ELEMENTARY EQUIVALENCE VERSUS ISOMORPHISM III

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1 ELEMENTARY EQUIVALENCE VERSUS ISOMORPHISM III FLORIAN POP Abstract. For every finitely generated field K with dim(k) = 3 and char(k) 2, we give sentences ϑ K in the language of fields which describe the isomorphy type of K among finitely generated fields, i.e., for any such field L one has: ϑ K holds in L if and only if K = L as fields. This extends earlier results by Rumely [10] and Pop [8, 9] thus proving the so called Pop s Conjecture in general, see [Sc]. 1 This also answers positively the long standing open Elementary equivalence versus Isomorphism Problem for the case dim(k) < 4 asserting that the first order theory of finitely generated fields encodes their isomorphy type. 1. Introduction We begin by recalling Rumely s result [Ru] showing that for every global field K there exists a sentence ϑk Ru which characterizes the isomorphy type of K among global fields, i.e., if L is any global field, then ϑk Ru holds in L if and only if L = K as fields. It is one of the main open questions in the first order theory of finitely generated fields whether a fact similar to Rumely s result mentioned above holds for all finitely generated fields K. 1 Finally we notice that the question above is related to, but much stronger than, the still open Elementary equivalence versus Isomorphy Problem, which asks whether the whole first order theory Th(K) encodes the isomorphy type of finitely generated fields K, see Pop [9] for details and literature about this. In the present note we show that the answer to the above question positively for fields K with Kronecker dimension dim(k) < 4 and characteristic char(k) 0; thus settle the Elementary equivalence versus Isomorphy Problem as well in this case. Main Result. For every finitely generated field K with dim(k) < 4 and char(k) 2, there exists a first order sentence in the language of fields ϑ K such that for all finitely generated fields L, one has: The sentence ϑ K holds in L if and only if L = K as fields. The key fact used to prove the Main Result above is the uniform first order definability of geometric prime divisors of finitely generated fields, which is the following. First, recall that the Kronecker dimension dim(ω) of arbitrary fields Ω is defined as follows: dim(f p ) = 0 for all p, dim(q) = 1, and if κ 0 Ω is the prime field of Ω, one defines dim(ω) = dim(κ 0 )+td(ω κ 0 ). Second, κ := Ω abs is the absolute subfield, or the constant subfield of Ω, i.e., the elements of Ω which are algebraic over the prime field κ 0 Ω. Further, given a valuation v of Ω, we 2010 Mathematics Subject Classification. Primary 11G30, 14H25; Secondary 03C62, 11G99, 12G10, 12G20, 12L12, 13F30. Key words and phrases. Elementary equivalence vs isomorphism, first order definability, e.g. of valuations, finitely generated fields, Milnor K-groups, Galois/étale cohomology, Kato s higher local-global Principles. 1 It was claimed [11] that the answer to this question is positive for all finitely generated fields. Unfortunately, the proof has a gap, see Erratum, JAMS 24 (2011), p Nevertheless, the work [11] reduces the problem of detecting the isomorphism type of K to describing the prime divisors of K. 1

2 denote by m v O v the valuation ideal, respectively valuation ring of v, by vω := Ω /O v the value group of v, and by κ(v) = Ωv := O v /m v the residue field of v. Let K be a finitely generated field, and κ K be its absolute subfield. The prime divisors of K are the valuations w of K satisfies the following equivalent conditions: i) There exists a normal Z-model X of K such that O w = O X,x, where x X is a point with codim(x) = dim(x) 1 = dim(k) 1. ii) w is non-trivial, and dim(kw) = dim(k) 1. A prime divisors w of K is called arithmetical, if w κ is nontrivial. In particular, K allows arithmetical prime divisors iff char(k) = 0 iff κ is a number field. Finally, given a subfield k K, e.g., k = κ, a prime divisor w of K is called a prime k-divisor of K, or a divisor of K k, if w is trivial on k. For w on K with w k trivial, the following conditions are equivalent: i) w is a discrete, and Kw is a finitely generated field with td(kw k) = td(k k) 1. ii) dim(kw) = dim(k) 1. In this manuscript, if not explicitly otherwise stated, when speaking about prime divisors of K, we will always mean prime κ-divisors, that is, prime divisors of K κ. The key technical tool in the proof of the above Main Result is the following: Theorem 1.1. The valuation rings of prime divisors of finitely generated fields of Kronecker dimension 3 and characteristic 2 are uniformly first order definable. The strategy of proof of Theorem 1.1 is based on refinements of ideas and results from [P2], and uses in an essential way the higher dimensional Hasse principles as initiated by Kato [Ka]. Here is the set up for the idea and strategy of the proof. By Pop [P1], there exist sentences ϕ d, d 0, such that for every finitely generated field K one has: dim(k) = d if and only if ϕ d holds in K. Further, by Poonen [Po], there exists a sentence ψ 0 such that ψ 0 holds in a finitely generated field K iff char(k) = 0, or equivalently, the constant field κ 0 K is a number field. In particular, considering the sentences td m (ψ 0 ϕ m+1 ) ( ψ 0 ϕ m ), m 0, the (absolute) transcendence degree td(k) := td(k κ) of K is td(k) = m if and only if td m holds in K. Further, there is a predicate ψ abs (x) which uniformly defines the constant subfield of finitely generated fields, i.e., if K is a finitely generated field, then κ = {x K ψ abs (x) holds in K}. Finally, there exist formulas ψ r (t 1,..., t r ), r > 0, with variables t 1,..., t r which characterize algebraic independence in finitely generated fields, i.e., if K is such a field, for t 1,..., t r K, one has: ψ r (t 1,..., t r ) holds in K iff t 1,..., t r are algebraically independent over the constant field κ K. Hence for algebraically independent elements t := (t 1,..., t r ) from K, the relative algebraic closure of k t of κ(t) in K is first order definable as follows: k t = {u K ψ r+1 (u, t 1,..., t r ) holds in K} Thus we conclude that the transcendence bases T of K κ are uniformly definable, and so are the (maximal) flags of relatively algebraically closed subfields of K starting with a global subfield k 0 K as follows k 0 k d 1 := K. 2

