Local-Global Principles for Zero-Cycles on Homogeneous Spaces over Arithmetic Function Fields
Jean-Louis Colliot-Thélène, David Harbater, Julia Hartmann, Daniel Krashen, R. Parimala, V. Suresh
aa r X i v : . [ m a t h . AG ] A p r LOCAL-GLOBAL PRINCIPLES FOR ZERO-CYCLES ONHOMOGENEOUS SPACES OVER ARITHMETIC FUNCTION FIELDS
J.-L. COLLIOT-THÉLÈNE, D. HARBATER, J. HARTMANN,D. KRASHEN, R. PARIMALA, AND V. SURESH
Abstract.
We study the existence of zero-cycles of degree one on varieties that are definedover a function field of a curve over a complete discretely valued field. We show that local-global principles hold for such zero-cycles provided that local-global principles hold for theexistence of rational points over extensions of the function field. This assertion is analogousto a known result concerning varieties over number fields. Many of our results are shown tohold more generally in the henselian case. Introduction
The study of rational points on varieties is a fundamental subject of arithmetic geometry.Local-global principles (and their obstructions) are one of the main tools in understandingwhether a rational point exists. A related object of study is the index of an algebraic variety,and one may ask whether it is equal to one, i.e., whether the variety admits a zero-cycle ofdegree one. Equivalently, for every prime ℓ , does there exist a point defined over a finitefield extension of degree prime to ℓ ?In this paper, we consider varieties over certain two-dimensional fields; in particular, one-variable function fields over complete discretely valued fields (so-called semi-global fields).These fields are amenable to patching methods. Local-global principles for rational pointson homogeneous spaces over such fields were studied for example in [CTPS12], [CTPS16],[HHK15a]. Here we exhibit several situations where such local-global principles imply cor-responding local-global statements for zero-cycles. Parallel results over number fields werefirst obtained by Liang (Prop. 3.2.3 in [Lia13]; see also [CT15], Section 8.2). However, oursituation involves substantial new difficulties to overcome.A semi-global field admits several natural collections of overfields with respect to whichlocal-global principles can be studied. After a choice of normal projective model of the semi-global field, one may consider two distinct collections of overfields: the first associated with patches on such a model (described in the beginning of Section 2), and the second consistingof overfields which are fraction fields of complete local rings at points of the closed fiber of themodel. Finally, as in the number field case, one may also work with the set of completionswith respect to discrete valuations.Consider a scheme Z of finite type over a semi-global field F . We show (Theorem 3.4) thatin the case of overfields coming from patching, a local-global principle for rational points for Date : September 17, 2018.
Mathematics Subject Classification (2010): Primary 14C25, 14G05, 14H25; Secondary 11E72, 12G05,12F10.
Key words and phrases.
Linear algebraic groups and torsors, zero-cycles, local-global principles,semi-global fields, discrete valuation rings. he base change Z E to all finite separable field extensions E/F implies a local-global princi-ple for separable zero-cycles of degree one over F (or analogously, of degree prime to somegiven prime ℓ ). This also gives a local-global principle for the (separable) index of Z (Corol-lary 3.5). The analogous results hold when the collection of overfields under considerationcomes from points of the closed fiber of a model as described above (Theorem 3.9). Moreover,these latter results extend to the case of function fields of curves over excellent henseliandiscrete valuation rings. In particular, we obtain local-global principles for zero-cycles inthat situation; see Proposition 3.12. The required local-global hypothesis for rational pointsover a semi-global field holds, for instance, if Z is a torsor under a linear algebraic groupsthat is connected and rational (see Corollary 3.10).In the situation where Z is a principal or projective homogeneous space under a linearalgebraic group, we also obtain local-global principles with respect to discrete valuations(Theorem 3.16) under additional hypotheses on F (e.g., when F is the function field ofa curve over a complete discretely valued field with algebraically closed residue field ofcharacteristic zero). In certain cases, the existence of local zero-cycles of degree one alreadyimplies the existence of a global rational point (Theorem 3.23, Theorem 3.27).The results are obtained using a combination of methods. Many of the local-global state-ments rely on descent results that we prove for finite field extensions and for the existenceof rational points, in context of a pair of fields L ⊆ L ′ and a finite separable field extension E ′ /L ′ . In the former type of descent result (e.g., Proposition 2.3), we find a finite separablefield extension E/L such that E ⊗ L L ′ ∼ = E ′ . In the latter type (e.g., Proposition 2.10),given an L -scheme Z with an E ′ -point, we find an appropriate finite separable field exten-sion E/L such that Z has a point over E ⊗ L L ′ . Descent of extensions of fields arising in thecontext of patching is also studied in [HHKPS17], which builds on the results here. For ourlocal-global principles with respect to discrete valuations, we also use structural propertiesof linear algebraic groups, as well as results about nonabelian cohomology in degree two.The manuscript is organized as follows. Section 2 contains descent results. The first twosubsections concern fields occurring in patching. We show that finite separable field exten-sions of some of these overfields descend to the semi-global field F (Subsection 2.1); in othercases it is still possible to descend the existence of points (Subsection 2.2; in particular,see Prop. 2.10). Subsection 2.3 contains local descent results. Local-global principles forzero-cycles are proven in Section 3; this is first done with respect to patches and points onarithmetic curves over complete discrete valuation rings (Subsection 3.1), and then general-ized to excellent henselian valuation rings (Subsection 3.2). In certain cases we obtain local-global principles with respect to discrete valuations (Subsections 3.3 and 3.4), for principaland projective homogeneous spaces over certain 2-dimensional fields, including semi-globalfields.Research on this subject was partially carried out during a visit of the authors at theAmerican Institute of Mathematics (AIM). We thank AIM for the productive atmosphereand wonderful hospitality. 2. Descent results
This section contains descent results which will be essential in proving the local-global prin-ciples in the following section. We first consider collections of fields coming from patching.We recall the following notation, which was established in [HH10], [HHK09], and [HHK15b]: efinition 2.1. Let K be a discretely valued field with valuation ring T , uniformizer t ,and residue field k . Let F be a one-variable function field over K ; i.e., a finitely generatedfield extension of transcendence degree one in which K is algebraically closed. A normalmodel of F is an integral T -scheme X with function field F that is flat and projective over T of relative dimension one, and that is normal as a scheme. If in addition the scheme X isregular, we say that X is a regular model . The closed fiber of X is X k := X × T k .In the case that the discrete valuation ring T is complete, we call F a semi-global field .(We will also often consider the more general case that T is excellent and henselian.)The following notation will be used throughout this manuscript. Notation 2.2.
In the context of Definition 2.1, let X be a normal model for F . If P is a(not necessarily closed) point of the closed fiber X of X , let R P be the local ring of X at P ;let b R P be its completion with respect to the maximal ideal m P ; and let F P be the fractionfield of b R P . In the case that P is a closed point of X , the branches of X at P are the heightone prime ideals of b R P that contain t . We write R ℘ for the local ring of b R P at a branch ℘ .This is a discrete valuation ring. We write b R ℘ for its completion, and F ℘ for the fractionfield of b R ℘ .If U is a non-empty connected affine open subset of X , then we write R U for the subringof F consisting of rational functions that are regular at each point of U . We let b R U be the t -adic completion of R U . This is an integral domain by [HHK15b, Proposition 3.4], and welet F U be the fraction field of b R U . If P ∈ U ⊆ U ′ , then b R U ′ ⊆ b R U ⊂ b R P and F U ′ ⊆ F U ⊂ F P .The spectra of the rings b R P and b R U above are thought of as “patches” on X . For the sakeof readers who are familiar with rigid geometry, we remark that our fields F P and F U are thesame as the rings of meromorphic functions on the corresponding rigid open and rigid closedaffinoid sets obtained by deleting the closed fiber from the patches; see [Ray74], concerningthe relationship between formal schemes and rigid analytic spaces.2.1. Descent of field extensions.
The following two statements generalize Proposition3.5 of [HHK15b]; by the trivial étale algebra (of degree n ) over a field L we mean the directproduct of n copies of L . Proposition 2.3.
Let X be a normal model of a semi-global field F , let P be a closed pointof X , let ℘ be a branch of the closed fiber X at P , and let E ℘ be a finite separable fieldextension of F ℘ . Then there exists a finite separable field extension E P of F P such that E P ⊗ F P F ℘ ∼ = E ℘ as extensions of F P , and such that E P induces the trivial étale algebra over F ℘ ′ for every other branch ℘ ′ at P .Proof. Let f ℘ ( x ) ∈ F ℘ [ x ] be the minimal polynomial of a primitive element of E ℘ . For eachother branch ℘ ′ at P , let f ℘ ′ ( x ) ∈ F ℘ ′ [ x ] be a separable polynomial of the same degree thatsplits completely over F ℘ ′ and thus defines the trivial étale algebra over F ℘ ′ . The field F P is dense in Q F ℘ ′ by Theorem VI.7.2.1 of [Bou72], where the product ranges over all thebranches at P (including ℘ ). So applying Krasner’s Lemma (e.g., [Lan70, Prop. II.2.4]) tothe above polynomials yields the desired extension of F P , which is a field since E ℘ is. (cid:3) Proposition 2.4.
