p c q r u in K p m q F . l Corollary.
Assume that r F p ω q : F s is coprime to m . Then K p m q F is stronglylinked if and only if any finite subgroup of K p m q F is contained in the kernel of thenatural map K p m q F Ñ K p m q F p m ? a q for some a P F ˆ .Proof: This follows directly from the definition of strong linkage for K p m q F to-gether with Theorem 3.5. l KARIM JOHANNES BECHER AND PARUL GUPTA
Corollary.
Assume that ω P F . Then K p m q F is strongly linked if and only ifif every finite number of central simple F -algebras of degree m is split by a cyclicfield extension of F of degree dividing m .Proof: The statement follows from Proposition 3.3, Theorem 3.4 and Corol-lary 3.6. l The unramified Brauer group
In this section we look at the unramified Brauer group of a field and recallsome statements from the literature. The purpose is to reformulate results onthe unramified Brauer group in the way we will use them later.Let F be a field and let Ω F be the set of all Z -valuations on F . Let m P N bea positive integer. For v P Ω F we have v p m q “ if and only if char p κ v q does notdivide m . We set Ω p m q F “ t v P Ω F | v p m q “ u . Let v P Ω p m q F . There is a well-known ramification homomorphism (see [31,Chapter 10, page 68]) B v : Br m p F q Ñ Hom p Gal p κ v q , Z { m Z q , where Gal p κ v q denotes the absolute Galois group of κ v , which is endowed withthe profinite topology, and where Hom p Gal p κ v q , Z { m Z q is the set of all continuoushomomorphisms Gal p κ v q Ñ Z { m Z . Let F v denote the completion of F withrespect to v and let ˆ v denote the unique unramified extension of v to F v . Themap B v factors as Br m p F q Ñ Br m p F v q B ˆ v Ñ Hom p Gal p κ v q , Z { m Z q , where the first map is the scalar extension homomorphism. In particular, ker p Br m p F q Ñ Br m p F v qq Ď ker pB v q . For the definition of Br p O v q , the Azumaya-Brauer group of the discrete valua-tion ring O v , we refer to [31, Chapter 3]. Here we only need this group to link twoimportant statements in the literature. The following statement characterizes the m -torsion part Br m p O v q of Br p O v q as the kernel of the ramification map B v .4.1. Theorem (Auslander-Brumer) . For v P Ω p m q F the following natural sequenceis exact: Ñ Br m p O v q Ñ Br m p F q B v Ñ Hom p Gal p κ v q , Z { m Z q Ñ Proof:
The statement follows from [3, page 289]. l We say that α P Br m p F q is unramified at v if B v p α q “ . Consider now anarbitrary subset Ω Ď Ω p m q F . We set Br nrm p F, Ω q “ t α P Br m p F q | B v p α q “ for all v P Ω u . TRONG LINKAGE FOR FUNCTION FIELDS OF SURFACES 9
We further abbreviate Br nrm p F q “ Br nrm p F, Ω p m q F q . Obviously, we have that Br nrm p F q Ď Br nrm p F, Ω q for any Ω Ď Ω p m q F .4.2. Proposition.
Let Ω Ď Ω p m q F be such that Br m p κ v q “ for all v P Ω . Then Br nrm p F, Ω q “ ker ˜ Br m p F q Ñ ź v P Ω Br m p F v q ¸ . Proof:
Consider v P Ω . By Theorem 4.1 and [4, Corollary 8.5] we have that ker pB ˆ v q “ Br m p O ˆ v q » Br m p κ v q “ . Hence ker pB v q “ ker p Br m p F q Ñ Br m p F v qq .Using this for all v P Ω , we obtain the statement. l Proposition.
Assume that ω P F . Let Ω Ď Ω p m q F be such that Br nrm p F, Ω q “ .Then the homomorphism B Ω : K p m q F Ñ à v P Ω K p m q κ v , α ÞÑ pB v p α qq v P Ω is injective.Proof: We need to verify that B Ω has trivial kernel. Consider α P K p m q F such that B v p α q “ for all v P Ω . By the injectivity of the map Φ ω : K p m q F Ñ Br m p F q fromTheorem 3.4, in order to show that α “ it suffices to verify that Φ ω p α q “ ,and since Br nrm p F, Ω q “ it is enough to check that B v p Φ ω p α qq “ for all v P Ω .Consider v P Ω and fix π P F ˆ with v p π q “ . In view of the bilinearity of thesymbol map F ˆ ˆ F ˆ Ñ K p m q F , we can write α “ ř ri “ t a i , b i u ` t π, c u with r P N and a , b , . . . , a r , b r , c P O v ˆ . Then t c u “ B v p α q “ in K p m q κ v , whereby c P κ ˆ mv .As v P Ω Ď Ω p m q we have v p m q “ , whereby char p κ v q does not divide m . Since F v is complete, we conclude by Hensel’s Lemma that c P F ˆ mv . Therefore t π, c u “ in K p m q F v and Φ ω p α q “ Â ri “ rp a i , b i q F v ,ω s P Br m p O ˆ v q “ ker pB ˆ v q . Since B v factorsover B ˆ v , we obtain that B v p Φ ω p α qq “ . l Function fields of curves with trivial Brauer group
Let E be a field. By an algebraic function field over E we mean a finitelygenerated field extension F { E of transcendence degree . In this section weoutline a strategy to show strong linkage for K p m q F for an algebraic function field F { E and for m P N not divisble by char p E q . This strategy will be applicablewhen the base field E satisfies a very strong property, which is given for examplewhen E is a finite field.For a field extension F { E we set Ω F { E “ t v P Ω F | v p E ˆ q “ u . We fix a positive integer m which is not a multiple of char p E q and denote by ω aprimitive m th root of unity contained in an algebraic extension of E .5.1. Lemma.