3 Coming back to definability of valuations and prime divisors, recall that for global fields K, i.e., finitely generated fields K with d = dim(k) = 1, such formulae were given by Rumely [10], and the case d = dim(k) = 2 was dealt with in Pop [8]. For dim(k) = 3, let k 0 k K be maximal flag of subfields as above. Then k 0 is a maximal global subfield of K, hence k 0 = κ if char(k) = 0, and k 0 κ is a global function field otherwise. Second, td(k k) = 1, and the prime k-divisors of K k are the non-trivial k-valuations v of K. Equivalently, K = k(c) is a the function field of an projective normal irreducible k-curve C, and the valuation rings O v of prime divisors v of K k are precisely the local rings O x with x C closed points. The following hold: a) Since κ is perfect, there are (many) separable transcendence bases T of K κ, and T = (t 0, t 1, t 2 ) if κ is finite, T = (t 1, t 2 ) if κ is a number field. Thus setting T = (T 1, t 2 ), the relative algebraic closure k K of κ(t 1 ) satisfies dim(k) = 2, and td(k k) = 1. b) If T is in general position, then k = κ(t 0, t 1 ) if κ is finite, respectively k = κ(t 1 ) if κ is a number field. Further, there exists an absolutely irreducible polynomial f K (T, U) over κ, monic in the variable U, such that f K (T, u) = 0 and K = κ(t )[u]. Hence one has: Fact 1.2. The isomorphy type of K is given by the following data: 1) A transcendence basis T = (T 1, t 2 ) in general position of K κ, i.e., satisfying: k = κ(t 1 ) is relatively algebraically closed in K. 2) An absolutely irreducible polynomial f K (T, U) over κ, monic in U, such that one has: f K (T, u) = 0 for some u K and K = κ(t )[u]. 3) Viewing K k as a function field in one variable, the degree deg(u) = [K : k(u)] of the function u K satisfies deg(u) = deg U (f K ). c) In particular, if the prime divisors of both the (relatively algebraically closed) subfields k K with dim(k) = 2, and the prime divisors of K k are uniformly first order definable, then the isomorphy type of K k, thus that of K as a finitely generated field, is described by a sentence in the language of fields. [That sentence is as concrete as the formulae describing the prime divisors of k and those of K k.] Theorem 1.3. Let k 0 k be finitely generated fields as above. Then the prime divisors of function fields of geometrically integral k-curves C are uniformly first order describable. In particular there exist formulas deg N (u), ψ Ru (u,u ), ψ(u,u ), with parameters u, u such that for every function field K k with td(k k) = 1 and k relatively algebraically closed in K, and u K the following hold: a) deg N (u) is true in K iff u has degree N as a function of K k. b) R u := {u K ψ Ru (u, u ) is true in K} is the integral closure of k[u] in K. c) k[u] = {u K ψ(u, u ) is true in K}. We mention that the formulas deg N (u), ψ Ru (u,u ), ψ(u,u ), and ψ p (u 1,u 0 ) are quite explicit, see section 4, in particular, so is Theorem 5.1. The main technical tool in the proof are some higher dimensional Hasse local-global principles, as initiated by Kato. Thanks: I would like to thank the participants at several activities, e.g., AWS 2003, AIM Workshop 2004, INI Cambridge 2005, HIM Bonn 2009, ALANT III in 2014, for the debates on the topic and suggestions concerning this problem. Special thanks are due to Bjorn Poonen and Jakob Stix for discussing technical 3

4 aspects of the proofs, and to Uwe Jannsen and Moritz Kerz for discussions concerning the higher dimensional Hasse local-global principles. The author would also like to thank the Universities of Heidelberg and Bonn for the excellent working conditions during his visits in as a A.v. Humboldt Preisträger. 2. Higher dimensional Hasse local-global principles A) Notations and general facts For a valuation v of K (possibly the trivial one), we denote by m v O v K the valuation ideal, respectively valuation ring of v. Further, we denote by U 1 v := 1 + m v O v =: U v the groups of principal v-units, respectively v-units in K. Finally, vk = K /U v is the value group of v, and Kv := O v /m v is the residue field of v. The space of all the (equivalence classes of) valuations on K is called the Riemann space of K, denoted Val K. For valuations v, w Val K, the following are equivalent: i) O w O v, or equivalently, w(x) 0 v(x) 0. ii) m w m v, or equivalently, w(x) > 0 v(x) > 0. iii) U w U v, or equivalently, v(x) = 0 w(x) = 0. iv) U 1 w U 1 v, or equivalently, w(1 x) > 0 v(1 x) > 0. If v, w satisfy the equivalent conditions above, we say that w v, or that w is coarser than v, respectively that v is finer than w, or that v is a refinement of w. It turns out that if w v, then m w O v is a prime ideal, and O v /m w Kw = O w /m w is a valuation ring of Kw. The corresponding valuation is called the (valuation theoretical) quotient of v by w on Kw, denoted v/w. It is obvious that v/w has valuation ideal m v /m w, hence O v/w /m v/w = O v /m v = Kv, and U v/w = O v/(1 + m w ) = U v /U 1 w. Hence there are canonical exact sequences: 0 (v/w)(kw) vk wk 0, 0 m v/w O v/w Kv 0. Notations/Remarks 2.1. Let K be a finitely generated infinite field. Let k K be a subfield, and Val K k Val K be the set of all valuations v Val K which are trivial on k. 1) A model of K is a separated finite type Z-scheme X with function field κ(x ) = K. Further, a k-model of K is any k-variety X with function field k(x) = K. 2) Let a Z-model X of K, and v Val K be given. We say that x X is the center of v on X if O x O v, that is, O x O v and m x = m v O x. By the valuation criterion one has: Since X is separated, every v Val K has at most one center on X. And X is proper iff then every valuation v Val K has a unique center on X. 3) Let a k-model X of K and v Val K be given. We say that x X is the center of v on X, if O x O v. If so, then v Val K k. By the valuation criterion one has: Since X is separated over k, every v has at most one center on X. And X is a proper k-variety iff every k-valuation v Val K k has a unique center on X. 4) In the above notation, let DK 1 D1 X be the spaces of prime divisors of K, respectively the ones defined by the prime Weil divisors of a normal model X of K. Further, let DK k 1 D1 X be the spaces of prime divisors of K k, respectively the ones defined by the prime Weil divisors of a normal k-model X of K k. 5) A valuation v of K is called a generalized r-prime divisor if there exists a chain of valuations v 1 v r of K such that v 1 is a prime divisor, and for 1 < i r, inductively one has: v i /v i 1 is a prime divisor of Kv i 1. By mere definitions, in the above 4