Let X be a normal model of a semi-global field F , let U be a non-emptyconnected affine open subset of the closed fiber of X , and let E U be a finite separable field xtension of F U . Then there is a finite separable field extension E of F such that E ⊗ F F U ∼ = E U as extensions of F U .Proof. Let ¯ U be the closure of U in the closed fiber X of X . After blowing up X at thepoints of P U := ¯ U r U (which changes the model X but does not change F , U , or F U ), wemay assume that ¯ U is unibranched at each point of P U (see [Lip75], Lecture 1). Let P be afinite set of closed points of X that satisfies P ∩ ¯ U = P U , and which contains at least onepoint on each irreducible component of X . Let U be the set of connected components of X r P . Then U ∈ U , and each element of U is affine.For each point P ∈ P U , consider the unique branch ℘ at P on U . Then E ℘ := E U ⊗ F U F ℘ is a finite direct product of finite separable field extensions E ℘,i of F ℘ . By Proposition 2.3,for each i there is a finite separable field extension E P,i of F P such that E P,i ⊗ F P F ℘ ∼ = E ℘,i and such that E P,i ⊗ F P F ℘ ′ is a trivial étale algebra over F ℘ ′ for every other branch ℘ ′ at P . So the direct product of the fields E P,i (ranging over i ) is a finite étale F P -algebra E P satisfying E P ⊗ F P F ℘ ∼ = E ℘ and such that E P ⊗ F P F ℘ ′ is the trivial étale algebra of degree n := [ E U : F U ] over F ℘ ′ . Here E P is well defined for each P ∈ P U , since ℘ is unique given P .For every P ∈ P that is not in P U , let E P be the trivial étale algebra of degree n over F P .Similarly, for every U ′ ∈ U other than U let E U ′ be the trivial étale algebra of degree n over F U ′ , and for every branch ℘ at a point that is not in ¯ U let E ℘ be the trivial étale algebra ofdegree n over F ℘ . Thus for every branch ℘ at a point P ∈ P lying on some U ′ ∈ U (includingthe case U ′ = U ), we have isomorphisms E P ⊗ F P F ℘ ∼ = E ℘ ∼ = E U ′ ⊗ F U ′ F ℘ . But patchingholds for finite separable algebras in this context; see Proposition 3.7 and Example 2.7 in[HHK15b]. So there is a finite étale F -algebra E that compatibly induces all the algebras E P , E U ′ , E ℘ . Since E U is a field, so is E . (cid:3) The above two propositions suggest analogous statements in which the roles of P and U are interchanged: The analog of Proposition 2.3 would assert that if ℘ is a branch at a point P in ¯ U r U , then every finite separable field extension of F ℘ would be induced by a finiteseparable field extension of F U . The analog of Proposition 2.4 would say that every finiteseparable field extension of F P is induced by a finite separable field extension of F . Thisdoes not hold in general, as the following example shows. Example 2.5.
Let T denote the complete discrete valuation ring k [[ t ]] , where k is a fieldof characteristic p > . Let X be the projective x -line over T , and let P be the origin onthe projective k -line X . Then F P equals k (( t, x )) , the fraction field of k [[ t, x ]] . Consider thefield extension E P /F P generated by the solutions of y p − y = αt , where α is a transcendentalpower series (i.e., a power series in x transcendental over F = k (( t ))( x ) ). Then one can showthat E P is not induced by an extension of F in the above sense. In fact, this example is aspecial instance of [HHKPS17], Lemma 2.14, to which we refer the reader for a proof.In [HHKPS17, Section 2], it is shown that versions of descent for field extensions with theroles of P and U interchanged do hold when the residue field k of K has characteristic zero,and that as a consequence there are local-global principles in that situation. To treat themore general case, we consider a different type of descent in the next subsection.2.2. Descent of existence of points.
In order to prove a local-global principle (withrespect to patches) for zero-cycles in arbitrary characteristic, we prove a descent result forthe existence of points instead of field extensions (Proposition 2.10 below). Specifically, let be an excellent henselian (e.g., complete) discrete valuation ring, and choose a normalmodel X of a one-variable function field F over the fraction field K of T . In the contextof Notation 2.2, we show that if E P /F P is a separable field extension whose degree is notdivisible by some prime number ℓ , and if Z is an F -scheme of finite type which has an E P -point, then there is a finite separable field extension E/F of degree prime to ℓ such that Z has a point over E ⊗ F F P . First, some preparation is needed.For P a point on the closed fiber of X as above, the henselization R h P of R P is the sameas the henselization at P of the coordinate ring of an affine open subset of X that contains P . Since that coordinate ring is of finite type over the excellent henselian discrete valuationring T , Artin’s Approximation Theorem (Theorem 1.10 of [Art69]) applies to a system ofpolynomial equations over R h P , and asserts that if there is a solution over b R P then there isa solution over R h P . By clearing denominators, the same assertion holds with b R P and R h P replaced by F P and F h P , where F h P is the fraction field of R h P (this being a separable fieldextension of F ). We use this in the proof of the next proposition.Note that we may pick a fixed algebraic closure ¯ F P of F P , and let ¯ F be the algebraicclosure of F in ¯ F P . Thus ¯ F is an algebraic closure of F that contains F h P . Proposition 2.6.
Let T be an excellent henselian discrete valuation ring with fraction field K . Let F be a one-variable function field over K (e.g., a semi-global field, if K is complete),and let X be a normal model of F . Let P be a (not necessarily closed) point of the closedfiber of X , and let Z be an F -scheme of finite type. Let ℓ be a prime number, and supposethat there is a finite separable field extension E P /F P of degree prime to ℓ such that Z ( E P ) isnon-empty. Then there is a finite separable field extension E ′ P /F h P of degree prime to ℓ suchthat Z ( E ′ P ) is non-empty.Proof. Let d be the smallest positive integer that is prime to ℓ such that there is a finiteseparable field extension E P /F P of degree d for which Z has an E P -point ξ . Since E P is separable over F P , there are exactly d distinct F P -embeddings σ , . . . , σ d of E P into analgebraic closure ¯ F P of F P . By the minimality of d and E P , the point ( σ ( ξ ) , . . . , σ d ( ξ )) ∈ Z d ( ¯ F P ) does not lie on the closed subset ∆ ⊂ Z d where two or more of the entries are equal.Consider the image D ⊂ S d ( Z ) of ∆ in the d -th symmetric power of Z ; i.e., S d ( Z ) r D =( Z d r ∆) /S d . The image ζ ∈ S d ( Z ) r D of ( σ ( ξ ) , . . . , σ d ( ξ )) is an F P -point on this F -scheme, corresponding to a morphism Spec( F P ) → S d ( Z ) r D . The image of this morphismis a point of (the underlying topological space of) S d ( Z ) r D , and this lies in some affineopen subset Spec( A ) ⊆ S d ( Z ) r D . This point corresponds to a solution over F P to a systemof polynomial equations over F that defines A .By Artin’s Approximation Theorem, there is a solution to this system of equations overthe field F h P . This corresponds to an F h P -point ζ ′ on S d ( Z ) r D . Pick a point on Z d r ∆ thatmaps to ζ ′ . Each entry lies in an algebraic closure of F h P , or equivalently of F . The d entriesare distinct, and this set of entries is stable under the absolute Galois group of F h P , since ζ ′ isdefined over F h P . Thus the entries form a disjoint union of orbits under this absolute Galoisgroup, say of orders d , . . . , d r , with P i d i = d . Since d is prime to ℓ , so is some d i . Let ξ ′ be an entry lying in the i -th orbit; this defines a point of Z , say with field of definition E ′ P .Then the field E ′ P is separable over F h P by the distinctness of the entries; the degree of E P over F h P is d i , which is prime to ℓ ; and ξ ′ is an E ′ P -point of Z . (cid:3) he above proof actually shows more, viz. that if [ E P : F P ] is minimal for the givenproperty, then E ′ P can be chosen so that [ E ′ P : F h P ] = [ E P : F P ] . This follows from the factthat at the end of the proof, d i must equal d (i.e., there is just one orbit) by minimality of d and because ξ ′ induces a point of Z over an extension of F P of degree at most d i . Lemma 2.7.
Let L ⊆ L ′ ⊆ E ′ be separable algebraic field extensions, where [ E ′ : L ′ ] isfinite. Let Z be an L -scheme of finite type such that Z ( E ′ ) is non-empty. Then there arefinite separable field extensions L ⊆ e L ⊆ e E such that e L ⊆ L ′ and e E ⊆ E ′ ; [ e E : e L ] = [ E ′ : L ′ ] ; Z ( e E ) is non-empty; and E ′ is the compositum of its subfields e E and L ′ .Proof. Let ξ ∈ Z ( E ′ ) . Since Z is an L -scheme of finite type, there is an affine open subsetof Z that contains ξ and is L -isomorphic to a Zariski closed subset Y of A nL for some n .Let y , . . . , y n ∈ E ′ be the coordinates of the image of ξ in Y . Let z be a primitive elementof the finite separable field extension E ′ /L ′ , say with minimal monic polynomial g over L ′ of degree d = [ E ′ : L ′ ] . Thus each y i is of the form P d − j =0 c ij z j with c ij ∈ L ′ . Let e L bethe subfield of L ′ generated over L by the coefficients of g and by the elements c ij ; this isfinite over L , and it is separable over L since L ′ is. The polynomial g is irreducible over e L because it is irreducible over L ′ . Thus e E := e L ( z ) ⊆ E ′ is separable and of degree d over e L , and e E ⊆ e EL ′ = L ′ ( z ) = E ′ . Also, each y i lies in e E , since z ∈ e E and c ij ∈ e L ⊆ e E . So ( y , . . . , y r ) ∈ Y ( e E ) and thus ξ ∈ Z ( e E ). (cid:3) Lemma 2.8.
