On some local–global principles for linear algebraic groups over infinite algebraic extensions of global fields

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Linear Algebra and its Applications www.elsevier.com/locate/laa

On some local–global principles for linear algebraic groups over infinite algebraic extensions of global fields Ngô Thi. Ngoan a , Nguyêñ Quôć Thˇańg b,∗ a

Department of Math. and Informatics, College of Science, Thai Nguyen University, Viet Nam b Institute of Mathematics, VAST, 18-Hoang Quoc Viet, Cau Giay, Hanoi, Viet Nam

a r t i c l e

i n f o

Article history: Received 19 January 2018 Accepted 25 May 2018 Available online xxxx Submitted by V. Futorny Dedicated to Vladimir Sergeichuk on his 70-th birthday MSC: 11E08 11E12 11E39 11E72 20G25 20G30

a b s t r a c t In this paper, we develop an arithmetic theory of quadratic and hermitian forms over infinite algebraic extensions of local and global fields. In particular, we prove that the cohomological Hasse principle for H1 holds for all semisimple simply connected algebraic groups defined over any infinite algebraic extension of any global field and we also show the validity of some local–global principles for (skew-)hermitian forms defined over such infinite extension fields. As applications, we deduce some analogs of well known results such as Landherr’s and Kneser’s Strong Hasse principle, Hasse–Maass–Schilling Norm Theorem, Albert–Brauer–Hasse–Noether Theorem and Hasse Norm Theorem over such fields. © 2018 Elsevier Inc. All rights reserved.

Keywords: Quadratic forms Hermitian forms Unitary groups Orthogonal groups Global fields

* Corresponding author. E-mail addresses: [email protected] (N.T. Ngoan), [email protected] (N.Q. Thˇ ańg). https://doi.org/10.1016/j.laa.2018.05.026 0024-3795/© 2018 Elsevier Inc. All rights reserved.

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Hasse principles Algebraic groups

1. Introduction

In the past, along with the study of finite extensions of local and global fields, there is a long and rich history of corresponding study of infinite algebraic extensions of such fields. In fact, this is indispensable, since there is a very close relation between the study of finite extensions of local or global fields and that of their infinite extensions. For example, the study of the maximal unramified extension of a given local field or the maximal abelian extension of a given global field show this connection. The main focus in such study was the divisorial theory (i.e. the arithmetic in the classical setting) of infinite global fields in general, and local and global class field theory, in particular, say as it was done in the works by Albert, Artin and Tate, Herbrand, Kawada, Krull, Moriya, Nakayama Schilling, Stiemke, Whaples etc., which dated back as early as 1926 (Stiemke). The readers can consult [37], [3], [15] and references there. Later on, there were also some investigations, notably on local fields and their infinite extensions (see [9] and references there) and on arithmetic and analytic theory of infinite extensions of global fields, say by Kurihara, Murty, Tsfasman and Vladuts (see [22], [25], [44]) etc. In general, the arithmetic theory of infinite local or global fields in particular and the corresponding theory of forms and algebraic groups over such fields in general are interesting and they are worth investigating. One of the famous local–global principles is the Hasse–Minkowski Theorem. To our knowledge, the validity of an analog of the Hasse–Minkowski Theorem for quadratic forms over infinite algebraic extensions of global fields was for the first time discussed in [21]. (For pseudo-global fields it was discussed in [2].) Here we note that the relation: global versus completions in the classical setting is replaced by another one: global versus localizations, where, for a place v of an infinite algebraic extension k of a global field L, the localization field is a certain subfield k(v) contained in the completion kv of k at v, see Section 2 below. In this paper, we will consider the Hasse principle in both the classical setting (where the completions are involved, i.e., the classical Hasse principle) and in the new setting. It was known that for quadratic forms in n ≥ 3 variables, the Hasse–Minkowski Theorem still holds in the new setting, but in general, it may fail for forms of fewer variables. It implies that in general, the classical Hasse–Minkowski Theorem may not hold for quadratic forms in the case of infinite extensions of global fields. It is natural to ask: What can be said about the validity of other known results for (skew-)hermitian forms over infinite extensions of local and global fields? In particular, what about the Hasse principles over infinite extensions of global fields?

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This paper is a sequel to our previous paper [27], where we consider some Hasse principles for algebraic groups over global fields. (It is also an expanded version of the short note [28].) In this paper we aim to answer the questions stated above. We consider here only infinite algebraic extensions of local and global fields. The case of transcendental extensions of such fields is also interesting and has been treated mostly in the case of p-adic curves (see for example [7], [30] and reference therein). More precisely, we study to what extent one can extend the known arithmetic results for (skew-)hermitian forms and also (cohomological) Hasse principles for related algebraic groups over ordinary global fields to their infinite algebraic extensions. Note that the arithmetic results obtained can be varying very differently depending on what kind of fields we are dealing with. For example, if k is just an algebraic closure of a global field, then the arithmetic results are almost trivial in this case. Our main results develop several aspects of arithmetic theory of (skew-)hermitian forms over infinite algebraic extensions of local and global fields in general and establish some local–global principles for (skew-)hermitian forms over such fields in particular. We show the validity of the cohomological Hasse principle for H1 for all semisimple simply connected algebraic groups defined over any infinite algebraic extension of a global field, both in the classical and the new setting and as applications, we derive the Hasse Norm Theorem and Albert– Brauer–Hasse–Noether Theorem for central simple algebras over any infinite global field. The plan of the paper is as follows. 1. 2. 3. 4. 5. 6. 7.

