Local-global principles for Galois cohomology

LOCAL-GLOBAL PRINCIPLES FOR GALOIS COHOMOLOGY arXiv:1208.6359v1 [math.NT] 31 Aug 2012 DAVID HARBATER, JULIA HARTMANN, AND DANIEL KRASHEN Abstract. This paper proves local-global principles for Galois cohomology groups over function fields F of curves that are defined over a complete discretely valued field. We show in particular that such principles hold for H n pF, Z{mZpn ´ 1qq, for all n ą 1. This is motivated by work of Kato and others, where such principles were shown in related cases for n “ 3. Using our results in combination with cohomological invariants, we obtain localglobal principles for torsors and related algebraic structures over F . Our arguments rely on ideas from patching as well as the Bloch-Kato conjecture. 1. Introduction In this paper we present local-global principles for Galois cohomology, which may be viewed as higher-dimensional generalizations of classical local-global principles for the Brauer group. These results then lead to local-global principles for other algebraic structures as well, via cohomological invariants. Recall that if F is a global field, the theorem of Albert-Brauer-Hasse-Noether says a central simple F -algebra is isomorphic to a matrix algebra if and only if this is true over each completion Fv of F . Equivalently, the natural group homomorphism ź BrpFv q BrpF q Ñ vPΩF is injective, where ΩF is the set of places of F . Kato suggested a higher dimensional generalization of this in [Kat86], drawing on the observation that the above result provides a local-global principle for the m-torsion part of the Brauer group BrpF qrms “ H 2 pF, Z{mZp1qq. (Here Z{mZpnq denotes µbn m , for m not dividing charpF q.) He proposed that the natural domain for higher-dimensional versions of local-global principles should be H n pF, Z{mZpn ´ 1qq, for n ą 1. Cohomological invariants (such as the Rost invariant) often take values in H n pF, Z{mZpn´1qq for some n ą 1; and thus such local-global principles for cohomology could be used to obtain local-global principles for other algebraic objects. In Theorem 0.8(1) of [Kat86], Kato proved such a principle with n “ 3 for the function field F of a smooth proper surface X over a finite field, both with respect to the discrete valuations on F that arise from codimension one points on X, and alternatively with respect to the set of closed points of X (in the latter case using the fraction fields of the complete local rings at the points). He also proved a related result [Kat86, Theorem 0.8(3)] for arithmetic Date: August 30, 2012. The first author was supported in part by NSF grant DMS-0901164. The second author was supported by the German Excellence Initiative via RWTH Aachen University and by the German National Science Foundation (DFG). The third author was supported in part by NSF grant DMS-1007462. 1 surfaces, i.e. for curves over rings of integers of number fields. The corresponding assertions for n ą 3 are vacuous in his situation, for cohomological dimension reasons; and the analogs for n “ 2 do not hold there (e.g. if the unramified Brauer group of the surface is non-trivial). Unlike the classical case of dimension one, in dimension two it is also meaningful to consider local-global principles for fields that are not global, e.g. kppx, yqq or kpptqqpxq. In [COP02, Theorem 3.8], the authors start with an irreducible surface over a finite field of characteristic not dividing m; and they take the fraction field F of the henselization of the local ring at a closed point. In that situation, they prove a local-global principle for H 3pF, Z{mZp2qq with respect to the discrete valuations on F . Also, while not explicitly said in [Kat86], it is possible to use Theorem 5.2 of that paper to obtain a local-global principle for function fields F of curves over a non-archimedean local field, with respect to H 3 pF, Z{mZp2qq. This was relied on in [CPS08, Theorem 5.4] and [Hu12] (cf. also [PS98, pp. 139 and 148]). 1.1. Results In this manuscript, we show that when F is the function field of a curve over an arbitrary complete discretely valued field K, local-global principles hold for the cohomology groups H n pF, Z{mZpn ´ 1qq for all n ą 1. In particular we obtain the following local-global principle with respect to points on the p of F over the valuation ring of K (where k is the residue field): closed fiber X of a model X Theorem (3.2.3). Let n ą 1 and let A be one of the following algebraic groups over F : (i) Z{mZpn ´ 1q, where m is not divisible by the characteristic of k, or (ii) Gm , if charpkq “ 0 and K contains a primitive m-th root of unity for all m ě 1. Then the natural map ź H n pF, Aq Ñ H n pFP , Aq P PX is injective, where P ranges through all the points of the closed fiber X. p at P . Here FP denotes the fraction field of the complete local ring of X We also obtain a local-global principle with respect to discrete valuations if K is equicharacteristic: Theorem (3.3.6). Suppose that K is an equicharacteristic complete discretely valued field of p is a regular projective T -curve with function field characteristic not dividing m, and that X F . Let n ą 1. Then the natural map ź H n pFv , Z{mZpn ´ 1qq H n pF, Z{mZpn ´ 1qq Ñ is injective. vPΩX x Here ΩXp is the set of discrete valuations on F that arise from codimension one points p Also, in the above results and henceforth, the cohomology that is used is Galois on X. cohomology, where H n pF, Aq “ H n pGalpF q, ApF sep qq for A a smooth commutative group scheme over F and n ě 0, with H 0 pF, Aq “ ApF q. (For non-commutative group schemes, we similarly have H 0 and H 1 .) 2 These results also yield new local-global principles for torsors under linear algebraic groups by the use of cohomological invariants such as the Rost invariant ([GMS03, p. 129]), following a strategy used in [CPS08] and [Hu12]. We list some of these applications of our local-global principles in Section 4. Note that although we also obtained certain local-global principles for torsors for linear algebraic groups in [HHK11a], the results presented here use a different set of hypotheses on the group. In particular, here we do not require that the group G be rational, unlike in [HHK11a]. 1.2. Methods Our approach to obtaining these local-global principles uses the framework of patching over fields, as in [HH10], [HHK09], and [HHK11a]. The innovation is that these principles derive from long exact Mayer-Vietoris type sequences with respect to the “patches” that arise in this framework. These sequences are analogous to those in [HHK11a] for linear algebraic groups that were not necessarily commutative (but where only H 0 and H 1 were considered for that reason). In Section 2, we derive Mayer-Vietoris sequences and local-global principles in an abstract context of a field together with a finite collection of overfields (Section 2.5). This allows us to isolate the combinatorial and cohomological properties of the fields and Galois modules which we need. The combinatorial data of the collection of fields we use is encoded in the notion of a Γ-field (Section 2.1), the group theoretic properties of our Galois modules we use we call “separable factorization” (Section 2.2), and the cohomological properties we require are formulated in the concept of global domination of Galois cohomology (Sections 2.3 and 2.4). An essential ingredient in our arguments is the Bloch-Kato conjecture. In Section 3, we apply our results to the situation of a function field over a complete discretely valued field. In Section 3.1 we obtain a local-global principle with respect to “patches.” This is used in Section 3.2 to obtain a local-global principle with respect to points on the closed fiber of a regular model. Finally, in Section 3.3, we obtain our local-global principle with respect to discrete valuations with the help of a result of Panin [Pan03] for local rings in the context of Bloch-Ogus theory. This step is related to ideas used in [Kat86]. In Section 4, we combine our local-global principles with cohomological invariants taking values in H n pF, Z{mZpn ´ 1qq, to obtain our applications to other algebraic structures. Acknowledgments. The authors thank Jean-Louis Colliot-Thélène, Skip Garibaldi, and Annette Maier for helpful comments on this manuscript. 2. Patching and local-global principles for cohomology This section considers patching and local-global principles for cohomology in an abstract algebraic setting, in which we are given a field and a finite collection of overfields indexed by a graph. The results here will afterwards be applied to a geometric setting in Section 3, where we will consider curves over a complete discretely valued field. In the situation here, we will obtain a new long exact sequence for Galois cohomology with respect to the given field and its overfields, which in a key special case can be interpreted as a Mayer-Vietoris sequence. In [HHK11a, Theorem 2.4], we obtained such a sequence for 3 linear algebraic groups that need not be commutative. Due to the lack of commutativity, the assertion there was just for H 0 and H 1 ; and that result was then used in [HHK11a] to obtain local-global principles for torsors in a more geometric context. In the present paper, we consider commutative linear algebraic groups, and so higher cohomology groups H n are defined. It is for these that we prove our long exact sequence, which we then use to obtain a local-global principle for Galois cohomology in the key case of H n pF, Z{mZpn ´ 1qq with n ą 1. This is carried out in Section 2.5. (Note that the six-term cohomology sequence in [HHK11a, Theorem 2.4] is used in our arguments here, in the proofs of Theorems 2.1.5 and 2.2.5.) 2.1. Γ-Fields and patching Our local-global principles will be obtained by an approach that formally emulates the notion of a cover of a topological space by a collection of open sets, in the special case that there are no nontrivial triple overlaps. In this case, one may ask to what extent one may derive global information from local information with respect to the sets in the open cover. We encode this setup combinatorically in the form of a graph whose vertices correspond to the connected open sets in the cover and whose edges correspond to the connected components of the overlaps (though we do not introduce an associated topological space or Grothendieck topology). In our setting the global space will correspond to a field F whose arithmetic we would like to understand, and the open sets and overlaps correspond to field extensions of F . This setup is formalized in the definitions below, which draw on terminology in [HH10] and [HHK11a]. 