Assume that r E p ω q : E s is coprime to m . Let F { E be an algebraicfunction field. Assume that Br nrm p F, Ω F { E q “ and Br nrm p F , Ω F { E q “ for everyfinite field extension F { F p ω q of degree m . Then the following hold: p a q If α P K p m q F and f P F ˆ is such that v p f q is coprime to m for every v P Ω F { E with B v p α q ‰ , then f is a slot of α . p b q K p m q F is strongly linked.Proof: Set Ω “ Ω F { E . p a q Let f P F ˆ and set S “ t v P Ω | v p f q coprime to m u . Let α P K p m q F besuch that B v p α q “ for all v P Ω z S . If S “ H then α P Br nrm p F, Ω q “ , andthus α “ t f, u . Suppose now that S ‰ H and set F “ F p m ? f q . Note that ω P F . We claim that r F : F p ω qs “ m . We choose any v P S and use that v p f q is coprime to m and v is unramified in F p ω q to conclude that r F : F p ω qs “ m .The hypothesis therefore yields that Br nrm p F , Ω F { E q “ . In order to show that f is a slot of α , it suffices by Theorem 3.5 to show that α F “ , and since Br nrm p F , Ω F { E q “ it is further sufficient by Proposition 4.3 to show that α F isunramified with respect to Ω “ Ω F { E .Consider w P Ω F { E . Then O w X F “ O v for a unique v P Ω F { E . If B v p α q “ ,then clearly B w p α F q “ . If B v p α q ‰ , then v P S , whereby v p f q is coprime to m , and as f P F m , it follows by Lemma 2.3 that B w p α F q “ . p b q Consider an arbitrary finite subset S Ď K p m q F . Note that, for any g P F ˆ ,there are only finitely many valuations v P Ω such that v p g q ‰ . Hence, thesubset S “ t v P Ω | B v p α q ‰ for some α P S u of Ω is finite. By the WeakApproximation Theorem [13, Theorem 2.4.1], there exists f P F ˆ such that v p f q “ for all v P S . Then f is a slot of every element of S , by p a q . This showsthat K p m q F is strongly linked. l Example.
Let E be a finite field which contains a primitive m th root ofunity. Let F { E be an algebraic function field. It follows by the Albert-Brauer-Hasse-Noether Theorem (see [29, Theorem 32.11]) and by Proposition 4.2 that Br nrm p F q “ . Hence it follows by Lemma 5.1 that K p m q F is strongly linked.We now apply the same method to show strong linkage for algebraic functionfields over PAC-fields. Recall that a field E is called pseudo algebraically closed or a PAC-field if every absolutely irreducible algebraic variety over E has an E -rational point. For a general discussion of these fields we refer to [16, Chapter11]. Note that, if E is a PAC-field, then by [16, Corollary 11.5.5] the value groupof any valuation on E is divisible, so in particular Ω E “ H .5.3. Theorem (Efrat) . Let E be a PAC-field such that char p E q does not divide m .Then Br nrm p F q “ for any algebraic function field F { E . TRONG LINKAGE FOR FUNCTION FIELDS OF SURFACES 11
Proof:
Let F { E be an algebraic function field. By [12, Corollary 3.2] and [12,Remark 3.5(a)], the natural homomorphism Br m p F q Ñ ‘ v P Ω F Br m p F v q is injective.Since E is a PAC-field, we have Ω E “ H and therefore Ω F “ Ω F { E . For v P Ω F ,then the residue field κ v is a finite extension of E and hence itself a PAC-field,whereby Br m p κ v q “ . Furthermore, the hypothesis on E yields that Ω F “ Ω p m q F .Hence we obtain by Proposition 4.2 that Br nrm p F q “ . l Theorem.