Introduction. Local–global principle for quadratic forms. The Quadratic Hasse Norm Theorem and Albert–Brauer–Hasse–Noether Theorem. Local theory of (skew-)hermitian forms. Cohomological Hasse principle for simply connected groups. Applications. Strong Hasse principle and global classification. Injectivity of restriction maps and counter-examples to the Hasse principle for connected groups.

Conventions and notation. A field, which is an infinite algebraic extension of a local (resp. global) field is called briefly infinite local field (resp. infinite global field). If k is a field, Br(k) denotes the Brauer group of k, n Br(k) denotes the n-th torsion of Br(k), Vk denotes the set of all places of k. If D is a division algebra, Mn (D) denotes the full matrix ring of n ×n-matrices with entries in D. If a, b ∈ k, then we denote by (a, b/k) the quaternion algebra over k with a k-basis {1, i, j, ij}, satisfying i2 = a, j 2 = b, ij = −ji. For a ring A, A∗ denotes the set of invertible elements of A and for a set S, |S| denotes the cardinality of S. If G is an affine algebraic group defined over a field k, then Hiflat(k, G) denotes the i-th flat cohomology of G (where i ≤ 1, if G is non-commutative). If G is smooth, then it is well known that this cohomology is isomorphic to Hi (k, G), the i-th Galois cohomology of G. For a natural number n, μn denotes the algebraic group of n-th roots of unity. If

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K/k is a finite extension and G is a K-group, then we denote by RK/k (G) the Weil’s (1) restriction of scalars of G, and in the case G = Gm , denote by RK/k (Gm ) the kernel of the norm map N : RK/k (G) → Gm . If S ⊂ Vk is a subset of places of a field k, a k-group G has weak approximation  property with respect to S, if, via the diagonal embedding, G(k) is dense in v∈S G(kv ) in the product topology, where G(kv ) is endowed with the v-adic topology. If S = Vk , we say G has weak approximation over k. A k-group G satisfies the cohomological Hasse principle in dimension i in classical  setting if the natural map Hiflat (k, G) → v∈Vk Hiflat (kv , G) has trivial kernel, whenever it makes sense. We use standard terminology and notation regarding (skew-)hermitian forms as in [35]. All quadratic forms and (skew-)hermitian forms are supposed to be non-degenerate. If k is a field then we denote by W (k) the Witt group of k and if D is a division k-algebra with an involution J, we denote by W (D, J) the Witt group of the pair (D, J). For a quadratic form q = a1 , ..., an  in its diagonal form over k, the Brauer class s(q) of  (ai , aj /k) in Br(k) is the Hasse invariant of q. For a J-hermitian form h defined over the k-vector space Dn , we denote by U(h), (resp. SU(h)) the unitary (resp. special unitary) algebraic k-group corresponding to h. Especially, if q is a quadratic form over k, O(q) (resp. SO(q)) denotes the orthogonal (resp. special orthogonal) group of q. We say that h is of Dynkin type A if J is an involution of the second kind and h is of Dynkin type C or D, if J is of the first kind and the corresponding algebraic k-group SU(h) has Dynkin type C or D in the sense of Tits classification [43], respectively. 2. Local–global principle for quadratic forms 2.1. Localization fields Let k be an infinite global field, which is an infinite algebraic extension of a global field F . Let v be a place of k and let kv be the completion of k at v. Many results in the arithmetic theory of global fields do not hold when one passes to the infinite algebraic extensions. One of the main obstacles is that a place w on F can have infinitely many extensions to k and some criteria for the finiteness have been found (see [37, p. 109]). In general, Schilling (see [37, Chapter IV]) formulated four axioms for an infinite global field that ensure the classical divisorial theory. The restrictions of v to finite extensions L/F contained in k give rise to the completions Lv of such extensions in kv . The collection of all these Lv is denoted by C. We say after [26, Chapter 1], [21], that a field k is a localization of k at v, if k is the direct limit of all extensions from C and we denote it by k(v). We call such fields as localization fields. Such fields were first considered by Moriya (see [23], [24]) and then by Moriya and Schilling, Moriya and Nakayama (see [37, Chapter VI, Sec. 11] and reference there). Alternatively, we may describe k(v) by the following simple lemma.