2.1.1. Graphs and Γ-fields By a graph Γ, we will always mean a finite multigraph, with a vertex set V and an edge set E; i.e. we will permit more than one edge to connect a pair of vertices. But we will not permit loops at a vertex: the two endpoints of an edge are required to be distinct vertices. By an orientation on Γ we will mean a choice of labeling of the vertices of each edge e P E, with one chosen to be called the left vertex lpeq and the other the right vertex rpeq of e. This choice can depend on the edge (i.e. a vertex v can be the right vertex for one edge at v, and the left vertex for another edge at v). Definition 2.1.1. Let Γ be a graph. A Γ-field F‚ consists of the following data: (1) For each v P V, a field Fv , (2) For each e P E, a field Fe , (3) An injection ιev : Fv Ñ Fe whenever v is a vertex of the edge e. ś ś e We will write F pVq “ vPV Fv and F pEq “ ePE Fe . Often we will regard ιv as an inclusion, and not write it explicitly in the notation if the meaning is clear. A Γ-field F‚ can also be interpreted as an inverse system of fields. Namely, the index set of the inverse system is the disjoint union V \ E; and the maps consist of inclusions of fields ιev : Fv ãÑ Fe as above. Conversely, consider any finite inverse system of fields whose index set can be partitioned into two subsets V \ E, such that for each e P E there are exactly two elements in v, v 1 P V 4 having maps Fv ãÑ Fe and Fv1 ãÑ Fe in the inverse system; and such that there are no other maps in the inverse system. Then such an inverse system of fields, called a factorization inverse system in [HHK11a, Section 2], gives rise to a graph Γ and a Γ-field F‚ as above. Given a Γ-field F‚ , we may consider the inverse limit FΓ of the fields in F‚ , with respect to the associated inverse system, in the category of rings. Equivalently, FΓ “ ta‚ P F pVq | ιev av “ ιew aw for each e incident to v and wu . We may also regard FΓ as a subring of F pEq, by sending an element a‚ “ pav qvPV to pae qePE , where ae “ ιev av “ ιew aw if e is incident to v and w. Note that if F‚ is a Γ-field, then we may regard each field Fv , Fe naturally as an FΓ -algebra in such a way that all the inclusions ιev are FΓ -algebra homomorphisms. Lemma 2.1.2. If F‚ is a Γ-field, then FΓ is a field if and only if Γ is connected. Proof. If Γ is disconnected, there are elements a‚ of the inverse limit FΓ such that aξ “ 0 for all ξ P V \ E that lie on one connected component of Γ, but aξ “ 1 for all ξ on another component. Hence FΓ has zero-divisors and is not a field. Conversely, if FΓ is not a field, then there is a zero-divisor a‚ . The set of vertices and edges ξ such that aξ “ 0 forms an open subset of Γ, since ιev av “ ae “ ιew aw whenever v, w are the vertices of an edge e. This open subset is neither empty nor all of Γ, since a‚ is a zero divisor. Hence Γ is disconnected.  Notation 2.1.3. We will say for short that F‚ is a Γ{F -field if Γ is a connected graph, F is a field, and F‚ is a Γ-field with FΓ “ F . 2.1.2. Patching Problems Given a Γ{F -field F‚ , and a finite dimensional vector space V over F , we obtain an inverse system VFξ “ V bF Fξ of finite dimensional vector spaces over the fields Fξ (for ξ P V \ E). Conversely, given such an inverse system, we can ask whether it is induced by an F -vector space V . More precisely, let VectpF q be the category of finite dimensional F -vector spaces; define a vector space patching problem V‚ over F‚ to be an inverse system of finite dimensional Fξ -vector spaces; and let PPpF‚ q be the category of vector space patching problems over F‚ . There is then a base change functor VectpF q Ñ PPpF‚ q. If it is an equivalence of categories, we say that patching holds for finite dimensional vector spaces over the Γ{F -field F‚ . We may consider analogous notions for other objects over F . In particular let A be a group scheme over F (which we always assume to be of finite type). Let TorspAq denote the category of A-torsors over F ; the objects in this category are classified by the elements in the Galois cohomology group H 1 pF, Aq. An object T in TorspAq induces an A-torsor patching problem T‚ over F‚ , i.e. an inverse system consisting of AFξ -torsors Tξ for each ξ P V \ E, together with isomorphisms φev : pTv qFe Ñ Te for v a vertex of an edge e. These patching problems form a category PPpF‚ , Aq, whose morphisms correspond to collections of morphisms of torsors which commute with the maps φev . (Once we choose an orientation on the graph Γ, an A-torsor patching problem can also be viewed as collection of A-torsors Tv for v P V, together with a choice of isomorphism pTlpeq qFe Ñ pTrpeq qFe for every edge e P E. This isomorphism corresponds to multiplication by an element of ApFe q if Te is trivial.) As before, we obtain a base change functor TorspAq Ñ 5 PPpF‚ , Aq; and we say that patching holds for A-torsors over the Γ{F -field F‚ if this is an equivalence of categories. For short we say that patching holds for torsors over F‚ if it holds for all linear algebraic groups A over F . (Our convention is that a linear algebraic group over F is a smooth closed subgroups A Ď GLn,F for some n.) 2.1.3. Local-global principles and simultaneous factorization Local-global principles are complementary to patching. Given a Γ{F -field F‚ , and a group scheme A over F , we say that A-torsors over F satisfy a local-global principle over F‚ if an Atorsor T is trivial if and only if each induced Fv -torsor Tv :“ T ˆF Fv is trivial. In [HHK11a], criteria were given for patching and for local-global principles in terms of factorization. Before recalling them, we introduce some terminology and notation. If F‚ is a Γ{F -field, and if Γ is given an orientation, then there are induced maps πl , πr : F pVq Ñ F pEq defined by pπl paqqe “ alpeq and pπr paqqe “ arpeq for a “ paś v qvPV P F pVq. Similarly, if A is a group scheme over F , there are induced maps ś ś πl , πr : vPV ApFv q Ñ ePE ApFe q given by the same expressions, for a “ pav qvPV P vPV ApFv q. We say that a group scheme A over F satisfies simultaneous factorization over ś a Γ{F -field F‚ (or ś for short, ´1 is factorizable over F‚ ) if the map of pointed sets πl ¨ πr : vPV ApFv q Ñ ePE ApFe q, ´1 defined by a ÞÑ πl paqπr paq , is surjective. In other words, if we are given a collection of elements ae P ApFe q for all e P E, then there exist elements av P ApFv q for all v P V such that ae “ alpeq arpeq for all e, with respect to the inclusions Flpeq , Frpeq ãÑ Fe . Note that this factorization condition does not depend on the choice of orientation, ś since if we reverse the orientation on an edge e then we may consider the element a1 P ePE ApFe q such that 1 a1e “ a´1 e and where the other entries of a are the same as for a. 2.1.4. Relations between patching, local-global principles and factorization Theorem 2.1.4. Let Γ be a connected graph, F a field, and F‚ a Γ{F -field. Then the following conditions are equivalent: (i) GLn is factorizable over F‚ for all n ě 1. (ii) Patching holds for finite dimensional vector spaces over F‚ . (iii) Patching holds for torsors over F‚ . Proof. It was shown in [HHK11a, Proposition 2.2] that (i) is equivalent to (ii); and in [HHK11a, Theorem 2.3] it was shown that (ii) implies (iii). It remains to show that (iii) implies (i). Fix an orientation for Γ and let g “ pge qePE P GLn pF pEqq. We wish to show that there exists h P GLn pF pVqq such that g “ πl phqπr phq´1 . Consider the patching problem for GLn -torsors over F‚ that is given by trivial torsors over Fe for each e P E, and such that the transition function pTlpeq qFe Ñ pTrpeq qFe is given by ge P GLn pFe q, for each e P E. By hypothesis (iii), there is a GLn -torsor T over F that induces this patching problem. But T is trivial, since H 1 pF, GLn q “ 0 by Hilbert’s Theorem 90 ([KMRT98, Theorem 29.2]). The transition functions TFv Ñ Tv are given by elements hv P GLn pFv q. Since T induces the given patching problem, we have hlpeq “ ge hrpeq for every e P E. Therefore g “ πl phqπr phq´1 , with h “ phv qvPV P GLn pF pEqq, as desired.  6 Theorem 2.1.5. Let Γ be a connected graph, F a field, and F‚ a Γ{F -field. Assume that patching holds for finite dimensional vector spaces over F‚ . Then a linear algebraic group A over F is factorizable over F‚ if and only if A-torsors over F satisfy a local-global principle over F‚ . Proof. This assertion is contained in the exactness of the sequence given in [HHK11a, Theoś ´1 1 rem 2.4]; i.e. πl ¨ πr is surjective if and only if H pF, Aq Ñ vPV H 1 pFv , Aq is injective.  Note that the hypothesis of Theorem 2.1.5 does not imply that the equivalent conditions in the conclusion of that theorem necessarily hold. (In particular, in Example 4.4 of [HHK09] there is a non-trivial obstruction to a local-global principle, by Corollaries 5.6 and 5.5 of [HHK11a]). Thus patching need not imply factorization over F‚ for all linear algebraic groups over F . But as shown in the next section (Corollary 2.2.6), patching does imply factorization for all linear algebraic groups if we are allowed to pass to the separable closure of F . This will be useful in obtaining local-global principles for higher cohomology. 2.2. Separable factorization As asserted in Theorems 2.1.4 and 2.1.5, there are relationships between factorization conditions on the one hand, and patching and local-global properties on the other. Below, in Theorem 2.2.5 and Corollary 2.2.6, we prove related results of this type, concerning “separable factorization”, which will be needed later in applying the results of Section 2. We also prove a result (Proposition 2.2.4) that will be used in obtaining our long exact sequence in Section 2.5, and hence our local-global principle there. 2.2.1. The Galois module Aξ To obtain our results, we will want to relate the cohomologies H n pF, Aq and H n pFξ , Aq for ξ a vertex or edge of Γ. One difficulty with this in general is the potential difference between the absolute Galois groups of F and Fξ . To bridge this gap, in Notation 2.2.1 we introduce a new Galois module Aξ whose cohomology H n pF, Aξ q is meant to approximate the cohomology H n pFξ , Aq. Here and below, we write GalpF q for the absolute Galois group GalpF sep {F q. Notation 2.2.1. Let A be an F -scheme, and let F‚ be a Γ{F -field. For ξ a vertex or edge, ś we define Aξ to ś be the GalpF q-module given by Aξ “ ApFξ bF F sep q. We write ApVq “ vPV Av and ApEq “ ePE Ae . Note that the Galois module Aξ is not the same as AFξ , which is the Fξ -scheme A ˆF Fξ obtained by base change from F to Fξ ; nor the same as ApFξ q. In the above situation we have morphisms of GalpF q-modules ApF sep q Ñ Av for each vertex v P V, and Av Ñ Ae when v P V is a vertex of Γ on the edge e P E. These are induced by the inclusions F sep Ñ Fv bF F sep and Fv bF F sep Ñ Fe bF F sep . If A is an F -scheme, and L Ď L1 are field extensions of F , then the natural map ApLq Ñ ApL1 q is an inclusion. (This is immediate if A is affine, and then follows in general.) In particular, given a Γ{F -field F‚ as above, the maps ApF q Ñ ApFv q and ApFv q Ñ ApFe q are injective for v a vertex of an edge e in Γ. 7 2.2.2. Separable factorization If we choose an orientation for the graph Γ, then as in Section 2.1.3 we may define maps πl , πr : ApVq Ñ ApEq by pπl paqqe “ alpeq and pπr paqqe “ arpeq . Lemma 2.2.2. Consider an affine F -scheme A, a graph Γ with a choice of orientation, and πl a Γ{F -field F‚ . Then the maps ApF sep q / ApVq πr / / ApEq form an equalizer diagram of sets. Proof. The hypothesis that F equals FΓ is equivalent to having an exact sequence of F -vector spaces 0 Ñ F Ñ F pVq Ñ F pEq, given by πl ¨ πr´1 on the right. Since F sep is a flat F -module, we have an exact sequence 0 Ñ F sep Ñ F sep b F pVq Ñ F sep b F pEq. This in turn tells us that in the category of rings, F sep “ lim F sep b Fξ . ÐÝ ξPV\E Write A “ SpecpRq. By the inverse limit property above, it follows that a homomorphism R Ñ F sep is equivalent to a homomorphism φ : R Ñ F sep b F pVq such that πl φ “ πr φ, where πl , πr : F sep b F pVq Ñ F sep b F pEq are the two projections. This gives an equalizer diagram ApF sep q / / ApF sep b F pVqq / ApF sep b F pEqq ApVq ApEq as desired.  