Let E be a PAC-field such that a finite extension of E of degreecoprime to m contains a primitive m th root of unity. Then K p m q F is stronglylinked for every algebraic function field F { E .Proof: The hypotheses on E imply that Ω E “ H and Ω F “ Ω F { E “ Ω p m q F for every algebraic function field F { E . Hence, by Theorem 5.3 we have that Br nrm p F , Ω F { E q “ Br nrm p F q “ for every algebraic function field F { E . Thereforethe statement follows by Lemma 5.1. l Ramification on two-dimensional regular local rings
In this section we assemble the notions and results from commutative algebraand algebraic geometry which will be essential in the proofs of our main results.A regular local ring is a commutative noetherian local ring R whose maximalideal is generated by d elements where d is the Krull dimension of R . By [24,Theorem 19.3], the localisation of a regular local ring at a prime ideal is regular.We call a commutative ring R regular if every localisation of R at a prime idealis a regular local ring.6.1. Theorem (Auslander-Buchsbaum) . Every regular semi-local ring is a uniquefactorization domain.Proof:
For regular local rings this is proven in [2]; see also [23, Theorem 20.3].The extension to regular semi-local rings is given in [32, Lemma 0.12]. l A domain is called normal if it is integrally closed inside its field of fractions.Unique factorization domains are normal. A one-dimensional noetherian localdomain is normal if and only if it is a discrete valuation ring (see [1, Proposition9.2]).For a regular local ring R with maximal ideal m , a set of generators of m ofcardinality equal to the dimension of R is called a parameter system of R , andan element π P R is called a parameter if it occurs in a parameter system of R .It is easy to see that any parameter of a regular local ring is a prime element.Let R be a two-dimensional regular local ring. We denote by P R the set ofall height-one prime ideals of R . Let p P P R . Then R { p is a one-dimensionalnoetherian local domain. Note that κ p “ R p { p R p , which is the fraction fieldof R { p . For x P R z p let l R p R {p p ` Rx qq denote the length of the R -module R {p p ` Rx q . By [22, Lemma 7.1.26], the rule p x ` p q ÞÑ l R p R {p p ` Rx qq for x P R z p defines a group homomorphism mult p : κ ˆ p Ñ Z . Fix now a positive integer m which is invertible in R . For p P P R , compositionof mult p with the residue map Z Ñ Z { m Z determines a homomorphism r p : K p m q κ p Ñ Z { m Z , t x u ÞÑ mult p p x q ` m Z . Compiling these maps for all p P P R , we obtain a group homomorphism r R : à p P P R K p m q κ p Ñ Z { m Z , p a p q p P P R ÞÑ ÿ p P P R r p p a p q . On the other hand, any p P P R gives rise to the ramification homomorphism B p “ B v p : K p m q F Ñ K p m q κ p given by the Z -valuation v p . Compilation of the maps B p for all p P P R yields a homomorphism B R “ à p P P R B p : K p m q F Ñ à p P P R K p m q κ p . The following reciprocity law is formulated in [30, Lemma 1.1] in terms of Galoiscohomology, but the proof essentially uses the defining relations for K p m q F . Westate it in terms of K -theory and include the proof.6.2. Proposition (Saltman) . Let R be a two-dimensional regular local ring with m P R ˆ . The homomorphism r R ˝ B R : K p m q F Ñ Z { m Z is trivial.Proof: We set
B “ B R and r “ r R . As R is a unique factorization domain, K p m q F is generated by symbols of the following three types: t a, b u , t π, b u and t π, δ u ,where a, b P R ˆ and π, δ are non-associated prime elements of R . Thus it sufficesto show that p r ˝ Bqp α q “ , where α is a symbol of one of these three types.Case 1: If α “ t a, b u with a, b P R ˆ , then Bp α q “ and hence p r ˝ Bqp α q “ .Case 2: Let α “ t π, b u with b P R ˆ and a prime element π of R . Then B p p α q “ in K p m q κ p for all p P P R zt Rπ u and B Rπ p α q “ t b u in K p m q κ Rπ . As πR ` bR “ R ,we have l R p R {p Rπ ` Rb qq “ and thus r Rπ pt b uq “ . Hence p r ˝ Bqp α q “ .Case 3: Let α “ t π, δ u with two non-associated prime elements π and δ of R .Then B p p α q “ for all p P P R zt Rπ, Rδ u . We further have p r ˝ B Rπ qp α q “ p r Rπ ˝ B Rπ qp α q “ r Rπ p ¯ δ q “ l R p R {p Rπ ` Rδ qq and p r ˝ B Rδ qp α q “ p r Rδ ˝ B Rδ qp α q “ r Rδ p ¯ π ´ q “ ´ l R p R {p Rπ ` Rδ qq . Thus p r ˝ Bqp α q “ in this case, too. l Corollary (Saltman) . Let R be a two-dimensional regular local ring with m P R ˆ . Let α P K p m q F and let π be a parameter of R and p “ Rπ . Assume that B q p α q “ for all q P P R zt p u . Then B p p α q “ t ¯ u u in K p m q κ p for some u P R ˆ . TRONG LINKAGE FOR FUNCTION FIELDS OF SURFACES 13
Proof:
Let u P R z p be such that B p p α q “ t ¯ u u in K p m q κ p and let δ P R be suchthat p π, δ q is a parameter system of R . Then δ ` p is a uniformizer of the discretevaluation ring R { p , so u ” δ l b mod p for some b P R ˆ and l P N .Using the hypothesis and Proposition 6.2, we obtain that p r p ˝ B p qp α q “ in Z { m Z . Hence l “ r p pt ¯ δ l ¯ b uq “ r p pt ¯ u uq “ in Z { m Z , whereby m divides l . Hence t u u “ t ¯ b u in K p m q κ p , which shows that we mayreplace u by b to achieve that u P R ˆ . l Given a field F and a subring R of F , we say that a Z -valuation v of F is centered on R if R Ď O v , and we set Ω F { R “ t v P Ω F | R Ď O v u . (This extends the notation Ω F { E for a field extension F { E given in the previoussection.) Note that if F is the field of fractions of R and v is a nontrivial valuationon F which is centered on R , then m v X R is a nonzero prime ideal of R . Givena prime ideal p of R , we say that a valuation is v is centered in p if R Ď O v and m v X R “ p .We will need the following consequence of a purity result on the Brauer groupdue to Auslander-Goldman, which is translated here to K -groups.6.4. Lemma.