5 notations one has that v r K = Z r ordered lexicographically, and by general valuation theory, it follows that Kv r is a finitely generated field satisfying dim(kv r ) = dim(k) r. In particular, every flag of generalized prime divisors has length bounder by dim(k). Define correspondingly the generalized r-prime k-divisors of K which are, by definition, the same as the generalized r-prime divisors of K k. By the remarks above, if v r is a generalized r-prime divisor of K k, then one has td(kv r k) 0, hence r td(k k). 6) Finally, let DK r Dr X be the spaces of r-prime divisors v r of K, respectively the ones having a center x r with codim(x r ) = r on some (normal) model X of K. Further, let DK k r Dr X be the spaces of r-prime divisors of K k, respectively the ones having a center x r with codim(x r ) = r on a normal k-model X of K k. B) Local-global principles Recall first the famous Hasse Brauer Noether LGP for the Brauer group of a global field k. Let P(k) be the set of non-trivial places of k. For v P(k), we denote by k v the completion of k with respect to v. Then k v is a locally compact (non-discrete) field, and the Brauer group Br(k v ) of k v satisfies: There exists a canonical embedding inv v : Br(k v ) Q/Z, called the invariant (isomorphism), satisfying the following: a) If k v = C, then Br(k v ) = 0 and inv v is the trivial map. b) If k v = R, then inv v : Br(k v ) 1 Z / Z Q/Z is an isomorphism. 2 c) In the remaining cases, inv v : Br(k v ) Q/Z is an isomorphism. Let n ( ) denotes the n-torsion in an Abelian group, thus n (Q/Z) = Z/n canonically. Then the Hasse Brauer Noether LGP asserts that one has a canonical sequence which is exact: 0 n Br(k) v n Br(k v ) Z/n 0, where, the first map is the direct sum of all the canonical restriction maps n Br(k) n Br(k v ); thus implicitly, for every division algebra D over k there exist only finitely many v such that D k k v is not a matrix algebra; and the second map is the sum of the invariant morphisms. It is a fundamental observation by Kato [Ka] that the above local-global principle has higher dimensional variants as follows: First, for every positive integer n, say n = mp r with p the characteristic, and an integer twist i, one sets Z/n(0) = Z/n, and defines in general Z/n(i) := µ i m W r Ω i log [-i], where W r Ω log is the logarithmic part of the de Rham Witt complex on the étale site, see Illusie [Ill] for details. With these notations, for every (finitely generated) field K one has: H 1( K, Z/n(0) ) = Hom cont (G K, Z/n), H 2( K, Z/n(1) ) = n Br(K), where G K is the absolute Galois group of K. Thus the cohomology groups H 1( K, Z/n(0) ) and H 2( K, Z/n(1) ) have particular arithmetical significance. Noticing that K is a global field iff dim(k) = 1, Kato had the fundamental idea that for finitely generated fields K with dim(k) = d, there should exist similar local-global principles for H d+1( K, Z/n(d) ). The Kato cohomological complex (KC) We briefly recall Kato s cohomological complex (similar to complexes defined by the Bloch Ogus) which is the basis of the higher dimensional Hasse local-global principles, see Kato 5

6 [Ka], 1, for details. First let L be an arbitrary field, and recall the canonical isomorphism (generalizing the classical Kummer Theory isomorphism) h 1 : L /n H 1( L, Z/n(1) ). As explained in [Ka], 1, the isomorphism h 1 gives rise canonically for all q 0 to morphisms, which by the (now proven) Milnor Bloch Kato Conjecture are actually isomorphism: h q : K M q (L)/n H q( L, Z/n(q) ), {a 1,..., a q }/n h 1 (a 1 )... h 1 (a q ) =: a 1... a q. Further, let v be a discrete valuation of L. Then one defines the boundary homomorphism v : H q+1( L, Z/n(q + 1) ) H q( Lv, Z/n(q) ) by a a 1... a q v(a) a 1... a q for a L arbitrary and v-units a 1,..., a q O v. Now let X be an excellent integral scheme, and denote: First, X i X is the set of all the points x i of dimension i, i.e., such that the closure X xi := {x i } X has dimension i. Hence X 0 X are the closed points, and if X dim(x) consists of the generic point η X X only. Second, X i X is the set of all the points of cxdimension i, i.e., such that dim(o x ) = i. For y X i+1, it follows that X i y X i consists of all the points x X i which lie in the closure of X y := {y}. Since X is excellent, the normalization X y X y of X y is a finite morphism. Hence for every x X y, the fiber X y, x := { x X y x x under X y X y } is finite, and the following hold: The local rings O x κ(y) of all x X i y are discrete valuations rings of the residue field L := κ(y), say with valuation v x, and κ( x) κ(x) are finite field extensions. Then for n > 1 invertible on X, one gets a sequence of the form: (KC)... xi+1 X i+1 Hi+2( κ(x i+1 ), Z/n(i + 1) ) xi X i Hi+1( κ(x i ), Z/n(i) )... where the component xi+1 x i : H i+2( κ(x i+1 ), Z/n(i + 1) ) H i+1( κ(x i ), Z/n(i) ) is defined by xi+1 x i := x X xi+1, x i cor κ( x) κ(xi ) v x Theorem 2.2 (Kato [Ka], Proposition 1.7). In the above notations, suppose that X is an excellent scheme satisfying the following: For every x i X i one has [κ(x i ) : κ(x i ) p ] p i, where p is the characteristic exponent of κ(x i ). Then (KC) is a complex. This being said, the Kato Conjectures are about aspects of the fact that in arithmetically significant situations, the complex (KC) above is exact, excepting maybe for i = 0, where the homology of (KC) is perfectly well understood. And Kato proved himself several forms of the above local-global Principles in the case X is an arithmetic scheme of dimension dim(x) = 2 and having further properties. Among other things, one has: Theorem 2.3 (Kato [Ka], Corollary, p.145). Let X be a proper regular integral Z-scheme, dim(x) = 2, and K = κ(x) having no orderings. Then one has an exact sequence: 0 H 3( K, Z/n(2) ) x1 X 1 H 2( κ(x 1 ), Z/n(1) ) x0 X 0 H 1( κ(x 0 ), Z/n ) Z/n 0. Finally, notice that in Theorem 2.3 above, K is finitely generated with dim(k) = 2. Unfortunately, for the time being, the above result is not known to hold in the same form in higher dimensions d := dim(k) > 2, although it is conjectured to be so. There are nevertheless partial results concerning the local-global principles involving H d+1( K, Z/n(d) ). From those results, we pick and choose only what is necessary for our goals, see below. 6