This lemma, and the notion of factorizability in the previous section, motivate the following definition. Definition 2.2.3. Let F‚ be a Γ{F -field, and suppose that A is a group scheme over F . We say that A is separably factorizable (over F‚ ) if the pointed set map πl ¨ πr´1 : ApVq Ñ ApEq is surjective for some (hence every) orientation on Γ. Proposition 2.2.4. Let F‚ be a Γ{F -field, and let A be a group scheme over F . Choose any orientation on Γ, and take the associated maps πl , πr . Then A is separably factorizable if and only if 0 / ApF sep q / ApVq πl ¨πr´1 / ApEq / 0 is an exact sequence of pointed GalpF q-sets (and in fact an exact sequence of Galois modules in the case that A is commutative). Proof. Since the above maps preserve the Galois action and take distinguished elements to distinguished elements, and since the composition ApF sep q Ñ ApVq Ñ ApEq of the two maps sends every element of ApF sep q to the distinguished element of ApEq, the above sequence is a complex of pointed GalpF q-sets. In the commutative case, the maps are group homomorphisms, and so it is a complex of abelian groups, and hence of Galois modules. By Lemma 2.2.2, ApF sep q is the kernel of πl ¨ πr´1 as a map of pointed sets. Thus the sequence is exact if and only if πl ¨ πr´1 : ApVq Ñ ApEq is surjective; i.e. if and only if A is separably factorizable.  8 2.2.3. Patching and separable factorization The following theorem and its corollary complement Theorems 2.1.4 and 2.1.5. Theorem 2.2.5. Let Γ be a connected graph, F a field, and F‚ a Γ{F -field. Then the following conditions are equivalent: (i) GLn is factorizable over F‚ , for all n ě 1. (ii) GLn is separably factorizable over F‚ , for all n ě 1. (iii) Every linear algebraic group over F is separably factorizable over F‚ . Proof. We begin by showing that (i) implies (iii). Fix an orientation for Γ. Let A be a linear algebraic group over F , and suppose we are given g P ApEq. We wish to show that there exists h P ApVq such that g “ πl phqπr phq´1 . Since E is finite, there is a finite separable field extension L{F such that g is the image of g 1 P ApL bF F pEqq. Let A1 “ RL{F AL , the Weil restriction of AL “ A ˆF L from L to F (see Section 7.6); this is a linear algebraic group over F . We may then view ś [BLR90], 1 1 g P ePE A pFe q. Since GLn is factorizable over F‚ by condition (i), Theorem 2.1.4 implies that patching holds for finite-dimensional vector spaces over F‚ . Thus [HHK11a, Theorem 2.4] applies, giving us a six-term cohomology sequence for A1 ; and we may consider the image of g under ś the coboundary map ePE A1 pFe q Ñ H 1 pF, A1 q. This image defines an A1 -torsor T 1 over F (viz. the solution to the patching problem that consists of trivial torsors over each Fv and for which the transition functions are given by g 1 ). But H 1 pF, A1 q may be identified with H 1 pL, Aq by Shapiro’s Lemma ([Ser97], Corollary to Proposition I.2.5.10), since A1 pF sep q is the Galois module induced from ApF sep q via the inclusion GalpLq Ñ GalpF q. So T 1 corresponds to an A-torsor T over L. There is then a finite separable field extension E{L over which T becomes trivial. After replacing L by E, we may assume that T and hence T 1 is trivial. Hence by the exactness of the six-term sequence in [HHK11a, Theorem 2.4], g 1 is ś the image of an element h1 P vPV A1 pFv q “ ApL b F pVqq under πl ¨ πr´1 . The image h P ApVq of h1 is then as desired, proving that condition (iii) holds. Condition (iii) trivially implies condition (ii). It remains to show that condition (ii) implies condition (i). If condition (ii) holds, then Proposition 2.2.4 yields a short exact sequence of GalpF qmodules 0 / GLn pF sep q / GLn pVq πl ¨πr´1 / GLn pEq / 0. This in turn yields an exact sequence of pointed sets in Galois cohomology that begins ź ź H 0 pF, pGLn qe q Ñ H 1 pF, GLn q. H 0 pF, pGLn qv q Ñ 0 Ñ H 0 pF, GLn q Ñ ePE vPV But the last term vanishes by Hilbert’s Theorem 90. The remaining short exact sequence is then equivalent to the condition that GLn is factorizable over F‚ , i.e. condition (i).  Corollary 2.2.6. Let Γ be a connected graph, F a field, and F‚ a Γ{F -field. Then patching holds for finite dimensional vector spaces over F‚ if and only if every linear algebraic group over F is separably factorizable over F‚ . 9 Proof. This is immediate from Theorem 2.1.4 and Theorem 2.2.5, which assert that these two conditions are each equivalent to GLn being factorizable over F‚ for all n ě 1.  2.3. Globally dominated field extensions and cohomology To carry out the strategy outlined at the beginning of Section 2.2.1, we will need to relate Galois cohomology groups of the field F to Galois cohomology groups of the fields Fξ . For this, we will introduce and study the notion of “global domination.” The condition that a Galois module has globally dominated cohomology provides an important ingredient in demonstrating the existence of Mayer-Vietoris type sequences and local-global principles for its Galois cohomology groups. These applications are developed in Section 2.5. 2.3.1. Globally dominated extensions Definition 2.3.1. Fix a field F . For any field extension L{F , with separable closure Lsep , let Lgd denote the compositum of L and F sep taken within Lsep . If E{L is a separable algebraic field extension, we say that E{L is globally dominated (with respect to F ) if E is contained in Lgd . Thus a separable algebraic field extension E{L is globally dominated if and only if E is contained in some compositum E 1 L Ď Lsep , where E 1 {F is a separable algebraic field extension. Also, the subfield Lgd Ď Lsep can be characterized as the maximal globally dominated field extension of L. Since the extension F sep {F is Galois with group GalpF q, it follows that the extension Lgd {L is Galois and that Galgd pLq :“ GalpLgd {Lq can be identified with a subgroup of GalpF q. Lemma 2.3.2. Let L{F be a field extension, and let A be a commutative group scheme defined over F . Then we may identify: H n pGalpF q, ApL bF F sep qq “ H n pGalgd pLq, ApLgd qq. Proof. We may identify the group H n pGalpF q, ApL b F sep q) as a limit of groups H n pGalpE{F q, ApL b Eqq, and the group H n pGalgd pLq, ApLgd qq as a limit of groups H n pGalpLE{Lq, ApLEqq, where both limits are taken over finite Galois extensions E{F , and where LE is a compositum of L and E. Therefore the result will follow from a (compatible) set of isomorphisms H n pGalpE{F q, ApL b Eqq – H n pGalpLE{Lq, ApLEqq. ś Write LbE “ m i“1 E śi for finite Galois field extensions Ei {L. We can also choose LE “ E1 . We have ApL b Eq “ i ApEi q. Let G “ GalpE{F q and let G1 be the stabilizer of E1 (as a set) with respect to the action of G on L bF E. Then we may identify the G-modules ApL b Eq and IndG G1 ApE1 q. We therefore have n n H n pG, ApL b Eqq – H n pG, IndG G1 ApE1 qq – H pG1 , ApE1 qq “ H pGalpLE{Lq, ApLEqq 10 by Shapiro’s Lemma ([Ser97], Corollary to Proposition I.2.5.10), as desired.  2.3.2. Globally dominated cohomology It remains to compare the cohomology with respect to the maximal globally dominated extension and the full Galois cohomology. For this we make the following Definition 2.3.3. Let A be a commutative group scheme over F and L{F a field extension. We say that the cohomology of A over L is globally dominated (with respect to F ) if H n pLgd , Aq “ 0 for every n ą 0. Proposition 2.3.4. Let A be a commutative group scheme over F and L{F a field extension. Suppose that the cohomology of A over L is globally dominated. Then we have isomorphisms: H n pGalpF q, ApL b F sep qq “ H n pGalgd pLq, ApLgd qq “ H n pGalpLq, ApLsep qq for all n ě 0. Proof. The identification of the first and second groups was given in Lemma 2.3.2, and it remains to prove the isomorphism between the second and third groups. By the global domination hypothesis, H n pGalpLgd q, ApLsep qq “ H n pLgd , Aq “ 0 for all n ą 0. Hence from the Hochschild-Serre spectral sequence H p pGalgd pLq, H q pGalpLgd q, ApLsep qqq ùñ H p`q pGalpLq, ApLsep qq for the tower of field extensions L Ď Lgd Ď Lsep (viz. by [HS53, Theorem III.2]), the desired isomorphism follows.  The notion of globally dominated cohomology can also be described just in terms of finite extensions of fields. First we prove a lemma. Lemma 2.3.5. Suppose that a field E0 is a filtered direct limit of subfields Ei , each of which is an extension of a field E. Let A be a commutative group scheme over E, and let α P H n pE, Aq for some n ě 0. If the induced element αE0 P H n pE0 , Aq is trivial, then there is some i such that αEi P H n pEi , Aq is trivial. Proof. Since αE0 P H n pE0 , Aq is trivial, we may find some finite Galois extension L{E0 such that αE0 may be written as a cocycle in Z n pL{E0 , ApLqq and such that it is the coboundary of a cochain in C n´1 pL{E0 , ApLqq. Now the Galois extension L{E0 is generated by finitely many elements of L, and the splitting cochain is defined by an additional collection of elements in ApLq, each of which is defined over some finitely generated extension of E (since A is of finite type over E). So we may find finitely many elements a1 , . . . , ar P E0 such that αEpa1 ,...,ar q “ 0. But since E0 is the filtered direct limit of the fields Ei , there is an i such that a1 , . . . , ar P Ei ; and then αEi “ 0 as desired.  Proposition 2.3.6. Let A be a commutative group scheme over F and L{F a field extension. Then the cohomology of A over L is globally dominated if and only if for every finite globally dominated field extension L1 {L, every n ą 0, and every α P H n pL1 , Aq, there exists a finite globally dominated extension E{L1 such that αE “ 0. 11 Proof. First suppose that the cohomology of A over L is globally dominated, and let α P H n pL1 , Aq for some finite globally dominated field extension L1 {L and some n ą 0. Then αLgd “ 0 by hypothesis; and so by Lemma 2.3.5 there is some finite globally dominated extension E{L1 such that αE “ 0, as desired. Conversely, suppose that the above condition on every α P H n pL1 , Aq holds. Let α P n H pLgd , Aq. Then α is in the image of some element α r P H n pL1 , Aq for some finite extension L1 {L that is contained in Lgd . Now L1 is globally dominated, so by hypothesis there exists a finite globally dominated field extension E{L1 such that α rE “ 0. Thus α “ α rLgd “ pr αE qLgd “ 0. This shows that H n pLgd , Aq is trivial, so the cohomology is globally dominated.  