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Lemma 2.1. Let F be a global field, k = ∪n Ln the union of an increasing sequence of finite extensions over F contained in k F = L0 ⊂ L1 ⊂ · · · ⊂ Ln ⊂ · · · ⊂ k, and let v be a place of k. Fix an embedding k → kv and let vn = v|Ln be the restriction of v to Ln . Then k(v) is a henselian field and it is the union of the increasing sequence of finite subextensions over Fv Fv = L0,v ⊂ L1,v1 ⊂ · · · ⊂ Ln,vn ⊂ · · · ⊂ kv . In particular, if v is a real (resp. complex) place, then k(v) = kv R (resp. k(v) = kv C). Proof. Let F = K0 ⊂ K1 ⊂ · · · ⊂ Kn ⊂ k be another sequence of finite extensions and let wi be the restriction of v to Ki . Then for each i (resp. n), there is an index m(i) (resp. j(n)) such that Ki ⊂ Lm(i) (resp. Ln ⊂ Kj(n) ). Then we have Ki,wi ⊂ Lm(i),vm(i) (resp. Ln,vn ⊂ Kj(n),wj(n) ) and eventually, ∪n Ln,vn = ∪i Ki,wi . Let On (v) be the ring of v-integers of Ln,vn with the maximal ideal mn (v), O(v) the ring of v-integers of k(v) with the maximal ideal m(v), κn (v) the residue field of Ln,vn and let κ(v) be the residue field of k(v). Then it is clear that we have increasing sequences O0 (v) ⊂ · · · ⊂ On (v) ⊂ · · · ⊂ O(v), m0 (v) ⊂ · · · ⊂ mn (v) ⊂ · · · ⊂ O(v), and O(v) = ∪n On (v), m(v) = ∪n mn (v). Since mm (v) ∩ On (v) = mn (v) for n ≤ m, we also have natural embeddings of residue fields κ0 (v) ⊂ · · · ⊂ κn (v) ⊂ · · · ⊂ κ(v) and we may set κ(v) = ∪n κn (v). Let f, g, h be monic polynomials in one variable T with ¯ g¯, h ¯ modulo m(v) coefficients from O(v) and degree ≥ 1 such that for their reductions f, ¯ ¯ we have f = g¯h. We may choose n sufficiently large such that all coefficients of f, g and h belong to On (v). Since each Ln,vn is henselian, from above sequences we may lift the ¯ (modulo mn (v)) to actual factorization f = gh in On (v)[T ], thus factorization f¯ = g¯h also to a factorization in O(v)[T ]. Therefore k(v) is henselian. The assertion regarding the real place is clear. If v is complex, i.e., kv C then for some n, we have Ln,vn C, thus k(v) C. 2 Definition 2.2. Let S ⊂ Vk be a subset of places of an infinite global field k. (a) We say that a k-group G has the weak approximation property (in new setting)  with respect to S, if, via the diagonal embedding, G(k) is dense in v∈S G(k(v)) in the product topology, where G(k(v)) is endowed with the v-adic topology. We say that G

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has the weak approximation with respect to localizations, if the above is true for any set S ⊂ Vk . (b) We say that the cohomological Hasse principle (in new setting) holds in degree i for a k-group G if the natural map Hiflat (k, G) →



Hiflat (k(v), G)

v∈Vk

has trivial kernel, whenever Hi makes sense. Remark 2.3. In general, it is well known that an infinite algebraic extension of a local field is not complete, thus the localization field k(v) introduced above in general does not coincide with kv . 2.2. Local classification of quadratic forms over infinite algebraic extensions of local or finite fields Let k be an infinite algebraic extension of a finite (resp. local) field F of characteristic = 2. An important result due to Springer allows one to relate the classification of quadratic forms on a henselian field with respect to a discrete valuation to similar problem on the residue field (see [35, Chapter 6, 2.6]). (The original statement of Springer’s Theorem was stated for complete fields, but it was noticed that the result is still valid for henselian fields.) As a consequence, we derive the following statement, the proof of which is essentially the same as in the finite (resp. local) field case. The following theorem follows directly from Springer’s result mentioned above and from [35, Chapter II, Theorem 14.5]. We keep the notation as above. Theorem 2.4. (1) If F is algebraic over a finite field of characteristic = 2 then any quadratic form q in three or more variables is isotropic over F. Thus any two quadratic forms over F are equivalent over F if they have the same dimension and determinant. (2) If k is either a localization field or the completion of such field with respect to a nonarchimedean place of an infinite global field of characteristic = 2 then any quadratic form q in five or more variables is isotropic over k. (3) Any two quadratic forms over k as in (2) with the same dimension, determinant and Hasse invariant are equivalent over k. Proof. (1) Trivial. (2) First assume that k is a localization field and q = a1 , ..., an  is a quadratic form in five or more variables in its diagonal form over k. We may assume that k = ∪n km , where L = k0 ⊂ · · · ⊂ km ⊂ · · · is an increasing sequence of local fields contained in k.

