Distinguishing every finitely generated field of characteristic \neq2 by a single field axiom
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DISTINGUISHING EVERY FINITELY GENERATED FIELD OFCHARACTERISTIC = 2 BY A SINGLE FIELD AXIOM † THE STRONG ELEMENTARY EQUIVALENCE VS ISOMORPHY PROBLEM IN CHARACTERISTIC = 2 FLORIAN POP
Abstract.
We show that the isomorphy type of every finitely generated field K withchar( K ) = 2 is encoded by a single explicit axiom ϑ K in the language of fields , i.e., forall finitely generated fields L one has: ϑ K holds in L if and only if K ∼ = L as fields. Thisextends earlier results by Julia Robinson, Rumely, Poonen, Scanlon , the author, and others. Introduction
We begin by recalling that a sentence , or an axiom in the language of fields is any formulain the language of fields which has not free variables. One denotes by Th ( K ) the set of allthe sentences in the language of fields which hold in a given field K . For instance, by meredefinitions, the field axioms are part of Th ( K ) for every field K ; further the fact that K isalgebraically closed, as well as char( K ) are encoded in Th ( K ). Namely, K is algebraicallyclosed iff K satisfies the scheme of axioms of algebraically closed fields (asserting that everynon-zero polynomial p ( T ) over K has a root in K ); respectively one has char( K ) = p > K satisfies the char = p scheme of axioms (asserting: char = p > p i=1 n i=1 = 0 for all n ). On the other hand, if K := Q ( t ) is the rational functionfield in the variable t over Q , then the usual way to say that t is transcendental over Q ,namely “ p ( t ) = 0 for all non-zero polynomials p ( T ) over Q ” is not a scheme of axioms inthe language of fields (because t is not part of the language of fields).Two fundamental general type results in algebra are the following:- Algebraically closed fields K, L have Th ( K ) = Th ( L ) iff char( K ) = char( L ).- Arbitrary fields K, L have Th ( K ) = Th ( L ) iff there are isomorphic ultra-powers ∗ K ∼ = ∗ L .Restricting to fields which are at the center of (birational) arithmetic geometry, namelythe finitely generated fields K , which are the function fields of integral schemes of finitetype, the elementary theory Th ( K ) is both extremely rich and mysterious. The so called Elementary equivalence vs Isomorphism Problem (EEIP) is about five decades old, and askswhether Th ( K ) encodes the isomorphy type of K in the class of all the finitely generated 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/´etale cohomology, Kato’s higher local-global Principles † Note : This manuscript is a revised version of
Pop [P5]. The main result of this manuscript was alsoannounced by
Dittmann [Di] (using the same technical tools, but expending rather on
Poonen [Po],
Pop [P4]). elds; or equivalently, whether there exists a system of axioms in the language of fields whichcharacterizes K among all the finitely generated fields. On the other hand, building on JuliaRobinson [Ro1], [Ro2] methods and ideas,
Rumely [Ru] showed at the end of the 1970’s thatfor every global field K there exists a sentence ϑ Ru K which characterizes the isomorphy type of K as a global field, i.e., if L is a global field, then ϑ Ru K holds in L iff K ∼ = L as fields. In otherwords, the isomorphy type of K as a global field is characterized by a single explicit axiom ϑ Ru K in the language of fields . This goes far beyond the EEIP in the class of global fields!Arguably, it is the main open question in the elementary (or first-order) theory offinitely generated fields whether a fact similar to Rumely ’s result [Ru] holds for all finitelygenerated fields K , namely whether there is a field axiom ϑ K which characterizes the iso-morphy type of K in the class of all finitely generated fields; this question is also calledthe strong EEIP . We notice that the (strong) EEIP is open in general; see Pop [P2], [P3],for more details and references on the EEPI both over finitely generated fields and functionfields over algebraically closed base fields. A first attempt towards tackling the strong EEIPwas
Scanlon [Sc], and that reduces the strong EEIP for each K to first-order defining “suf-ficiently many” divisorial valuations of K . Finally,
Pop [P4] tackles the strong EEIP forfinitely generated fields which are function fields of curves over global fields. In the presentnote we generalize that result to all finitely generated fields of characteristic = 2. Main Theorem.
For every finitely generated field K with char( K ) = 2 , there exists asentence ϑ K in the language of fields such that for all finitely generated fields L one has: ϑ K holds in L if and only if L ∼ = K as fields. The Main Theorem above will be proved in Section 5. One can give three (by some stan-dards similar) proofs. A first-proof follows simply from
Scanlon , by invoking Theorem 1.1below for the definability of geometric prime divisors (thus circumventing the gap in the proofof defining divisorial valuations in Section 3 of loc.cit.). A second proof reduces the MainTheorem above to results by
Aschenbrenner–Kh´elif–Naziazeno–Scanlon [AKNS], by showingthat finitely generated integrally closed subdomains in finitely generated fields of character-istic = 2 are uniformly first-order definable. Among other things, these proofs show thatfinitely generated fields of characteristic = 2 are bi-interpretable with arithmetic , seee.g. [Sc], Section 2, and/or [AKNS], Section 2, for a detailed discussion of bi-interpretablilitywith arithmetic. Third, a more direct proof based on Pop [P2],
Poonen [Po1], and conse-quences of
Rumely [Ru] (namely that the number fields are bi-interpretable with arithmetic).The main step and technical key point in the proof of the Main Theorem is to give formulae val d , all d >
0, in the language of fields, which uniformly first-order define the geometric primedivisors of finitely generated fields K with char( K ) = 2 and dim( K ) = d . That is the contentof Theorem 1.1 below, which could be viewed as the main result of this note.To make these assertions more precise, let us introduce notation and mention a few fun-damental facts about finitely generated fields, to be used throughout the manuscript.For arbitrary fields Ω, let κ ⊂ Ω denote their prime fields. Recall that the Kronecker di-mension of Ω is dim(Ω) = dim( κ ) + td(Ω | κ ), where td(Ω | κ ) denotes the transcendencedegree, and dim( F p ) = 0, dim( Q ) = 1. We denote by κ := Ω abs the constant subfield of Ω,i.e., the elements of Ω which are algebraic over the prime field κ ⊂ Ω, and set ˜Ω := Ω[ √− . See the discussion below for more about this. or aa := ( a ,..., a r ) with a i ∈ Ω × we consider the r -fold Pfister form q aa ( x ) in the variables x = ( x ,..., x r ) and for field extensions Ω ′ | Ω define the image of Ω ′ under q aa as being q aa (Ω ′ ) := { q aa ( x ′ ) | x ′ ∈ Ω ′ r , x ′ = } . Next we recall that, using among other things the Milnor Conjecture, by Pop [P2] thereare sentences ϕ d , and by Poonen [Po1] there is a predicate ψ abs ( x ), and formulas ψ r ( t ) withfree variables t := ( t ,..., t r ) such that for all finitely generated fields K and κ = K abs ⊂ K ,setting ˜ K := K [ √− K ) = d iff ϕ d holds in K . Actually, ϕ d ≡ (cid:0) ( ϕ d ∧ ∨ ( ϕ d +1 ∧ = 0) (cid:1) , where ϕ r ≡ (cid:0) ∃ aa = ( a ,..., a r ) s.t. 0 q aa ( ˜ K ) (cid:1) & (cid:0) ∀ aa = ( a ,..., a r +1 ) one has 0 ∈ q aa ( ˜ K ) (cid:1) . - κ is defined by ψ abs ( x ) inside K , i.e., one has κ = { x ∈ K | ψ abs ( x ) holds in K } .- t ,..., t r ∈ K are algebraically independent over κ iff ψ r ( t ,..., t r ) holds in K .In particular, for algebraically independent elements tt r := ( t , . . . , t r ) of K , the relativealgebraic closure k tt r of κ ( tt r ) in K is uniformly first-order definable, hence so are the maximalglobal subfields k ⊂ K of K , as well as the transcendence bases T := ( t , . . . , t d K ) of K | κ .A prime divisor of K is (the valuation ring of) a valuation v whose residue field Kv satisfiesdim( Kv ) = dim( K ) − . It turns out that prime divisors v of finitely generated fields are discrete valuations, and Kv is a finitely generated fields as well. A prime divisor v of K is called arithmetic , if v isnon-trivial on κ = K abs — in particular κ must be a number field, respectively geometric ,if v is trivial on κ . Recall that Rumely [Ru] gives formulae val which uniformly first-orderdefine the prime divisors of global fields, and Pop [P4] gives formulae val which uniformlyfirst-order define the geometric prime divisors in the case dim( K ) = 2. The focus of thisnote is to give similar formulae val d for fields • satisfying: (H) • is finitely generated, d := dim( • ) >
2, char( • ) = 2 Theorem 1.1.
There is an explicit procedure that, given an integer d > , produces a first-order formula val d that in any finitely generated field K of characteristic char( K ) = 2 andKronecker dimension dim( K ) = d defines all the geometric prime divisors of K . For the proof see Section 4, Theorem 4.2, and Recipe 4.8 for the concrete form of val d .We conclude the Introduction with the following remarks.First, in the early version [P5], the case of finitely generated fields of characteristic zerowas considered/dealt with. The methods and techniques of [P5] are unchanged, except a keytechnical point of the procedure of giving val d , namely the old Proposition 3.5, whose newvariant Proposition 3.4 below works for all finitely generated fields satisfying Hypothesis (H).Second, although the formulae val d are completely explicit, see Recipe 4.8, it is an openquestion whether these formulae are optimal in any concrete sense; in particular, the for-mulae val d do not address the question about the complexity of (uniform) definability of (some or all) the prime divisors. The complexity of definability of valuations deserves See e.g.