2.4. Criteria for global domination In the case of cyclic groups, the condition for cohomology to be globally dominated will be made more explicit here, using the (now proven) Bloch-Kato conjecture to reduce to consideration of just the first cohomology group. Specifically, we will rely on the following assertion, which is a consequence of Bloch-Kato, and which is well-known to the experts. Proposition 2.4.1. Let F be a field and let m be a positive integer not divisible by charpF q. Then for every n ě 1, every element of H n pF, Z{mZpnqq is a sum of n-fold cup products of elements of H 1 pF, Z{mZp1qq. Proof. For a fixed m, consider the norm residue homomorphism of graded rings K˚M pF q Ñ H ˚ pF, Z{mZp˚qq from Milnor K-theory to Galois cohomology, with n-th graded piece hn,m : KnM pF q Ñ H n pF, Z{mZpnqq for n ě 1. Since every element in KnM pF q is by definition a sum of n-fold products of elements of K1M pF q, the proposition follows from the assertion that the maps hn,m are surjective, with kernel mKnM pF q. In the case that m is prime, this was shown in the proof of the Bloch-Kato conjecture (i.e. the norm residue isomorphism theorem) in [Voe11] and [Wei09]. This in turn implies the case of general m by induction on n, via a reduction result of Tate (see [GS06], Proposition 7.5.9).  2.4.1. Global domination for cyclic groups Proposition 2.4.2. Let L{F be a field extension, and m an integer not divisible by the characteristic of F . Then the following are equivalent: (i) The cohomology of Z{mZ over L is globally dominated. (ii) For every finite globally dominated field extension L1 {L and every positive integer r dividing m, every Z{rZ-Galois field extension of L1 is globally dominated. (iii) The multiplicative group pLgd qˆ is m-divisible; i.e. ppLgd qˆ qm “ pLgd qˆ . Proof. (i) ñ (ii): A Z{rZ-Galois field extension of L1 corresponds to an element α P H 1 pL1 , Z{rZq. Let β be the image of α in H 1 pLgd , Z{rZq. It suffices to show that β “ 0. In the long exact cohomology sequence associated to the short exact sequence of constant groups 0 Ñ Z{rZ Ñ Z{mZ Ñ Z{pm{rqZ Ñ 0, the map H 0 pLgd , Z{mZq Ñ H 0 pLgd , Z{pm{rqZq is surjective, so the map H 1 pLgd , Z{rZq Ñ H 1 pLgd , Z{mZq is injective. But the latter group is trivial, by hypothesis. Hence β “ 0. 12 (ii) ñ (iii): Given a P pLgd qˆ , we wish to show that a P ppLgd qˆ qm . Let ζ be a primitive m-th root of unity in F sep Ď Lgd , and let L1 “ Lpζ, aq Ď Lgd . Thus L1 {L is finite and separable. The field E “ L1 pa1{m q Ď Lsep is Galois over L1 , with Galois group cyclic of order r for some r dividing m. Thus the extension E{L1 is globally dominated, by (ii); i.e. E Ď L1 gd “ Lgd . Hence a P ppLgd qˆ qm . (iii) ñ (i): Since charpF q does not divide m, the field Lgd “ LF sep contains a primitive m-th root of unity, and Z{mZ “ Z{mZpjq over Lgd for all j. So H 1 pLgd , Z{mZq “ H 1 pLgd , Z{mZp1qq “ pLgd qˆ {ppLgd qˆ qm , which is trivial by (iii). It remains to show the triviality of H n pLgd , Z{mZq “ H n pLgd , Z{mZpnqq řsfor n ą 1. So let α lie in this cohomology group. By Proposition 2.4.1, we may write α “ i“1 αi , with each αi having the form αi “ αi,1 Y ¨ ¨ ¨ Y αi,n , where αi,j P H 1 pLgd , Z{mZq. But this H 1 is trivial. Hence each αi is trivial, and α is trivial, as desired.  2.4.2. Global domination for commutative group schemes Using the above result, the question of global domination for the cohomology of a finite commutative group scheme A can be reduced to the case of cyclic groups of prime order. We restrict to the case that the characteristic of F does not divide the order of A (equivalently, AF sep is a finite constant group scheme of order not divisible by charpF q). Corollary 2.4.3. Let L{F be a field extension, and S a collection of prime numbers unequal to charpF q. Suppose that the cohomology of the finite constant group scheme Z{ℓZ over L is globally dominated for each ℓ P S. Then for every finite commutative group scheme A over F of order divisible only by primes in S, the cohomology of A over L is globally dominated. Proof. We wish to show that H n pLgd , Aq “ 0 for n ą 0. Since A is a finite étale group scheme defined over F , it becomes split (i.e. a finite constant groups scheme) over F sep and hence over Lgd “ LF sep . In particular, the base change of A to Lgd is a product of copies of cyclic groups Z{mZ, where each prime dividing m lies in S. Since cohomology commutes with taking products of coefficient groups, we are reduced to the case that A – Z{mZ for m as above. The result now follows from condition (iii) of Proposition 2.4.2, since a group is m-divisible if it is ℓ-divisible for each prime factor ℓ of m.  In characteristic zero, we also obtain a result in the case of group schemes that need not be finite. First we prove a lemma. If A is a group scheme over a field E and m ě 1, let Arms denote the m-torsion subgroup of A, i.e. the kernel of the map A Ñ A given by multiplication by m. Thus there is a natural map H n pE, Armsq Ñ H n pE, Aq. Lemma 2.4.4. Let A be a group scheme over a field E of characteristic zero, and let n ě 1. Then every element of H n pE, Aq is in the image of H n pE, Armsq Ñ H n pE, Aq for some m ě 1. Proof. The group H n pE, Aq is torsion by [Ser97, I.2.2 Cor. 3], and so for every α P H n pE, Aq there exists m ě 1 such that mα “ 0 (writing A additively). Since charpEq “ 0, there is a short exact sequence 0 Ñ Arms Ñ A Ñ A Ñ 0 of étale sheaves which yields an 13 exact sequence H n pE, Armsq Ñ H n pE, Aq Ñ H n pE, Aq of groups, where the latter map is multiplication by m. Thus α is sent to zero under this map, and hence it lies in the image of H n pE, Armsq.  Proposition 2.4.5. Assume that charpF q “ 0, and let L{F be a field extension. Suppose that the cohomology of the finite constant group scheme Z{ℓZ over L is globally dominated for every prime ℓ. Then for every smooth commutative group scheme A over F , the cohomology of A over L is globally dominated. Proof. Let α P H n pLgd , Aq, for some n ą 0. We wish to show that α “ 0. By Lemma 2.4.4, α lies in the image of H n pLgd , Armsq for some m ě 1. But Corollary 2.4.3 asserts that the cohomology of the finite commutative group scheme Arms over L is globally dominated, since charpF q “ 0; i.e. H n pLgd , Armsq “ 0. So α “ 0.  2.5. Mayer-Vietoris and local-global principles We now use the previous results to obtain our long exact sequence, which in particular gives the abstract form of our Mayer-Vietoris sequence, and we then prove the abstract form of a local-global principle for Galois cohomology. Theorem 2.5.1. Given an oriented graph Γ, fix a Γ{F -field F‚ and consider a separably factorizable smooth commutative group scheme A over F . Suppose that for every ξ P V \ E, the cohomology of A over Fξ is globally dominated. Then we have a long exact sequence of Galois cohomology: ś ś 0 0 / H 0 pF, Aq / / 0 H pF , Aq v vPV ePE H pFe , Aq GF  BC H 1 pF, Aq / ś 1 vPV H pFv , Aq ś / ePE H 1 pFe , Aq / ¨¨¨ Proof. By hypothesis, the cohomology of A over Fξ is globally dominated. By Proposition 2.3.4, with L “ Fξ , we may identify H n pF, Aξ q – H n pFξ , Aq. Since A is separably factorizable, by Proposition 2.2.4 we have a short exact sequence of GalpF q-modules 0 / ApF sep q / ApVq πl ¨πr´1 / ApEq / 0. This induces a long exact sequence in Galois cohomology over F . Applying the above identification to the terms of this sequence, we obtain the exact sequence asserted in the theorem.  Corollary 2.5.2. Given a separably factorizable smooth commutative group scheme A over F and a Γ{F -field F‚ , the long exact sequence in Theorem 2.5.1 holds in each of the following cases: (i) A is finite; and for every ξ P V \ E, and every prime ℓ dividing the order of A, the cohomology of Z{ℓZ over Fξ is globally dominated. (ii) F is a field of characteristic zero; and for every ξ P V \ E, and every prime number ℓ, the cohomology of Z{ℓZ over Fξ is globally dominated. 14 Proof. By Theorem 2.5.1 it suffices to show that the cohomology of A over Fξ is globally dominated. In these two cases, this condition is satisfied by Corollary 2.4.3 and Proposition 2.4.5 respectively.  An important case is that of a graph Γ that is bipartite, i.e. for which there is a partition V “ V0 \ V1 such that for every edge e P E, one vertex is in V0 and the other is in V1 . Given a bipartite graph Γ together with such a partition, we will choose the orientation on Γ given by taking lpeq and rpeq to be the vertices of e P E lying in V0 and V1 respectively. Corollary 2.5.3 (Abstract Mayer-Vietoris). In the situation of Theorem 2.5.1, assume that the graph Γ is bipartite, with respect to a partition V “ V0 \ V1 of the set of vertices. Then the long exact cohomology sequence in Theorem 2.5.1 becomes the Mayer-Vietoris sequence 0 / ApF q 1 GF  H pF, Aq 2 GF  H pF, Aq ∆ ∆ ∆ / ś / / ś vPV0 ApFv q ˆ ś vPV1 ApFv q H 1 pFv , Aq ˆ ś vPV1 H 1 pFv , Aq vPV0 ´ ´ / / ś ś ApFe q ePE BC ePE H 1 pFe , Aq BC ¨¨¨ where the maps ∆ and ´ are induced by the diagonal inclusion and by subtraction, respectively. Theorem 2.5.4 (Abstract Local-Global Principle). Fix a Γ{F -field F‚ , and fix a positive integer m not divisible by the characteristic of F . Suppose that the following conditions hold: (i) Γ is bipartite, with respect to a partition V “ V0 \ V1 of the set of vertices; (ii) for every ξ P V \ E, the cohomology of Z{mZ over Fξ is globally dominated; (iii) given v P V0 , and elements ae P Feˆ for all e P E that are incident to v, there exists a P Fvˆ such that a{ae P pFeˆ qm for all e (where we identify Fv with its image iev pFv q Ď Fe ). Then for all n ą 0, the natural local-global maps ź σn : H n`1 pF, Z{mZpnqq Ñ H n`1pFv , Z{mZpnqq vPV are injective. Proof. Given hypothesis (i), as above we choose the orientation on Γ such that lpeq P V0 and rpeq P V1 for all e P E. Consider the homomorphisms: ź ź ρi,j , ρi,j H i pFv , Z{mZpjqq Ñ H i pFe , Z{mZpjqq, 0 : vPV where for α P ś vPV ePE H i pFv , Z{mZpjqq the e-th entries of ρi,j pαq, ρi,j 0 pαq are given by ρi,j pαqe “ pαlpeqq qFe ´ pαrpeq qFe , ρi,j 0 pαqe “ pαlpeq qFe . 15 Using hypothesis (ii), Theorem 2.5.1 allows us to identify the kernel of σn with the cokernel of ź ź H n pFv , Z{mZpnqq Ñ H n pFe , Z{mZpnqq. ρn,n : vPV ePE n,n Thus it suffices to show that ρ is surjective for n ě 1. This in turn will follow from showing is contained in that of ρn,n (using that Γ is is surjective, since the image of ρn,n that ρn,n 0 0 bipartite, and setting αv “ 0 for all v P V1 ). : HV‚ Ñ HE‚ is a homomorphism of graded rings, where we write HV‚ “ Note that ρ‚,‚ 0 ś ś 1,1 ‚ ‚ ‚ vPV H pFv , Z{mZp‚qq and HE “ ePE H pFe , Z{mZp‚qq. By hypothesis (iii), ρ0 is surjective, since H 1 pE, Z{mZp1qq “ E ˆ {pE ˆ qm for any field E of characteristic not dividing m. By Proposition 2.4.1, every element in HEn is a sum of n-fold products of elements in HE1 , for n ě 1. But since the map ρ0‚,‚ is a ring homomorphism, and ρ01,1 is surjective, it follows that ρn,n is surjective as well for all n ě 1.  