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∗ For m sufficiently large, ai ∈ km for all i = 1, ..., n. Then as is well known, q is isotropic over km , thus also over k. Now let k = Kv be the completion of an infinite global field K (thus also the completion of the localization K(v)). Since K(v) is v-adically dense in k, we may choose bi ∈ K(v)∗ , i = 1, ..., n such that bi is close to ai in v-adic topology for all i. Since k has characteristic = 2, so if the approximation is good enough, then we have bi = c2i ai , where ci ∈ k∗ , for all i. Thus q  := b1 , ..., bn  q over k. Since q  is isotropic over K(v), it is so over k, thus so is q over k. (3) Follows from [35, Chapter II, Theorem 14.5]. 2

Notice that the Springer Theorem mentioned above can be extended to a more general case of (skew-)hermitian forms over division algebras (see [34] or Section 4 below). We define the Hilbert symbol in k as usual (see [39, Chapter III]). We set, for a, b ∈ k∗ , (a, b) = 1 if and only if the equation z 2 − ax2 − by 2 = 0 has a non-trivial solution in k3 , and (a, b) = −1 otherwise. We mention that (see [21]) if k is a localization of an infinite global field as above, K be as above, then [k∗ : NK/k (K ∗ )] ≤ 2. Lemma 2.5. Let D be a non-trivial quaternion division algebra over a non-archimedean localization field k (resp. the completion kv of k). Then D is unique up to an isomorphism. Proof. Case 1. D is a k-algebra. We write D = (a, b/k). By definition, k = ∪n kn , where {kn } is an increasing sequence of local fields contained in k. It is clear that D is defined over a local field, say L = kn , i.e., a, b ∈ L∗ . Then this pair (a, b) defines D uniquely up to L-isomorphisms. Assume that D is another non-trivial quaternion division algebra over k, D = (a , b /k), a , b ∈ k, and one may assume that D is defined over km , i.e., a , b ∈ km . If r = max{n, m}, then D, D are still non-trivial quaternion division algebra over kr , so D D over kr , due to the uniqueness (up to isomorphism) of quaternion algebras over local fields. A fortiori, D D over k. Case 2. D is a kv -algebra. Assume there exists another quaternion division kv -algebra D . By the general local class field theory (see [37, Chapter 6, Section 11, Lemma 49]), there exists a quaternion division k-algebra Q (resp. Q ) such that D = Q ⊗k kv (resp. D = Q ⊗k kv ). From Case 1, we know Q Q , hence D D over kv . 2 2.3. Classification of quadratic forms over infinite global fields of characteristic = 2. The strong Hasse principle Let k be an infinite global field and let L be the set of all localizations of k. For certain property P , which can be defined over k and over all of its localizations ∈ L, we say that the Hasse principle holds for P , if P holds over k, whenever P holds over all of the localizations. (Then we say also that P holds locally everywhere.) Further, whenever it says that the Hasse principle holds, it is always understood in the new setting (i.e. in the sense “global versus localizations”). Here we are interested in the Hasse principle for

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a quadratic form q over k. We have the following assertion, where (1)–(3) are due to Koziol and Kula (see [21]) and the last statement (4) follows from (3). Theorem 2.6 (Koziol–Kula). Let k be an infinite global field of characteristic = 2 and let q a quadratic form over k of dimension n. (1) If n ≥ 3, then the Hasse principle holds for q, i.e., if q represents 0 locally everywhere, then it does so over k. (2) If n ≥ 2 and a ∈ k, then q represents a over k if and only if it does so locally everywhere. (3) There exist infinite global fields k of characteristic 0 and elements d ∈ k∗ such that locally everywhere, d is a square in k(v)∗ , but it is not so globally. (4) For any natural number n, there are infinite global fields k of characteristic 0 and quadratic forms of dimension n, which are equivalent over k(v) (thus also kv ) for all places v of k, but not equivalent over k. 2 Remark 2.7. By Theorem 2.6, the Weak Hasse principle (in the new setting) for quadratic forms may break down. In particular, the cohomological Hasse principle (in new setting) may not hold for H1flat (k, μ2 ) or H1flat (k, O(q)) for some forms q. Since the groups μ2 and O(q) satisfy the cohomological Hasse principle in degree 1 (in classical sense) over global fields, Theorem 2.6 implies that there are quadratic forms (resp. finite group schemes) which satisfy the (Weak or Strong) classical Hasse (resp. the cohomological Hasse) principle over any global fields, but the classical (Weak or Strong) Hasse (resp. cohomological Hasse) principle may fail for them over infinite global fields. 3. The quadratic Hasse norm theorem and Albert–Brauer–Hasse–Noether theorem Let k be an infinite global field with the set Vk of all places of k. In this section we discuss two of the most important local–global principles in the classical arithmetic of the global fields, namely the Quadratic Hasse Norm Theorem and the Albert–Brauer– Hasse–Noether Theorem. 3.1. The quadratic Hasse norm theorem We have the following extension of the (quadratic) Hasse Norm Theorem to the case of infinite global fields. A full analog of the Hasse Norm Theorem for cyclic extensions of infinite global fields will be given later (see Theorem 5.13). Proposition 3.1. Let k be an infinite global field of characteristic = 2. √ (1) Let K = k( a) be a quadratic extension of k. Then an element b ∈ k is a norm from K if and only if it is so locally everywhere.