Pfister [Pf1], Ch. 2, for basic facts. Proved by
Vojevodsky, Orlov–Vishik–Vojevodsky , and
Rost , see e.g. the survey articles [Pf2], [Kh]. urther special attention, because among other things it ties in with previous first-orderdefinability results of valuations (of finitely generated fields and more general fields) by Eisentr¨ager [Ei],
Eisentr¨ager–Shlapentokh [E-S],
Kim-Roush [K-R],
Koenigsmann [Ko1, Ko3],
Miller-Shlapentokh [M-Sh],
Poonen [Po2],
Shlapen- tokh [Sh1], [Sh2], and others. The focus ofthe aforementioned results and research is yet another open problem in the theory of finitelygenerated fields and function fields, namely the generalized Hilbert Tenth Problem — whichfor the time being is open over all number fields, e.g. Q , C ( t ), etc.Third, it is strongly believed that the (strong) EEIP should hold for the function fields K | k over “reasonable” base fields k ; in particular, since finitely generated fields of characteristic = 2 are nothing but function fields K over prime fields with char = 2, the Main Theoremabove asserts that Q and F p , p = 2, are “reasonable.” If k is an algebraically closed field,facts proved by Durr´e [Du],
Pierce [Pi],
Vidaux [Vi] for td( K | k ) = 1, respectively Pop [P2]for td( K | k ) arbitrary, are quite convincing partial results supporting the possibility thatalgebraically closed fields are “reasonable.” Finally, Koenigsmann [Ko2],
Poonen-Pop [P-P] giveevidence for the fact that the much more general large fields k , as introduced in Pop [P1], e.g. k = R , Q p , PAC, etc., should be “reasonable” base fields. These partial/preliminary resultsover large fields (including the algebraically closed ones) do not involve prime divisors of K | k . Two fundamental open questions arise: First, is it possible to recover prime k -divisorsof functions fields K | k over large fields k , at least in the case of special classes of large fields,e.g. local fields, or quasi-finite fields? Second, are there alternative approaches (which donot involve prime divisors) for recovering the isomorphy type of K from Th ( K )? 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, MFO Oberwolfach in 2016, IHP Paris in 2018, forthe debates on the topic and suggestions concerning this problem. Special thanks are due to Bjorn Poonen,Thomas Scanlon, Jakob Stix, and Michael Temkin for discussing technical aspects of the proofs, and to UweJannsen and Moritz Kerz for discussions concerning Kato’s higher dimensional Hasse local-global principles.The author would also like to thank the University of Heidelberg and the University of Bonn for the excellentworking conditions during his visits in 2015 and 2016 as a
Humbodt Preistr¨ager. Higher dimensional Hasse local-global principles A) Notations and general facts
For a (possibly trivial) valuation v of K , let m v ⊂ O v ⊂ K be its valuation ideal andvaluation ring ring, vK := K × /U v be its (canonical) value group, and Kv := κ ( v ) := O v / m v be its residues field. We denote by V K the Riemann space of K , i.e., the space of all the(equivalence classes of) valuations of K .Let X be a scheme of finite type over either Z or a field k . For x ∈ X , let X x := { x } ⊂ X be the closure of x in X , and recall that dim( x ) := dim( X x ). Following Kato [Ka], define: X i := { x ∈ X | dim( x ) = i } , X i := { x ∈ X | codim( x ) = i } , and recall that if X is projective normal integral, then for all 0 i dim( X ) one has:dim( X ) = codim( x ) + dim( x ) , and therefore: x ∈ X i ⇔ x ∈ X dim( X ) − i Notations/Remarks 2.1.
Let K be a finitely generated field, and k ⊂ K be a subfield.1) A model of K is a separated scheme of finite type X with function field κ ( X ) = K . Anda k -model of K is a k -variety, i.e., a separated k -scheme of finite type, with k ( X ) = K . ) Let a model X of K , and v ∈ V K be given. We say that v has center x ∈ X 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 ∈ V K has at most one center on X , respectively: X isproper iff every valuation v ∈ V K has a center on X (which is then unique).3) Let a k -model X of K and v ∈ V K be given. We say that x ∈ X is the center of v on X , if O x ≺ O v . If so, then v ∈ V K | k . By the valuation criterion one has: Since X isseparated over k , every v has at most one center on X , respectively that X is a proper k -variety iff every k -valuation v ∈ V K | k has a center on X (which is then unique).4) A prime divisor of K is any v ∈ V K satisfying the following equivalent conditions:i) dim( Kv ) = dim( K ) − v is discrete, and Kv is finitely generated and has dim( Kv ) = dim( K ) − v is defined by a prime Weil divisor of a projective normal model X of K .5) A prime k -divisor of K is any v ∈ V K | k satisfying the following equivalent conditions:i) td( Kv | k ) = td( K | k ) − v is a prime divisor of K which is trivial on k .iii) v is defined by a prime Weil divisor of a projective normal model X of K | k .6) Let D K ⊃ D X be the spaces of prime divisors of K , respectively the ones defined bythe prime Weil divisors of a quasi-projective normal model X of K . Further define D K | k ⊃ D X correspondingly, where X is a quasi-projective normal k -model of K .7) In the above notation, let X and X be projective. Then one has canonical identifications: D X ↔ X = X dim( K ) − , D X ↔ X = X td( K | k ) − .B) Local-global principles (LGP)Let us first recall the famous Hasse–Brauer–Noether LGP. Let k be a global field, P ( k )be the set of non-trivial places of k , and for v ∈ P ( k ), let k v be the completion of k withrespect to v . Denoting by n ( ) the n -torsion in an Abelian group, e.g. n ( Q / Z ) ∼ = Z /n , theHasse–Brauer–Noether LGP asserts that one has a canonical exact sequence:0 → n Br( k ) → ⊕ v n Br( k v ) → Z /n → n Br( k ) → n Br( k v ),and the second map is the sum of the invariants P v inv v .It is a fundamental observation by Kato [Ka] that the above local-global principle hashigher dimensional variants as follows: First, following
Kato loc.cit, for every positive inte-ger n , say n = mp r with p the characteristic exponent and ( m, p ) = 1, an integer twist i , onesets Z /n (0) = Z /n , and defines in general Z /n ( i ) := µ ⊗ im ⊕ W r Ω i log [ − i ], where W r Ω log is thelogarithmic part of the de Rham–Witt complex on the ´etale site, see Illusie [Ill] for details.With these notations, for every (finitely generated) field K one has:H (cid:0) K, Z /n (0) (cid:1) = Hom cont ( G K , Z /n ), H (cid:0) K, Z /n (1) (cid:1) = n Br( K ),where G K is the absolute Galois group of K . Thus the cohomology groups H i +1 (cid:0) K, Z /n ( i ) (cid:1) have a particular arithmetical significance, and in these notation, the Hasse–Brauer–NoetherLGP is a local-global principle for the cohomology group H (cid:0) K, Z /n (1) (cid:1) . Noticing that K isa 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 (cid:0) K, Z /n ( d ) (cid:1) . The Kato cohomological complex (KC)
We briefly recall Kato’s cohomological complex (similar to complexes defined by theBloch–Ogus) which is the basis of the higher dimensional Hasse local-global principles,see
Kato [Ka], §
1, for details. Let L be an arbitrary field, and recall the canonical isomor-phism (generalizing the classical Kummer Theory isomorphism) h : L × /n → H (cid:0) L, Z /n (1) (cid:1) .As explained in [Ka], §