7 Notations/Remarks 2.4. Let K be a finitely generated field with constant field κ. We supplement Notations/Remarks 2.1 as follows: 1) n > 1 is a prime number char(k), and µ n K. 2) k 0 K is a relatively algebraically closed subfield with dim(k 0 ) = 1. Hence k 0 is a global field, and let P(k 0 ) be the set of places of k 0. For v P(k 0 ), consider: a) The v-completion k 0v of k 0 at v. b) If v is a non-archimedean place, let R v k 0v the Henselization of O v inside k 0v. Localizing the global field k 0 In the above notations, for every v P(k 0 ), consider the compositum K v := Kk 0v (in some fixed universe). Then via the restriction functor(s), one gets canonical localization maps H d+2( K, Z/n(d) ) H d+2( K v, Z/n(d) ). Theorem 2.5 (Jannsen [Ja], Theorem 0.4). Recalling that n char(k), in the above notations, the localization maps give rise to an embedding Local-global principles over R v, v P(k 0 ) H d+2( K, Z/n(d) ) v P(k0 )H d+2( K v, Z/n(d) ). In the above notations, for every non-archimedean place v P(k 0 ), the Henselian ring R v is a DVR with finite residue field κ(v). This being said, one has the following: Theorem 2.6 (Kerz Saito [K S], Theorem 8.1). Let R is either a finite field or a Henselian discrete valuation ring with finite residue field. Suppose that n be invertible in R and µ n R. Let X be a proper regular flat S-scheme, where S = Spec R. Then the complex (KC) for X has trivial homology. 3. Consequences/applications of the local-global principles In this section we give a few consequences of the higher Hasse local-global principles mentioned above, as well as an arithmetical interpretation of these consequences. But first, we supplement the above Notations 2.1, 2.4 as follows: Notations/Remarks 3.1. Recall that K is a finitely generated field, and k 0 K is a relatively algebraically closed global subfield, and K separably generated over k 0. Further, n char(k) is a prime number such that µ n K, hence µ n k 0. 1) Let S 0 be the canonical model of k 0, that is: - If κ is a number field, then S 0 = Spec O κ. - If κ is a finite field, then S 0 is the unique projective smooth κ-curve with κ(s 0 ) = k 0. Further notice the following: a) For every non-empty subset Σ 0 P fin (k 0 ), the complement U 0 := S 0 \Σ 0 is actually an affine open subset of S 0. b) For non-zero δ k 0 = κ(s 0 ), let δ P fin (k 0 ) be the support of the divisor of δ. Notice that since k 0 is a global field, for very open subset U 0 S 0 there exists δ k 0 such that U 0 = S 0 \ δ. In these notation, consider: - δ k 0 is the set of all a k 0 such that a δ = O and a non-empty. 7

8 - nδ := { a δ v(a 1) > 2v(n), provided v(n) > 0 }. Next let K be finitely generated with dim(k) = 3, hence td(k k 0 ) 1 = dim(k) 2 = 1. 2) Let t K\k 0, and let k K be the relative algebraic closure of k 0 (t) in K, hence k is the function field of some (projective normal) k 0 -curve S, and Dk k 1 0 is in a canonical bijection with the closed points of S. For every δ 1 k let δ 1 := {w D k k0 w(δ 1 ) 0} be the support of the divisor of δ 1. Further define δ1 := {a k 0 w D 1 k k 0 one has: w(t a) > 0 w(δ 1 ) = 0} and let δ 1 := {a = (a, a ) 2 δ 1 a a } be the complement of the diagonal. 3) Since td(k k) = 1, there exists a unique projective normal k-curve C with k(c) = K, and the closed points P C are in bijection with the prime divisors D K k. For P C, let O P be the local ring at P, and v P be the k-valuation of K with O vp = O P. ( ) For functions f K\k, set D f := {P C v P (f) n v P (K)}. 4) Let f, t, δ 1, δ 1 be as above. For a := (a, a ) δ 1 and every δ k 0 we set: H f,t,a, δ := { } a b u f a nδ, b δ, u = ( t a t a H 4 K, Z/n(3) ) Theorem 3.2. In the above notations, the following assertions are equivalent: i) δ 1 k a δ 1 such that δ k 0, the set H f,t,a, δ is non-zero. ii) The set D f is non-empty. The proof of Theorem 3.2 is quite technical, and will be accomplished in several steps. A) Invariance under finite purely inseparable extension K K In this subsection we show the both conditions i) and ii) from Theorem 3.2 are invariant under finite purely inseparable extensions of K. The precise assertion is as follows: Let K K denote any finite extensions of K of degree [ K : K] prime to n, and k k and k 0 k 0 be the corresponding relative algebraic closures in K. Then letting C C be the normalization of the k 0 -curve in the field extension K K, one has K = k( C), and C is a projective normal k-curve. Further, C C gives rise to a surjective map f K f K between the support of the divisors of f in C and C. Finally, for every P C, we denote by C P C the fiber of C C above P C, and as at Notations/Remarks 3.1, 2) above define: - Df := { P C v P ( f) n v P ( K) } f. - H f,t,a, δ H 4( K, Z/n(3) ) for f, t, a and δ k 0 be correspondingly defined. Proposition 3.3. In the above notation, let K K denote finite extensions of K of degree [ K : K] prime to n. In the above notations, the following hold: 1) C C defines surjective maps f K f K, Df D f. Hence D f O iff D f O. 2) H f,t,a, δ is non-zero for all δ iff H f,t,a, δ is non-zero for all δ. Therefore, in order to prove the equivalence of i), ii) from Theorem 3.2, one can always replace K by any finite purely inseparable extension K K. 8

9 Proof. To 1): Recall that by the fundamental equality for the valuations v P extension K K, one has: in the field [ K : K] = P CP e( P P )f( P P ) = P CP [ K P : K P ]. In particular, since [ K : K] is prime to n, it follows that for every closed point P C, there exists some P C P such that the local degree e( P P )f( P P ) = [ K P : K P ]. is not divisible by n. Therefore, P D f iff v P (f) is not divisible by n if and only if v P (f) = e( P P )v P (f) is prime to n iff v P (f) = v P ( f) is prime to n iff P D f, as claimed. To 2): Since cor res is the multiplication by [ K : K], which is relatively prime to n, it follows that the restriction map res : H 4( K, Z/n(d) ) H 4( ) K, Z/n(d) is injective. Next let δ k 0 be given, and Σ S 0 be the image of δ under S 0 S 0. Then for every δ k 0 with Σ δ, it follows by mere definitions that res maps H f,t,a, δ into H f,t,a, δ, hence if the RHS is non-zero, so is the RHS. Conversely, for δ k 0, let δ k 0 be the image of δ under k 0 k 0, and suppose that H f,t,a, δ H 4( ) K, Z/n(d) is non-zero. Then for ã b u f 0 from H f,t,a,δ one has: Let m := p e be sufficiently large power of the characteristic (exponent) of k 0 such that a := ã m and b := b m lie in k 0. (Note that such an m exists by the fact that k 0 k 0 is purely inseparable.) Then one clearly has a nδ and b δ, and since m = p e and n are relatively prime, one has res(a b u f) = m 2 ã b u f 0. Hence a b u f is non-zero, concluding that H f,t,a, δ non-zero. The upshot of Proposition 3.3 is that after replacing K by finite purely inseparable extension K K, we can suppose that the embeddings k 0 k K satisfy: Notations/Remarks 3.4. The following hold: 1) There exist finite purely inseparable fields extensions k 0 k 0, k k with k 0 k such that setting K = Kk one has: a) K k 0 is the function field of a projective smooth k 0-surface X and k k 0 is the function field of a projective smooth k 0-curve S. Further, there exists a dominant morphism of k 0 varieties X S defining the field k 0-embedding k K. b) K k is the function field of a projective smooth k-curve C, and C = X S k is the generic fiber of X S. Hence replacing K by K, etc, mutatis mutandis, without loss of generality, in proving Theorem keythm, we can and will suppose the following: Hypothesis: There exists a dominant morphism of projective smooth k 0 -varieties X S defining the k 0 -embedding k 0 (S) = k K = k 0 (X). Further, the generic fiber of X S is a projective smooth k-curve C = X S k, thus K = k(c). 2) Since being smooth is an open condition, there exists open subsets U S and U 0 S 0 such that the following hold: 9