Let R be a two-dimensional regular local ring with m P R ˆ . Let F be the fraction field and assume that F contains a primitive m th root of unity.Let α P K p m q F be such that B v p α q “ for all v P Ω F which are centered in aheight-one prime ideal of R . Then B v p α q “ holds for all v P Ω F { R .Proof: Let Ω denote the set of Z -valuations on F which are centered in a height-one prime ideal of R . Note that Ω Ď Ω p m q F . By [4, Proposition 7.4], we havethat Br p R q “ č v P Ω Br p O v q . Hence, for the m -torsion part we obtain by Theorem 4.1 that Br m p R q “ Br nrm p F, Ω q . Let ξ “ Φ ω p α q where Φ ω is the homomorphism in Theorem 3.4. It follows bythe hypothesis on α that ξ P Br nrm p F, Ω q “ Br m p R q . Let v P Ω F { R . Since R Ď O v it follows that Br m p R q Ď Br m p O v q . Hence ξ P Br m p O v q and therefore B v p ξ q “ .This means that B v p α q “ . l We need also the following well-known consequence of the previous statements.
Lemma.
Let R be a two-dimensional regular local ring with fraction field F and residue field κ . Assume that char p κ q does not divide m and that κ ˆ “ κ ˆ m .Assume that F contains a primitive m th root of unity. Let α P K p m q F . Assumethat there exists a parameter π of R such that B q p α q “ for all q P P R zt Rπ u .Then B v p α q “ for all v P Ω F { R centered in the maximal ideal of R .Proof: Let p “ Rπ . Since B q p α q “ for all q P P R zt p u it follows by Corollary 6.3that B p p α q “ t ¯ u u in K p m q κ p for some u P R ˆ . Set β “ α ´ t π, u u . Then B R p β q “ . By Lemma 6.4, we have that B v p β q “ for every v P Ω F { R .Consider now a valuation v P Ω F { R which is centered in the maximal ideal m of R . Then m “ m v X R and κ Ď κ v . Since u P R ˆ Ď O ˆ v and ¯ u P κ ˆ “ κ ˆ m Ď κ ˆ mv ,we obtain that B v pt π, u uq “ v p π qt ¯ u u “ in K p m q κ v , whereby B v p α q “ B v p β q ` B v pt π, u uq “ in K p m q κ v . l Ramification on surfaces
Our standard reference for results and terminology from algebraic geometry is[22].Let X be a noetherian integral separated scheme of finite dimension d and let F be the function field of X . For a point x on X let O X ,x denote the stalk ofthe structure sheaf O X at x , which is a local domain with fraction field F .We call a point x of X regular (resp. normal ) or we say that X is regular (resp. normal ) at x if the local ring O X ,x is regular (resp. normal). We say that X is regular (resp. normal ) if X is regular (resp. normal) at every point of X .For i P N we denote by X i the set of all points of codimension i of X . Note thatif R is a two-dimensional noetherian domain and X is the corresponding affinescheme Spec p R q , then R is regular (resp. normal) if and only if X is regular(resp. normal), and X “ P R .A prime divisor on X is an irreducible closed subset of X of codimension one.For x P X the closure t x u is a prime divisor on X . For x, y P X we have t x u “ t y u if and only if x “ y . Let Div p X q be the free abelian group generatedby the prime divisors of X . An element of Div p X q is called a divisor on X .Let D P Div p X q . For x P X we denote by n x p D q the coefficient of the primedivisor t x u in D . We set Supp X p D q “ t x P X | n x p D q ‰ u . A prime divisor D is called a component of D if Supp X p D q Ď Supp X p D q .Assume now that X is normal. For every x P X the local domain O X ,x is normal and hence a discrete valuation ring of F . For x P X we denote thecorresponding Z -valuation on F by v x and its residue field by κ x . TRONG LINKAGE FOR FUNCTION FIELDS OF SURFACES 15
For f P F ˆ the set t x P X | v x p f q ‰ u is finite (see [22, Lemma 7.2.5]), andwe set p f q X “ ÿ x P X v x p f q ¨ t x u and call this the principal divisor given by f .7.1. Example.
Let R be unique factorization domain and X the affine scheme Spec p R q . Then X is normal. Since every height-one prime ideal of R is principal,we obtain that every divisor on X is principal.Let U be an open subscheme of X . Note that U Ď X . We have a surjectivegroup homomorphism Div p X q Ñ Div p U q which maps the prime divisor t x u on X to the prime divisor t x u on U for every x P X X U and which maps t x u to for every x P X z U . For D P Div p X q theimage of D in Div p U q is denoted by D U .Let x P X . There is a canonical morphism Spec p O X ,x q Ñ X which factorsvia any affine open neighbourhood of x (see [22, Example 2.3.16]). This inducesa group homomorphism Div p X q Ñ Div p Spec p O X ,x qq , which factors via Div p U q for any affine open neighbourhood U of x . For D P Div p X q the image of D in Div p Spec p O X ,x qq is denoted by D x .Let x be a regular point of X and let D P Div p X q . Then O X ,x is a regularlocal ring, hence a unique factorization domain. By Example 7.1 we obtain that D x “ p f q Spec p O X ,x q in Div p Spec p O X ,x qq for some f P F ˆ , uniquely determined upto a unit factor in O X ,x ; we call such an element f a local equation of D at x .By a surface we mean an integral separated noetherian scheme of dimension .Assume that X is a normal surface. Let D P Div p X q and let x P X . We saythat D contains x or that x lies on D if there is a component of D that contains x .Note that if x P X , then t x u is the only prime divisor on X that contains x ,and hence x lies on D if and only if n x p D q ‰ .We call a point x P X a crossing point of D if x lies on more than onecomponent of D . Let x P X be a regular point of X . We say that D has normal crossing at x if a local equation of D at x can be given in the form π i δ j with a parameter system p π, δ q of O X ,x and some i, j P N . We say that D hasnormal crossings on X if D has normal crossing at every point x P X .Let R be a noetherian domain of dimension at most . A surface over R is an R -scheme X which is a surface.Our arguments in Section 8 use the following well-known geometrical method.7.2. Proposition.