1, the isomorphism h gives rise canonically for all q = 0 to morphisms,which by the (now proven) Milnor–Bloch–Kato Conjecture are actually isomorphisms: h q : K M q ( L ) /n → H q (cid:0) L, Z /n ( q ) (cid:1) , { a , . . . , a q } /n h ( a ) ∪ ... ∪ h ( a q ) =: a ∪ ... ∪ a q . Further, let v be a discrete valuation of L . Then one defines the boundary homomorphism ∂ v : H q +1 (cid:0) L, Z /n ( q + 1) (cid:1) → H q (cid:0) Lv, Z /n ( q ) (cid:1) , defined by a ∂ v v ( a ) if q = 0, a ∪ a ∪ ... ∪ a q ∂ v v ( a ) · a ∪ ... ∪ a q for a ∈ L × , a ,..., a q ∈ U v if q > X be an excellent integral scheme, with generic point η X , and recall the notations X i , X i ⊂ X ; hence X ⊂ X are the closed points, and X dim( X ) = { η X } . By mere definitions,for every x i +1 ∈ X i +1 , one has that X x i +1 , ⊂ X i consists of all the points x i ∈ X i which liein the closure of X x i +1 := { x i +1 } . Since X is excellent, the normalization ˜ X x i +1 → X x i +1 of X x i +1 is a finite morphism. Hence for every x i ∈ X x i +1 , , there a finitely many ˜ x ∈ ˜ X x i +1 such that ˜ x x i under ˜ X x i +1 → X x i +1 , and the following hold: The local rings O ˜ x of all˜ x x i are discrete valuations rings of the residue field κ ( x i +1 ), say with valuation v ˜ x , andthe residue field extensions κ (˜ x ) | κ ( x i ) are finite field extensions. Then for every integer n >
1, which is invertible on X , letting 0 i < dim( X ), one gets a sequence of the form: (KC) ... → ⊕ x i +1 ∈ X i +1 H i +2 (cid:0) κ ( x i +1 ) , Z /n ( i +1) (cid:1) → ⊕ x i ∈ X i H i +1 (cid:0) κ ( x i ) , Z /n ( i ) (cid:1) → ... where the component H i +2 (cid:0) κ ( x i +1 ) , Z /n ( i + 1) (cid:1) → ⊕ x i ∈ X i H i +1 (cid:0) κ ( x i ) , Z /n ( i ) (cid:1) is defined by P ˜ x ∈ ˜ X xi +1 , ˜ x x i cor κ (˜ x ) | κ ( x i ) ◦ δ v ˜ x Theorem 2.2 ( Kato [Ka], Proposition 1.7) . Suppose that X is an excellent scheme suchthat for all p dividing n and x i ∈ X i one has: If p = char (cid:0) κ ( x i ) (cid:1) , then [ κ ( x i ) : κ ( x i ) p ] ≤ p i .Then (KC) is a complex. In particular, if n is invertible on X , then (KC) is a complex. That being said, the
Kato Conjectures are about aspects of the fact that in arithmeticallysignificant situations, the complex (KC) above is exact, excepting maybe for i = 0, where thehomology of (KC) is perfectly well understood. And Kato proved himself several forms of theabove local-global Principles in the case X is an arithmetic scheme of dimension dim( X ) = 2and 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: → H (cid:0) K, Z /n (2) (cid:1) → ⊕ x ∈ X H (cid:0) κ ( x ) , Z /n (1) (cid:1) → ⊕ x ∈ X H (cid:0) κ ( x ) , Z /n (cid:1) → Z /n → . 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 formin higher dimensions d := dim( K ) >
2, although it is conjectured to be so. There arenevertheless partial results concerning the local-global principles involving H d +1 (cid:0) K, Z /n ( d ) (cid:1) .From those results, we pick and choose only what is necessary for our goals, see below. otations/Remarks 2.4. Let K be a finitely generated field with constant field κ . Wesupplement Notations/Remarks 2.1 as follows:1) n > K ).2) k ⊂ K is global subfield, and S be the canonical model of k , i.e., S = Spec O k if k is a number field, respectively S is a projective smooth curve if κ is finite.3) Let P fin ( k ) be the set of finite places of k . For v ∈ P ( k ), consider/denote:- The Henselization R v of O v . Hence R v is a Henselian DVR with finite residue field.- The Henseliazation k v = Quot( R v ) of k at v . • Localizing the global field k In the above notations, for every v ∈ P ( k ), consider the compositum K v := Kk v of K and k v (in some fixed algebraic closure K ). Then via the restriction functor(s) in cohomology,one gets canonical localization maps H d +1 (cid:0) K, Z /n ( d ) (cid:1) → H d +1 (cid:0) K v , Z /n ( d ) (cid:1) . Theorem 2.5 ( Jannsen [Ja], Theorem 0.4) . In the above notations, suppose that char( K ) does not divide n . Then the localization maps give rise to an embedding H d +1 (cid:0) K, Z /n ( d ) (cid:1) → ⊕ v ∈ P ( k ) H d +1 (cid:0) K v , Z /n ( d ) (cid:1) . • Local-global principles over R v , v ∈ P ( k )In the above notations, for every non-archimedean place v ∈ P ( k ), let R v ⊂ k v be the(unique) Henselization of the valuation ring O v inside k v , hence recall that R v is a Henseliandiscrete valuation ring with residue field κ ( v ) finite. This being said, one has the following: Theorem 2.6 ( Kerz–Saito [K–S], Theorem 8.1) . Suppose that R is either (i) a finite field,or (ii) a Henselian discrete valuation ring with finite residue field, such that n is invertiblein R , and µ n ⊂ R . Then for every projective regular flat R -scheme X , the complex (KC) for X is exact, with the only exception of the homology group H (KC) = Z /n in the case (i) . Consequences/applications of the local-global principles
In this section we give a few consequences of the higher Hasse local-global principles men-tioned above, as well as an arithmetical interpretation of these consequences.A)
A technical result for later use
Notations/Remarks 3.1.
Let L be a field satisfying Hypothesis (H) from the Introduction.Let k ⊂ L be a global subfield, n = char( L ) be a prime number, and suppose that µ n ⊂ k .1) U = Spec R ⊂ S are open subsets with n ∈ R × , and set:∆ U := { u ′ ∈ k × | v ( u ′ − > · v ( n ) , ∀ v U } .2) For f := ( f ,..., f r ) with f i ∈ L × , and dense open subsets U ⊂ S , denote:H U , f := (cid:10) u ′ ∪ f ∪ ... ∪ f r | u ′ ∈ ∆ U (cid:11) ⊂ H r +1 (cid:0) L, Z /n ( r ) (cid:1) . Proposition 3.2.
In the above notation, let Z be a projective smooth U -variety with genericfiber Z := Z × U k , and function field L = k ( Z ) with dim( L ) = r . The following hold: The map H U , f → ⊕ z ∈ Z H r (cid:0) κ ( z ) , Z /n ( r − (cid:1) from the Kato complex (KC) is injective. Let α = u ′ ∪ f ∪ ... ∪ f r ∈ H U , f and z ∈ Z satisfy ∂ z ( α ) = 0 , and w be the primedivisor of L | k with O w = O z . Then there is an f i such that w ( f i ) n · wL . roof. For α ∈ H U , f non-trivial, proceed as follows:First, by Jannsen ’s Theorem 2.5 above, there exists v ∈ P ( k ) such that α is non-trivialover L v = Lk v . In particular, u ′ is not an n th power in k v , hence u ′ ∈ ∆ U . In particular,letting R := O hv be the Henselization of O v , the base change Z R = Z × U R is a smooth R -variety (because Z was a smooth U -variety). Set Spec R = { η , m } .Second, by the Kerz–Saito
Theorem 2.6 above, there are points z R ∈ Z R such that one has:0 = α z R := ∂ z R ( α ) ∈ H r (cid:0) κ ( z R ) , Z /n ( r − (cid:1) . Hence setting k R := Quot( R ) = k v , one has: If z R η under Z R → Spec R , then z R lies in the generic fiber Z k R = Z R × R k R of Z R . Henceletting z R z under Z R → Z , one gets: Since Z k R = Z × k k R , one has z ∈ Z . Second, since κ ( z ) ֒ → κ ( z R ), it follows that 0 = α z := ∂ z ( α ) ∈ H r (cid:0) κ ( z ) , Z /n ( r − (cid:1) , as claimed. Nextsuppose that z R m under Z R → Spec R . Since Z R is a projective smooth R -variety, itsspecial fiber Z m is reduced and has projective smooth integral κ ( m )-varieties as connectedcomponents, Z z R = { z R } being such one. Since 0 = α z := ∂ z ( α ) ∈ H r (cid:0) κ ( z ) , Z /n ( r − (cid:1) ,by Kerz–Saito ’s Thm 2.6, there is y ∈ Z z with 0 = ∂ y (cid:0) ∂ z R ( α ) (cid:1) ∈ H r − (cid:0) κ ( y ) , Z /n ( r − (cid:1) .On the other hand, dim( Z z R ) = dim( Z R ) −
1, hence Z z R ⊂ Z R . Hence codim Z R ( y ) = 2, and y m under Z R → Spec R . Let Z ( y ) := (cid:8) z ′ R ∈ Z R | y ∈ { z ′ R } (cid:9) . The Kato complex (KC) forthe projective smooth R -scheme Z R implies: P z ′ R ∈Z ( y ) ∂ y (cid:0) ∂ z ′ R ( α ) (cid:1) = 0. Since z R ∈ Z ( y ) and ∂ y (cid:0) ∂ z R ( α ) (cid:1) = 0, there must exist points z ′ R ∈ Z ( y ) satisfying: z ′ R = z R , = ∂ y (cid:0) ∂ z ′ R ( α ) (cid:1) ∈ H r − (cid:0) κ ( y ) , Z /n ( r − (cid:1) . We claim that all z ′ R ∈ Z ( y ), z ′ R = z R must satisfy: z ′ R η ∈ Spec R under Z R → Spec R .By contradiction, suppose that z ′ R m , or equivalently, z ′ R ∈ Z m . Arguing as above aboutas we did for z R , it follows that Z z ′ R = { z ′ R } is a connected component of Z m . Since theconnected components Z z R and Z z ′ R are either identical or disjoint, and y ∈ Z z R ∩ Z z ′ R , itfollows that z ′ R = z R , contradiction! (cid:3) Notations/Remarks 3.3.