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(2) Let D = (a, b/k) be a quaternion algebra over k. Then D is trivial if and only if it is trivial locally everywhere. Proof. (1) It follows from Theorem 2.6(2), applied to the form f = x2 − ay 2 − bz 2 . (2) The same argument, applied to the norm form x2 − ay 2 − bz 2 + abt2 of D. 2 3.2. Infinite trees and König’s lemma We recall the following notion of infinite trees and other related notions (see [20]). A graph (Γ, V (Γ), ≤) with a partial order ≤ on the set V (Γ) of its vertices is called a locally finite (rooted) tree, if there exists a unique minimal vertex, v0 , called the root of the tree, each vertex has finite degree and for each vertex v ∈ V (Γ), the set A(v) := {s ∈ V (Γ) | s ≤ v} is a finite directed (or linearly ordered) graph. A path is a connected linearly ordered subset, and a branch is a maximal path. Then the following holds (see [19, Section 3, Theorem E, p.120] or [20, Chapter XIII]). Theorem 3.2 (König). If Γ is an infinite locally finite rooted tree, then there exists an infinite branch. Remark 3.3. This theorem (usually called as König’s Lemma) has many applications and one can apply it to prove Theorem 2.6 above. Further, we will make an extensive use of this theorem to prove our local–global principles. 3.3. Central simple algebras. The Albert–Brauer–Hasse–Noether theorem One of the celebrated results of global class field theory is the Albert–Brauer–Hasse– Noether Theorem. One way of stating it is as follows. Theorem 3.4 (Albert–Brauer–Hasse–Noether). Let k be a global field. (1) If A a central simple algebra of dimension n2 over its center k, then for almost all places v ∈ Vk , Av := A ⊗k kv is trivial, i.e., Av Mn (kv ).  (2) There is an exact sequence 0 → Br(k) → v Br(kv ) → Q/Z → 0. Here we prove a partial extension of this theorem to the case of infinite global fields. Though a general case will be proved in Section 5, after some general results are given, we give a proof here, since it serves as a model for other proofs where König’s Lemma is used. Theorem 3.5 (Hasse principle for Brauer groups). Let k be an infinite global field. Then  the canonical homomorphism Br(k) → v Br(k(v)) is injective.

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Proof. Assume that A is a central simple algebra over k of degree n, which is locally trivial (i.e., trivial over k(v) for all v ∈ Vk , that is Av Mn (k(v))). Since A is of finite dimension over k, A is in fact defined over some global field L, contained in k, i.e., A = B ⊗L k, where B is a central simple algebra over L. Further we apply an argument involving König’s Lemma as in the proof of Theorem 4 of [21]. We need to show that A itself is trivial, i.e., A Mn (k). Assume the contrary, i.e., A is not trivial over k. Then by the Artin–Wedderburn Theorem, A Mm (D), where D is a non-trivial division algebra of center k. We are going to show that there exists v ∈ Vk such that A is non-trivial over k(v). Take any infinite tower of extensions L = L0 ⊂ L1 ⊂ · · · ⊂ Ls ⊂ · · · ⊂ k, where each Ls /L is a finite extension, such that k = ∪s Ls . Since A is not trivial over k, it remains so over all Ls . Therefore, by the usual Albert–Brauer–Hasse–Noether Theorem above, A remains non-trivial over at least one completion of each Ls . Also, by the same theorem, A is non-trivial over only a finite number ns > 0 of completions of the given Ls . After a renumbering, we may assume that these are Ls,v(s)1 , ..., Ls,v(s)n(s) . Consider the following graph Γ, where the set of vertices is V (Γ) := {the completions Ls,v(s)n of Ls , | s ≥ 0, 1 ≤ n ≤ n(s)}, and the set of arrows (oriented edges) is A(Γ) := {the arrows Ls,v(s)n → Lt,w(t)m | s ≤ t, w(t)m is an extension of v(s)n to Lt }. We say that Ls < Lt if s < t, Ls,v(s)n → Lt,w(t)m , and w(t)m is an extension of v(s)n . It is clear that (Γ, <) satisfies all the requirements of König’s Lemma, so by the lemma, there exists an infinite branch. Let this branch B of the tree be consisting of vertices Ls,v(s)is , is ∈ [1, ns ], ∀ s ≥ 0. Let Lv be the root of the branch of the tree, where v is a place of L. Denote by v also the extension of v to k, which restricting to each Ls gives v(s)is . It is clear that the union of all these completions Ls,v(s)is of the branch is equal to k(v) (see Lemma 2.1), since for all s, each restriction to L of v(s)is should be the same. By our assumption, A is trivial over k(v) and we have k(v) = ∪s Ls,v(s)is , so it implies that A also is trivial over a finite extension K of Lv , contained in k(v). It follows that A is trivial over (clearly infinitely many) those Lt,v(t)n which contains K, which is contradicting to the choice of Ls,v(s)is , that A is non-trivial over such fields. Therefore A is non-trivial over k(v), which again contradicts the initial assumption on A. 2 As an immediate consequence, we have the following Corollary 3.6. Let k be an infinite global field. If G = μn or G is a k-torus defined (1) either by RK/k (Gm ), or RK/k (Gm ), where K/k is a finite separable extension, then the cohomological Hasse principle in dimension 2 holds for G.