10 a) There are a projective smooth U 0 -surface X with generic fiber X = X U0 k 0 and a projective smooth U 0 -curve S with generic fiber S = S U0 k 0. Further, there is a U 0 -morphism X S whose base change to k 0 is the given X S. b) The base change X U = X S U U is a projective smooth U-curve with generic fiber C = X U U k = X S k. Hence the special fiber X s := X U κ(s), s U, is a projective smooth irreducible κ(s)-curve, and let x s X s be its generic point. The local ring O X,xs = O XU,x s is the valuation ring of the canonical constant reduction w s of K k at the valuation v s D k k0. And w s is a prime divisor of K k 0, having κ(w s ) = k 0 (X s ). Finally, after properly shrinking U, we can and will suppose that: - f is a regular function at w s, i.e., f is a w s -unit, and the residue f s κ(w s ) of f at w s is non-constant, and deg(f) = deg(f s ). - Let P s Div(X s ) be the divisor reduction of P f. Then P s is reduced, and if P s,i, i I P, are the irreducible components of P s, then P s = i P s,i. - If P, P f are distinct, then P s, P s have disjoint supports, and one has: deg(p ) = i deg(p s,i), v P (f) = v Ps,i (f s ) for all i I P ( ) In particular, P D f iff P s,i D fs for i I P. 3) In the above notations, let δ U0 k 0, δ U k satisfy S 0 = U 0 δ U0 and S = U δ U. B) A local-global principle for H δu0,n In the Notations/Remarks 2.1, and δ U0 as in Notations/Remarks 3.4, 3), define: H δu0,n := a b f 1 f 2 a nδu0, b δu0, f 1, f 2 K H 4( K, Z/n(3) ). Further recall that for the (projective smooth, thus) normal model X of K k 0, we denote by D 1 X the set of prime divisors of K k 0 which are defined by the Weil prime divisors of X, i.e., w D 1 X iff O w = O X, x for some point x X with dim(x) = dim(x) 1. Proposition 3.5. In the above notations, suppose that k 0 has no orderings, and µ n K. Then the composition of the canonical maps below is injective: H δu0,n H 4( K, Z/n(3) ) w D 1 X H 3( κ(w), Z/n(2) ). Proof. Let α 0 in H δu0,n H 4( K, Z/n(3) ) be given. Step 1. First, since α 0, by Jannsen s Theorem 2.5 mentioned above, there exists some v P(k 0 ) such that α 0 over K v = Kk 0v. And since k 0 allows no orderings, it follows that v P(k 0 ) must be a non-archimedean place. [Indeed, by contradiction, suppose that v is an archimedean place of k 0. Then k 0 is a number field, and k 0v = C. Hence k 0v being algebraically closed, K v = Kk 0v has cohomological dimension equal to td(k v k 0v ) = dim(k) 2, see e.g., Serre [Se]; thus concluding that H 4( K v, Z/n(3) ) = 0, contradiction!] Let O v k 0 be the valuation ring of v. Claim. v(n) = 0, and in particular, n is invertible in O v. Indeed, for every generator a b f 1 f 2 of H δu0,n one has: The element a nδ satisfies: v(a 1) > 2v(n), provided v n δ U0. Thus by Hensel Lemma over k 0v one has: Since v(a 1) > 2v(n), it follows that a is an n th power in the completions k 0v. Hence setting 10

11 a = c n for some c k 0v, one has a b f 1 f 2 = n (c b f 1 f 2 ) = 0 over K v. Since this holds for all the generators of the group H δu0,n, it follows that the image of H δu0,n in H 4( K v, Z/n(3) ) is trivial. Thus we conclude: If α 0 over K v, then v n δ U0, and therefore, v(n) = 0 and n is invertible in O v, as claimed. Step 2. By the conclusion of the Step 1 above, and in Notations/Remarks 3.4 above, it follows that α 0 over K v implies v U 0. In particular, the base change X Ov := X U0 O v of X U 0 under Spec O v U 0 is a projective smooth O v -schemes, because X U 0 was a projective smooth U 0 -scheme. Let R k 0v be the (unique) Henselization of O v inside k 0v. Setting κ R := Quot(R), consider the compositum K R := Kκ R K v, and the base change X R := X Ov Ov R X Ov defined by the canonical morphism O v R. Since X Ov is a projective smooth O v -scheme, and Spec R Spec O v is a pro-étale morphism, it follows that the base change X R is a projective smooth R-scheme, and X R X Ov is a pro-étale morphism as well. Hence one has: a) The fibers of X R are: i) The special fiber is X v = X R R κ(v). ii) The generic fiber is X κr = X R R κ R. b) Concerning points and their behavior under X R X, one has the following: First, since X R is a projective smooth, thus regular flat R-scheme, the local ring O xr of a point x R X R with dim(x R ) = 3 is a DVR with residue field κ(x R ) satisfying dim ( κ(x R ) ) = dim(x R ) 1 = dim(k R ) 1 = dim(k) 1 = 2. Further, for a point x R X R with dim(x R ) = 2, the following are equivalent: i) O xr dominates R, or equivalently, x R lies in the special fiber X v. ii) x R = η XR is the generic point of the special fiber X v. If so, then the image x R x X of x R under X R X is the generic point η Xv of the special fiber X v of X at v U 0. d) The following are equivalent: i) O xr does not dominate R, or equivalently, κ R O xr. ii) x R is the generic point of a (Weil) prime divisor of the generic fiber X κr. If so, then the image x R x X of x R under X R X is the generic point of a (Weil) prime divisor of the generic fiber X of X S 0. ( ) Hence we conclude that if x R X R has dim(x R ) = 2, then O xr is a DVR of K R, say with canonical valuation w xr, such that the restriction w := w xr K is a prime divisor of K. Moreover, if x R x under X R X, then dim(x) = 2, and O w = O x. Finally, the following conditions re equivalent: i) w is trivial on k 0. ii) R O xr, or equivalently, x R is not the generic point of the special fiber X v. 11