In Notations/Remarks 2.4, let k ⊂ K be a (relatively alge-braically closed) subfield with td( K | k ) = 1 and k ⊂ k . Then there exists a unique projectivenormal (or equivalently regular) k -curve C such that K = k ( C ). The closed points P ∈ C are in canonical bijection with the prime divisors v of K | k via O P = O v .Let f ∈ K \ k be given. Our aim in this subsection is to give a criterion — which for n = 2and char = 2 turns out to be first-order — to express the following: The set D f := { P ∈ C | v P ( f ) n · v P ( K ) } is non-empty. In order to do so, we supplement the previous notations and remarks as follows:1) Let tt := ( t ,..., t e ) denote transcendence bases of k | k such that t i are n th powers in K .2) A t i ⊂ P t i are the S -affine/projective t i -lines, and set A tt := × i A t i ⊂ × i P t i =: P tt .3) For aa = ( a , . . . , a e ) ∈ k e , set uu = ( u i ) i = ( t i − a i ) i , and for U , aa , f as above, consider:H U , aa ,f := (cid:10) u ′ ∪ u ∪ u ∪ ... ∪ u e ∪ f | u ′ ∈ ∆ U , u ∈ k × (cid:11) ⊂ H d +1 (cid:0) K, Z /n ( d ) (cid:1) . Proposition 3.4.
In the above notations, the following are equivalent: i) D f is non-empty. ii) ∃ U tt ⊂ A tt Zariski open dense such that ∀ aa ∈ U tt ( k ) , ∀ U one has H U , aa ,f = 0 . iii) ∀ U tt ⊂ A tt Zariski open dense ∃ aa ∈ U tt ( k ) such that ∀ U one has H U , aa ,f = 0 . roof. To i) ⇒ ii): The proof of this implication is “easy” and requires just standard facts.Let X → P f, tt be the normalization of P f, tt := P f × P tt in the field extension k ( f, tt ) ֒ → K .Let P ∈ D f be given, hence by mere definitions we have v P ( f ) = m ∈ Z , and m is notdivisible by n in v P K . Then the residue map ∂ P : H d +1 (cid:0) K, Z /n ( d ) (cid:1) → H d (cid:0) κ ( P ) , Z /n ( d − (cid:1) restricted to H U , aa ,f is ∂ P ( u ′ ∪ u ∪ u ∪ ... ∪ u e ∪ f ) = ( − d m · u ′ ∪ u ∪ u ∪ ... ∪ u e .Let k ( tt ) ֒ → l be the separable part of k ( tt ) ֒ → κ ( P ), and S P → S → P tt be the normaliza-tion of P tt in the finite field extensions κ ( P ) ← ֓ l ← ֓ k ( tt ). Then S → P tt is ´etale above a denseopen subset U tt ⊂ A tt . Since A tt is regular, the preimages s aa ∈ S of aa = ( a ,..., a e ) ∈ U tt ( k )are regular points, uu := ( u ,..., u e ) = ( t − a ,..., t e − a e ) is a system of regular parametersat s aa , and κ aa := κ ( s aa ) is a finite separable extension of k . Hence letting R, m be the localring O s aa , m s aa of s aa , the m -adic completion of R is nothing but b R = κ aa [[ u , . . . , u e ]], whichobviously embeds into b l := κ aa (( u )) . . . (( u e )). Since k ֒ → κ aa is an extension of global fields,the image of H U = { u ′ ∪ u | ( u ′ , u ) ∈ ∆ U } under res : H (cid:0) k , Z /n (1) (cid:1) → H (cid:0) κ aa , Z /n (1) (cid:1) is non-trivial. And if u ′ ∪ u is non-trivial over κ aa , and easy induction on e shows that β := u ′ ∪ u ∪ u ∪ ... ∪ u e is non-trivial over b l , thus over l ⊂ b l . Finally, since κ ( P ) | l is purelyinseparable, β is non-trivial over κ ( P ), hence 0 = β ∪ f = u ′ ∪ u ∪ u ∪ ... ∪ u e ∪ f ∈ H U , aa ,f .To ii) ⇒ iii): Clear!To iii) ⇒ i): The (quite involved) proof is by induction on e . For every non-empty subset I ⊂ I e := { ,..., e } , set tt I = ( t i ) i ∈ I , k I := k ( tt I ), and denote A I := × i ∈ I A t i ⊂ × i ∈ I P t i =: P I .For J ⊂ I ⊂ I e , the canonical projections P I ։ P J are open surjective and define k J ֒ → k I . Construction I . Let I = ( I ν ) ν be a chain of subsets of I e with | I ν | = ν , I = ∅ . Setting k ν := k Iν , the canonical projections P tt = P Ie ։ ... ։ P I define k ( tt ) = k e ← ֓ ... ← ֓ k .Given Zariski open dense subsets U ν := U Iν ⊂ P Iν with U e → ... → U , and the preimages P f, U ν ։ P U ν of U ν under P f, tt ։ P tt ։ P Iν , one has canonical open k -immersions:( ∗ ) I U e = P U e ֒ → ... ֒ → P U ֒ → P Ie = P tt , P f, U e ֒ → ... ֒ → P f, U ֒ → P f, tt = P f × P tt .Let X → P f, tt be any projective morphism of k -varieties defining k ( f, tt ) ֒ → K . Using prime to n alterations , see [ILO], Expos´e X, Theorem 2.1, there is a projective smooth k -variety ˜ X and a projective surjective k -morphism ˜ X → X defining a field extension K =: K := k ( X ) ֒ → k ( ˜ X ) =: ˜ K of degree prime to n . Proceed as follows:Step 1. Generic part I . Let X → P k be the generic fiber of the P I -morphisms ˜ X → P tt .Then X is a projective regular k -variety, and K := k ( X ) = ˜ K . By [ILO], loc.cit., thereis a projective smooth k -variety ˜ X and a projective k -morphism ˜ X → X defining afield extension K = k ( X ) ֒ → k ( ˜ X ) =: ˜ K of degree prime to n . Inductively on ν , forthe already constructed projective smooth k ν − -variety ˜ X ν − , let X ν → P k ν be the genericfiber of the P I ν -morphism ˜ X ν − → P k ν − , thus X ν is a projective regular k ν -variety. By [ILO],loc.cit., there is a projective smooth k ν -variety ˜ X ν and a projective k ν -morphism ˜ X ν → X ν such that the field extension K ν = k ν ( X ν ) ֒ → k ν ( ˜ X ν ) =: ˜ K ν has degree prime to n .Notice that the generic fibers ˜ C ν → C ν → P f,k e of the P k ν -morphisms ˜ X ν → X ν → P f,k ν are finite morphisms of projective regular k e -curves, hence flat morphisms.Step 2. Deformation part I . Since being a projective smooth and/or finite flat morphismis an open condition on the base, there are Zariski dense open subsets U ν ⊂ A I ν ⊂ P I ν suchthat ( ∗ ) I above holds, there are projective smooth U ν -varieties ˜ X ν , 1 ν e , such that X ν → U ν has as generic fiber the projective smooth k ν -variety ˜ X ν . And setting ˜ X := ˜ X ,and X ν := ˜ X ν − × P Ie P U ν for 1 ν e , there is a projective morphism of U ν -varieties˜ X ν → X ν with generic fiber ˜ X ν → X ν , thus defining the field embedding ˜ K ν ← ֓ K ν of degreeprime to n degree. Note that Construction I realizes the morphisms above in dependence on I = ( I ν ) ν , that is, one should rather speak about ˜ X I ,ν → X I ,ν → U I ,ν , etc. One the otherhand, for all sufficiently small Zariski open dense subsets U ⊂ A I e the following is satisfied:( † ) For all I = ( I ν ) ν e one has: U ⊂ U I ,e and ˜ X I ,ν → X I ,ν → P U I ,ν are flat above U . From now on U tt := U ⊂ A I e always satisfies condition ( † ), and we replace U I ,ν by the image U ν := U I ν of U under P I e → P I ν , etc. Hence the objects above satisfy the following: Hypothesis 3.5. U tt ⊂ A I e is a Zariski open subset such that for all I = ( I ν ) ν e , andthe resulting open surjective projections U = U e ։ ... ։ U , and the open immersions U = P U ֒ → ... ֒ → P U ֒ → P Ie = P tt and P f, U ֒ → ... ֒ → P f, U ֒ → P f, tt = P f × P tt , there are/one has:1) Projective smooth U ν -varieties ˜ X ν generic fiber the projective smooth k ν -variety ˜ X ν .2) Projective morphisms of U ν -varieties ˜ X ν → X ν defining the field embedding ˜ K ν ← ֓ K ν ,where X ν := ˜ X ν − × P Ie P U ν , and ˜ X := ˜ X , thus a canonical morphism ˜ X ν → ˜ X ν − .3) Setting ˜ X ν, U := ˜ X ν × P Uν U , one has projective flat U -morphisms of regular U -curves:˜ X e,U → ... → ˜ X ,U → ˜ X ,U → P f,U defining ˜ K e ← ֓ ... ← ֓ ˜ K ← ֓ ˜ K ← ֓ k ( f, tt ) . In particular, for all I as above, and aa := ( a ,..., a e ) ∈ U ( k ), aa ν := ( a i ) i ∈ I ν ∈ U ν ( k ), thefibers ˜ X aa ν of ˜ X ν → U ν at aa ν ∈ U ν is a projective smooth k -varieties. Hence if U ⊂ S is asufficiently small Zariski open dense subset (depending on aa ), one has:( ‡ ) aa For all I = ( I ν ) ν , ˜ X and all the k -varieties ˜ X aa ν have projective smooth U -models. Returning to the proof of implication iii) ⇒ i), we proceed by proving: Lemma 3.6.