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Proof. The first two cases follow directly form Theorem 3.5. The last one follows from Shapiro lemma. Let 1 → G → RK/k (Gm ) → Gm → 1 be the exact sequence defining G. From this we deduce the following commutative diagram with rows being exact sequences of Galois cohomology 0

→

→

H2flat (k, G)

H2flat (k, RK/k (Gm ))

↓α 0

→

 v∈Vk

↓β

H2flat (k(v), G)

→

 v∈Vk

H2flat (k(v), RK/k (Gm ))

For each place v of K, we consider the restriction of v to k (which is denoted also by v). Then we have a natural embedding k(v) ⊂ K(v), thus K ⊗k(v) = K(v). By Theorem 3.5,  we have an injection γK : H2flat (K, Gm ) → v∈VK H2flat (K(v), Gm ). By Shapiro Lemma we have H2flat (k, RK/k (Gm )) = H2flat (K, Gm ) and for each v as above, it is easy to see that we have H2 (k(v), RK/k (Gm )) = H2flat (K(v), Gm ). By grouping those different places v = w which have the same restriction to k, the map γK can be written as   γK : H2flat (K, Gm ) → u∈Vk v∈VK ,v|k =u H2 (K(v), Gm ) =

 u∈Vk

 v∈VK ,v|k =u

H2flat (k(v), RK/k (Gm ))

Therefore, it follows from the injectivity of γK that β is also injective, and we conclude that so is α. 2 It is interesting to see if similar results for central simple algebras over local and global fields are true over infinite local or global fields. As Proposition 3.7 below shows, the outcome is that some of them may not be so and it is of interest to have a systematic treatment of these questions. We have the following (see [1] for infinite number fields) Proposition 3.7 (Albert). (1) Let k be an infinite local or global function field and let A be a central simple algebra with the center k. If A is a division algebra, then A is cyclic and the index of A is equal to its exponent. (2) There exists a central simple algebra over an infinite number field which is not cyclic. Proof. The proof of [1] works verbatim also over local fields or global function fields. 2 Now we consider central simple algebras with involutions over infinite local or global fields. From Proposition 3.7 we deduce the following.

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Proposition 3.8. Let k be an infinite local (or global) field and let A be a central simple k-algebra with a non-trivial involution J. (1) If J is of the first kind then A Mn (D), where D is a division k-algebra of degree ≤ 2. (2) If k is an infinite non-archimedean local field and J is of the second kind, then A is trivial. Proof. (1) It is well known (see for example [35, Theorem 8.4, p. 306]) that if A has involutions of the first kind, then its order is ≤ 2 in Br(k). Let A = Mn (D), where D is a division algebra over k. Then D has its exponent 2 in Br(k), and by Proposition 3.7, D is either a quaternion algebra over k or is equal to k. (2) If k is an infinite non-archimedean local field, K/k a separable quadratic extension, √ and J is a K/k-involution (i.e., k = K J the fixed points of J in K). Set K = k( α) and let L be a local field contained in k such that (D, J) is defined over L. Hence A is also √ defined over L, J is a L( α)/L-involution and by a classical result (see [35, Theorem 2.2, p. 353]), A is trivial over L, hence also over k. 2 Remark 3.9. (1) A true extension of the classical Albert–Brauer–Hasse–Noether Theorem will be given in Section 5, as an application of a general theorem on cohomogical Hasse principle for semisimple groups. As another application, one may derive Corollary 3.6 from Theorem 5.7 below. (2) In Theorem 3.4, any central simple algebra over a global field k is trivial over kv almost everywhere, i.e., except for a finite set of places. If k is an infinite global function field and if for some global subfield L ⊂ k the extension k/L is purely inseparable, then it is known (see [37, Chapter II, Corollary, p. 57]) that each place v ∈ VL has a unique extension to k, thus each central simple k-algebra is trivial outside a finite set of places. We show that the phenomenon of being trivial “almost everywhere” does not hold in the infinite global field case. Proposition 3.10. Not every central simple algebra over an infinite global field k is almost trivial locally everywhere (over k(v)) (i.e., trivial over k(v) outside a finite set of places v). Proof. We start with a number field L, a non-empty finite set SL of places of L with |SL | being an even number ≥ 4, and a quaternion division algebra D over L, such that Dv is trivial if and only if v ∈ SL . This is possible due to [35, 6.8, p. 225]. For each v ∈ SL we choose an arbitrary positive odd number nv and let n be the least common multiple of all nv , v ∈ SL . Then by global class field theory (see [3, Theorem 5, p. 105]), there exists a cyclic extension K/L of degree n such that for all v ∈ SL and any extension w to K of v, the degree [Kw : Lv ] = nv . We denote by SK the set of all places of K where