Let ℓ be a prime number, and let L ⊆ e L ⊆ e E be finite separable field extensionssuch that [ e E : e L ] is prime to ℓ . Let b E be the Galois closure of e E/L . Then for every Sylow ℓ -subgroup S of Gal( b E/L ) , there is some σ ∈ Gal( b E/L ) such that the compositum e L b E S ⊆ b E contains σ ( e E ) .Proof. The intersection
Gal( b E/ e L ) ∩ S ⊆ Gal( b E/L ) is an ℓ -subgroup of Gal( b E/ e L ) , and so it iscontained in a Sylow ℓ -subgroup S ∗ of Gal( b E/ e L ) . Let S ′ be a Sylow ℓ -subgroup of Gal( b E/ e E ) .Since [ e E : e L ] is prime to ℓ , the group S ′ is also a Sylow ℓ -subgroup of Gal( b E/ e L ) . Thus S ∗ , S ′ are conjugate subgroups of Gal( b E/ e L ) , say by an element σ ∈ Gal( b E/ e L ) ⊆ Gal( b E/L ) .Since S ′ ⊆ Gal( b E/ e E ) , its conjugate S ∗ = ( S ′ ) σ is contained in Gal( b E/σ ( e E )) . Thus b E S ∗ contains σ ( e E ) . So e L b E S = b E Gal( b E/ e L ) b E S = b E Gal( b E/ e L ) ∩ S , which contains b E S ∗ and hence contains σ ( e E ) . (cid:3) Recall that if L is a field, Z is an L -scheme of finite type, and A/L is a finite direct productof field extensions L i /L , an A -point on Z is a collection of points in Z ( L i ) for each i . Inparticular, Z ( A ) is nonempty if and only if Z ( L i ) is nonempty for all i . Proposition 2.9.
Let ℓ be a prime number, and let L ⊆ L ′ ⊆ E ′ be separable algebraic fieldextensions, where [ E ′ : L ′ ] is finite and prime to ℓ . Let Z be an L -scheme of finite type suchthat Z ( E ′ ) is non-empty. Then there is a finite separable field extension E/L of degree primeto ℓ such that Z ( E ⊗ L L ′ ) is non-empty.Proof. Let e L and e E be as in Lemma 2.7; in particular, [ e E : e L ] = [ E ′ : L ′ ] is prime to ℓ . Let b E be the Galois closure of e E over L ; let S be a Sylow ℓ -subgroup of Gal( b E/L ) ; and let E be the fixed field b E S . We will show that E has the desired properties. ince S is a Sylow ℓ -subgroup of Gal( b E/L ) , the degree of E over L is prime to ℓ . In orderto show that Z ( E ⊗ L L ′ ) is non-empty, it is sufficient to show that Z ( E ⊗ L e L ) is non-empty,because L ⊆ e L ⊆ L ′ . Since E/L is a separable field extension of finite degree, E ⊗ L e L is afinite separable algebra over e L , and hence is a finite direct product Q e E i of finite separablefield extensions of e L . It therefore suffices to show that Z ( e E i ) is non-empty for all i . Hereeach e E i is isomorphic to a compositum of e L and E with respect to some L -embeddings ofthose two fields into the common overfield b E (since b E/L is Galois).After conjugating, we may assume that the above L -embedding of e L into b E is the givenone (i.e., the original composition e L ֒ → e E ֒ → b E ), while allowing the L -embedding of E into b E to vary. The images of E in b E under the various L -algebra embeddings are just the Galoisconjugates σ ( E ) for σ ∈ Gal( b E/L ) . Since E is the fixed field b E S , its Galois conjugatesare the fixed fields of the conjugates of S , viz. the fields b E S ′ where S ′ varies over the Sylow ℓ -subgroups of Gal( b E/L ) . So it suffices to show that for each S ′ , there is a point of Z definedover the compositum e L b E S ′ ⊆ b E .So consider any S ′ . By Lemma 2.8 (which applies since [ e E : e L ] is prime to ℓ ), e L b E S ′ contains τ ( e E ) for some τ ∈ Gal( b E/L ) . But Z is an L -variety that has an e E -point ξ , byLemma 2.7. Hence Z also has a τ ( e E ) -point, viz. τ ( ξ ) . But Z ( τ ( e E )) ⊆ Z ( e L b E S ′ ) , since τ ( e E ) ⊆ e L b E S ′ . So indeed Z has a point defined over e L b E S ′ . (cid:3) Proposition 2.10.
Let T be an excellent henselian discrete valuation ring with fraction field K . Let F be a one-variable function field over K (e.g., a semi-global field, if K is complete),and let X be a normal model of F . Let P be a (not necessarily closed) point of the closedfiber of X , and let Z be an F -scheme of finite type. Let ℓ be a prime number, and supposethat there is a finite separable field extension E P /F P of degree prime to ℓ such that Z ( E P ) is non-empty. Then there is a finite separable field extension E/F of degree prime to ℓ suchthat Z ( E ⊗ F F P ) is non-empty.Proof. By Proposition 2.6, there is a finite separable field extension E ′ P /F h P of degree primeto ℓ such that Z ( E ′ P ) is non-empty. Applying Proposition 2.9 with L = F , L ′ = F h P , and E ′ = E ′ P , we obtain a finite separable field extension E/F of degree prime to ℓ such that Z ( E ⊗ F F h P ) is non-empty. But F h P is contained in F P , and so Z ( E ⊗ F F P ) is non-empty. (cid:3) Local descent results.
In this subsection, we establish local descent results which willbe used to prove local-global principles with respect to discrete valuations in Subsection 3.3.Let A be a complete regular local ring of dimension 2 with field of fractions F and residuefield k . For any prime π of A , let F π denote the completion of F with respect to the discretevaluation associated to π , and let k ( π ) denote the residue field.The following lemma is proved in ([PPS16, Lemma 5.1]) for Galois extensions, and asimilar proof gives the general case. Lemma 2.11.
Let A be a complete regular local ring of dimension two, F its field of fractionsand k its residue field. Let π, δ ∈ A generate the maximal ideal and let E π /F π be a finiteseparable unramified field extension. If char( k ) does not divide [ E π : F π ] , then there exists afinite separable field extension E/F such that E ⊗ F F π ≃ E π and the integral closure of A in E is a complete regular local ring with fraction field E and maximal ideal ( π ′ , δ ′ ) , where π ′ and δ ′ generate the unique primes lying over π and δ , respectively. roof. The residue field F ( π ) of F π is the field of fractions of A/ ( π ) . Since A is a completeregular local ring, A/ ( π ) is a complete discrete valuation ring, k is the residue field of A/ ( π ) ,and the image ¯ δ of δ is a uniformizer in A/ ( π ) .Let E ( π ) be the residue field of E π ; this is a complete discretely valued field. Let L ( π ) be the maximal unramified field extension of F ( π ) contained in E ( π ) , and let L π be thesubextension of E π /F π whose residue field is L ( π ) . Let κ be the residue field of E ( π ) (orequivalently, of L ( π ) ) at its discrete valuation. Since [ E π : F π ] is coprime to char( k ) , sois [ κ : k ] , and thus κ is a finite separable field extension of k . Write κ = k [ t ] / ( f ( t )) forsome monic separable polynomial f ( t ) ∈ k [ t ] . By lifting the polynomial f ( t ) to a monicpolynomial over A , we obtain a finite étale A -algebra B ; this is a complete regular local ringwith maximal ideal ( π, δ ) at which the residue field is κ . The residue field of L (i.e., of B )at π is L ( π ) . Since the same is true for L π , it follows that the complete discretely valuedfields L ⊗ F F π and L π are isomorphic over F π . Note that the image ¯ δ ∈ B/ ( π ) ⊂ L ( π ) of δ is a uniformizer for L ( π ) .Now E ( π ) /L ( π ) is totally ramified of degree d prime to char ( k ) . So by [Lan70, PropositionII.5.12], E ( π ) = L ( π )( d √ v ¯ δ ) for some unit v ∈ B/ ( π ) , the valuation ring of L ( π ) . Let u ∈ B be a lift of v ∈ B/ ( π ) and let E = L ( d √ uδ ) . Then E is a finite separable field extension of F , and E ⊗ F F π ≃ E ⊗ L L ⊗ F F π ≃ E ⊗ L L π ≃ L π ( d √ uδ ) ≃ E π . The integral closure of B in E (or equivalently, of A in E ) is a two-dimensional complete local ring with maximal ideal ( π, d √ uδ ) , and hence it is regular. Its fraction field is E , and the ideals in this ring that aregenerated by π and d √ uδ are the unique prime ideals lying over the ideals ( π ) and ( δ ) in A .So E is as asserted. (cid:3) The above lemma can be used to obtain a descent statement for not necessarily unramifiedextensions.