Let R be a noetherian domain of dimension at most . Let X be a regular surface over R such that X Ñ Spec p R q is projective and let F be the function field of X . Let D P Div p X q . Then there exists a divisor D P Div p X q which does not contain any crossing point of D and such that Supp X p D q X Supp X p D q “ H and D ` D “ p f q X for some f P F ˆ . Proof:
We fix a finite set P of closed points of X containing all crossing pointsof D and at least one point on each component of D . By the ApproximationTheorem [13, 2.4.1], there exists g P F ˆ such that v x p g q “ n x p D q for every x P Supp X p D q . Let D “ p g q X ´ D .Since X Ñ Spec p R q is projective, by [22, Proposition 3.3.36 p b q ] there existsan affine open subset U of X containing P . Let A “ O X ,U . Let M be the set ofthe maximal ideals of A corresponding to points in P . We set S “ A zp Ť m P M m q and obtain that A S is a regular semi-local ring and F is the fraction field of A S . By Theorem 6.1, we have that A S is a unique factorization domain, soin particular every divisor on Spec p A S q is principal, by Example 7.1. Hence D Spec p A S q “ p h q Spec p A S q for some element h P F ˆ .Set D “ D ´ p h q X . Then the divisor D on X does not contain any point of P . In particular Supp X p D q X Supp X p D q “ H . Then f “ gh ´ is an elementof F ˆ with the desired properties. l Let R be a henselian local noetherian domain with fraction field F . Let X be an R -scheme with structure morphism η : X Ñ Spec p R q . Consider a point x P Spec p R q . We denote by X x the κ x -scheme X ˆ Spec p R q Spec p κ x q , called the fiber of η over x . By [22, Proposition 3.1.16], the underlying topological space of X x is naturally homeomorphic to η ´ p x q .Let s be the closed point of Spec p R q . Then X s is called the special fiber of η .Note that η ´ p s q is a closed subset of X . There is a unique induced reducedsubscheme structure on η ´ p s q . We denote the closed reduced subscheme η ´ p s q by ¯ X s .7.3. Proposition.
Let R be a henselian local noetherian domain of dimension oneor two. Let X be a surface over R whose structure morphism η : X Ñ Spec p R q is proper and surjective. Then all closed points of X lie on X s . Furthermore, dim p X s q ď and X s is connected. In particular, X s is either a closed point of X or a finite union of one-dimensional irreducible subsets of X .Proof: Since η is proper, for any closed point x of X , η p x q is closed in Spec p R q ,whereby η p x q “ s . Thus X s contains all closed points of X .We have that Spec p κ s q Ñ Spec p R q is a closed immersion, by [22, Lemma 2.3.17],and thus of finite type, by [22, Proposition 3.2.4 p a q ]. Since for morphisms theproperty to be of finite type is stable under base change, by [22, Proposition3.2.4 p c q ], we get that X s Ñ X is of finite type. Since X is noetherian, X s isnoetherian as well, by [22, Exercise 3.2.1]. Hence X s has finitely many irreduciblecomponents.As η is surjective, we have that η ´ p s q ‰ X . Since X is irreducible and X s is closed, by [22, Proposition 2.5.5] we get that dim X s ď ă dim X . As R is ahenselian local ring and η is proper, by [10, Page 135, Proposition 18.5.19], X s is connected. If dim X s “ , then X s is a closed point on X . If dim X s “ , thenit follows that all components of X s are of dimension . l TRONG LINKAGE FOR FUNCTION FIELDS OF SURFACES 17
In the situation of Proposition 7.3, if dim p X s q “ , then we write Supp X p X s q for the finite set of points x P X for which the closure t x u is a component of X s .7.4. Theorem (Colliot-Thélène-Ojanguren-Parimala) . Let R be a henselian lo-cal domain of dimension one or two. Assume that the residue field κ is eitherseparably closed or finite and that char p κ q does not divide m . Let X be a regularprojective surface over R such that X Ñ Spec p R q is surjective and let F be thefunction field of X . Then Br nrm p F, Ω X q “ , where Ω X “ t v x | x P X u .Proof: By Proposition 7.3, the special fiber of X has dimension at most one.Then Br nrm p F, Ω X q Ď Br m p X q (See [19, Theorem 6.1 p b q ] or [8, Theorem 1.2]).Since X is regular, we obtain by [7, Corollary 1.10 and Corollary 1.11] that Br m p X q “ Br p X q “ . l Let X be a surface. An X -scheme X is called a model of X if the structuremorphism X Ñ X is birational and proper. A desingularisation of X is a pair p X , η q where X is a regular scheme and η : X Ñ X is a proper birationalmorphism; in this case we also call X a regular model of X .We refer to [22, Definition 8.2.35] for the definition of excellence for a ringas well as for a scheme, and also for the fact that for any excellent noetheriancommutative ring R the scheme Spec p R q is excellent. Given an excellent surfacethe existence of a regular model is ensured by a result due to Lipman.7.5. Theorem (Lipman) . Let X be an excellent surface. There exists a desin-gularisation p X , η q of X Moreover, given an effective divisor D on X , p X , η q can be chosen such that the divisor η ´ p D q has normal crossings on X .Proof: See [21, p. 193] or [34, Theorem, p. 38 and Remark 2, p. 43]. l The following examples cover the situations where we will apply Theorem 7.5.7.6.
Examples.