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DK is not trivial. We have a natural map of restriction rK/L : VK → VL and it is clear that on the one hand we have rK/L (SK ) ⊂ SL . On the other hand, since nv are odd for all v ∈ SL , it implies that for any extension w of v as above, D ⊗ Kw is not trivial, so w ∈ SK and we have rK/L (w) = v. Thus rK/L (SK ) = SL . Also, we may choose nv s so that |SK | > |SL |. Keep iterating this process, we finally arrive at the following infinite tower of extensions of L L = L0 ⊂ L1 ⊂ · · · ⊂ Ln ⊂ · · · , such that |SLn | > |SLm |, if n > m. We set k = ∪n Ln . By the very construction, we see that for any v ∈ SL with an extension v˜ to k, D is not trivial over k(˜ v ); in fact, if this were not so, D is trivial over k(˜ v ), then it is also trivial over some Ln,w , for some n, where w is an extension of v ∈ SL to Ln . This is a contradiction, since [Ln,w : Lv ] is odd. Also we find that rk/Ln (Sk ) = SLn for all n. It implies that Sk is infinite. 2 4. Local theory of (skew-)hermitian forms We consider (skew-)hermitian forms over division algebras over localization fields. The archimedean case is classical and well known, so we consider only non-archimedean localization fields. 4.1. General results We need the following important result due to Scharlau (see [34, Satz 3.6]), which extends the Springer Theorem (see [35, Chapter 6, Corollary 2.6]) to the case of division algebras. Let K be a field which is henselian with respect to a discrete valuation v, R the ring of v-integers of K with an uniformizing element π, maximal ideal p = (π) and residue field κ := R/p of characteristic = 2. Let V be a finitely dimensional right vector space over a division algebra D of center K with a hermitian form h on V with respect to an involution J of D. Let D± := {x ∈ D | xJ = ±x}. We denote by the same symbol v the unique extension of the valuation v to D, and denote by RD (reps. pD ) the ring of integers (reps. ¯ := RD /pD be the corresponding residue algebra. Consider maximal ideal) of D. Let D (skew-)J-hermitian forms with values in D, where J is an involution of D. Let K0 be the set of J-fixed elements of K. For each J-symmetric (reps. J-skew-symmetric) element d ∈ D∗ , we set Jd : x → dxJ d−1 . Then Jd is an involution again. If J is of the first (reps. second) kind, then K = K0 (reps. K/K0 is a separable quadratic extension) and ¯ We consider the following two exceptional cases: J induces an involution J¯ on D. (1) K/K0 is a ramified quadratic extension, J is of the second kind and D = K; (2) D = (a, π/K) is a quaternion algebra over K = K0 , J is of the first kind and dim(D+ ) = 1.

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Then in all other cases one may choose a symmetric uniformizing element t ∈ D such that v(t) > 0 and is minimal. For such t we consider the involution Jt as above. Then we have the following extension of the Springer Theorem due to Scharlau (see [34, Proposition 3.3, Bemerkung]). Proposition 4.1 (Scharlau). Let notation be as above and let the residue field κ have characteristic = 2. Let a1 , ..., an be J-(skew-)symmetric elements of RD \ pD . Then 

xJi ai yi is isotropic on D if and only ¯ ¯ x, y¯) :=  if the reduction (skew-)hermitian form h(¯ ¯Ji a ¯i y¯i is isotropic over 1≤i≤n x ¯ J). ¯ (D, (2) Except for two exceptional cases above, there exists an isomorphism of Witt groups of hermitian forms

(1) The (skew-)hermitian form h(x, y) :=

1≤i≤n

¯ J) ¯ W (D, J) W (D,



¯ J¯t ), W (D,

¯1, h ¯ 2 ), where h ¯ i is the i-th-residue form of h and t is by means of the map h → (h chosen as above. (3) In the exceptional cases, we have an isomorphism ¯ J) ¯ W (D, J) W (D, ¯ 1. by means h → h From this we derive immediately the following ¯ of Corollary 4.2. With notation and assumption as in Proposition 4.1, if every form h dimension n +1 is isotropic, then so is every form h of dimension 2n +1 and if there is an ¯ of dimension n, so is there an anisotropic form of dimension 2n. 2 anisotropic form h Remark 4.3. The original statement of [34, 3.3, 3.6], was stated for complete fields, but it was remarked there (see [34, Bemerkung, p. 206]) that the result is still valid for henselian fields. 4.2. Local theory for forms of type A Let k be a non-archimedean localization field of an infinite global field of characteristic = 2. From Proposition 3.8 we derive the following result similar to that of [18, Theorem, p. 62], in the case of infinite local fields, which will be needed in the sequel. Theorem 4.4. Let k be a non-archimedean localization field of characteristic = 2, D a division k-algebra with the center K and an involution J of the second kind, non-trivial