12 Coming back to the proof of the Proposition 3.5, we have: Since α is not-trivial over K v, and K R K v, it follows that the image α R H 4( K R, Z/n(3) ) of α over K R is non-trivial. Hence by the Kerz Saito Theorem 2.6 mentioned above, there exist x R X R, dim(x R ) = 2, such that α xr := xr (α R ) H 3( κ(x R ), Z/n(2) ) is nontrivial. Second, since α xr := xr (α R ) H 3( κ(x R ), Z/n(2) ) is non-trivial, it follows by mere definitions that α is non-trivial over the completion K wxr of K R. Hence setting w := w xr K, one has K w K wxr, and therefore we conclude: α is non-trivial over K w. One has: Case 1. w is trivial on k 0. Then w is a prime divisor of K k 0, and the proof of Proposition 3.5 is complete. Case 2. w is non-trivial on k 0. Then by the discussion at the beginning of Step 2, the center v S 0 of w under k 0 K lies in U 0. Hence the fiber X v of X U 0 at v is a projective smooth geometrically integral κ(v)-surface, having function field κ(x v ) = κ(w) = κ(x R ). Since α xr H 3( κ(x R ), Z/n(2) ) is non-trivial, by the Kato LGP applied to the projective smooth κ(v)-surface X v, there exists a point x v X v with dim(x v ) = 1 such that 0 xv (α xr ) H 2( κ(x v ), Z/n(1) ). Let X (x v ) X R be the points x R with dim(x R ) = 2 such that x v {x R }. Recalling the complex (KC) for the projective regular flat R-scheme X R, it follows that x R X (xv) x R, xv x R (α) = 0 On the other hand, x R X (x v ), and one has that xr,x v xr (α) = xr, x v (α xr ) 0. Hence there exist points x R X R, x v satisfying the following: - x R x R and x v {x R } {x R}. - α x R := x R (α R ) H 3( κ(x R ), Z/n(2)) is non-trivial. Let w x R be the canonical valuation of the DVR O x R, and set w := w x R K. Claim. w is a prime divisor of K k 0. Indeed, by the discussion above, we have to show that κ R O x R, or equivalently, that x R is not the generic point of the special fiber X v of X R. This is more or less obvious by the fact that x R x R and that x R = η Xv is the generic point of the special fiber of X R. Hence we conclude that w K is a prime divisor of K k 0 such that α is non-trivial over the completion K w. This completes the proof of Proposition 3.5. C) Proof of Theorem 3.2 The implication i) ii): Choose an arbitrary δ 1 k such that for all v 1 DK k 1 0 one has: v 1 (δ U ) v 1 (δ 1 ) 0, where δ U k is as in Definitions/Remarks 3.4, 3). Let a δ 1 be such that for all δ k 0 the set H f, t,a,δ1 is non-zero, and let α := a b u f H f,t,a,δ1 be non-zero. Then by Proposition 3.5, there exists a prime divisor w DX 1 such that α is non-zero over the completion K w, or equivalently, α w := w (α) H 3( κ(w), Z/n(2) ) is non-zero. One has the following case-by-case analysis: 12

13 Case 1. w k is trivial. Then w is a prime divisor of K k, hence w = v P for some P C, and α w = v P (f) a b u 1 H 3( κ(p ), Z/n(2) ). Hence α w 0 implies that v P (f) cannot be divisible by n, thus P D f, and D f is non-empty. Case 2. w k is not trivial, hence w k = v s for some closed point s S. Subcase 2.1. v s (u) = 0. Then the image u κ(v s ) of u in the residue field κ(v s ) κ(w) is algebraic over k 0 (by the fact that k k 0 is a function field of the k 0 -curve S). Hence one gets: α w = w(f) a b u H 3( κ(v s ), Z/n(2) ) H 3( κ(w), Z/n(2) ), and since κ(v s ) k 0 is a finite field extension, it follows that α w = 0 by the fact that the global field κ(v s ) has cohomological dimension 2 (because k 0 κ(v s ) allows no orderings). This contradicts the fact that α w 0, hence Subcase 2.1 is not possible. Subcase 2.2. v s (u) 0. Recall that, by definition, one has: u = (t a )/(t a ) with a a and a, a δ1. We claim that v s (t) = 0. Indeed, case-by-case analysis shows: First, if v s (t) < 0, then v s (t a ) = v s (t a ) < 0, hence v s (u) = 0, contradiction! Second, if v s (t) > 0, then v s (t a ) = v s (t a ) = 0, hence v s (u) = 0, contradiction! Therefore one has: Since v s (t) = 0 and v s (a ) = 0 = v s (a ), one has v s (t a ), v s (t a ) 0; finally, since 0 v s (u) = v s (t a ) v s (t a ), it follows that at least one of the values v s (t a ), v s (t a ) is strictly positive. Without loss of generality, let v s (t a ) > 0. Then one has: First, since t a = (t a ) + a with a = a a 0 in k 0, hence v s (a) = 0, one has v s (t a ) = 0. Second, since v s (t a ) > 0, and by hypothesis one has a δ1, it follows by definitions, see Definitions/Remarks 3.1, 2), that v s (δ 1 ) = 0. Hence by the choice of δ 1 as done at the beginning of the proof above, one has that v s (δ U ) = 0 as well, hence s U. Therefore, by the discussion from Notations/Remarks 3.4, 2), b), one has the following: First, recall that v s := w k is the restriction of w DX 1 to k. Let x X be the point of co-dimension one with w = w x, that is, the center of w on X. Since the center of w on S under k 0 (S) = k K = k 0 (X) is s U, it follows that x s under X S. And since s U, and X U := X S U, it follows that x X U lies in the fiber X s := X S κ(s). Thus one must necessarily have that x = x s X s is the generic point of the special fiber X s of X S at s U. Hence by Definitions/Remarks 3.4, b), one has: f is a v s -unit, and recalling that f s is the image of f in κ(w) = k 0 (X s ), one has: 0 α w = v s (u 1 ) a b f s H 3( κ(w), Z/n(2) ). We thus conclude that for every δ k 0, there exist a nδ, b δ such that the symbol a b f s H 3( κ(v s ), Z/n(2) ) is non-zero. Hence by Pop [8], Lemma 3.3, it follows that D fs O. Thus by Notations/Remarks 3.4, 2), ( ), it follows that D f O as well. The implication ii) i): Let P D f be fixed. Then k := κ(p ) is a finite extension of k of degree [κ(p ) : k] = deg(p ), hence k k 0 (t) is a finite extension is well. Let l k 0 (t) be the separable part of k k 0 (t). Then only finitely many places of k 0 (t) ramify in k P k 0 (t), and therefore, there exists a finite subset Σ k 0 such that for all a k 0 \Σ one has: The (t a)-adic valuation v a of k 0 (t) is not ramified in l k 0 (t). Therefore, if ṽ a is any prolongation of v a to k, it follows that e(ṽ a v a ) is a power of the characteristic exponent of k 0, hence prime to n. Hence there exists δ P k 0 (t) k satisfying: If ṽ is a prime divisor of D k k0 k k 0 such that ṽ(δ P ) = 0, then the ramification index of ṽ in the field extension k k 0 (t) is not divisible by n. With this choice/hypothesis, let a = (a, a ) δ 1 be given, and for 13