By [22, Theorem 8.2.39, Corollary 8.2.40], every complete noe-therian local ring as well as every Dedekind domain of characteristic zero is ex-cellent. Furthermore, by [22, Theorem 8.2.39 p c q ], given an excellent noetherianring R , any projective surface over Spec p R q is excellent.7.7. Corollary.
Let E be a field and let v be a discrete valuation on E . If char p E q ‰ then assume that v is complete. Let F be an algebraic function fieldover E . Then F is the function field of a regular projective surface X over O v such that X Ñ Spec p O v q is surjective.Proof: By [22, Proposition 7.3.13], there exists a unique normal projective curve C over E with function field F . Using [22, Example 10.1.4] and the first steps of[22, Proposition 10.1.8], we obtain a projective flat surface Y over O v with func-tion field F . In particular Y Ñ Spec p O v q is surjective. Note that O v is a localDedekind ring and either of characteristic zero or complete. Since Y Ñ Spec p O v q is projective, we obtain by Examples 7.6 that Y is excellent. By Theorem 7.5, itfollows that there exists a desingularisation p Y , η q of Y . Now by [18, Corollaire X of Y . Since X Ñ Y is pro-jective and birational, it follows that X Ñ Spec p O v q is projective and surjectiveand F is the function field of X . l Corollary.
Let R be a two-dimensional excellent noetherian local domain.Then there exists a regular projective model of Spec p R q .Proof: In view of Theorem 7.5, there exists a regular surface X over R suchthat X Ñ Spec p R q is proper and birational. Now by [18, Corollaire 5.6.2] thereexists a regular surface X over R such that X Ñ Spec p R q is projective andbirational. l Function fields of surfaces
In this section we consider function fields of surfaces over a complete localdomain of dimension one or two with algebraically closed residue field. Typicalexamples are the fields C pp t qqp X q and C pp X, Y qq and their finite extensions. It wasshown in [14] and [15] that any central simple algebra over such a field is cyclic.In [7, Theorem 2.1], using techniques developed in [30], a simpler proof of themain result of [15, Theorem 1.6] was given.Here we will show that any such field F is strongly linked in any degree coprimeto the residue characteristic. The proof mainly follows the method of [7, Theorem2.1].Consider a normal surface X and let F be the function field of X . Let m bea positive integer. Any x P X gives rise to a Z -valuation v x on F and thereforeto a ramification homomorphism B v x : K p m q F Ñ K p m q κ x , which we denote by B x .For α P K p m q F , we set Supp X p α q “ t x P X | B x p α q ‰ u . We set Ω F { X “ t v P Ω F | v centered on O X ,x for some x P X u . Remark.
Consider v P Ω F { X . Let x P X be such that O X ,x Ď O v . Then m v X O X ,x is a nonzero prime ideal of O X ,x , hence either a height-one primeideal or equal to m X ,x . In particular, if m v X O X ,x ‰ m X ,x , then x P X and O v is the localisation of O X ,x at m v X O X ,x and therefore equal to O X ,x for some x P X . Hence, in any case we may choose x P X with O X ,x Ď O v in such waythat m v X O X ,x “ m X ,x .The following lemma is distilled from the proof of [7, Theorem 2.1].8.2. Lemma.
Let X be an excellent regular surface and let F be its functionfield. Assume that F contains a primitive m th root of unity. Assume that, forevery closed point x P X , char p κ x q does not divide m and κ ˆ x “ κ ˆ mx . Let f P F ˆ and let D , D P Div p X q be such that p f q X “ D ` D and Supp X p D q X Supp X p D q “ H . TRONG LINKAGE FOR FUNCTION FIELDS OF SURFACES 19
Assume that D has normal crossings on X , n x p D q “ for every x P Supp X p D q and D does not contain any crossing point of D . Let α P K p m q F be such that Supp X p α q Ď Supp X p D q . Then F p m ? f q{ F splits the ramification of α at every v P Ω F { X .Proof: Let F “ F p m ? f q . Let v P Ω F { X . Consider an arbitrary Z -valuation w on F with O w X F “ O v . We need to show that B w p α F q “ in K p m q κ w .If B v p α q “ in K p m q κ v , then this is obvious. We assume now that B v p α q ‰ . ByRemark 8.1 we may choose x P X such that O X ,x Ď O v and m v X O X ,x “ m X ,x .Assume first that x P X . Then v “ v x and x P Supp X p α q Ď Supp X p D q .We obtain that v p f q “ v x p f q “ n x p D q “ . It follows by Lemma 2.3 that B w p α F q “ .Assume now that x P X . Set R “ O X ,x . Since v is centered on R and B v p α q ‰ , we obtain by Lemma 6.4 that B R p α q ‰ . Hence there exists a primeelement π of R such that B Rπ p α q ‰ . Since D has normal crossings on X and Supp X p α q Ď Supp p D q , we obtain that π is a parameter of R . Set p “ Rπ .Since B v p α q ‰ , it follows by Lemma 6.5 that B q p α q ‰ for some q P P R zt p u .In particular, x is a crossing point of D . By the hypotheses on D and D , itfollows that x does not lie on D . We set δ “ π ´ f . Since the divisor D hasnormal crossings and all its components have multiplicity , we obtain that p π, δ q is a parameter system of R and q “ Rδ .Fix λ, µ P R zt u such that B p p α q “ t ¯ λ u in K p m q κ p and B q p α q “ t ¯ µ u in K p m q κ q .We have that R { p and R { q are discrete valuation rings of κ p and κ q with uni-formisers ¯ δ and ¯ π , respectively; we denote the corresponding Z -valuations by w ¯ δ and w ¯ π . Set r “ w ¯ π p ¯ µ q and s “ w ¯ δ p ¯ λ q . In particular, there exist ν, ν P R ˆ suchthat λ ” νδ s mod π and µ ” ν π r mod δ . Using Proposition 6.2, we obtain that r ` s ” m Z . Set β “ α ´ t π, ν u ´ t δ, ν u ´ s t π, δ u . We obtain that B R p β q “ . It follows by Lemma 6.4 that B v p β q “ . Note thatthe symbols t π, ν u and t π, π u lie in ker pB p q for any p P P R zt p u . Similarly, t δ, ν u lies in ker pB p q for any p P P R zt q u . We obtain by Lemma 6.5 that B v pt π, ν uq “ B v pt π, π uq “ B v pt δ, ν uq “ in K p m q κ v . We conclude that B w p β F q “ B w pt π, ν uq “ B w pt π, π uq “ B w pt δ, ν uq “ in K p m q κ w . Since f “ πδ , we have t π, δ u “ t π, f u ´ t π, π u , and as t π, f u “ in K p m q F , we conclude that B w pt π, δ uq “ in K p m q κ w . Thisshows that B w p α F q “ in K p m q κ w . l Theorem.