0 3. Curves over complete discrete valuation rings We now apply the previous general results to the more specific situation that we study in this paper: function fields F over a complete discretely valued field K. In Section 3.1 we will obtain a Mayer-Vietoris sequence and a local-global principle in the context of finitely many overfields Fξ of F (“patches”). This can be compared with Theorem 3.5 of [HHK11a]. We will afterwards use that to obtain local-global principles with respect to the points on the closed fiber of a model (in Section 3.2), and with respect to the discrete valuations on F or on a regular model of F (in Section 3.3). These will later be used in Section 4 to obtain applications to other algebraic structures. 3.0.1. Notation We begin by fixing the standing notation for this section, which follows that of [HH10], [HHK09], and [HHK11a]. Let T be a complete discrete valuation ring with fraction field p be a projective, integral and normal K and residue field k and uniformizer t, and let X p We let X be the closed fiber of X, p and we choose T -curve. Let F be the function field of X. a non-empty collection of closed points P Ă X, containing all the points at which distinct irreducible components of X meet. Thus the open complement X r P is a disjoint union of finitely many irreducible affine k-curves U. Let U denote the collection of these open sets U. pP be the completion For a point P P P, we let RP be the local ring OX,P at P , and we let R p pP . For a component U P U, we let RU at its maximal ideal. Let FP be the fraction field of R p that are regular at the points of be the subring of F consisting of rational functions on X U, i.e. for all Q P Uu. RU “ tf P F |f P OX,Q p pU be the t-adic completion of RU , and we let FU be the field of fractions of We also let R pU . Here R pP and R pU are Noetherian integrally closed domains (because X p is normal), and R in particular Krull domains. For a point P P P and a component U P U, we say that P and U are incident if P is contained in the closure of U. Given P P P and U P U that are incident, the prime ideal 16 red p induces a (not necessarily prime) ideal IP sheaf I defining the reduced closure U of U in X pP . We call the height one prime ideals of R pP containing IP the in the complete local ring R branches on U at P . We let B denote the collection of branches on all points in P and all pP at ℘ is a discrete components in U. For a branch ℘ on P P P and U P U, the local ring of R p℘ be its ℘-adic (or equivalently t-adic) completion, and let F℘ be valuation ring R℘ . Let R p℘ . Note that this is a complete discretely valued field containing the field of fractions of R FU and FP (see [HH10], Section 6, and [HHK09], page 241). p and our choice of points P, we define a reduction graph Γ “ Γ p Associated to the curve X X,P whose vertex set is the disjoint union of the sets P and U and whose edge set is the set B of branches. The incidence relation on this (multi-)graph, which makes it bipartite, is defined by saying that an edge corresponding to a branch ℘ P B is incident to the vertices P P P and U P U if ℘ is a branch on U at P . We choose the orientation on Γ that is associated p to the partition P \ U of the vertex set. We will consider the Γ-field F‚ “ F‚X,P defined by p FξX,P “ Fξ for ξ P P, U, B. 3.1. Mayer-Vietoris and local-global principles with respect to patches Using the results of Section 2.5, we now obtain the desired Mayer-Vietoris sequence for the Γ-field F‚ that is associated as above to the function field F and a choice of points P on the p (see Theorem 3.1.3). In certain cases we show that this seclosed fiber of a normal model X quence splits into short exact sequences, possibly starting with the H 2 term (Corollaries 3.1.6 and 3.1.7). Related to this, we obtain a local-global principle for H n pF, Z{mZpn ´ 1qq, in this patching context. Theorem 3.1.1. With F and F‚ as above, F‚ is a Γ{F -field, and patching holds for finite dimensional vector spaces over F‚ . Thus every linear algebraic group over F is separably factorizable over F‚ . Proof. According to [HHK11a, Corollary 3.4], the fields Fξ for ξ P P \ U \ B form a factorization inverse system with inverse limit F . That is, F‚ is a Γ{F -field. That result also asserts that patching holds for finite dimensional vector spaces over F‚ . The assertion about being separably factorizable then follows from Corollary 2.2.6.  3.1.1. Global domination and Mayer-Vietoris The following result relies on a form of the Weierstrass Preparation Theorem that was proven in [HHK11b], and which extended related results in [HH10] and [HHK09]. Another result that is similarly related to Weierstrass Preparation appears at Lemma 3.1.4 below. Theorem 3.1.2 (Global domination for patches). If ξ P U \ P \ B and if m is a positive integer not divisible by charpkq, then the cohomology of Z{mZ over Fξ is globally dominated. Proof. By Proposition 2.4.2, it suffices to show that pFξgd qˆ “ ppFξgd qˆ qm . So let a P pFξgd qˆ . p1 Ñ X p be the Thus a P Fξ F 1 Ď Fξsep for some finite separable extension F 1 {F . Let X p in F 1 , so that X p 1 is a normal projective T -curve with function field normalization of X ś F 1 . Using [HH10, Lemma 6.2], we may identify Fξ bF F 1 with ξ1 Fξ11 , where ξ 1 ranges 17 p 1 . We also see through the points, components or branches, respectively, lying above ξ on X by this description that for each ξ 1 , the field Fξ11 is the compositum of its subfields Fξ and p 1 and the field F 11 , and again F 1 . Applying [HHK11b, Theorems 3.3 and 3.7] to the curve X ξ 2 1 p p using [HH10, Lemma 6.2], it follows that there is an étale cover X of X such that a “ bcm p2 for some b P F 2 Ď F sep and c P Fξ22 “ Fξ F 2 Ď Fξgd ; here F 2 is the function field of X p 2 that lies over ξ 1 on X p 1 . Now and ξ 2 is any point, component or branch, respectively, on X gd b P pF sep qˆ , and charpF q does not divide m, so b P ppF sep qˆ qm . Thus a P ppFξ qˆ qm .  Theorem 3.1.3 (Mayer-Vietoris for Curves). Let A be a commutative linear algebraic group over F . Assume that either (i) A is finite of order not divisible by the characteristic of k; or (ii) charpkq “ 0. Then we have a long exact Mayer-Vietoris sequence: ś ś ∆ / ApF q / 0 P PP ApFP q ˆ U PU ApFU q 1 GF  H pF, Aq 2 GF  H pF, Aq ∆ ∆ ś / / P PP H 1 pFP , Aq ˆ ś U PU H 1 pFU , Aq ´ ´ / / ś ℘PB ś ℘PB ApF℘ q BC H 1 pF℘ , Aq BC ¨¨¨ Proof. Let Γ be the bipartite graph ΓX,P as above. By Theorem 3.1.1, A is separably p factorizable over F‚ . Now for each prime ℓ unequal to the characteristic of k, and each ξ P U \ P \ B, the cohomology of Z{ℓZ over Fξ is globally dominated, by Theorem 3.1.2. The conclusion now follows from Corollaries 2.5.2 and 2.5.3.  3.1.2. Local-global principles with respect to patches Lemma 3.1.4. Let m be a positive integer that is not divisible by charpKq. Let P be a closed point of X, let ℘1 , . . . , ℘s be the branches of X at P , and let ai P F℘ˆi . Then there exists a P FPˆ such that a{ai P pF℘ˆi qm for every i. Proof. As before, let t be a uniformizer of the valuation ring T of K. After multiplying the elements ai by tN m for some sufficiently large positive integer N, we may assume that p℘ , the complete local ring of R pP at ℘i , for all i. By the assumption on the ai lies in R i pP ; let ri ě 0 be its ℘i -adic valuation. Since the localization characteristic, m is non-zero in R p p℘ , there exists a1i P pR p P q℘ Ă R p℘ such that pRP q℘i is ℘i -adically dense in its completion R i i i a1i ” ai modulo ℘i2ri `1 . pP q℘ Ă FP , so by the Approximation Theorem [Bou72, VI.7.3, Theorem 2] Now ai P pR i p℘ for each i, and there exists a P FP such that a ” a1i modulo ℘i2ri `1 for all i. Thus a P R i 2ri `1 p℘ and in particular a ‰ 0. a ” ai modulo ℘i ; so a{ai is a unit in R i p℘ rY s. Thus f 1 p1q “ m, and fi p1q ” 0 modulo For i “ 1, . . . , s, let fi pY q “ Y m ´ a{ai P R i i 2ri `1 2 ℘i “ m ℘i . So the hypotheses of the strong form of Hensel’s Lemma are satisfied (see 18 p℘ such that f pci q “ 0 [Bou72, III.4.5, Corollary 1 to Theorem 2]). Hence there exists ci P R i m ˆ m and ci ” 1 modulo m℘i . Thus a{ai “ ci P pF℘i q for all i.  p be a normal projective curve over a comTheorem 3.1.5 (Local-Global Principle). Let X plete discrete valuation ring T with residue field k, let P be a non-empty finite subset of the closed fiber X that includes the points at which distinct irreducible components of X meet, and let U be the set of components of X r P. Suppose that m is an integer not divisible by the characteristic of k. Then for each integer n ą 1, the natural map ź ź H n pFU , Z{mZpn ´ 1qq H n pF, Z{mZpn ´ 1qq Ñ H n pFP , Z{mZpn ´ 1qq ˆ P PP U PU is injective. Proof. The graph ΓX,P is bipartite, with the set of vertices V partitioned as V0 \V1 with V0 “ p P and V1 “ U. So hypothesis (i) of Theorem 2.5.4 holds. Hypothesis (ii) of that theorem, concerning global domination, also holds, by Theorem 3.1.2. Finally, hypothesis (iii), in this case concerning the lifting of elements of the F℘ˆi ’s to an element of FPˆ modulo m-th powers, holds by Lemma 3.1.4. Thus Theorem 2.5.4 applies, and the conclusion follows.  In some cases we can allow arbitrary Tate twists, and as a result the Mayer-Vietoris sequence splits into shorter exact sequences: Corollary 3.1.6. Let m be an integer not divisible by the characteristic of k, and suppose that the degree rF pµmq : F s is prime to m (e.g. if m is prime or F contains a primitive m-th root of unity). Let r be any integer. Then the Mayer-Vietoris sequence in Theorem 3.1.3 for A “ Z{mZprq splits into exact sequences ś ś ś / ApF q / / 0 P PP ApFP q ˆ ℘PB ApF℘ q U PU ApFU q 1 GF  BC H pF, Aq and / 0 / H n pF, Aq for all n ą 1. / ś P PP H 1 pFP , Aq ˆ ś U PU H 1 pFU , Aq ś P PP H n pFP , Aq ˆ ś U PU H n pFU , Aq ś / / ℘PB ś ℘PB H 1 pF℘ , Aq H n pF℘ , Aq 0 / / 0 Proof. If F contains a primitive m-th root of unity, then A “ Z{mZ “ Z{mZpn ´ 1q over F and its extension fields, for all in the Mayer-Vietoris sequence in Theorem 3.1.3(i), ś n. Hence ś n n n the maps ιF : H pF, Aq Ñ P PP H pFP , Aq ˆ U PU H pFU , Aq are injective for all n ą 1, by Theorem 3.1.5. The result now follows in this case. More generally, let F 1 “ F pµm q and similarly for FP and FU . As above, ιF 1 is injective. Using the naturality of ιF with respect to F , we have kerpιF q Ď kerpιF 1 ˝ resF 1 {F q. Further, by the injectivity of ιF 1 , kerpιF 1 ˝ resF 1 {F q “ kerpresF 1 {F q Ď kerpcorF 1 {F ˝ resF 1 {F q. But cor ˝ res : H n pF, Aq Ñ H n pF, Aq is multiplication by rF 1 : F s ([GS06], Proposition 3.3.7), which is injective since |A| “ m and rF 1 : F s is prime to m. Thus these kernels are all trivial, and again the result follows.  