14 a {a, a }, consider the valuations ṽ a v a of k k 0 (t) as above. Then for δ k 0 and a nδ, b δ, and α = a b u f one has: α v P v P (f) a b u ṽa v P (f) ṽ a (u) a b H 2( κ(ṽ a ), Z/n(1) ) In particular, since v P (f) and ṽ a (u) are not divisible by n, it follows that α 0, provided a b H 2( κ(ṽ a ), Z/n(1) ) is non-zero. On the other hand, κ(ṽ a ) k 0 is a finite extension. Hence arguing word-by-word as in Pop [8], 3, C), it follows that for the finite extension κ(ṽ a ) k 0 and every δ k 0 there exist many a nδ, b δ such that the image of a b H 2( k 0, Z/n(1) ) in H 2( κ(ṽ a ), Z/n(1) ) is non-zero. Hence finally H f,t,a,δ is non-zero. The proof pf Theorem 3.2 is complete. 4. Definability of the k-valuations of K k In this section we work in the context and notations of the previous sections, but suppose that n = 2. In particular, dim(k) = 3, k 0 K is a global subfield, relatively algebraically closed in K, and k K is the relative algebraic closure of k 0 (t) in K, where t K \k 0 is arbitrary. Our final aim in this section is to show that the prime divisors of K k are uniformly first order definable. Precisely, we will give formulas val( t, u, γ 1 k, a δ 1, γ k 0, a nδ, b δ ; x ) which interpreted as a predicate in the variable x define all the valuation rings O w K of prime k-divisors K k. These formulas involve among other things the formulas which uniformly describe the prime divisors of finitely generated fields of Kronecker dimension 2, as given in Pop [8]; which themselves involve the formulas which uniformly describe the prime divisors of global fields given by Rumely [10]. Finally, we should noticed that the case char(k) = 2 is still open, and the methods of this manuscript do not work so far in that case. The generic type of result we will prove in section is therefore Theorem 4.1. There exist formulas which uniformly describe the prime divisors of finitely generated fields K with char(k) 2 and dim(k) 3. Before proceeding, we notice that describing uniformly the prime divisors of finitely generated fields K with char(k) 2 and dim(k) 3 is the same as uniformly describing the prime divisors of such fields K with µ 4 K. Therefore, from now on we will tacitly assume that µ 4 K, hence in particular, K admits no orderings. Further, we should mention that the formulas under discussion are completely explicit; but for the moment it does not appear that these formulas are optimal, in particular, we do make any claims about the complexity of uniform defininability of some or all the prime divisors. A) The definable subsets U f, t, M f, t, m f, t, U f, t, O f, t K Notations/Remarks 4.2. We supplement Notations/Remarks 3.1 and 3.4 as follows. 1) For ε K, let K ε := K[ n ε ], and consider the restriction map res ε : H 4( K, Z/n(3) ) H 4( K ε, Z/n(3) ). 2) Let C ε C, P ε P, be the normalization of C in the field extension K ε K, and denote by D f, ε the set of all P ε C ε such that v Pε (f) n v Pε (K ε ). 14

15 3) For every P C, let U P := O P be the P -units, and let U P O vp be the v P -units. Then for ε K the following are equivalent: i) ε U P K n. ii) v P (ε) n v P (K) Indeed, i) ii) is clear. For ii) i), let v P (ε) n v P (K), and η K be such that v P (ε) = n v P (η) = v P (η n ). Then ε/η n U P, hence one has ε U P K n. Lemma 4.3. In the above notations, the assertions below are equivalent: i) δ 1 k a δ 1 such that δ k 0 the set res ε (H f, t, a, δ ) is non-zero. ii) ε P Df U P K n. In particular, the set U f, t := P Df U P K n K is first oder definable in K. Proof. The implication i) ii): Consider any ε P Df U P K n that is, ε U P K n for all v D f. Then by Notations/Remarks 4.2, 3), it follows that v P (ε) n v P (K) for all P D f. Therefore, for all P D f, and any prolongation P ε to K ε one has e(p ε P ) = n, and therefore, v Pε (f) = e(p ε P ) v P (f) n v Pε (K ε ). Further, since v P (f) n v P (K) for P D f, one has v Pε (f) n v Pε (K ε ) for P ε P D f. Thus we finally conclude that v Pε (f) n v Pε (K ε ) for all P ε C ε. On the other hand, by hypothesis i), applying Theorem 3.2 to K ε, it follows that D f, ε is non-empty, contradiction! ii) i): Let ε P Df U P K n be given, and P D f be such that ε U P K n. Then P is unramified in the extension K ε K. Hence if C ε C is the normalization of C in the field extension K ε K, it follows that v Pε (f) = v P (f) is prime to n. Further, w.l.o.g., we can suppose that ε U P, and is so, then κ(p ε ) = κ(p )[ n ε ], where ε is the residue of ε in κ(p ). Further, let δ 1 k be chosen such that it contains the discriminant of function field extension κ(p ε ) κ(p ) which are finite extensions of k. That is, if v 1 Dκ(P 1 ) k 0 and v 1 (δ 1 ) = 0, then v 1 is not ramified in κ(p ε ) κ(p ). Equivalently, for all prolongations v 1,ε v 1 of v 1 to κ(p ε ), then the ramification index satisfies e(v 1,ε v ε ) = 1. Therefore, for any α = a b u f H f,t, a,δ, and β = res ε (α) H 4( K ε, Z/n(3) ), one has: β v Pε v Pε a b u v 1,ε v Pε v 1,ε (u) a b H 2( κ(v 1,a ), Z/n(1) ) In particular, if a b H 2( κ(p ε ), Z/n(1) ) is non-zero, then β H 4( K ε, Z/n(3) ) is nonzero. Arguing as at the end of the proof of Theorem 3.2, conclude that there exist many a nδ, b δ such that a b H 2( κ(v 1,ε ), Z/n(1) ) is non-zero, hence β is non-zero. Notations/Remarks 4.4. In the notations from Lemma 4.3 above, we have the following: 1) Let ε K \ U f, t be given. Then by Notations/Remarks 4.2, 3), it follows that for all P D f one has: v P (ε) n v Pε (K ε ). In particular, v P (ε) 0, and therefore one has: - If v P (ε) > 0, then 1 + ε 1 + m P, hence finally 1 + ε U f, t. - If v P (ε) < 0, then v P (1 + ε) = v P (ε) n v P (K). Thus by the discussion at Notations/Remarks 4.2, 3), it follows that 1 + ε U P K n. 2) Hence we conclude that for ε K one has: v P (ε) < 0 for all P D f and v P (ε) n v P (K) for all P D f iff ε, 1 + ε U f, t. 3) In particular, M f, t := {η K η 1, 1 + η 1 U f, t } K is definable, and one has: η M f, t iff P D f one has: v P (η) > 0, v P (η) n v P (K). 15