Let R be an excellent henselian local domain of dimension oneor two with separably closed residue field of charactersitic not dividing m . Let X be a projective surface over R such that X Ñ Spec p R q is surjective. Let F bethe function field of X . Then K p m q F is strongly linked.Proof: It follows from the hypothesis that the residue field contains a primitive m th root of unity.Let S Ď K p m q F be a finite subset. To show that K p m q F is strongly linked, weneed to show that all elements S have a common slot.Since X Ñ Spec p R q is projective and since R is excellent, it follows from [22,Theorem 8.2.39 (c)] that X is excellent as well. We set Supp X p S q “ ď α P S Supp X p α q . By Proposition 7.3,
Supp X p X s q is finite. Hence Z “ Supp X p S q Y Supp X p X s q isa finite set. By Theorem 7.5, there exists a desingularisation p X , η q of X suchthat η ´ ´ř z P Z t z u ¯ has normal crossings on X .Let Z “ Supp X p S q Y Supp X p X s q and set D “ ÿ x P Z t x u . By Proposition 7.2, there exists f P F ˆ such that p f q X “ D ` D for some D P Div p X q with Supp X p D q X Supp X p D q “ H and such that D does not contain any crossing point of D . We claim that such an element f is aslot of every element of S .Consider α P S . Since Supp X p α q Ď Supp X p D q , it follows by Lemma 8.2 that F p m ? f q splits the ramification of α over Ω F { X . Let Y be the normalization of X in F p m ? f q . Since F p m ? f q{ F is a finite separable field extension, it follows by[22, Proposition 4.1.25] that Y Ñ X is a finite morphism. Hence Y is excellent,by [22, Theorem 8.2.39 p c q ]. By Theorem 7.5, there exists a regular model Y of Y . Since restriction of a valuation in Ω F p m ? f q{ Y on F is equivalent to a valuationin Ω F { X , it follows by Lemma 8.2 that B v y p α F p m ? f q q “ for all y P Y . Since Y Ñ X and X Ñ Spec p R q are projective and surjective, we have that Y is aregular projective surface over R such that Y Ñ Spec p R q is surjective. Thus byTheorem 7.4, we have that Br nrm p F p m ? f q , Ω Y q “ . It follows by Proposition 4.3that α F p m ? f q “ . We conclude that f is a slot of α , by Theorem 3.5. l Corollary.
Let E be the fraction field of an excellent henselian discrete val-uation ring with separably closed residue field of characteristic not dividing m .Let F { E be an algebraic function field. Then K p m q F is strongly linked. TRONG LINKAGE FOR FUNCTION FIELDS OF SURFACES 21
Proof:
Let R denote the valuation ring. By Corollary 7.7, there exists a regularprojective surface X over R such that X Ñ Spec p R q is surjective and such that F is the function field of X . Hence the result follows from Theorem 8.3. l Examples.
Let m P N be a positive integer. Consider the following cases: p q E “ k pp X qq for an algebraically closed field k of characteristic coprime to m . p q E is the maximal unramified extension of the field of Q p for a prime number p not dividing m .In each of these cases, E is a complete discretely valued field and the correspond-ing discrete valuation ring is excellent, by Examples 7.6. Hence, it follows byCorollary 8.4 that K p m q F is strongly linked for any algebraic function field F { E .8.6. Corollary.
Let F be the fraction field of a two-dimensional excellent henselianlocal domain with separably closed residue field of characteristic not dividing m .Then K p m q F is strongly linked.Proof: Let R denote the domain. As R is excellent, Corollary 7.8 asserts theexistence of a regular projective model of Spec p R q . Now the result follows fromTheorem 8.3. l Example.