Lemma 2.12.
Let A be a complete regular local ring of dimension 2, F its field of fractionsand k its residue field. Let π, δ ∈ A generate the maximal ideal and let E π /F π be a finiteseparable field extension. Suppose char( k ) does not divide [ E π : F π ] . Let ℓ be a prime number.If [ E π : F π ] is prime to ℓ , then there exists a finite separable field extension E/F such that • [ E : F ] is prime to ℓ ; • E ⊗ F F π is a field; • E π is isomorphic to a subfield of E ⊗ F F π ; • the integral closure of A in E is a complete regular local ring with fraction field E and maximal ideal ( π ′ , δ ′ ) , where π ′ and δ ′ generate the unique primes lying over π and δ , respectively.Proof. Let L π /F π be the maximal unramified extension contained in E π . By Lemma 2.11,there is a finite separable extension L/E such that L ⊗ F π ≃ L π ; the integral closure of A in L is regular with maximal ideal ( π ′ , δ ′ ) ; and such that ( π ′ ) and ( δ ′ ) are the unique primeslying over the primes ( π ) and ( δ ) of A . Thus, replacing F π by L π and F by L , we mayassume that E π /F π is totally ramified. Since char( k ) does not divide n := [ E π : F π ] , neitherdoes the characteristic of the residue field k ( π ) of F π . Hence E π = F π ( n √ uπ ) for some u ∈ F π which is a unit at the discrete valuation of F π ([Lan70, Section II.5, Proposition 12]). Let ¯ u be the image of u in k ( π ) . Since k ( π ) is the field of fractions of A/ ( π ) , we have ¯ u = ¯ v ¯ δ i forsome i and some unit ¯ v ∈ A/ ( π ) with preimage v ∈ A . Thus u − vδ i lies in the valuation ringof F π and is congruent to mod π . By Hensel’s Lemma, this element has an n -th root in F π , nd so E π = F π ( n √ vδ i π ) . Let E = F ( n √ δ, n √ vπ ) . Since n is not divisible by either char( k ) or ℓ , the field extension E/F is separable and of degree prime to ℓ . Moreover, E ⊗ F F π isisomorphic to the field F π ( n √ δ, n √ vπ ) , which contains E π as a subfield. Since E/F is finiteand since A is a complete local ring with fraction field F , the integral closure B of A in E isa complete local ring with fraction field E . By ([PS14, Lemma 3.2]), B is a regular local ringwith maximal ideal ( n √ δ, n √ vπ ) . Since ( n √ δ ) and ( n √ vπ ) are the unique primes of B lyingover the primes ( δ ) and ( π ) of A , the result follows. (cid:3) Finally, we require a simultaneous descent result:
Lemma 2.13.
Let A be a complete regular local ring of dimension , F its field of fractions, ( π, δ ) its maximal ideal, and k its residue field. Suppose that char( k ) = 0 . Let E π /F π and E δ /F δ be finite field extensions. Let ℓ be a prime. Suppose that the degrees of E π /F π and E δ /F δ are prime to ℓ . Then there exists a finite (separable) field extension E/F such that • [ E : F ] is prime to ℓ ; • the integral closure of A in E is a complete regular local ring; • E ⊗ F F π and E ⊗ F F δ are fields; • E π is isomorphic to a subfield of E ⊗ F F π ; • E δ is isomorphic to a subfield of E ⊗ F F δ .Proof. By Lemma 2.12, there exists a finite (separable) field extension ˜ E/F such that [ ˜ E : F ] is prime to ℓ , ˜ E ⊗ F F π is a field which contains an isomorphic copy of E π as a subfield, andthe integral closure ˜ B of A in ˜ E is a regular local ring with maximal ideal ( π ′ , δ ′ ) , with π ′ and δ ′ lying over π and δ , respectively. Moreover, the ideals ( π ′ ) and ( δ ′ ) are uniquelydetermined by π and δ . Since δ ′ is a prime lying over δ , F δ ⊆ ˜ E δ ′ . Since E δ /F δ is a separablefield extension, E δ ⊗ F δ ˜ E δ ′ is a product of field extensions E i / ˜ E δ ′ . Since [ E δ : F δ ] prime to ℓ , [ E i : ˜ E δ ′ ] is prime to ℓ for some i .Again by Lemma 2.12 (this time applied to the complete regular local ring e B and E i / e E δ ′ ),there exists a finite field extension E/ e E of degree prime to ℓ with the following properties: E ⊗ e E e E δ ′ is a field containing E i as a subfield; the integral closure B of e B in E is a completeregular local ring having fraction field E and maximal ideal of the form ( π ′′ , δ ′′ ) ; and ( π ′′ ) and ( δ ′′ ) are the unique primes of B that lie over the primes ( π ′ ) and ( δ ′ ) of e B . Note thatthe uniqueness of ( δ ′ ) implies that ˜ E δ ′ = ˜ E ⊗ F F δ . Since E δ is isomorphic to a subfield of E i , E δ is isomorphic to a subfield of E ⊗ ˜ E ˜ E δ ′ = E ⊗ ˜ E ( ˜ E ⊗ F F δ ) = E ⊗ F F δ as claimed.Similarly, E π is a subfield of ˜ E ⊗ F F π which is a subfield of E ⊗ F F π by base change. Theuniqueness of ( π ′′ ) implies that the latter is a field. Since [ E : ˜ E ] and [ ˜ E : F ] are prime to ℓ , [ E : F ] is prime to ℓ . Since the integral closure of A in E is B , the assertion follows. (cid:3) Local-global principles for zero-cycles
Let F be a field, and let Z be an F -scheme. A zero-cycle on Z is a finite Z -linearcombination P n i P i of closed points P i of Z . Its degree is P n i deg( P i ) , where deg( P i ) is thedegree of the residue field of P i over F . We say that a closed point P of Z is separable if itsresidue field is separable over F . A zero-cycle P n i P i is called separable if each P i is.We will prove local-global principles for zero-cycles in different settings. First, we considercollections of overfields coming from patching and from points on the closed fiber of a model. econd, we will use this to obtain local-global principles with respect to discrete valuations.Many of the statements in this section rely on the descent results in the previous section. Inour results, we have to assume a local-global principle for points (in the respective setting)over the function field F as well as over all of its finite separable field extensions. This isanalogous to results in the number field case; see [Lia13], Prop. 3.2.3, and [CT15], Section 8.2.Example 3.7 shows that this hypothesis is actually necessary. Corollary 3.10 below exhibitssituations in which the assumption is satisfied.3.1. Local-global principles with respect to patches and points.
In this subsection weprove that certain local-global principles for the existence of rational points imply analogouslocal-global principles for zero-cycles, for varieties over semi-global fields. This will be provenin two contexts, one where the overfields come from patching, and one where they correspondto points on the closed fiber of a model.
Notation 3.1.
Let X be a normal model of a one-variable function field F over a discretelyvalued field K , and let X denote the closed fiber. Let P be a finite nonempty set of closedpoints of X that meets each irreducible component of X , and let U be the set of connectedcomponents of the complement of P in X . Let B be the set of branches of X at points of P .We let Ω X be the collection of field extensions F P /F where P is a (not necessarily closed)point of X , and let Ω X , P denote the collection of field extensions F ξ /F where ξ ∈ P ∪ U .(See Notation 2.2.) Notation 3.2.
Let X be a normal model for F as above, and let E/F be a finite fieldextension. We let X E denote the normalization of X in E . (This is a normal model for E .)If P is a finite set of closed points of X , we let P E denote its preimage under the naturalmap X E → X . Definition 3.3.
Let F be a field, let Z be an F -scheme of finite type, and let Ω be acollection of overfields of F . We use the following terminology: • The pair ( Z, Ω) satisfies a local-global principle for rational points if it has the fol-lowing property: Z ( F ) = ∅ if and only if Z ( L ) = ∅ for every L ∈ Ω . • The pair ( Z, Ω) satisfies a local-global principle for closed points of degree prime to ℓ if it has the following property: Z has a closed point of degree prime to ℓ if and onlyif the base change Z L has a closed point of degree prime to ℓ for every L ∈ Ω . • The pair ( Z, Ω) satisfies a local-global principle for zero-cycles of degree one if andonly if it has the following property: Z has a zero-cycle of degree one if and only if Z L has a zero-cycle of degree one for all L ∈ Ω .Similarly, one can speak of a local-global principle for separable closed points or separablezero-cycles.Below we consider the case where F is a semi-global field (i.e., K is complete). Note thata finite separable field extension E/F is again a semi-global field.
Theorem 3.4.