16 Proposition 4.5. In the above notation, one has the following: m f, t := P Df m P = M f, t M f, t, hence m f, t K is definable. Therefore, the subsets of K below are definable as well: R f, t := P Df O P = { θ K θ m f, t m f, t }, U f, t := P Df U P = {ɛ K ɛ m f, t = m f,t } Proof. Since M f m f and the latter is closed w.r.t. addition, it follows that M f M f m f. For the converse inclusion, let θ m f be arbitrary. Since m f = P Df m P, it follows that v P (θ) > 0 for all P D f. Hence by the weak approximation lemma, it follows that there exists η K such that v P (η θ) = 1 and v P (η) = 1 for all P D f. Hence η, η := η θ M f, and obviously one has: θ = η η M f M f. Finally,the definability of the last two subsets follows by mere definitions, and the equalities involved are well known basic valuation theoretical facts (which follow, e.g. using the weak approximation lemma). B) Defining the k-valuation rings of K k In the notations and hypotheses the previous sections, recall that K = k(c) for some projective normal k-curve C. By Riemann-Roch we have: Let p be the characteristic exponent of K. then for every closed point P C and m, e >> 0, there exist functions f K such that (f) = p e m P. Hence choosing m to be prime to n = 2, we have P D f. Thus by Proposition 4.5, it follows that P D f and R f, t = O P R 0f, where R 0f = P D f O P with P P the zeros of f which lie in D f. For f as above, we set g := f+1, and notice that (g) = p e mp = (f), etc., and obviously, f and g have no common zeros. We repeat the constructions above with f replaced by g, and get R g, t = O P R 0g, where R 0g = Q D g O Q with Q P the zeros of g from D g. Since div(f) div(g) = {P }, by the weak approximation lemma one has: and finally one recovers m P as being R f, t R g, t = O P m P = {θ K θ O P, θ 1 O P } Hence we have the following first oder recipe to define all the k-valuation rings of K k. Recipe 4.6. Let K be a finitely generated field with dim(k) = 3 and char(k) 2. Recall that defining the prime divisors of K is equivalent to defining the prime divisors of K k, where K := K[µ 4 ], k := k[µ 4 ]. Thus without loss of generality, supposing that µ 4 K, consider the following steps: 1) The global subfields k 0 K, which are relatively algebraically closed in K are defined by the Poonen predicate ψ 1 (s). 2) For every such global subfield k 0 K as above, and t K\k 0, using Poonen s predicate ψ(t, t ), let k K be the relative algebraic closure of k 0 (t) in K. ( ) Recall that the valuations of k 0 are uniformly first order definable by Rumely [10], and that the prime divisors of k k 0 are uniformly first order definable by Pop [8]. 3) Consider all f K such that δ 1 k a δ 1 and u = (t a )/(t a ) such that δ k 0 a nδ, b δ with H f, t, a,δ non-zero. Note that the latter condition is expressible as a condition on the Pfister froms q f, t, u,δ. 16

17 4) For f, t, g := f + 1 as above, and the definable subrings R f,t, R g, t K, check whether O := R f, t R g, t is a valuation ring of K. If so, O = O w for a prime divisor w of K k. 6) Finally note that all the prime divisors O w, m w of the finitely generated field K are obtained by the recipe above for various k 0 K, t K \k 0, f K \k. 5. Proof of Main Theorem and Theorem 1.3 A) Concluding the proof of Theorem 1.3 First, the above Recipe 4.6 provides a uniform first order definability of the k-valuation rings of function fields of complete normal k-curves, where k are finitely generated fields with dim(k) = 2, char(k) 2. One proceeds as in the last part of Pop [8]. For reader s sake, we repeat here the quite standard arguments. In order to give the formula deg N (t), we first recall that k is a Hilbertian field. Hence for every non-constant function t K there exist (infinitely many) specializations t a k such that the point P X with t = a in κ(p ) is unique, hence [κ(p ) : k] = [K : k(t)] is the degree of t. Thus one possibility for the formula def N (t) would be the following: ( O, m: t k + m [O/m : k] N ) & ( O, m: t k + m and [O/m : k] = N ) For the formula ψ R (t, t ) we recall that one has: t O k[t] O R t O. Further, R t is actually the intersection of all the O which contain k[t], or equivalently, which contain t. Thus the formula ψ R (t, t ) could be: O, m: t O t O Finally, for the formula ψ(t, t ), recall that t K lies in k[t] if and only if t R t and for all O, m one has: If the residue t O/m lies in k O/m, then the residue t O/m lies in k O/m. (Indeed, this follows again from the fact that k is Hilbertian.) Thus one possibility for the formula ψ(t, t ) could be: O, m: t O t O & t m + k t m + k This completes the proof of Theorem 1.3. B) Proof of Main Theorem This follows along the lines of Fact 1.2 in Introduction. Let namely T = (T 1, t 2 ) be a transcendence basis of K κ in general position, and f K (T, U) be as in loc.cit. Then the isomorphy type of K as a finitely generated field is given by that of k := κ(t 1 ) and the polynomial f K (T, U). Now by the main results of Pop [7], combined with the description of the absolute constants by Poonen [6], and the main results from [8], one has: 1) There exists a sentence ϕ 0 K ( T u ϕk) 0 such that ϕ 0 K holds in K, and if K is a finitely generated field in which ϕ 0 K holds, then there exists an embedding of fields ı : K K such that the following are satisfied: i) ı(κ) = κ, i.e., K and K have the same absolute subfields. ii) ( T 1, t 2 ) := ı(t 1, t 2 ) is transcendence basis of K κ in general position. 2) In particular, setting ϑ K ϕ 0 K & deg N(u), where N := [K : k(u)], one has: If ϑ K holds in K, then N = [ K : k(ũ)]. Therefore, [K : k(u)] = [ K : k(ũ)], hence the embedding K K is an isomorphism. 17