Under Hypothesis 3.5, let aa ∈ U tt ( k ) , and U satisfy ( ‡ ) aa . Then for each = α ∈ H U , aa ,f , there is P ∈ C such that ∂ P ( α ) is non-trivial, and v P ( f ) n · v P ( K ) .Proof of Lemma 3.6. The proof is by induction on e as follows: Since [ ˜ K : K ] is prime to n ,the restriction res : H d +1 (cid:0) K, Z /n ( d ) (cid:1) → H d +1 (cid:0) ˜ K , Z /n ( d ) (cid:1) is injective, hence ˜ α := res( α ) isnontrivial over ˜ K . Since ˜ X has a projective smooth U -model, by Proposition 3.2 there is apoint x ∈ ˜ X with codim ˜ X ( x ) = 1 and β := ∂ x ( ˜ α ) is nontrivial in H d (cid:0) κ ( x ) , Z /n ( d − (cid:1) .Hence the prime divisor w := w x of ˜ K | k satisfies:Case 1. w ( u i ) = 0 for all i = 1 ,..., e . Then setting u i u i under O w = O x → κ ( x ), itfollows that ∂ x ( ˜ α ) = u ′ ∪ u ∪ u ∪ ... ∪ u e = 0 in H d (cid:0) κ ( x ) , Z /n ( d − (cid:1) . Hence u ,..., u e mustbe algebraically independent over k , or equivalently, w must be trivial on k ( u ,..., u e ).Thus w | K = v P for some P ∈ C such that ∂ P ( α ) = over κ ( P ), and v P ( f ) n · v P K (because w ( f ) n · w ˜ K ). Thus finally implying that D f is non-empty, as claimed.Case 2. The set I := { i | w ( u i ) = 0 } is non-empty. Recalling that t i is an n th powerin K , say t i = t n , and u i = t i − a i , we claim that w ( t i ) = 0. Indeed, by contradiction, let w ( t i ) = 0. First, if w ( t i ) >
0, then u i = a i , hence u ′ ∪ u ∪ a i ∈ H (cid:0) k , Z /n (2) (cid:1) = 0 is asub-symbol of ∂ x ( ˜ α ), implying that ∂ x ( ˜ α ) = 0, contradiction! Second, if w ( t i ) <
0, then t i − a i = t i (1 − a i /t i ) = t n u ′ i with u ′ i ∈ m w . Hence u ′ i = 1, and α = α ′ , where the atter symbol is obtained by replacing u i by u ′ i in the symbol α . Then u ′ ∪ u ∪ ∂ x ( ˜ α ), thus ∂ x ( ˜ α ) = 0, contradiction! Thus conclude that w ( t i ) = 0, hence w ( t i − a i ) = 0 ⇒ w ( t i − a i ) >
0. In particular, replacing f by f − if necessary, w.l.o.g., w ( f ) >
0. Thus finally u ,..., u e , f ∈ O x , and u i ∈ m x iff i ∈ I .Consider the images x x f, aa I x aa I aa I := ( a i ) i ∈ I under ˜ X → P f, tt → P I e → P I .Then aa I ∈ P U I ( k ), and x aa I ∈ P tt is defined by tt I = aa I , hence td (cid:0) κ ( x aa I ) | k (cid:1) e − | I | .Further, x aa I ∈ U (because x aa I is a generalization of aa ). Hence e = td (cid:0) κ ( x ) | k (cid:1) , togetherwith ˜ X → P f, tt → P tt being flat at x x f, aa I x aa I ∈ U e , imply: e − (cid:0) κ ( x ) | k (cid:1) − (cid:0) κ ( x f, aa I ) | k (cid:1) − (cid:0) κ ( x aa I ) | k (cid:1) e − | I | , hence | I | = 1,and td (cid:0) κ ( x f, aa I ) | k (cid:1) − (cid:0) κ ( x aa I ) | k (cid:1) implies f / ∈ m x f, aa I ⊂ m x , thus f, u i ∈ O × x , i / ∈ I .Reasoning as in Case 1), the non-triviality of ∂ x ( ˜ α ) implies that the residues f, u i ∈ κ ( x )of f, u i , i / ∈ I are algebraically independent over k . Equivalently, the residues f, t i , i / ∈ I are algebraically independent over k , hence x f, aa I ∈ P f, tt is the generic point of the fiber of P f, tt → P I at aa I ∈ P I ( k ). Since x x f, aa I x aa I aa I under ˜ X → P f, tt → P I e → P I , one has:( ∗ ) x is a generic point of the fiber ˜ X , aa I of ˜ X → P I at aa I ∈ U I ( k ) . After renumbering ( t ,..., t e ), w.l.o.g., I = { e } . Considering all the chains I = ( I ν ) ν with I = { e } , and viewing ˜ X ν → P U ν and ˜ X ν → X ν → P f,U ν as U -morphisms via U ν → U , let˜ X ν, aa → X ν, aa → P U ν , aa → U ν, aa be the fibers of ˜ X ν → X ν → P U ν → U ν at aa ∈ U ( k ).Let w is the prime divisor of ˜ K defined by x ∈ ˜ X . Then Hypothesis 3.5, 3) implies:The sequences ( w ν ) ν e of prolongations of w to the tower of extensions ( ˜ K ν ) ν satisfying w ν +1 | ˜ K ν = w ν are in canonical bijection with the sequences ( x ν ) ν e of generic points x ν ∈ ˜ X ν, aa with x ν +1 x ν for all 0 ν < e . Hence denoting e ν +1 := e ( w ν +1 | w ν ), f ν +1 := f ( w ν +1 | w ν ) = [ κ ( x ν +1 ) : κ ( x ν )] the ramification index, respectively the residuedegree of w ν +1 | w ν , one has: Since [ ˜ K ν +1 : ˜ K ν ] is prime to n , by the fundamental equality, forevery w ν there is a prolongation w ν +1 with e ν +1 f ν +1 prime to n . We will call such sequences( x ν ) ν and/or ( w ν ) ν prime to n compatible . Notice that letting ˜ X x ν ⊂ ˜ X ν be the Zariskiclosure of x ν , the morphism ˜ X ν +1 → ˜ X ν gives rise to a morphism ˜ X x ν +1 → ˜ X x ν defining thefinite extension ˜ L ν := κ ( x ν ) ֒ → κ ( x ν +1 ) =: ˜ L ν +1 of degree f ν +1 prime to n . Further, ˜ X x is an irreducible component of the projective smooth k -variety ˜ X aa = ˜ X , aa , hence ˜ X x isitself a projective smooth k -variety. And by assumption ( ‡ ) aa , ˜ X x has a projective smooth U -model. Finally, since ˜ L = κ ( x ) ֒ → κ ( x ) = ˜ L has degree prime to n , and β = ∂ x ( ˜ α )is non-trivial over ˜ L = κ ( x ), it follows that β = res( β ) is non-trivial over ˜ L . Hence byProposition 3.2, there is a point y ∈ ˜ X x such that ∂ y ( β ) = 0. Let ˜ X ( y ) ⊂ ˜ X be the setof points x ′ ∈ ˜ X with codim ˜ X ( x ′ ) = 1 and y ∈ { x ′ } . Then reasoning as in the proof ofProposition 3.2, it follows that there is x ′ ∈ ˜ X ( y ) such that, first, ∂ x ′ ( ˜ α ) = 0 over κ ( x ′ ),and second, x ′ is not contained in any fiber ˜ X aa ′ of ˜ X → U . Equivalently, x ′ lies in thegeneric fiber X of ˜ X → U , and the prime divisor w ′ of ˜ K | k defined by x ′ is trivial on k .Case e = 1: One has tt = ( t ), hence k = k ( tt ). Therefore w ′ | K = v P for some P ∈ C ,and conclude by reasoning as in Case 1 above. ase e >
1: Setting µ := ν − J µ := I ν \ I , the inclusions V µ := U ν, aa ⊂ P ν, aa = P J µ are openimmersions, where V := Spec k =: P J . Given a prime to n compatible sequence ( x ν ) ν e ,for every ν > Y µ := ˜ X x ν ⊂ ˜ X ν, aa , and recall that ˜ L µ = k ( ˜ Y µ ) = κ ( x ν ). Since˜ X ν → U ν is projective smooth, its fiber ˜ X ν, aa → U ν, aa is projective smooth, hence so is itsirreducible component ˜ Y µ ⊂ ˜ X ν, aa . Further, since ˜ X ν → X ν = ˜ X ν − × P tt P U ν is a projective U ν -morphism, the transitivity of base change implies that Y µ := X ν, aa = ˜ Y µ − × P tt ′ P V µ ,and the resulting V µ -morphism ˜ Y µ → Y µ is projective defining ˜ L µ ← ֓ ˜ L µ − . ThereforeHypothesis 3.5 holds mutatis mutandis in the context below, after replacing e by e ′ , namely:For the canonical open projections V := V e ′ ։ ... ։ V and the resulting open immersions V e ′ = P V e ′ ֒ → ... ֒ → P V ֒ → P J e ′ and P f,V e ′ ֒ → ... ֒ → P f,V ֒ → P f,J e ′ , the following hold:1) ′ ˜ Y µ → V µ is a projective smooth integral