19 In Corollary 3.1.6, the initial six terms need not split into two three-term short exact sequences; i.e. the map on H 1 pF, Aq need not be injective. In fact, for A “ Z{mZ with m ą 1, a necessary and sufficient condition for splitting is that the reduction graph Γ is a tree ([HHK11a], Corollaries 5.6 and 6.4). But in the next result, there is splitting at every level. Corollary 3.1.7. Suppose that charpkq “ 0 and that K contains a primitive m-th root of unity for all m ě 1. Then the Mayer-Vietoris sequence in Theorem 3.1.3(ii) for Gm splits into exact sequences ś ś ś 0 / H n pF, Gm q / P PP H n pFP , Gm q ˆ U PU H n pFU , Gm q / ℘PB H n pF℘ , Gm q / 0 for all n ě 0. Proof. By Theorem 3.1.3(ii), it suffices to prove the injectivity of the maps H n pF, Gm q Ñ ś ś n n P PP H pFP , Gm q ˆ U PU H pFU , Gm q for all n ě 1. The case n “ 1 follows from the 1 vanishing of H pF, Gm q by Hilbert’s Theorem 90. It remains to show injectivity for n ą 1. Since K contains all roots of unity, for each m we may identify the Galois module Gm rms “ µm with Z{mZ and Z{mZpn ´ 1q. By Theorem 3.1.3(ii), the desired injectivity will follow from the surjectivity of the map ź ź ź H n´1 pF℘ , Gm q. H n´1pFU , Gm q Ñ H n´1 pFP , Gm q ˆ U PU P PP ℘PB ś So let α P ℘PB H n´1 pF℘ , Gm q, and write α “ pα℘ q℘PB , with α℘ P H n´1 pF℘ , Gm q. For each ℘ P B, the element α℘ is the image of some α r℘ P H n´1 pF℘ , µm℘ q for some m℘ ě 1, by Lemma 2.4.4. Since B is finite, we may ś let m be the least common multiple of the integers m℘ . Thus α is the image of α r “ pr α℘ q P ℘PB H n´1 pF℘ , µm q. By Theorem 3.1.5 and Theorem 3.1.3(i), the map ź ź ź H n´1 pFP , Z{mZpn ´ 1qq ˆ H n´1pFU , Z{mZpn ´ 1qq Ñ H n´1pF℘ , Z{mZpn ´ 1qq P PP U PU ℘PB is surjective. So by the identification µm “ Z{mZpn ´ 1q, it follows that α r is the image of ś ś n´1 n´1 r some element β P P PP H pFP , µm q ˆ U PU H pFU , µm q. Let β be the image of βr in ś ś n´1 pFP , Gm q ˆ U PU H n´1 pFU , Gm q. Since the diagram P PP H ś ś βr P P PP H n´1 pFP , µm q ˆ U PU H n´1 pFU , µm q βP ś  n´1 pFP , Gm q ˆ P PP H ś n´1 pFU , Gm q U PU H commutes, β maps to α, as desired. / / ś ś ℘PB H n´1 pF℘ , µm q Q α r  n´1 pF℘ , Gm q Q α ℘PB H  Note that Corollaries 3.1.6 and 3.1.7 also provide patching results for cohomology, in addition to local-global principles. Namely, for n ‰ 1 in Corollary 3.1.6, or any n in Corollary 3.1.7, those assertions show the following. Given a collection of elements αξ P H n pFξ , Aq for all ξ P P \ U such that αP , αU induce the same element of H n pF℘ , Aq whenever ℘ is a 20 branch on U at P , there exists a unique α P H n pF, Aq that induces all the αξ . In the situation of Theorem 3.1.3, where splitting is not asserted, a weaker patching statement still follows: given elements αξ as above, there exists such an α, but it is not necessarily unique. 3.2. Local-global principles with respect to points In this section we will investigate how to translate our results into local-global principles p rather than in terms of patches. Extending in terms of the points on the closed fiber X of X, our earlier notation, if P P X is any point (not necessarily closed), we let FP denote the p p . In particular, if η is the generic point of pP :“ O fraction field of the complete local ring R X,P an irreducible component X0 of the closed fiber X, then Fη is a complete discretely valued field, and it is the same as the η-adic completion of F . 3.2.1. The field Fηh In order to bridge the gap between the fields FU and the fields Fη , where η is the generic point of U, we will consider a subfield Fηh of Fη that has many of the same properties but is much smaller. pV , where V ranges Namely, with notation as above, let Rηh be the direct limit of the rings R over the non-empty open subsets V of X0 that do not meet any other irreducible component of X. Equivalently, we may fix one such non-empty open subset U, and consider the direct pη ; and we let F h limit over the non-empty open subsets V of U. Here Rηh is a subring of R η be its fraction field. Thus Fηh is a subfield of Fη . Lemma 3.2.1. Let X0 Ď X be an irreducible component with generic point η, and let U Ă X0 be a non-empty open subset meeting no other component. Then Rηh is a Henselian discrete valuation ring with respect to the η-adic valuation, having residue field kpUq “ kpX0 q. Its fraction field Fηh is the filtered direct limit of the fields FV , where V ranges over the non-empty open subsets of U. Proof. Each FV is contained in Fηh , and every element of Fηh is of the form a{b with a, b in some common FV . So Fηh is the direct limit of the fields FV . The fields FV each have a discrete valuation with respect to η, and these are compatible. So Fηh is a discretely valued field with respect to the η-adic valuation. We wish to show that the valuation ring of Fηh is Rηh , with residue field kpUq. pV is contained in the η-adic valuation ring of FV , it follows that Rηh is contained Since R in the valuation ring of Fηh . To verify the reverse containment, consider a non-zero element α P Fηh with non-negative η-adic valuation. Thus α P FVˆ for some V ; and so α “ a{b with pV non-zero and vη paq ě vη pbq. Since R pV is a Krull domain, the element b P R pV a, b P R has a well defined divisor, which is a finite linear combination of prime divisors; and other pV q, each of them has a locus that meets this than the irreducible closed fiber V of SpecpR closed fiber at only finitely many points. After shrinking V by deleting these points, we pV rt´1 s. But also vη pa{bq ě 0; and thus b P R pˆ . So the may assume that b is invertible in R V h h p element α “ a{b P FV actually lies in RV , and hence in Rη as desired. Thus Rη is indeed the valuation ring of Fηh . Since the valuations on the rings RV are compatible and induce 21 pV of R pV ; that of Rηh , the maximal ideal ηRηh of Rηh is the direct limit of the prime ideals η R pV . and so the residue field of Rηh is the residue field kpUq “ kpV q of the ring R h It remains to show that Rη is Henselian. Let S be a commutative étale algebra over Rηh , together with a section σ : η Ñ SpecpSq of π : SpecpSq Ñ SpecpRηh q over the point η. To show that Rηh is Henselian, we will check that σ may be extended to a section over all of SpecpRηh q. pV -algebra SV for Now since S is a finitely generated Rηh -algebra, it is induced by an étale R pV q that induces π and has a some V , together with a morphism πV : SpecpSV q Ñ SpecpR pV q. Here σ 0 section σV0 : η Ñ SpecpSV q over the generic point η of the closed fiber of SpecpR V defines a rational section over V , and hence a section over a non-empty affine open subset of V . So after shrinking V , we may assume that σV0 is induced by a section σV : V Ñ SpecpSV q. pV is η-adically complete (or equivalently, t-adically complete); so by a version But the ring R of Hensel’s Lemma (Lemma 4.5 of [HHK09]) the section σV over V extends to a section of pV q. This in turn induces a section of π over SpecpRηh q that extends σ, πV , over all of SpecpR thereby showing that Rηh is Henselian.  Proposition 3.2.2. Let η be the generic point of an irreducible component X0 of X, and let U be a non-empty affine open subset of X0 that does not meet any other irreducible component of X. Let A be a smooth commutative group scheme defined over the complete discrete valuation ring T . Suppose α P H n pFU , Aq satisfies αFη “ 0. Then there is a Zariski open neighborhood V of η in U such that αFV “ 0. Proof. The map H n pFU , Aq Ñ H n pFη , Aq factors through H n pFηh , Aq. But both Fη and Fηh are Henselian local rings with residue field kpUq, in the latter case using the first part of Lemma 3.2.1. So by [Art62, Theorem III.4.9], we have H n pFηh , Aq “ H n pkpUq, Aq “ H n pFη , Aq. Thus αFηh “ 0. The conclusion now follows from Lemma 2.3.5, since Fηh is the filtered direct limit of the fields FV , by the second part of Lemma 3.2.1.  3.2.2. Local-global principles with respect to points We now obtain a local-global principle in terms of points on the closed fiber X. Theorem 3.2.3. Let A be a commutative linear algebraic group over F and let n ą 1. Assume that either (i) A “ Z{mZprq, where m is an integer not divisible by charpkq, and where either r “ n ´ 1 or else rF pµm q : F s is prime to m; or (ii) A “ Gm , charpkq “ 0, and K contains a primitive m-th root of unity for all m ě 1. Then the natural map ź H n pF, Aq Ñ H n pFP , Aq P PX is injective, where P ranges through all the points of the closed fiber. Proof. Let α P H n pF, Aq be an element of the above kernel. Consider the irreducible components Xi of X, and their generic points ηi P Xi Ď X. Thus αFηi “ 0 for each i (taking P “ ηi ). By Proposition 3.2.2, we may choose a non-empty Zariski affine open subset Ui Ă Xi , not meeting any other component of X, such that αFUi is trivial. Let U be the collection of these 22 sets Ui , and let P be the complement in X of the union of the sets Ui . Then α is in the kernel of the map on H n pF, Aq in Theorem 3.1.5, Corollary 3.1.6, or Corollary 3.1.7 respectively. Since that map is injective, it follows that α “ 0.  3.3. Local-global principles with respect to discrete valuations Using the previous results, we now investigate how to translate our results into local-global principles involving discrete valuations on our field F , and in particular those valuations p Our main result here is Theorem 3.3.6, arising from codimension one points on model X. which parallels Theorem 3.2.3(i), and asserts the vanishing of the obstruction Xn pF, Aq to such a local-global principle, for n ą 1 and A an appropriate twist of Z{mZ. In the case n “ 1, a related result appeared at [HHK11a, Corollary 8.11], but with different hypotheses and for different groups. In fact, for a constant finite group A, the obstruction p of F is a tree (see X1 pF, Aq is non-trivial unless the reduction graph Γ of a regular model X [HHK11a], Proposition 8.4 and Corollary 6.5). p is regular. For the remainder of this section we make the standing assumption that X Lemma 3.3.1. Let P be a point of X and let v be a discrete valuation on FP . Then the restriction v0 of v to F is a discrete valuation on F . Moreover if v is induced by a codimension pP q (or equivalently, a height one prime of R pP ), then v0 is induced by a one point of SpecpR p whose closure contains P . codimension one point of X Proof. The first assertion is given at [HHK11a, Proposition 7.5]. For the second assertion, if pP , then R pP is contained in the valuation ring of v. v is induced by a height one prime of R Hence so is the local ring RP , which is then also contained in the valuation ring of v0 . Thus v0 is induced by a codimension one point of SpecpRP q, and so by a codimension one point p whose closure contains P . of X  Given a field E, let ΩE denote the set of discrete valuations on E. For v P ΩE , write Ev for the v-adic completion of E. If A is a commutative group scheme over E, let ˙ ˆ ź n n n H pEv , Aq . X pE, Aq “ ker H pE, Aq Ñ vPΩE Similarly, given a normal integral scheme Z with function field E, let ΩZ Ď ΩE denote the subset consisting of the discrete valuations on E that correspond to codimension one points on Z. If A is as above, let ˙ ˆ ź n n n H pEv , Aq . X pZ, Aq “ ker H pE, Aq Ñ vPΩZ We will be especially interested in the case that E “ F , the function field of a regular p over our complete discrete valuation ring T ; and where Z is either X p or projective curve X pP q for some closed point P P X. p SpecpR 23 3.3.1. Relating local-global obstruction on a regular model to obstructions at closed points A key step in relating our patches to discrete valuations is the following lemma. Here Xp0q denotes the set of closed points of X. Lemma 3.3.2. Let n, m, r be as in Theorem 3.2.3(i). Then the map in that result induces an injection ź pP q, Z{mZprqq, p Z{mZprqq Ñ Xn pSpecpR ι : Xn pX, P PXp0q p Z{mZprqq and projection onto the factors where P P Xp0q . by restriction to Xn pX, p Z{mZprqq in H n pFP , Z{mZprqq Proof. It follows from Lemma 3.3.1 that the image of Xn pX, pP q, Z{mZprqq. Thus we obtain a well-defined group homomorphism ι as lies in Xn pSpecpR in the assertion. We wish to show that ι is injective. p Z{mZprqq be any element in the kernel of this map. Then the image Let α P Xn pX, n pP q, Z{mZprqq Ď H n pFP , Z{mZprqq is trivial for every closed point P of α in X pSpecpR on the closed fiber X. Meanwhile, for any non-closed point η of X (viz. the generic point of an irreducible component of X), the image of α in H n pFη , Z{mZprqq is also trivial, by p Hence α lies in the kernel the definition of Xn , since η is a ś codimension one point of X. n n of the map H pF, Z{mZprqq Ñ P PX H pFP , Z{mZprqq. But this map is injective, by Theorem 3.2.3(i). So α is trivial, and this proves the desired injectivity.  