Let F be a semi-global field with normal model X . Let P be a finite nonemptyset of closed points that meets every irreducible component of the closed fiber X of X . Let Z be an F -scheme of finite type. Assume that for all finite separable field extensions E/F , ( Z E , Ω X E , P E ) satisfies a local-global principle for rational points. Then for every prime num-ber ℓ , ( Z, Ω X , P ) satisfies a local-global principle for separable closed points of degree primeto ℓ . roof. Let U be as in Notation 3.1. We need to prove that if Z F ξ has a separable closedpoint z ξ of degree prime to ℓ for every ξ ∈ P ∪ U , then Z has a separable closed point z ofdegree prime to ℓ . For each ξ , let E ξ /F ξ be the residue field extension at z ξ .By Proposition 2.4, for each U ∈ U , there is a finite separable field extension A U of F that induces E U (i.e., A U ⊗ F F U is isomorphic to E U ), and hence has degree prime to ℓ . Inparticular, each Z ( A U ⊗ F F U ) is non-empty. By Proposition 2.10, for each P ∈ P , there isa finite separable field extension A P of F of degree prime to ℓ such that Z ( A P ⊗ F F P ) isnon-empty. Let A be the tensor product over F of all the fields A P and A U . This is an étale F -algebra of degree prime to ℓ , and it is the direct product of finite separable field extensions A i of F , each of which is a compositum of the fields A P and A U . Since the degree of A is thesum of the degrees of the fields A i over F , at least one of those fields A i has degree prime to ℓ . Write E for this field. So E/F is separable of degree prime to ℓ , and for each ξ ∈ P ∪ U , Z ( E ⊗ F F ξ ) is non-empty.Let X E , P E be as in Notation 3.2, let X E be the closed fiber, and let U E be the set ofconnected components of the complement of P E in X E . For each P ∈ P , E ⊗ F F P is the directproduct of the fields E P ′ , where P ′ runs over the points of P E that lie over P ; and similarlyfor each U ∈ U . Hence for each ξ ′ ∈ P E ∪ U E , Z ( E ξ ′ ) is non-empty. By assumption, thisimplies Z ( E ) is nonempty; i.e., Z has a point defined over a finite separable field extensionof F of degree prime to ℓ . (cid:3) The hypothesis in the above theorem cannot be weakened to consider merely F instead ofall finite separable field extensions; see Example 3.7 below.Given a variety V over a field k , the index (resp. separable index ) of V is the greatestcommon divisor of the degrees of the finite (resp. finite separable) field extensions of k overwhich V has a rational point. This is the same as the smallest positive degree of a zero-cycle(resp. separable zero-cycle) on V . In this terminology, Theorem 3.4 says that if the separableindex of each Z F ξ is prime to ℓ , then so is the separable index of Z ; i.e., if each Z F ξ has azero-cycle of degree prime to ℓ , then so does Z . Corollary 3.5.
Let F be a semi-global field with normal model X , and let X denote theclosed fiber. Let P be a finite nonempty set of closed points of X which meets every irreduciblecomponent of X . Let Z be an F -scheme of finite type such that for every finite separableextension E/F , ( Z E , Ω X E , P E ) satisfies a local-global principle for rational points. Then theprime numbers that divide the separable index of Z are precisely those that divide the separableindex of some Z L for L ∈ Ω X , P . In particular, the separable index of Z is equal to one ifand only if the separable index of each Z L is equal to one. Corollary 3.6.
In Corollary 3.5, if char F = 0 , or if Z is regular and generically smooth,then the assertion also holds with the separable index replaced by the index. In particular, ( Z, Ω X , P ) satisfies a local-global principle for zero-cycles of degree one.Proof. This is trivial in the former case. In the latter case it follows from the fact that underthose hypotheses the index is equal to the separable index (Theorem 9.2 of [GLL13]). (cid:3)
The above results (and their proofs) show that if ( Z E , Ω X E , P E ) satisfies a local-globalprinciple for rational points for all finite separable E/F , the index (resp. separable index) of Z divides some power of the least common multiple of the indices (resp. separable indices)of the fields Z L for L ∈ Ω X , P . It would be interesting to obtain a bound on the exponent, interms of F and Z . (See Section 2.2 of [HHKPS17] for a partial result in this direction.) xample 3.7. This example shows that in Theorem 3.4 and in Corollaries 3.5 and 3.6,we cannot simply assume that Z satisfies a local-global principle over F , but must alsoconsider finite field extensions of F . Let k be a field of characteristic zero, let T = k [[ t ]] , let K = k (( t )) , let F = K ( x ) , and let X be the projective x -line over T . On the closed fiber of X , take P to be the set { P , P } consisting of the two points on the closed fiber where x = 0 and where x = 1 . The complement of P in the closed fiber is a connected affine open set U ;take U = { U } .Let E = F [ y ] / ( y − x ( x − t )( x − x − − t )) , a degree two separable field extension of F that splits over F U but whose base change to F P i is a degree two field extension for i = 0 , .So the (separable) index of Y := Spec( E ) is two over F and over each F P i , but is one over F U . The normalization Y of X in E is a degree two branched cover of X , whose fiber overthe generic point of X is Y , and whose fiber over the closed point of Spec( T ) consists oftwo copies of the projective k -line that meet at two points. Its reduction graph containsa loop, and so by [HHK15a, Proposition 6.2] there is a degree three connected split cover Z → Y (i.e.,
Z × Y Q consists of three copies of Q for every point Q ∈ Y other than thegeneric point). The fiber Z of Z over the generic point of X is the spectrum of a degree sixseparable field extension L of F , and hence the index of Z over F is six. But since Z → Y is a split cover, the index of Z over each F P i is two and over F U is one (the same as for Y ).Since the index of Z over F (resp. over F P i , F U ) is six (resp. two, one), it follows that Z ( F P i ) = ∅ and hence ( Z, Ω X , P ) trivially satisfies a local-global principle for rational points;but the conclusions of Theorem 3.4 and its corollaries fail here if we take ℓ = 3 . Theexplanation is that ( Z E , Ω Y , P E ) does not satisfy a local-global principle for rational points,due to the above index computations for Z .We next prove analogous results in which all the points on the closed fiber X of X areused. That is, instead of considering a collection of overfields of the form Ω X , P for somefinite set P as before, we consider the collection Ω X of all overfields of F of the form F P ,where P ranges over all (not necessarily closed) points of X . Proposition 3.8.
Let F be a semi-global field with normal model X , and let P be a finitenonempty set of closed points of X which meets every irreducible component of the closedfiber. Let Z be an F -scheme of finite type. Then the following are equivalent:(1) ( Z, Ω X ) satisfies a local-global principle for rational points.(2) For every choice of P as in Notation 3.1, ( Z, Ω X , P ) satisfies a local-global principlefor rational points.Proof. First suppose that ( Z, Ω X ) satisfies a local-global principle for rational points. Let P and U be as in Notation 3.1. Assume that Z has a rational point over F ξ for each ξ ∈ P ∪ U .Then every point P of X that is not contained in P lies in some U ∈ U , and hence F U ⊆ F P .Thus Z has a rational point over F P for every point P ∈ X (including the generic points ofirreducible components). Hence by hypothesis, Z has an F -point. Thus ( Z, Ω X , P ) satisfiesa local-global principle for rational points.Conversely, suppose that ( Z, Ω X , P ) satisfies a local-global principle for rational points forevery choice of P as above. Assume that Z ( F P ) is non-empty for every (not necessarilyclosed) point P ∈ X . Given an irreducible component X i of X , consider its generic point η i . By Proposition 5.8 of [HHK15a], since Z ( F η i ) is non-empty it follows that Z ( F U i ) is alsonon-empty for some non-empty affine open subset U i ⊂ X i that does not meet any other rreducible component of X . Let U be the collection of these disjoint open sets U i of X ,and let P be the complement of their union in X ; note that this is a non-empty finite set ofpoints. By hypothesis, Z has an F -point (note that Z ( F P ) is non-empty for each P ∈ P ).So ( Z, Ω X ) satisfies a local-global principle for rational points. (cid:3) Using Proposition 3.8, we obtain the following result, which parallels Theorem 3.4, Corol-lary 3.5, and Corollary 3.6:
Theorem 3.9.