Let m P N be a positive integer. Consider F “ k pp X, Y qq where k is an algebraically closed field of characteristic coprime to m . Then F is thefraction field of the two-dimensional complete regular local domain k rr X, Y ss ,which is excellent by Examples 7.6. Hence it follows by Corollary 8.6 that K p m q F is strongly linked.9. Discretely valued quasi-finite fields
A field is called quasi-finite if it has a finite field extension of every degree andall its finite field extensions are cyclic. Equivalently, a field is quasi-finite if itis perfect and its absolute Galois group is isomorphic to the procyclic group ˆ Z .Finite fields are quasi-finite. Another natural example of a quasi-finite field is C pp t qq , the field of Laurent series in one variable over C .Let m be a positive integer. Recall that K p m q F is strongly linked for anyglobal field F , hence in particular when F is an algebraic function field overa finite field. We further have seen in Examples 8.5 that the same holds foralgebraic function fields over C pp t qq . It is well-known that quasi-finite fields canhave different properties relative to solvability of certain systems of polynomialequations. For example, while finite fields and C pp t qq are so-called C -fields, in[5] J. Ax constructed a quasi-finite field which is not a C -field. The observationthat strong linkage holds for algebraic function fields over finite fields (by [20])and over C pp t qq (by Examples 8.5) motivates the following question.9.1. Question.
Let E be a quasi-finite field. Is K p m q F strongly linked for everyalgebraic function field F { E and every positive integer m ? We do not know the answer to this question even for m “ . However, wecan give a positive answer in the case where E carries a discrete valuation. Thisrelies on the following characterisation of this situation.9.2. Theorem.
Let E be a field and let v be a Z -valuation on E . Then E is quasi-finite if and only if v is henselian, κ v is algebraically closed and char p κ v q “ . Most of the implications contained in this statement follow immediately fromclassical results in valuation theory. We include a complete elementary proof.
Proof:
Let π P E ˆ be such that v p π q “ .Assume that κ v is algebraically closed with char p κ v q “ and that v is henselian.Then char p E q “ , hence E is perfect. It follows by Hensel’s Lemma that E contains all roots of unity. Since v p π q “ , for any positive integer n , the extension E p n ? π q{ E has degree n . Let L { E be a finite field extension and d “ r L : E s .Since v is henselian, v has a unique extension w to L . Since κ v is algebraicallyclosed, the residue field of w is equal to κ v , and hence by [13, Theorem 3.3.5] wehave that r w p L ˆ q : Z s “ r L : E s “ d . We fix δ P L such that w p δ q “ d . Then δ d π P O ˆ w . Since κ v is algebraically closed of characteristic zero, the assumptionthat w is henselian yields that the polynomial X d ´ δ d π has a root in O w r X s .Hence there exists u P O w with δ d “ πu d . We conclude that L “ E p d ? π q . Hencefor any positive integer d the unique field extension of E of degree d is given by E p d ? π q , which proves that E is quasi-finite.Assume now that E is quasi-finite. Since E is perfect and carries a Z -valuation,we have that char p E q “ . For any positive integer d , since X d ´ π is irreducible in E r X s , we obtain a field extension of degree d of E in which v is totally ramified.As E has a unique extension of any degree, we obtain that v is totally ramified inevery finite field extension of E . Therefore κ v is algebraically closed and v extendsuniquely to any finite field extension of E . This shows that v is henselian.Suppose that char p κ v q “ p for a prime number p . As v p π q “ and v p ` π q “ ,the classes of π and ` π in E ˆ { E ˆ p do not generate the same subgroup. Since char p E q “ , it follows that X p ´ π and X p ´ ´ π do not have the same rootfield over E . Since the root field of X p ´ π over E is the unique field extension of E of degree p , we conclude that X p ´ ´ π splits over E . Hence ` π “ ξ p forsome ξ P E . Let ϑ “ ξ ´ . Since char p κ v q “ p it follows that v p ϑ q ą . On theother hand, π “ p ` ϑ q p ´ “ ř pi “ ` pi ˘ ϑ i . Since v p p q ě and v p ϑ q ě while v p π q “ , we have a contradiction. This proves that char p κ v q “ . l Corollary.
Let E be a quasi-finite field and m a positive integer. If E carriesa Z -valuation, then K p m q F is strongly linked for any algebraic function field F { E .Proof: Let v be a Z -valuation on E . By Theorem 9.2, O v is a henselian discretevaluation ring and κ v is algebraically closed with char p κ v q “ . In particular char p O v q “ . Hence O v is excellent, by Examples 7.6. Therefore K p m q F isstrongly linked, by Corollary 8.4. l TRONG LINKAGE FOR FUNCTION FIELDS OF SURFACES 23
We point out a type of examples of quasi-finite fields for which we do not knowthe answer to Question 9.1.9.4.
Example.
Let k be a quasi-finite field of characteristic zero (e.g. k “ C pp s qq ,the Laurent series field in one variable s over C ). Consider E “ Ť r “ k pp t { r qq ,the field of Puiseux series in t over k . This field E carries a henselian valuation v with value group Q and residue field k . Since the value group of v is divisible, v is unramified in every finite extension of E . On the other hand, since v ishenselian, for every positive integer d , any finite extension of k lifts uniquely toan unramified extension of E of the same degree. Since the field k has a uniquefield extension of degree d for any positive integer d , we conclude that the sameholds for E . Hence E is quasi-finite. Note that E does not carry any Z -valuation,so Theorem 9.2 does not help to answer Question 9.1 in this case. Acknowledgments.
The authors express their gratitude to Jean-Louis Colliot-Thélène, Arno Fehm, David Grimm, Gonzalo Manzano Flores, R. Parimala,Suresh Venapally and Jan Van Geel for various answers, discussions, suggestions,simplifications and other valuable input related to this article.
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TRONG LINKAGE FOR FUNCTION FIELDS OF SURFACES 25
Institut für Algebra, Technische Universität Dresden, 01062 Dresden, Ger-many.
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