V µ -variety of dimension e ′ + 1 − µ for 1 µ e ′ .2) ′ ˜ Y µ → Y µ := ˜ Y µ − × P Je ′ P V µ is a projective surjective V µ -morphism for 1 ν e ′ .3) ′ Setting ˜ Y µ,V := ˜ Y µ × P Jµ V for 0 µ e ′ , one gets canonically a sequence of projectivesurjective flat morphisms of V -curves ˜ Y e ′ ,V → ... → ˜ Y ,V → ˜ Y ,V → P f,V .Further, aa ′ := ( a ,..., a e ′ ) ∈ V ( k ), aa ′ µ := ( a i ) i ∈ J µ ∈ V µ ( k ), and the fiber ˜ Y aa µ of ˜ Y µ → V µ at aa ′ µ ∈ V µ is a projective smooth k -variety. Moreover, since each ˜ Y µ, aa ′ µ is an irreduciblecomponent of the projective smooth k -variety ˜ X ν, aa ν , the condition ( ‡ ) aa above implies:( ‡ ) aa ′ For all J = ( J µ ) µ , ˜ Y and all the k -varieties ˜ Y aa ′ µ have projective smooth U -models. In particular, since e ′ < e , we can apply the induction hypothesis for β = ∂ x ( ˜ α ) as follows:Recall that the canonical projective morphism ˜ Y := ˜ X , aa → ˜ X , aa =: Y is the fiber of˜ X → ˜ X ,U at aa ∈ U ( k ), and the corresponding extension L = κ ( x ) ֒ → κ ( x ) = ˜ L hasdegree f ( x | x ) prime to n . Hence the image ˜ β := res( β ) of β over ˜ L is non-trivial. Let C ′ be the generic fiber of ˜ Y ,V → V , hence of ˜ Y → P tt ′ , and set uu ′ := tt ′ − aa ′ = ( t i − a i ) i e ′ .Then C ′ is a projective regular k ( tt ′ )-curve, and the images ( f, uu ′ ) ( f, uu ′ ) ( f ′ , uu ′ )under O x → κ ( x ) ֒ → ˜ L satisfy: ˜ β = u ′ ∪ u ∪ u ′ ∪ ... ∪ u ′ e ′ ∪ f ′ . Hence by the inductionhypothesis, there is P ′ ∈ C ′ such that ∂ P ′ ( ˜ β ) = 0 and v P ′ ( f ′ ) / ∈ n · v P ′ ( ˜ L ). In particular, y := P ′ ∈ C ′ ⊂ ˜ Y is a point of codimension one. Recalling that ˜ Y = ˜ X , aa = ˜ X aa isthe fiber of ˜ X at aa ∈ U ( k ), it follows that y ∈ ˜ X is a point of codimension two.By the discussion before Case e = 1 above, there exists x ′ ∈ ˜ X ( y ) with ∂ x ( ˜ α ) = 0 and k ( t e ) ⊂ O x ′ . On the other hand, k ( tt ′ ) ⊂ κ ( y ) ⊂ κ ( x ), hence finally, tt = ( tt ′ , t e ) consists of w x -units. Conclude as in Case 1. (cid:3) B) A criterion for D f to be non-empty Notations/Remarks 3.7.
Recalling that v denotes the finite places of k , we supplementthe previous Notatations/Remarks 3.1 as follows. For every δ, δ , . . . , δ e ∈ k × we define:1) U δ := { a ∈ k × | v ( δ ) = 0 or v ( n ) > ⇒ v ( a − > · v ( n ) ∀ v } , the n,δ -units .2) Σ δ = { a ∈ k × | v ( δ ) = 0 ⇒ v ( a ) = 0 } , the non- δ -units . We notice the following:a) U δ is a subgroup of k × , and Σ δ is a U δ -set, i.e., U δ · Σ δ = Σ δ .b) For every finite set A ⊂ k there exist “many” δ ∈ k × such that A ∩ Σ δ = /O.3) U • δ := U δ × k × , and Σ δ := × i Σ δ i for δ := ( δ , . . . , δ e ). ) For aa = ( a i ) i ∈ Σ δ , set uu := tt − aa , and for δ ∈ k × and uu := ( u ′ , u ) ∈ U • δ , define α uu , aa ,f := u ′ ∪ u ∪ u ∪ ... ∪ u e ∪ f ∈ H d +1 (cid:0) K, Z /n ( d ) (cid:1) and consider the subgroup H δ , aa ,f := (cid:10) α uu , aa ,f | uu ∈ U • δ (cid:11) ⊂ H d +1 (cid:0) K, Z /n ( d ) (cid:1) . Key Lemma 3.8.
In the above notations, the following are equivalent: i) D f is non-empty. ii) ∃ δ ∀ a ∈ Σ δ ... ∃ δ e ∀ a e ∈ Σ δ e ∀ δ ∃ ( u ′ , u ) ∈ U • δ such that α uu , aa ,f = 0 .Proof. The proof follows easily from Proposition 3.4 along the following lines:To i) ⇒ ii): Given i), by Proposition 3.4, ∃ U tt ⊂ A tt Zariski open dense s.t. H U , aa ,f = 0 forall aa ∈ U tt ( k ), U ⊂ S . Set tt i = ( t ,..., t i ). Then ̟ i : A tt → A tt i defined by k [ tt i ] ֒ → k [ tt ]are Zariski open maps. Hence Σ i := ̟ i (cid:0) U tt ( k ) (cid:1) ⊂ k i is Zariski open dense, and one has:a) If aa i ∈ Σ i , and U tt , aa i ⊂ A tt , aa i ֒ → A tt are the fibers of U tt ⊂ A tt at aa i under ̟ i : A tt → A tt i ,then at the level of k -rational points, one has canonical identifications U tt , aa i ( k ) ⊂ A tt , aa i ( k ) = ̟ − i ( aa i ) = aa i × k ( e − i ) ⊂ k e = A tt ( k ).b) In particular, U tt , aa i ( k ) ⊂ aa i × k ( e − i ) is a Zariski open dense subset for all aa i ∈ U i .Proceed by induction on i = 1 , . . . , e as follows:Step 1. i = 1: Since Σ := ̟ (cid:0) U tt ( k ) (cid:1) ⊂ k is Zariski open dense, A := k \ Σ is finite.Hence ∃ δ ∈ k × such that Σ ,δ ∩ A = /O, thus Σ ,δ ⊂ Σ . Then ∀ a ∈ Σ δ , set aa := ( a ).Step 2. i ⇒ i + 1: Suppose that aa i = ( a , . . . , a i ) ∈ ̟ i (cid:0) U tt ( k ) (cid:1) is inductively constructed.Let ̟ i +1 ,i : A tt i +1 → A tt i be the canonical projection. Then ̟ i = ̟ i +1 ,i ◦ ̟ i +1 , and all theprojections involved are open surjective. Further, by the discussion above, one has that U aa i := U tt , aa i ( k ) ⊂ aa i × k ( e − i ) is Zariski open dense, and therefore ̟ i +1 ,i (cid:0) U aa i ) ⊂ aa i × k is adense open subset. Hence there exists Σ i +1 ⊂ k cofinite such that aa i × Σ i +1 ⊂ ̟ i +1 ,i (cid:0) U aa i ),thus aa i × Σ i +1 ⊂ U aa i ⊂ aa i × k is a Zariski open dense subset. In particular, A i +1 := k \ Σ i +1 is finite. Hence ∃ δ i +1 ∈ k × such that Σ δ i +1 ∩ A i +1 = /O, and in particular, Σ δ i +1 ⊂ Σ i +1 . Then ∀ a i +1 ∈ Σ δ i +1 , setting aa i +1 := ( aa i , a i +1 ), one has: aa i +1 ∈ ̟ i +1 (cid:0) U tt , aa i ( k ) (cid:1) ⊂ ̟ i +1 (cid:0) U tt ( k ) (cid:1) .This completes the proof of the induction step, thus of the implication i) ⇒ ii).To ii) ⇒ i): Let U tt ⊂ A tt be a Zariski dense open subset. Then condition ii) of the KeyLemma 3.8 above, implies that there is aa ∈ U tt ( k ) such that ∀ U ⊂ S one has H U , aa ,f = 0.Hence condition iii) from Proposition 3.4 is satisfied, concluding that D f = /O. (cid:3) Uniform definability of the geometric prime divisors of K In this section we work in the context and notation of the previous sections, but specializeto the case n = 2 = char. In particular, for aa := ( a ,..., a r ) with a i ∈ K × , by the MilnorConjecture, a ∪ ... ∪ a r ∈ H r (cid:0) K, Z /n ( r − (cid:1) is trivial iff 0 ∈ q aa ( K ). Therefore: a ∪ ... ∪ a r = 0 is first-order expressible by ∃ ( x ,..., x r ) = 0 s.t. q aa ( x ,..., x r ) = 0 . Let K satisfy Hypothesis (H), k ⊂ K be a (relatively algebraically closed) global subfield, e = td( K | k ) − >
0, ( f, t , . . . , t e ) be algebraically independent functions over k , such thateach t i is an n th power in K . Then K is the function field of a projective normal curve C over k ( tt ), and f is a non-constant function on C . Finally, recalling the context of the Key emma 3.8, let xx := ( x ,..., x d +1 ) = (0 ,...,
0) be a system of 2 d +1 variables, and considerthe following uniform first-order formula: ϕ ( f, tt ) ≡ ∃ δ ∀ a ∈ Σ δ . . . ∃ δ e ∀ a e ∈ Σ δ e ∀ δ ∃ ( u ′ , u ) ∈ U • δ ∀ xx : q uu , uu ,f ( xx ) = 0. Key Lemma (revisited) 4.1.