Let F be a semi-global field with normal model X . Let Z be an F -schemeof finite type such that for every finite separable field extension E/F , ( Z E , Ω X E ) satisfies alocal-global principle for rational points. Then(a) For every prime number ℓ , ( Z, Ω X ) satisfies a local-global principle for separable closedpoints of degree prime to ℓ .(b) The prime numbers that divide the separable index of Z are precisely those that dividethe separable index of Z L for some L ∈ Ω X . In particular, the separable index of Z is equal to one if and only if the separable index of each Z L is equal to one.(c) If char( F ) = 0 or if Z is regular and generically smooth, the previous assertion alsoholds with the separable index replaced by the index. In particular, ( Z, Ω X ) satisfiesa local-global principle for zero-cycles of degree one.Proof. We begin by proving (a). Suppose that for every P in the closed fiber X of X , Z has a point over a finite separable extension E P of F P having degree prime to ℓ . Foreach irreducible component X i of X with generic point η i , Proposition 2.10 yields a finiteseparable field extension E i of F of degree prime to ℓ such that Z ( E i ⊗ F F η i ) is non-empty.By Proposition 5.8 of [HHK15a], there is a non-empty affine open subset U i ⊂ X i that meetsno other irreducible component of X , such that Z ( E i ⊗ F F U i ) is non-empty. Since E i /F isa separable field extension of degree prime to ℓ , so is E U i /F U i for some direct factor E U i of E i ⊗ F F U i . Here Z ( E U i ) is non-empty, since Z ( E i ⊗ F F U i ) is non-empty.As in the proof of Proposition 3.8, we let U be the collection of open sets U i , and let P consist of the (finitely many) closed points of X that do not lie in any U i . For any finiteseparable field extension E/F , ( Z E , Ω X E , P E ) satisfies a local-global principle for rationalpoints, by the hypothesis of the theorem together with Proposition 3.8 applied to Z E . So byTheorem 3.4, ( Z, Ω X , P ) satisfies a local-global principle for separable closed points of degreeprime to ℓ . Since Z has a separable point of degree prime to ℓ over F ξ for each ξ ∈ P ∪ U ,it follows that Z has a separable point of degree prime to ℓ .The other two statements are then immediate, as in the analogous case for Ω X , P in Corol-laries 3.5 and 3.6. (cid:3) The following corollary exhibits some cases in which our theorem above applies, for ho-mogeneous spaces under linear algebraic groups that are connected and (retract) rational.Here, by a homogeneous space Z under a linear algebraic group G defined over a field F , wemean an F -scheme Z together with a group scheme action of G on Z over F such that theaction of the group G ( ¯ F ) on the set Z ( ¯ F ) is simply transitive. Corollary 3.10.
Let F be a semi-global field with normal model X . Let G be a linear alge-braic group over F (i.e., a smooth affine group scheme over F ), and let Z be a homogeneousspace for G . In each of the following cases, the prime numbers that divide the separable ndex of Z are exactly those that divide the separable index of some Z L , where L ∈ Ω X ; andmoreover ( Z, Ω X ) satisfies a local-global principle for separable zero-cycles of degree one.(1) G is connected and retract rational, and Z is a torsor under G .(2) G is connected and rational, and G ( E ) acts transitively on Z ( E ) for every field ex-tension E/F .The same conclusion holds without separability (i.e., for the index and for zero-cycles) if Z is smooth; e.g., this holds in case (1).Proof. In both cases, ( Z, Ω X ) satisfies a local-global principle for rational points. In the firstcase, this was shown in [Kra10] (see also Corollary 6.5 of [HHK15a] for the case when G is rational); in the second this follows from Corollary 2.8 of [HHK15a]. These local-globalprinciples also hold for Z E if E/F is a finite separable field extension of F , since the aboveassumptions on F are preserved under such a base change. Thus Theorem 3.9(b) applies. If Z is smooth (e.g., in case (1)), Theorem 3.9(c) also applies. (cid:3) Theorem 3.9 also applies in the case of certain other linear algebraic groups that need notbe rational; see Section 4.3 of [HHK14] for examples.3.2.
Local-global principles over excellent henselian discrete valuation rings.
Inthis subsection, T is assumed to be excellent henselian instead of complete. We now extendthe results from the previous subsection to that case via Artin Approximation.Let T be an excellent henselian discrete valuation ring, with residue field k and fractionfield K . Let X be a normal flat projective T -curve with closed fiber X and function field F ; i.e., X is a normal model of F . Every finite separable field extension of F is also thefunction field of a normal flat projective T -curve. The completion b T of T is a completediscretely valued field with the same residue field k ; its fraction field b K is the completion of K . Because T is henselian, the base change b X := X × T b T is a normal connected projective b T -curve with closed fiber X and function field b F := frac( F ⊗ K b K ) . The complete local ringsof b X and X at a point P ∈ X are naturally isomorphic, and so we may write b R P and F P without ambiguity, for those rings and their fraction fields. Similarly, for U an affine opensubset of the closed fiber, the notations b R U and F U are unambiguous, being the same for b X and X . If E/F is a finite separable field extension, we write b E = E ⊗ F b F = frac( E ⊗ K b K ) .This is the function field of the normalization X E of X in E , whose closed fiber is denotedby X E . Proposition 3.11.
In the above situation, let Z be an F -scheme of finite type, let E/F bea finite separable field extension. If Z ( b E ) is non-empty then so is Z ( E ) .Proof. By hypothesis, Z E has a point over b E . By Artin’s Approximation Theorem (Theo-rems 1.10 and 1.12 of [Art69]), Z E also has an E -point. Equivalently, Z has an E -point. (cid:3) Let F be as above with normal model X , and let Z be an F -scheme of finite type. Asin Definition 3.3, Ω X denotes the set of all overfields of the form F P . By Proposition 3.11above, ( Z, Ω X ) satisfies a local-global principle for rational points if and only if ( b Z, Ω b X ) satisfies such a local-global principle, where b Z := Z × F b F .Preserving the notation given just before Lemma 3.11, we have: roposition 3.12. The assertions of Theorem 3.9 remain true if F is the function field ofa curve over an excellent henselian discrete valuation ring T as above.In particular, assume that Z is an F -scheme of finite type which is regular and genericallysmooth, such that for every finite separable field extension E/F , ( Z E , Ω X E ) satisfies a local-global principle for rational points. Then ( Z, Ω X ) satisfies a local-global principle for zero-cycles of degree one.Proof. Let ℓ be prime and assume that for every P ∈ X there is a finite separable extension E P /F P of degree prime to ℓ such that Z ( E P ) is non-empty. By Proposition 2.10, for each P there is a finite separable field extension A P /F of degree prime to ℓ such that Z ( A P ⊗ F F P ) is non-empty. If η is the generic point of an irreducible component of X , then by applying[HHK15a, Proposition 5.8] to b Z A η we deduce that Z ( A η ⊗ F F U ) = b Z ( A η ⊗ F F U ) is non-empty for some dense open subset U of that component that meets no other component.Write A U = A η . Let U be the collection of these (finitely many) sets U , and let P be thecomplement in X of the union of these sets U .Proceeding as in the second half of the proof of Theorem 3.4 using the field extensions A P , A U of F (for P ∈ P and U ∈ U ), we obtain a finite separable extension E/F of degreeprime to ℓ such that Z ( E × F F ξ ) is non-empty for each ξ ∈ P ∪ U . Now every point P ∈ X outside of P lies on some U ∈ U , and hence F P contains F U ; so Z ( E × F F P ) is non-emptyfor every point P on the closed fiber of X . Thus by hypothesis, Z ( E ) is non-empty. Thisproves the analog of Theorem 3.9(a) in the henselian case.The henselian analogs of the remaining assertions are then immediate, as before. (cid:3) As a consequence, Corollary 3.10 remains valid if F is the function field of a curve overan excellent henselian discrete valuation ring.In the above situation, with T an excellent henselian discrete valuation ring and P a pointof X , let R h P be the henselization of the local ring R P at its maximal ideal, and let F h P beits fraction field. Also, for every discrete valuation v on F , let R v ⊂ F be the associatedvaluation ring, with henselization R h v and completion b R v , having fraction fields F h v and F v .Recall that a point Q of X is called the center of v if R v contains the local ring O Z,Q of Q on Z , and if the maximal ideal m Q of O Z,Q is the contraction of the maximal ideal m v of R v . Proposition 3.13.
Let T be an excellent henselian discrete valuation ring, and let F and X be as above. For every discrete valuation v on F , there is a point P on the closed fiber X of X such that F h P ⊆ F h v .Proof. We proceed as in the proof of [HHK15a, Proposition 7.4]. First note that v has acenter Q on X that is not the generic point of X . In the case that T is complete, this wasshown in [HHK15a, Lemma 7.3]; but the proof used the completeness hypothesis only to citeHensel’s Lemma in the proof of Lemma 7.1 there, and that holds by definition for henselianrings. If Q lies on X , then we may take P = Q . Otherwise, Q is a closed point of the genericfiber of X , of codimension one, and v is defined by Q ; so F Q = F v . Since T is henselian,the closure of Q in X meets X at a single closed point P . Every étale neighborhood of P in X also defines an étale neighborhood of Q , since the residue field of Q is a henseliandiscretely valued field whose residue field corresponds to P . Thus R h P ⊆ R h Q = R h v , and hence F h P ⊆ F h v . (cid:3) .3. Local-Global Principles with respect to discrete valuations in characteristic . In the previous subsections, we considered local-global principles with respect to a collectionof overfields arising from patching. More classically, local-global principles are stated withrespect to overfields that are completions at discrete valuations. The aim of this subsectionis to discuss when such local-global principles for rational points imply analogous principlesfor zero-cycles. Moreover, we show that in some cases, the existence of local zero-cycles ofdegree one even implies the existence of a global rational point.
Notation 3.14.
For a field F , let Ω F be the collection of overfields of F that are completionsof F at discrete valuations.Below we consider the function field F of a curve over an excellent henselian discretevaluation ring. In this situation, we begin by explaining how local-global principles withrespect to discrete valuations relate to local-global principles as studied in the previoussubsections. Proposition 3.15.