3.3.2. Local-global principles at closed points We will use the following statement of Panin which asserts a particular case of the analog of the Gersten conjecture in the context of the theory of Bloch and Ogus. Here κpzq denotes the residue field at a point z, and Z piq denotes the set of points of Z having codimension i. Theorem 3.3.3 ([Pan03, Theorem C]). Suppose that R is an equicharacteristic regular local ring with fraction field F , and let Z “ SpecpRq. Then for any positive integer m that is not divisible by the characteristic, and any m-torsion étale commutative group scheme A over R, the Cousin complex à n´2 à n´1 H pκpzq, Ap´2qq Ñ ¨ ¨ ¨ H pκpzq, Ap´1qq Ñ 0 Ñ H n pZ, Aq Ñ H n pF, Aq Ñ zPZ p2q zPZ p1q of étale cohomology groups is exact. Proposition 3.3.4. Under the hypotheses of Theorem 3.3.3, assume that R is complete. Then Xn pF, Aq “ Xn pZ, Aq “ 0 for n ě 1. Proof. Since Xn pF, Aq is contained in Xn pZ, Aq, it suffices to show the vanishing of the latter group. Let α P Xn pZ, Aq Ď H n pF, Aq. Consider the exact sequence in Theorem 3.3.3. For each z P Z p1q , the ramification map H n pF, Aq Ñ H n´1 pκpzq, Ap´1qq factors through the map to the completion H n pFz , Aq.ÀBut the image of α in H n pFz , Aq vanishes, since α P Xn pZ, Aq. Hence α maps to zero in zPZ p1q H n´1pκpzq, Ap´1qq, and thus it is induced by a class α rP H n pZ, Aq. 24 Let k 1 be the residue field of R at its maximal ideal (corresponding to the closed point of pσ for the completion of the Z). Let π, σ be a regular system of parameters in R, and write R pσ is a complete discrete valuation ring with local ring of R at the prime ideal pσq. Thus R uniformizer σ; let Fσ and κpσq denote its fraction field and residue field, respectively. Here κpσq is a complete discretely valued field with uniformizer π̄, the image of π in κpσq, and with residue field k 1 . Let Oκpσq denote its valuation ring. By the discrete valuation ring case of the Gersten conjecture ([CT95, sect. 3.6]; or by Theorem 3.3.3), the natural map pσ , Aq Ñ H n pFσ , Aq H n pR is an injection. The complete local rings Oκpσq and R each have residue field k 1 , so by [Art62, Theorem III.4.9] we may identify H n pZ, Aq “ H n pk 1 , Aq “ H n pOκpσq , Aq, via restriction to the closed point. We have the following commutative diagram: H n pOκpσq , Aq ♦♦♦ ♦♦♦ ♦ ♦ ♦ ♦♦♦ H n pk 1 , Aq❖ O pσ , Aq H n pR ❖❖❖ ❖❖❖ ❖❖❖❖ ❖❖ O α̃ P H n pZ, Aq H n pFσ , Aq / / O H n pF, Aq Q α It follows from this diagram that the composition of the vertical maps to H n pOκpσq , Aq is pσ , Aq Ñ H n pFσ , Aq is injective, and the an isomorphism. On the other hand, the map H n pR n n n image of α P X pZ, Aq Ď H pF, Aq in H pFσ , Aq is trivial since σ defines a codimension one point of Z. Thus α r “ 0, and hence α “ 0 as well.  pP arising from a regular model X, p Proposition 3.3.4 asserts: In our situation, with R “ R Corollary 3.3.5. Suppose that K is an equicharacteristic complete discretely valued field of p is regular. Then for every P P X and m-torsion characteristic not dividing m, and that X pP , Xn pFP , Aq “ Xn pSpecpR pP q, Aq “ 0. étale commutative group scheme A over R 3.3.3. Local-global principles for function fields Finally, we obtain our local-global principles over our field F with respect to discrete valuations: Theorem 3.3.6. Suppose that K is an equicharacteristic complete discretely valued field of p is regular. Let n ą 1. Then characteristic not dividing m, and that X p Z{mZpn ´ 1qq “ 0. Xn pF, Z{mZpn ´ 1qq “ Xn pX, p Z{mZprqq “ 0 for all r. If rF pµm q : F s is prime to m then also Xn pF, Z{mZprqq “ Xn pX, p Z{mZpn ´ 1qq holds by Corollary 3.3.5 and Lemma 3.3.2, Proof. The vanishing of Xn pX, since n ą 1. The vanishing of Xn pF, Z{mZpn ´ 1qq then follows since that group is contained 25 p Z{mZpn ´ 1qq. If rF pµmq : F s is prime to m then the same argument shows the in Xn pX, result asserted in that case, since the hypothesis of Theorem 3.2.3(i) is then satisfied and thus Lemma 3.3.2 again applies.  Remark 3.3.7. It would be interesting to know if Theorem 3.3.6 carries over to the mixed characteristic case, and also if it has an analog for split tori in characteristic zero as in Theorem 3.2.3(ii). But carrying over the above proof would require versions of Panin’s result [Pan03, Theorem C] in those situations. 4. Applications to torsors under noncommutative groups As an application of our results, in this section we give local-global principles for Gtorsors over F for certain connected noncommutative linear algebraic groups G, and for related structures. Our method is to use cohomological invariants in order to reduce to our local-global principles in Galois cohomology (viz. to Theorems 3.2.3(i) and 3.3.6). We preserve the notation and terminology established at the beginning of Section 3. In p particular, we write T for the valuation ring of K, and k for the residue field. We let X be a normal, integral projective curve over T , with closed fiber X and function field F . As before, we write ΩF for the set of discrete valuations on the field F , and write ΩXp for the subset of ΩF consisting of those discrete valuations that arise from codimension one points p on X. 4.1. Relation to prior results The basic strategy used in this section to obtain local-global principles for torsors was previously used in [CPS08, Theorem 5.4], to obtain a local-global principle for G-torsors over the function field F of a smooth projective geometrically integral curve over a p-adic field K, where G is a linear algebraic F -group that is quasisplit, simply connected, and absolutely almost simple without an E8 factor. There they used the local-global principle of Kato for H 3 together with the fact that the fields under their consideration were of cohomological dimension three. Our applications arise from our new local-global principles for higher cohomology groups, and hence do not require any assumptions on cohomological dimension. Local-global principles for G-torsors were also obtained in [HHK11a] (as well as in [HHK09], in the context of patches). But there the linear algebraic groups G were required to be rational varieties, whereas here there is no such hypothesis. On the other hand, here we will be looking at specific types of groups, such as E8 and F4 . Another difference is that in [HHK11a], in order to obtain local-global principles with respect to discrete valuations, we needed to make additional assumptions (e.g. that k is algebraically closed of characteristic p see [HHK11a, Corollary 8.11]). Here the zero, or that G is defined and reductive over X; only assumption needed for local-global principles with respect to discrete valuations is that K is equicharacteristic. (If we wish to consider only those discrete valuations that arise from p of F , then we also need to assume that X p is regular.) Thus even in the a given model X cases where the groups considered below are rational, the results here go beyond what was shown for those groups in [HHK11a]. 4.2. Injectivity vs. triviality of the kernel The local-global principles for G-torsors will be phrased in terms of local-global maps on H 1 pF, Gq. Because of non-commutativity, H 1 pF, Gq is just a pointed set, not a group. Thus there are two distinct questions that can be posed about a local-global map: whether the kernel is trivial, and whether the map is injective (the latter condition being ś stronger). And as in Section 3, there are actually several local-global maps: H 1 pF, Gq Ñ vPΩF H 1 pFv , Gq, ś ś H 1 pF, Gq Ñ vPΩ x H 1 pFv , Gq, and H 1 pF, Gq Ñ P PX H 1 pFP , Gq. Their kernels will be X p Gq “ X1 pX, p Gq, and X0 pX, p Gq respectively. denoted by XpF, Gq “ X1 pF, Gq, XpX, p Gq “ 0 for some model X p then XpF, Gq “ 0; and similarly for injectivity Note that if XpX, p Gq and X0 pX, p Gq. of the corresponding maps. So we will emphasize the cases of XpX, 4.3. Local-global principles via cohomological invariants The approach that we take here for obtaining our applications is to use cohomological invariants of algebraic objects. Recall that an invariant over F is a morphism of functors a : S Ñ H, where S : pFields{F q Ñ pPointed Setsq and H : pFields{F q Ñ pAbelian Groupsqp[GMS03], Part I, Sect. I.1). Most often, as in [GMS03], S will have the form SG given by SG pEq “ H 1 pE, Gq for some linear algebraic group G over F ; this classifies G-torsors over E, and also often classifies other types of algebraic structures over F . In practice, HpEq will usually take values in Galois cohomology groups of the form H n pE, Z{mZpn ´ 1qq. The simplest situation is described in the following general result, where we retain the standing hypotheses stated at the beginning of Section 3. Proposition 4.3.1. Let a : S Ñ H be a cohomological invariant over F , where HpEq “ H n pE, Z{mZprqq for some integers n, m, r with n, m positive, and where m is not divisible by charpkq. Assume either that r “ n ´ 1, or else that the degree rF pµmq : F s is prime to m. (a) If apF q : SpF q Ñ HpF q has trivial kernel, then so does the local-global map SpF q Ñ ś p is regular, then the same if K is equicharacteristic and X P PX SpFP q. Moreover, ś holds for SpF q Ñ vPΩ x SpFv q. X ś (b) If apF q : SpF q Ñ HpF q is injective, then so is the local-global map SpF q Ñ P PX SpFP q. p is regular, then SpF q Ñ ś If in addition K is equicharacteristic and X SpFv q is vPΩX x injective as well. Proof. Consider the commutative diagrams SpF q ś  P PX SpFP q apF q ś apFP q HpF q / / ś SpF q ś  P PX HpFP q  vPΩ x SpFv q X apF q ś apFv q / / ś HpF q  vPΩX x HpFv q. The result follows by a diagram chase, using the fact that the right-hand vertical map in the first diagram is injective by Theorem 3.2.3(i), and that the corresponding map in the 27 p is regular, by second diagram is injective in the case that K is equicharacteristic and X Theorem 3.3.6.  