In the above notation, the following are equivalent: i) D f is non-empty. ii) ϕ ( f, tt ) holds in ˜ K = K [ µ ] .Proof. As explained above, this is just a reformulation of Key Lemma 3.8. (cid:3)
Our final aim in this section is to show that the prime divisors of K | k are uniformly first-order definable. Precisely, recalling the ( d + 1)-fold Pfister forms q uu , uu ,f defined using f, tt asintroduced above, the Recipe 4.8 below gives a proof of the following: Theorem 4.2.
There exist explicit formulae val d which uniformly define the geometric primedivisors of finitely generated fields K with char( K ) = 2 and dim( K ) > as follows val d (cid:0) x ; f, tt , ξ , δδ , ( a i ) i ∈ ΣΣ δδ , δ , uu ∈ U • δ , q uu , uu ,f ( x ξ ) (cid:1) . The proof of Theorem 4.2 follows from the Recipe 4.8 below.A)
The uniformly definable subsets Θ f, tt , Θ f, tt and semi-local subrings a f, tt ⊂ R f, tt of K Notations/Remarks 4.3.
We supplement Notations/Remarks 3.1, 3.3, as follows.1) For ε ∈ K , set K ε := K [ n √ ε ] and consider res ε : H d +1 (cid:0) K, Z /n ( d ) (cid:1) → H d +1 (cid:0) K ε , Z /n ( d ) (cid:1) .2) Let C ε → C , P ε P , be the normalization of C in the field extension K ε | K , anddenote by D f, ε the set of all P ε ∈ C ε such that v P ε ( f ) n · v P ε K ε .3) For every P ∈ C , let U P := O × P be the P -units, and let U P ⊂ O v P be the v P -units.Then for ε ∈ K × one has: (i) ε ∈ U P · K · n iff (ii) v P ( ε ) ∈ n · v P K . Lemma 4.4.
In the above notations, ε ∈ ∪ P ∈ D f U P · K · n ⇔ ϕ ( f, tt ) holds in K ε .In particular, Θ f, tt := ∪ P ∈ D f U P · K · n ⊂ K are uniformly first-oder definable in K as follows: Θ f, tt = { ε ∈ K | ϕ ( f, tt ) holds in ˜ K ε = K ε [ µ ] } Proof. To ⇒ : Let ε ∈ ∪ P ∈ D f 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 thefield extension K ε ← ֓ K , it follow that v P ε ( f ) = v P ( f ) is prime to n . Hence D f, ε = /O, andtherefore, by Key Lemma 4.1 follows that condition ii) is satisfied over K ε . Let k ε = k ∩ K ε be the field of constants of K ε . Then k ε | k is a finite field extension, and therefore, for every δ ε ∈ k × ε there is δ ∈ k × such that for all v ε and v := v ε | k one has: v ε ( δ ε ) = 0 iff v ( δ ) = 0.In particular, Σ δ ε ∩ k = Σ δ . Therefore, condition ii) for K ε implies condition ii) for K .To ⇐ : Let ε
6∈ ∪ P ∈ D f U P · K · n that is, ε U P · K · n for all v ∈ D f . Then by Nota-tions/Remarks 3.7, 3), one has v P ( ε ) n · v P ( K ) for all P ∈ D f . Hence for all P ∈ D f , andany prolongation P ε to K ε one has e ( P ε | P ) = n , thus v P ε ( f ) = e ( P ε | P ) v P ( f ) ∈ n · v P ε ( K ε ).Further, since v P ( f ) ∈ n · v P ( K ) for P D f , one has v P ε ( f ) ∈ n · v P ε ( K ε ) for P ε P D f ,thus conclusing that v P ε ( f ) ∈ n · v P ε ( K ε ) for all P ε ∈ C ε . On the other hand, by hypothesis ii),applying Key Lemma 4.1 to K ε , it follows that D f, ε is non-empty, contradiction! (cid:3) otations/Remarks 4.5. In the notations from Lemma 4.4 above, we have the following:1) Let η ∈ K \ Θ f, tt be given. Then by Notations/Remarks 4.3, 3), it follows that for all P ∈ D f one has: v P ( η ) n · v P ( K ). In particular, v P ( η ) = 0, and therefore one has:- If v P ( η ) >
0, then η − ∈ m P − ⊂ U P ⊂ Θ f, tt , hence finally η − ∈ Θ f, tt .- If v P ( η ) <
0, then v P ( η −
1) = v P ( η ) n · v P ( K ). Therefore, by the discussion atNotations/Remarks 4.3, 3), it follows that η − U P · K · n .2) Conclude that for η ∈ K the conditions (i), (ii) below are equivalent:(i) η, η − Θ f, tt ; (ii) v P ( η ) < and v P ( η ) n · v P ( K ) for all P ∈ D f .
3) Hence Θ f, tt := (cid:8) ξ ∈ K | ξ , ξ − Θ f, tt (cid:9) are uniformly definable, and( ∗ ) ξ ∈ Θ f, tt iff ∀ P ∈ D f one has: v P ( ξ ) > , v P ( ξ ) n · v P ( K ) . Lemma 4.6.
In the above notation, one has a f, tt := ∩ P ∈ D f m P = Θ f, tt − Θ f, tt . Hence a f, tt ⊂ K is uniformly definable, thus so is the subring R f, tt = ∩ P ∈ D f O P = { r ∈ K | r · a f, tt ⊂ a f, tt } of K .Proof. We first prove the equality ∩ P ∈ D f m P = Θ f, tt − Θ f, tt . For the inclusion “ ⊂ ” notice thatΘ f, tt ⊂ m P , P ∈ D f by Notations/Remarks 4.3, 3) above. Hence Θ f, tt − Θ f, tt ⊂ m P − m P = m P , P ∈ D f , thus finally one has Θ f, tt − Θ f, tt ⊂ a f, tt . For the converse inclusion “ ⊃ ” let ξ ∈ a f, tt be arbitrary. Since a f, tt = ∩ P ∈ D f m P , it follows by Notations/Remarks 4.3, 3), above that v P ( ξ ) > P ∈ D f . Hence by the weak approximation lemma, it follows that thereexists ξ ′ ∈ K such that both ξ ′ and ξ ′′ := ξ ′ − ξ satisfy v P ( ξ ′ ) , v P ( ξ ′′ ) = 1. In particular, byNotations/Remarks 4.3, 3), one has ξ ′ , ξ ′′ ∈ Θ f, tt , hence ξ = ξ ′ − ξ ′′ ∈ Θ f, tt − Θ f, tt . Concerningthe assertions about R f, tt , the first row equalities are well known basic valuation theoreticalfacts (which follow, e.g. using the weak approximation lemma). (cid:3) Corollary 4.7.
In the above notation, let D f := { P ∈ D f | v P ( f ) > } and f ′ := f / ( f + 1) .Then R f, tt := ∩ P ∈ D f O P = R f, tt · R f ′ , tt := { rr ′ | r ∈ R f, tt , r ′ ∈ R f ′ , tt } is uniformly first-order definable.Proof. Noticing that f, f ′ have the same zeros and no common poles, the assertion of theCorollary 4.7 follows from Lemma 4.6 by the weak approximation lemma. (cid:3) B) Defining the k -valuation rings of K | k In the notations and hypotheses the previous sections, recall that K = k ( C ) for someprojective normal k -curve C . By Riemann-Roch, for every closed point P ∈ C there areintegers m ≫ n = 2 and functions h ∈ K × such that ( h ) ∞ = m P . Hence setting f = 1 /h , by Corollary 4.7 one has: O P = R f, tt = R f, tt · R f ′ , tt , m P = { r ∈ K | r ∈ O P , r −
6∈ O P } .Hence we have the following uniform first-oder recipe to define the prime k -divisors of K | k : Recipe 4.8.
Recall ϕ d , ψ abs ( x ), ψ r ( t ,..., t r ) from the Introduction. If dim( K ) = 1, thenthe prime divisors of K are uniformly first-order definable by the formulae val given by Rumely [Ru]; and if dim( K ) = 2, the geometric prime divisors of K | k are uniformly first-orderdefinable by the formulae val given by Pop [P4]. Next let K satisfying Hypothesis (H) fromIntroduction, k ⊂ K be (maximal) global subfields, and e := dim( K ) − K | k ) − > val d (cid:0) x ; f, tt , ξ , δδ , ( a i ) i ∈ ΣΣ δδ , δ , uu ∈ U • δ , q uu , uu ,f ( x ξ ) (cid:1) concretely along the steps: ) Consider the systems tt := ( t ,..., t e ), of k -algebraically independent elements of K with t i squares in K . These are uniformly definable using the formula ψ e ( t ,..., t e ) over k .2) P k , val := { ( tt , f ) ∈ K e +1 | R f, tt ( K is a valuation ring } is uniformly first-order definable.