Let T be an excellent henselian discrete valuation ring, and let F bethe function field of a normal flat projective T -curve X , with closed fiber X . Let Z be an F -scheme of finite type. If ( Z, Ω F ) satisfies a local-global principle for rational points then ( Z, Ω X ) satisfies a local-global principle for rational points.Proof. Suppose that Z has an F P -point for every point P ∈ X . We want to show that Z hasan F -point. By Proposition 3.13, for every discrete valuation v on F , there is a point P ∈ X such that F h P ⊆ F h v . By Artin’s Approximation Theorem ([Art69, Theorem 1.10]) applied tothe coordinate ring of an affine neighborhood of P in X , Z has an F h P -point. Hence Z has an F h v -point, and thus also an F v -point. By the local-global hypothesis, Z has an F -point. (cid:3) We next consider the case of torsors under a linear algebraic group G over a function field F . These are classified up to isomorphism by the first Galois cohomology set H ( F, G ) . Wedefine X ( F, G ) to be the kernel of the local-global map H ( F, G ) → Q v H ( F v , G ) , where v runs over all discrete valuations on F . Hence X ( F, G ) is trivial if and only if ( Z, Ω F ) satisfies a local-global principle for rational points for every G -torsor Z . Theorem 3.16.
Let T be an excellent henselian discrete valuation ring, K its field of frac-tions and k its residue field. Suppose that char( k ) = 0 . Let F be a one variable functionfield over K and let X be a regular model of F over T . Let G be a connected linear algebraicgroup over F which is the generic fiber of a reductive smooth group scheme over X . Let Z be a G -torsor over F . Suppose that for all finite separable field extensions E/F , ( Z E , Ω E ) satisfies a local-global principle for rational points. Then ( Z, Ω F ) satisfies a local-global prin-ciple for zero-cycles of degree one. In particular, this applies if X ( E, G E ) is trivial for allfinite separable field extensions E/F .Proof.
Let Z be a G -torsor over F . Let Y be a sequence of blow-ups of X such that Z isunramified except along a union of regular curves with normal crossings, i.e., Z extends toa torsor on the complement of the union of these curves. Since the underlying group schemeof G is smooth over X , it is also smooth over Y . Let Y be the closed fiber of Y .Let P ∈ Y be a closed point. Let R P be the regular local ring at P and b R P be thecompletion of R P at its maximal ideal. Since the ramification locus of Z is a union of regular urves with normal crossings, there exist π, δ ∈ R P such that the maximal ideal at P is ( π, δ ) and Z is unramified on R P except possibly at π and δ .Suppose that Z has a zero-cycle of degree one over F v for all discrete valuations v of F .Let ℓ be a prime. Then there exist field extensions E π /F P,π and E δ /F P,δ of degree prime to ℓ such that Z ( E π ) = ∅ and Z ( E δ ) = ∅ . Here F P,π := ( F P ) π and F P,δ := ( F P ) δ are associatedto F P as in the beginning of Subsection 2.3. By Lemma 2.13, there exists a field extension E P /F P of degree prime to ℓ such that the integral closure B P of b R P in E P is a completeregular local ring and E π (resp. E δ ) is isomorphic to a subfield of the field E P ⊗ F P F P,π (resp. E P ⊗ F P F P,δ ). Moreover, the maximal ideal of B P is of the form ( π ′ , δ ′ ) for unique primes π ′ and δ ′ lying over π and δ , respectively.Since Z ( E π ) and Z ( E δ ) are non-empty, Z ( E P ⊗ F P,π ) and Z ( E P ⊗ F P,δ ) are non-empty.Hence Z E P ⊗ F P,π ≃ G E P ⊗ F P,π and Z E P ⊗ F P,δ ≃ G E P ⊗ F P,δ . In particular Z is unramified at π ′ and δ ′ . But Z is unramified on R P except possibly at π and δ , thus it is indeed unramifiedat every height one prime ideal of B P . Hence the class of Z E P comes from a class ζ in H ( B P , G ) ([CTS79, Corollary 6.14]). Since ζ is trivial over the completion at π ′ , the imageof ζ in H ( k ( π ′ ) , G ) is trivial, where k ( π ′ ) is the residue field of E π . Since ζ ∈ H ( B P , G ) ,the image of ζ in H ( k ( π ′ ) , G ) comes from the image ζ ( π ′ ) of ζ in H ( B P / ( π ′ ) , G ) . Since B P / ( π ′ ) is a discrete valuation ring with field of fractions k ( π ′ ) and the image of ζ ( π ′ ) in H ( k ( π ′ ) , G ) is trivial, ζ ( π ′ ) is trivial ([Nis84]). Hence the image of ζ in H ( κ, G ) is trivial,where κ is the residue field of B P . Since B P is a complete regular local ring, Hensel’s Lemmaimplies that ζ is trivial in H ( B P , G ) , and hence its image is trivial in H ( E P , G ) . Thus Z ( E P ) = ∅ . Since gcd( ℓ, [ E P : F P ]) = 1 and since the prime ℓ is arbitrary, Z has a zero-cycleof degree one over F P .Since ( Z, Ω F ) satisfies a local-global principle for rational points, ( Z, Ω Y ) satisfies a local-global principle for rational points by Proposition 3.15, and similarly for finite separablefield extensions E/F . Since Z has a zero-cycle of degree one over F P for all P ∈ Y , Z has azero-cycle of degree one over F by Proposition 3.12. (cid:3) Remark 3.17.
As the proof shows, the above theorem holds under the weaker assumptionthat for every blow-up (i.e., birational projective morphism)
Y → X , and for every finiteseparable field extension
E/F , ( Z E , Ω Y E ) satisfies a local-global principle for rational points.This hypothesis is equivalent to the one stated in Theorem 3.16 in important special cases,e.g. when K is complete, or when k is algebraically closed and G is stably rational. Theformer case follows from [HHK15a, Theorem 8.10(ii)]. For the latter case, the henselization F h P of F at a point P on the closed fiber is algebraic over F , and so every discrete valuationon F h P restricts to a discrete valuation on F . Thus if Z ( F v ) = ∅ for every v ∈ Ω F , then Z (( F h P ) v ) = ∅ for every v ∈ Ω F h P . By [BKG04, Cor. 7.7], it follows that Z ( F h P ) = ∅ and hence Z ( F P ) = ∅ as required. Corollary 3.18.
Let T , F , X , and G be as in Theorem 3.16. Assume that the linearalgebraic group G is retract rational over F , and let Z be a G -torsor over F . Then ( Z, Ω F ) satisfies a local-global principle for zero-cycles of degree one.Proof. By Remark 3.17, it suffices to check that ( Z E , Ω Y E ) satisfies a local-global principlefor rational points, for all blow-ups Y of a regular model X and all finite separable fieldextensions E/F . That condition holds by [Kra10] (see also Corollary 3.10). (cid:3) .4. Local-global principles with respect to discrete valuations: case of an alge-braically closed residue field.
The remainder of this section is devoted to results abouttorsors and projective homogeneous spaces over certain 2-dimensional fields over an alge-braically closed field k of characteristic zero, including semi-global fields. More precisely: Hypothesis 3.19.
Let k be an algebraically closed field of characteristic zero. Let F be oneof the following:(a) The fraction field of a normal, 2-dimensional, excellent henselian local ring withresidue field k .(b) A one-variable function field over the fraction field of an excellent henselian discretevaluation ring with residue field k .Note that if F is a field as in Hypothesis 3.19, then finite field extensions of F are also ofthe same type. Proposition 3.20.
Let F be a field as in Hypothesis 3.19. Then F has the following prop-erties:(1) cd( F ) ≤ .(2) For central simple algebras over finite field extensions of F , period and index coincide.(3) For any semisimple simply connected group G over F , H ( F, G ) = 1 .(4) For any quasi-trivial torus P over F , the diagonal map H ( F, P ) → Y v H ( F v , P ) , is injective. (Here v runs through all discrete valuations on F .)Proof. We first assume that F is as in Hypothesis 3.19(a). For the first three properties, see[CTGP04, Thm. 1.4], whereas property (4) is an immediate consequence of [CTOP00, Cor.1.10].Next assume F is as in Hypothesis 3.19(b). For property (1), the fraction field K ofan excellent henselian discrete valuation ring with algebraically closed residue field is C by [Ser00, II §3.3(c)]; and hence a one-variable function field F over K is C by [Ser00,II §4.5 Example (b)]. Therefore cd( F ) ≤ by [Ser00, end of II §4.5], giving property (1).Property (2) is a special case of [HHK09, Corollary 5.6]. Property (4) follows from [CTOP00,Corollary 1.10(b)]. It remains to show property (3). The absolute Galois group of the fractionfield K of an excellent henselian discrete valuation ring with residue field k is pro-cyclic andin particular abelian, because the finite extensions of K are all obtained by taking roots ofthe uniformizer. Hence ¯ K ⊗ K F/F is a pro-cyclic field extension, of cohomological dimensionat most by Tsen’s theorem. But F ab contains ¯ K ⊗ K F since the absolute Galois group of K is abelian, and moreover this field extension is algebraic. Hence cd( F ab ) ≤ by [Ser00,II §4.1 Proposition 10]. The statement now follows from [CTGP04, Thm. 1.2 (v)] (see also[Par10, end of §6]). (cid:3) Remark 3.21.