Recall that a linear algebraic group G over F is quasi-split if it has a Borel subgroup defined over F . It is split if it has a Borel subgroup over F that has a composition series whose successive quotient groups are each isomorphic to Gm or Ga . If G is reductive, this is equivalent to G having a maximal torus that is split (i.e. a product Gnm ). Corollary 4.3.2. Let G be a simply connected linear algebraic group over F . Consider the Rost invariant RG : H 1 p˚, Gq Ñ H 3 p˚, Z{mZp2qq of G, and assume that the characteristic of p Gq “ 0; and if k does not divide the order m of RG . In each of the following cases, X0 pX, p Gq “ 0 for X p a regular model of F . K is equicharacteristic, then XpF, Gq “ 0 and XpX, ‚ G is a quasi-split group of type E6 or E7 . ‚ G is an almost simple group that is quasi-split of absolute rank at most 5. ‚ G is an almost simple group that is quasi-split of type B6 or D6 . ‚ G is an almost simple group that is split of type D7 . Proof. In each of these cases, the Rost invariant RG has trivial kernel. This is by [Gar01, Main Theorem 0.1] in the first case, and by [Gar01, Theorem 0.5] in the other cases. So the assertion follows from Proposition 4.3.1(a).  Corollary 4.3.3. Let m be a square-free positive integer that is not divisible by the characteristic of k, and let ś A be a1 central simple F -algebra of degree m. Then the local-global map 1 H pF, SL1 pAqq Ñ P PX H pFP , SL1 pAqq is injective. If in addition K is equicharacteristic ś 1 p is regular, then the map H 1 pF, SL1 pAqq Ñ and X vPΩ x H pFv , SL1 pAqq is injective. X Proof. By [MS82, 12.2] (see also [Ser95, 7.2]), given a division algebra A of degree m, there is a cohomological invariant a : H 1 p˚, SL1 pAqq Ñ H 3 p˚, Z{mZp2qq that is injective if m is square-free. So the result follows from Proposition 4.3.1(b).  In particular, X0 pF, SL1 pAqq and XpF, SL1 pAqq respectively vanish in the above situations. Also, via the identification of H 1 pF, SL1 pAqq with F ˆ { NrdpAˆ q, the above result gives a local-global principle for elements of F ˆ to be reduced norms from a (central) division algebra A; cf. also [Kat86, p. 146]. Other applications can be obtained by using a combination of cohomological invariants. This is done in the next results. Proposition 4.3.4. Let G be a simple linear algebraic group of type E8 over F . (a) Assume charpKq “ 0. Then the group G is split over some odd degree extension of F if and only if GFP is split over some odd degree extension of FP for every P P X. (b) Assume charpKq ‰ 2, 3, 5. Then the same holds for extensions of degree prime to five (rather than of odd degree) over F and each FP . p is regular, then the assertions in parts (a) and (b) (c) If K is equicharacteristic and X hold with the fields FP replaced by the fields Fv for all v P ΩXp . Proof. The forward implications are trivial, and we will show the reverse implications. Proof of (a) and the corresponding part of (c): 28 Let G0 be a split simple algebraic group over F of type E8 . Then H 1 pF, G0 q classifies simple algebraic groups of type E8 over F , since G0 “ AutpG0 q. Given a group G as in the proposition, let rGs be the class of G in H 1 pF, G0q, and let rG :“ RG0 prGsq be the associated Rost invariant, say with order m. For each P P X, the group G becomes split over some extension EP {FP of odd degree dP . Thus the Rost invariant of G over FP maps to zero in H 3 pEP , Z{mZp2qq, and hence it is dP -torsion in H 3 pFP , Z{mZp2qq by a standard restriction-corestriction argument. Let d be the least common multiple of the odd integers dP . Thus drG P H 3 pF, Z{mZp2qq has trivial image in H 3 pFP , Z{mZp2qq for all P . It follows from Theorem 3.2.3(i) that drG is trivial. Hence the order of the Rost invariant rG over F is odd. Let H 1 p˚, G0 q0 Ď H 1 p˚, G0 q be the subset consisting of classes α such that RG0 pαq has odd order. By the above, this contains rGs. Now by [Sem09, Corollary 8.7], since charpF q “ charpKq “ 0, there is a cohomological invariant u : H 1 p˚, G0 q0 Ñ H 5 p˚, Z{2Zq such that for any field extension E{F , the invariant uprGE sq vanishes if and only if G splits over a field extension of E of odd degree. By functoriality of u, the class uprGsq maps to uprGFP sq for every P P X. But for every P P X, GFP is split over an extension of odd degree and hence uprGFP sq is trivial in H 5 pFP , Z{2Zq. By Theorem 3.2.3(i), it follows that uprGsq is trivial in H 5 pF, Z{2Zq. The conclusion of (a) now follows from the defining property of u. The corresponding part of (c) is proved in exactly the same way, but with Fv replacing FP and with Theorem 3.3.6 replacing Theorem 3.2.3(i). Proof of (b) and the corresponding part of (c): By the main theorem in [Che94], since charpF q ‰ 2, 3, 5, the Rost invariant of G over a field extension E{F has trivial image in H 3 pE, Z{5Zp2qq if and only if G splits over some finite extension of E having degree prime to five. The desired assertion now follows from Proposition 4.3.1, taking SpEq to be the subset of H 1 pE, G0 q that consists of elements that split over some field extension of E having degree prime to five, and with a being the restriction to this subset of the Rost invariant modulo 5.  Proposition 4.3.5. Assume that charpKq ‰ 2, 3. Then Albert algebras over F have each of the following properties if and only if the respective properties hold after base change to FP for each P P X. ‚ ‚ ‚ ‚ The algebra is reduced. The algebra is split. The automorphism group of the algebra is anisotropic. Two reduced algebras are isomorphic. The same holds for base change to Fv for each v P ΩXp , in the case that K is equicharacteristic p is regular. and X Proof. Albert algebras are classified by H 1 pF, Gq, where G is a split simple linear algebraic group over F of type F4 . Moreover (see [Ser95, 9.2,9.3]) there are cohomological invariants f3 : H 1 pF, Gq Ñ H 3 pF, Z{2Zq, f5 : H 1 pF, Gq Ñ H 5 pF, Z{2Zq, g3 : H 1 pF, Gq Ñ H 3 pF, Z{3Zq, 29 where H 3 pF, Z{3Zq “ H 3 pF, Z{3Zp2qq. The properties of Albert algebras listed in the proposition are respectively equivalent to the following conditions involving these invariants (see [Ser95, 9.4]): ‚ ‚ ‚ ‚ The The The The invariant g3 vanishes on the algebra. invariants f3 and g3 each vanish on the algebra. invariants f5 and g3 are each non-vanishing on the algebra. two reduced algebras have the same pair of invariants f3 , f5 . By the injectivity of the local-global maps on H 3 pF, Z{2Zq, H 5 pF, Z{2Zq, and H 3 pF, Z{3Zp2qq (viz. by Theorems 3.2.3(i) and 3.3.6 respectively), and by the functoriality of the invariants f3 , f5 , g3 , the assertion then follows.  References [Art62] Michael Artin. Grothendieck Topologies. Dept. of Mathematics, Harvard University, Cambridge, MA, 1962. [BLR90] Siegfried Bosch, Werner Lütkebohmert, and Michel Raynaud. Néron Models. Ergebnisse der Mathematik, vol. 21. Springer-Verlag, Berlin and Heidelberg, 1990. [Bou72] Nicolas Bourbaki. Commutative Algebra. Hermann and Addison-Wesley, 1972. [Che94] V. I. Chernousov. A remark on the (mod 5)-invariant of Serre for groups of type E8 . (Russian.) Mat. Zametki 56 (1994), no. 1, 116–121, 157; English translation in Math. Notes 56 (1994), no. 1-2, 730–733 (1995). [CT95] Jean-Louis Colliot-Thélène. Birational invariants, purity and the Gersten conjecture. In: Ktheory and algebraic geometry: connections with quadratic forms and division algebras (Santa Barbara, CA, 1992), 1–64, Proc. Sympos. Pure Math. 58, Part 1, Amer. Math. Soc., Providence, RI, 1995. [COP02] J.-L. Colliot-Thélène, M. Ojanguren, and R. Parimala. Quadratic forms over fraction fields of two-dimensional Henselian rings and Brauer groups of related schemes. In: “Proceedings of the International Colloquium on Algebra, Arithmetic and Geometry”, Tata Inst. Fund. Res. Stud. Math., vol. 16, pp.185–217, Narosa Publ. Co., 2002. [CPS08] Jean-Louis Colliot-Thélène, R. Parimala, and V. Suresh. Patching and local-global principles for homogeneous spaces over function fields of p-adic curves. To appear in Comment. Math. Helv. Available at arXiv:0812.3099. [Gar01] Skip Garibaldi. The Rost invariant has trivial kernel for quasi-split groups of low rank. Comment. Math. Helv. 76 (2001), no. 4, 684–711. [GMS03] Skip Garibaldi, Alexander Merkurjev, and Jean-Pierre Serre. Cohomological invariants in Galois cohomology. University Lecture Series, vol. 28, AMS, Providence, RI, 2003. [GS06] Philippe Gille and Tamás Szamuely. Central Simple Algebras and Galois Cohomology. Cambridge Studies in Advanced Mathematics, vol. 101, Cambridge, UK, 2006. [HH10] David Harbater and Julia Hartmann. Patching over fields. Israel Journal of Math. 176 (2010), 61–107. [HHK09] David Harbater, Julia Hartmann, and Daniel Krashen. Applications of patching to quadratic forms and central simple algebras. Invent. Math. 178 (2009), 231–263. [HHK11a] David Harbater, Julia Hartmann, and Daniel Krashen. Local-global principles for torsors over arithmetic curves. 2011 manuscript. Available at arXiv:math/1108.3323. [HHK11b] David Harbater, Julia Hartmann, and Daniel Krashen. Weierstrass preparation and algebraic invariants. 2011 manuscript. Available at arXiv:math/1109.6362. [HS53] Gerhard Hochschild and Jean-Pierre Serre. Cohomology of Lie algebras. Ann. of Math. (2) 57 (1953), 591–603. 30 [Hu12] Yong Hu. Hasse principle for simply connected groups over function fields of surfaces. 2012 manuscript. Available at arXiv:math/1203.1075. [Jan09] Uwe Jannsen. Hasse principles for higher-dimensional fields. 2009 manuscript, available at arXiv:0910.2803. [Kat86] Kazuya Kato. A Hasse principle for two-dimensional global fields. With an appendix by JeanLouis Colliot-Thélène. J. Reine Angew. Math. 366 (1986), 142–183. [KMRT98] Max-Albert Knus, Alexander S. Merkurjev, Markus Rost, and Jean-Pierre Tignol. The Book of Involutions. American Mathematical Society, Providence, RI, 1998. [Lam05] Tsit-Yuen Lam. Introduction to quadratic forms over fields. American Mathematical Society, Providence, RI, 2005. [MS82] Alexander S. Merkurjev and Andrei A. Suslin. K-cohomology of Severi-Brauer varieties and the norm residue homomorphism (in Russian). Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), 10111046, 1135-1136. English translation: Math. USSR-Izv. 21 (1983), 307–340. [Pan03] Ivan Alexandrovich Panin. The equicharacteristic case of the Gersten conjecture. Tr. Mat. Inst. Steklova 241 (Teor. Chisel, Algebra i Algebr. Geom.) (2003), 169–178. Translation in: Proc. Steklov Inst. Math. 241, no. 2 (2003), 154–163. [PS98] R. Parimala and V. Suresh. Isotropy of quadratic forms over function fields of p-adic curves. Inst. Hautes Études Sci. Publ. Math. No. 88 (1998), 129–150 (1999). [Sem09] Nikita Semenov. Motivic construction of cohomological invariants. 2009 manuscript, available at arXiv:math/0905.4384. [Ser95] Jean-Pierre Serre. Cohomologie galoisienne: progrès et problèmes. Séminaire Bourbaki, vol. 1993/94. Astérisque 227 (1995), Exp. No. 783, 4, 229–257. [Ser97] Jean-Pierre Serre. Galois cohomology. Springer Monographs in Mathematics, 1997. [Voe11] Vladimir Voevodsky. On motivic cohomology with Z{ℓ-coefficients. Ann. of Math. (2) 174, (2011), no. 1, 401–438. [Wei09] Chuck Weibel. The norm residue isomorphism theorem. J. Topol. 2 (2009), no. 2, 346–372. Author information: David Harbater: Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104-6395, USA email: [email protected] Julia Hartmann: Lehrstuhl A für Mathematik, RWTH Aachen University, 52062 Aachen, Germany email: [email protected] Daniel Krashen: Department of Mathematics, University of Georgia, Athens, GA 30602, USA email: [email protected] 31