3) Finally, the above R f, tt are valuation rings of prime k -divisors of K | k , and conversely,for every prime k -divisor w of K | k there are pairs ( f, tt ) ∈ P k , val such that O w = R f, tt .Conclude that the geometric prime divisors of K are uniformly first-order definable via P k , val .5. Proof of the Main Theorem A) First proof : Using
Scanlon [Sc]A first proof follows simply from
Scanlon , Theorem 4.1 and Theorem 5.1, applied to thecase of characteristic = 2, using Theorem 1.1 for the definability of valuations (which isessential in both Theorem 4.1 and Theorem 5.1 of loc.cit.). This proof also shows thatfinitely generated fields of characteristic = 2 are bi-interpretable with the arithmetic. B) Second proof : Using
Aschenbrenner–Kh´elif–Naziazeno–Scanlon [AKNS]Recall that one of the main results of [AKNS] asserts that the finitely generated infinitedomains R are bi-interpretable with arithmetic, see Theorem in the Introduction of loc.cit.In particular, the isomorphism type of any such domain is encoded by a sentence ϑ R . Thusthe Main Theorem from the Introduction follows from the following stronger assertion: Theorem 5.1.
Let κ a prime field, char( κ ) = 2 , R κ ⊂ κ be its prime subring, and T = ( t , . . . , t r ) be independent variables. Then the integral closures R ⊂ K of R κ [ T ] infinite field extensions K | κ ( T ) are uniformly first-order definable finitely generated domains.Proof. First, by the Finiteness Lemma, R is a finite R [ T ]-module, hence finitely generated asring. The uniform definability of R is though more involved, and uses the uniform definabilityof generalized geometric prime divisors of K combined with Rumely [Ru].
Lemma 5.2.
Let A be an integrally closed domain, and V be a set of valuations of thefraction field K A := Quot( A ) such that A = ∩ v ∈V A O v . Let B be the integral closure of A inan algebraic extension K B | K A , and W be the prolongation of V to K B . Then B = ∩ w ∈W O w .Proof. Klar, left to the reader. (cid:3)
Let R be an integrally closed domain, L := Quot( R ), and V be a set of valuations of L such that R = ∩ v ∈V O v . Let L | L ( t ) be a finite field extension, R ⊂ ˜ R ⊂ L bethe integral closures of R [ t ] ⊂ L [ t ] in L . Then κ ( P ) are finite field extensions of L , P ∈ Max( ˜ R ) and let V P be the prolongation of V to κ ( P ). Finally let V be the set of allthe valuations of the form v := v P ◦ v P with v P the valuation of P ∈ Max( ˜ R ), and v P ∈ V P .Then v P = v /v P , v L = v P L P × Z lexicographically ordered, and L v = L P v P . Further,the canonical restriction map Val( L ) → Val( L ) gives rise to a well defined surjective maps: V → V P → V , v v P v := v P | L = v | L . Lemma 5.3.
In the above notation, one has R = ∩ v ∈V O v .Proof. Lemma 5.2 reduces the problem to the case L = L ( t ), R = R [ t ]. For v = v P ◦ v P , O v ⊂ O v P , hence ∩ v ∈V O v ⊂ ∩ P O v P = L [ t ]. Thus it is left to prove that f ( t ) ∈ L [ t ]satisfies: v ( f ) ≥ v ∈ V iff f ∈ R [ t ]. This easy exercise is left to the reader. (cid:3) emma 5.4. Suppose that all the valuation rings O P , P ∈ Max( ˜ R ) and O v P , v P ∈ V P are ( uniformly ) first-oder definable. Then so are O v , v ∈ V and R = ∩ v ∈V O v .Proof. For v = v P ◦ v P , one has O v = π − P ( O v P ), where π P : O P → κ ( P ) =: L P , etc. (cid:3) Finally, all of the above can be performed inductively for systems of variables T := ( t ,..., t r ), L r finite field extension of L ( T ), R r ⊂ ˜ R r ⊂ L r the integral closures of R r − [ t r ] ⊂ L r − [ t r ],thus leading to the corresponding sets of all valuations V r of L r the form v r = v Pr − ◦ v P ,where P ∈ Max( ˜ R r ) and v Pr − lies in the prolongation V Pr − of V r − to κ ( P ). Lemma 5.5.
In the above notation, one has R r = ∩ v r ∈V r O v r . Further, if all the valuationrings O P , P ∈ Max( ˜ R r ) and O v P , v P ∈ V Pr − are ( uniformly ) first-order definable, then soare the valuation rings O v r , v r ∈ V r and R r = ∩ v r ∈V r O v r .Proof. Induction on r reduces everything to r = 1. Conclude by using Lemmas 5.3, 5.4. (cid:3) Coming back to the proof of Theorem 5.1, one has R κ = F p , p > R κ = Z , and R ⊂ K is the integral closure of R κ [ T ] in K . Hence Theorem 5.1 follows from Lemma 5.5. (cid:3) C) Third proof : A direct proof involving
Rumely [Ru]We begin by mentioning that
Pop [P4], Theorem 1.2 holds in the following more generalform (which might be well known to experts, but we cannot give a precise reference). Namely,let K be a class of function fields of projective normal geometrically integral curves K = k ( C )such that k ⊂ K and the k -valuations rings O , m of K | k are (uniformly) first-order definable.Then for every non-zero t ∈ K , e >
0, the setsΣ t,e := {O , m | t ∈ m e , t m e +1 } are (uniformly) first-order definable subsets of the set of all the valuation rings O , m . Hencegiven N >
0, a function t ∈ K × has deg( t ) := [ K : k ( t )] = N iff the following hold:i) Σ t,N +1 = /O and | Σ t,e | ≤ N for all 0 < e N .ii) dim k O / m ≤ N for all O , m ∈ Σ t,e , and moreover: N = P Rumely [Ru], there exists a sentence ϑ k , Σ such that for all pairs k ′ , Σ ′ one has: ϑ k , Σ holdsin k ′ , Σ ′ if and only if there exists a field isomorphisms ı : k → k ′ with ı (Σ) = Σ ′ . Further,inductively on e , let ϑ k , tt e be the sentence asserting that if tt e = ( t ,..., t e ) are k -algebraicallyindependent in K | k , then k := k ( tt e ) satisfies k = k ∩ K . Finally, consider the sentence ϑ K defined as follows: ϑ K ≡ a) ∧ b) ∧ c) ∧ d) ∧ ϑ k , Σ fK ∧ ϑ k , tt e ∧ deg N fK ( u ). et L be a finitely generated field such that ϑ K holds in L . Then the assertions a), b), c), d)together with ϑ k , Σ fK imply that there exists a subfield l ⊂ L with l = l ∩ L and dim( l ) = 1,and an isomorphism ı : k → l such that setting g L := g L ( T e , T, U ) := ı (cid:0) f K ( T e , T, U ) (cid:1) ,one has: g L ( T e , T, U ) is irreducible over l , and there exist: First, a transcendence basis( tt ′ e , t ′ ) with tt ′ e := ( t ′ ,..., t ′ e ) of L | l , such that ϑ l , tt ′ e holds in L , hence l := l ( tt ′ e ) satisfies l = l ∩ L . Second, u ′ ∈ L such that both g L ( tt ′ e , t ′ , u ′ ) = 0 and deg f K ( u ′ ) hold in L | l , hence[ L : l ( u ′ )] = N f K . Therefore, ı : k → l together with ( tt e , t, u ) ( tt ′ e , t ′ , u ′ ) give rise to a fieldembedding ı K : K → L such that l = l ( tt ′ e ) is relatively algebraically closed in L , and setting L ′ := ı K ( K ), one has: L ′ | l ֒ → L | l is a finite extension of function fields of curves over l and[ L : l ( u ′ )] = N f K = deg U ( f K ) = deg U ( g L ) = [ L ′ : l ( u ′ )] . Hence we conclude that L = L ′ , thus L = l ( tt ′ e , t ′ , u ′ ). Therefore, the canonical embedding ı K : K ֒ → L is actually a field isomorphism prolonging ı : k → l . References [AKNS] Aschenbrenner, M., Kh´elif, A., Naziazeno, E. and Scanlon, Th., The logical complexity of finitelygenerated commutative rings, Int. Math. Res. Notices (to appear).[Di] Dittmann, Ph., Defining Subrings in Finitely Generated Fields of Characteristic Not Two, See: arXiv:1810.09333 [math.LO], Oct 22, 2018.[Du] Duret, J.-L., ´Equivalence ´el´ementaire et isomorphisme des corps de courbe sur un corps algebrique-ment clos, J. 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Purdue 1963,New York 1965, pp. 82–92.[Sh1] Shlapentokh, A., First Order Definability and Decidability in Infinite Algebraic Extensions of Ra-tional Numbers, Israel J. Math. (2018), 579–633.[Sh2] Shlapentokh, A., On definitions of polynomials over function fields of positive characteristic, See arXiv:1502.02714v1 [Vi] Vidaux, X., ´Equivalence ´el´ementaire de corps elliptiques, CRAS S´erie I (2000), 1-4. Department of Mathematics, University of PennsylvaniaDRL, 209 S 33rd Street, Philadelphia, PA 19104, USA E-mail address : [email protected] URL : http://math.penn.edu/~pophttp://math.penn.edu/~pop