As pointed out by J. Starr, Prop. 3.20(1)-(3) for fields of type (b) in Hy-pothesis 3.19 can also be deduced by a localization process from global results establishedusing the theory of rationally simply connected varieties (see [Sta17, Prop. 4.4]). et G be a connected reductive linear algebraic group over a field K of characteristic zero.By [BK00, Lemma 1.4.1] and [CT08, Prop. 4.1, Cor. 5.3], there exists a central extension → P → H → G → ∗ ) such that H is a connected reductive group, its derived group H ss is a (semisimple) simplyconnected group, the quotient H/H ss is a torus Q and the kernel P is a quasi-trivial torus.If moreover G is rational over K , there exists such a presentation of G for which the torus Q is a quasi-trivial torus. Proposition 3.22.
Let F be any field of characteristic zero and let G be a connected re-ductive linear algebraic group over F . Suppose that H ( F, M ) = 1 for all semisimple simplyconnected groups M over F . Let Z be a G -torsor over F .(a) If Z has a zero-cycle of degree one, then Z has a rational point.(b) If G is rational, then Z has a rational point if and only if the image of the class of Z under the boundary map H ( F, G ) → H ( F, P ) vanishes. (Here P is as in the aboveshort exact sequence.)Proof. For the proof of the first statement, let → P → H → G → be as in ( ∗ ) above.Since P is a quasi-trivial torus, H ( F, P ) is trivial and hence we have an exact sequence ofpointed sets → H ( F, H ) → H ( F, G ) → H ( F, P ) . Let [ Z ] ∈ H ( F, G ) denote the class of Z , and let ζ be the image of [ Z ] in H ( F, P ) .Suppose that Z has a zero-cycle of degree one. Then since P is a torus, using restriction-corestriction, it follows that ζ is the trivial element. Hence there exists an H -torsor e Z suchthat its class [ e Z ] ∈ H ( F, H ) maps to [ Z ] in H ( F, G ) . Since for any finite field extension L/F , the map H ( L, H ) → H ( L, G ) has trivial kernel and Z has a zero-cycle of degree one, e Z has a zero-cycle of degree one.Consider the exact sequence → H ss → H → Q → . Since H ss is semisimple simplyconnected, by the hypothesis H ( F, H ss ) is trivial. Hence the map H ( F, H ) → H ( F, Q ) has trivial kernel.Let ¯ Z be a torsor representing the image of [ e Z ] in H ( F, Q ) . Since e Z has a zero-cycle of degree one, ¯ Z has a zero-cycle of degree one. Since Q is a torus, once againthe restriction-corestriction argument gives that ¯ Z has a rational point. Since the map H ( F, H ) → H ( F, Q ) has trivial kernel, e Z is trivial. In particular Z is trivial and hencehas a rational point.To see the second assertion, assume that G is rational over F . Then by the discussionpreceding Proposition 3.22, we may assume that in the exact sequence → H ss → H → Q → , with H ss semisimple simply connected, Q is a quasi-trivial torus. We thus have H ( F, H ) =1 . Cohomology of the exact sequence → P → H → G → then gives the result. (cid:3) Theorem 3.23.
Let F be a field as in Hypothesis 3.19. Let G be a connected and rationallinear algebraic group over F . Let Z be a G -torsor over F . If Z has a zero-cycle of degreeone over F v for all discrete valuations v of F , then Z has a rational point over F . roof. Since F has characteristic zero, the kernel H ( F, R u ( G )) of the map H ( F, G ) → H ( F, G/R u ( G )) is trivial by [Ser00, III §2.1, Prop. 6]; here R u ( G ) denotes the unipotentradical of G . The analogous statement holds for all field extensions E/F . Hence we mayassume without loss of generality that G is reductive. The hypotheses of Proposition 3.22then hold, by Proposition 3.20(3).Let → P → H → G → be a central extension as in the discussion preceding Proposition 3.22. As in the proof ofthat proposition, this gives rise to an exact sequence of pointed sets → H ( F, H ) → H ( F, G ) → H ( F, P ) , and similarly, → H ( F v , H ) → H ( F v , G ) → H ( F v , P ) for each discrete valuation v of F . By assumption, Z F v has a zero-cycle of degree one for eachsuch v . By restriction-corestriction (as in the proof of Proposition 3.22), the class of Z F v maps to the trivial element in H ( F v , P ) , for each v . According to Proposition 3.20(4), thisimplies that the class of Z maps to the trivial element in H ( F, P ) . Hence Proposition 3.22(b)applies and implies the result. (cid:3) Remark 3.24.
For an alternative proof after reducing to the reductive case, first note that Z ( F v ) = ∅ by Proposition 3.22(a). The local case (when F is as in Hypothesis 3.19(a)) nowfollows from [BKG04, Corollary 7.7]. In the case of a function field of a normal curve X over an excellent henselian discrete valuation ring T as in Hypothesis 3.19(b), the local caseimplies that Z has a point over the fraction field of the henselization of the local ring of X at any closed point P . Hence Z also has a point over the fraction field of the complete localring at P . It also has a point over F η , for each generic point η of the closed fiber. Thus ithas a point over the function field of X × T b T by [HHK15a, Theorem 5.10]. This case of thetheorem then follows from Lemma 3.11 above.Below we study local-global principles for zero-cycles on projective homogeneous spaces.We use a criterion for the existence of rational points from [CTGP04]. We begin by recallingsome notation and facts (loc. cit., p. 333–335, which closely follows work of Borovoi [Bor93]).Let F be a field of characteristic zero. Let G be a connected reductive linear algebraicgroup over F and let Z be a projective homogeneous G -space. Let ¯ H be the isotropy groupof an ¯ F -point of Z ; note that since Z is projective, ¯ H is parabolic and hence connected. Asin [CTGP04], one can define an associated F -torus H tor (this is an F -form of the maximaltorus quotient of ¯ H ). Because Z is projective, the F -torus H tor is a quasi-trivial torus by[CTGP04, Lemma 5.6].As in [CTGP04], one may further define an F -kernel L = ( ¯ H, κ ) , a cohomology set H ( F, L ) , and a class η ( Z ) ∈ H ( F, L ) associated to Z . This class is a neutral class ifand only if Z comes from a G -torsor; i.e., if and only if there exists a G -torsor Y and a G -equivariant morphism Y → Z . There is a natural map t ∗ : H ( F, L ) → H ( F, H tor ) whichis functorial in the field F (loc. cit.), and which sends neutral classes in H ( F, L ) to thetrivial element in H ( F, H tor ) .The following proposition is an immediate consequence of [CTGP04, Prop. 5.4] and willbe the key ingredient in the proof of the theorem below. roposition 3.25. Suppose that F is a field which satisfies properties (1), (2), and (3)in Proposition 3.20. Let G be a semisimple simply connected linear algebraic group over F , and let Z be a projective homogeneous space under G . Then Z ( F ) = ∅ if and only if t ∗ ( η ( Z )) = 1 ∈ H ( F, H tor ) .Proof. If Z ( F ) contains a rational point x then the map G → Z given by the action of G on x shows that η ( Z ) is neutral, and thus t ∗ ( η ( Z )) = 1 ∈ H ( F, H tor ) . Conversely, if t ∗ ( η ( Z )) = 1 , then η ( Z ) is neutral by [CTGP04, Prop. 5.4]; i.e., it comes from a G -torsor.By assumption, H ( F, G ) consists of a single element, hence that torsor has an F -rationalpoint which maps to a point on Z , showing Z ( F ) = ∅ . (cid:3) Theorem 3.26.
Let F be a field of characteristic zero which satisfies properties (1), (2), (3)in Proposition 3.20. Let G be a connected linear algebraic group over F , and let Z be aprojective homogeneous G -space. If Z has a zero-cycle of degree one, then Z has a rationalpoint. In particular, this assertion holds whenever F is a field as in Hypothesis 3.19.Proof. Let R ( G ) be the radical of G , i.e., the maximal connected solvable subgroup of G .This is contained in any parabolic subgroup of G . Since Z is projective, the action of G on Z factors through G ss = G/R ( G ) . Let G sc → G ss be the simply connected cover of G ss . Thuswe may view Z as a projective homogenous G sc -space. Replacing G by G sc , we may thereforeassume that G is semisimple and simply connected (cf. proof of [CTGP04, Corollary 5.7]).If Z has a zero-cycle of degree one, there exist finite field extensions F i /F such that Z ( F i ) = ∅ for all i and the g.c.d. of the degrees of F i /F is . Since Z ( F i ) = ∅ , the imageof t ∗ ( η ( Z )) in H ( F i , H tor ) is trivial by Proposition 3.25. Using a restriction-corestrictionargument, we conclude that t ∗ ( η ( Z )) is trivial in H ( F, H tor ) . One then concludes that Z ( F ) = ∅ by a second application of Proposition 3.25. The last statement follows fromProposition 3.20. (cid:3) If F is the function field of a p -adic curve, there are projective homogeneous spaces undersimply connected groups which admit zero-cycles of degree one, but with no rational points.See [Par05]. Theorem 3.27.
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