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on K, and let h be a J-hermitian form of dimension n with values in D. Then D = K, thus h is Morita equivalent to a quadratic form of dimension 2n over k = K J . Corollary 4.5. With notation as in Theorem 4.4, every hermitian form h of dimension ≥ 3 is isotropic over k and hermitian forms are classified by their dimension and determinant. Proof. For, we know that h, D are in fact defined over a non-archimedean local field L contained in k. Then the hermitian form h corresponds to a quadratic form qh (called the trace form of h) (see [35, p. 348]) of dimension 2 dim(h) ≥ 6, and h is isotropic over L if and only if so is qh . Hence by h represents zero over the non-archimedean local field L and a fortiori also over k. The last assertion follows from above and from [35, Chapter II, Theorem 14.5]. 2 4.3. Local theory for forms of type C Let k be a non-archimedean localization of an infinite global field of characteristic = 2. Denote by D a division algebra with the center k, J an involution of the first kind of D. Let h be a J-hermitian form with values in D of type C. By Proposition 3.8, D is either equal to k or is a quaternion division k-algebra. The first case corresponds to the symplectic form h, which is well known, so we assume that D is a quaternion division k-algebra and J is the canonical involution of D. Then again, h, D are defined over a non-archimedean local field L contained in k. We have the following Proposition 4.6. Let k be a non-archimedean localization of an infinite global field of characteristic = 2 and let h be a J-hermitian form of type C with values in a quaternion division k-algebra D, where J is the canonical involution of D. If dim(h) ≥ 2, then h is isotropic. Hermitian J-forms are classified by their dimension. Proof. Let h = axxJ + byy J , a, b ∈ k(v)∗ , x, y ∈ D and let qh be the corresponding trace form (see [35, p. 352]). By Lemma 2.5 we know that dim(qh ) = 4 dim(h) = 8 > 5, thus qh is isotropic over L, where L is a local field as above. Therefore, so is h over L, thus also over k. The last statement follows from above and from [35, Chapter X, Example 1.8(ii)]. 2 4.4. Local theory for forms of type D Let k be a non-archimedean localization field of an infinite global field of characteristic = 2. Denote by D a division algebra with the center k, h a J-skew-hermitian form with values in D, where J is an involution of the first kind of D. Assume that h is of type D, thus J is of orthogonal type, which means that the space D+ of J-symmetric elements of D has dimension d(d − 1)/2, where d = deg(D). Since D has an involution of the first kind, and it has order 2 in the Brauer group Br(k), so the index of D is equal

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to its exponent (see Proposition 3.8), thus D is a quaternion algebra over k and J is the canonical involution of D. Then we may apply Theorem 4.1, Corollary 4.2 to our case and we have the following analogs of Tsukamoto’s result (see [35, p.363], for p-adic fields). Theorem 4.7. Let k be a non-archimedean localization field with residue field κ of characteristic = 2, D = (a, b/k) a quaternion division algebra with the canonical involution J and let h be a J-skew-hermitian form with values in D. (1) If dim(h) ≥ 4, then h is isotropic. (2) If dim(h) ≥ 3, and a ∈ D is a J-symmetric (resp. J-skew-symmetric) element of D∗ , then h represents a over D. Proof. Of course (1) implies (2), so we need only to prove (1). First proof. One may use the approach of Tsukamoto, which has been simplified by Scharlau (see [35, Chapter X, Theorem 3.6]), by combining with results in previous sections. Namely, the isotropicity of quadratic forms of dimension ≥ 5 (Theorem 2.4) (or the same, the universality of forms of dimension ≥ 4), and the uniqueness of a non-trivial quaternion division (Lemma 2.5) are all that is needed. The rest is the same as in [35]. (Here we do not need the assumption on characteristic of the residue field.) Second proof. Let k = ∪n kn is the union of the tower k0 ⊂ k1 ⊂ · · · ⊂ k, where each kn is a local field with residue field κn . Then we have κ = ∪n κn (see the proof of Lemma 2.1). We may assume that a, b ∈ k0 , so D is defined over k0 . Let pD be the maximal ideal of RD as in subsection 4.1. Since κ is an algebraic extension of a finite field, it is known that κ is of type C1 (see [35, Chapter 2, Theorem 15.2, Theorem 15.4]), so Br(κ) = 0. ¯ obtained by reduction modulo pD mentioned Therefore the central simple algebra D ¯ M2 (κ). By [18, p. 141], or [35, p. 361], there to h correspond above is trivial, D uniquely a symmetric bilinear form bh and quadratic form qh with dim(qh ) = 2 dim(h). The form h is isotropic if and only if the Witt index of qh is ≥ 2. Thus if dim(qh ) ≥ 8 (i.e., dim(h) ≥ 4) then the Witt index of qh is ≥ 2, hence by Corollary 4.2, h is isotropic. 2 The following extends Tsukamoto’s main result to infinite local field case (see Tsukamoto [45], [35, p. 363], for p-adic fields). Theorem 4.8. Let k be a non-archimedean localization field of characteristic = 2, D = (a, b/k) a quaternion division algebra with the canonical involution J and let h be a J-skew-hermitian form with values in D. (1) For any d ∈ k∗ , d ≡ −1(mod.k∗2 ), there is a J-skew-hermitian form with determinant d. For any n > 1 and d ∈ k∗ , there is a J-skew-hermitian form h of dimension n with determinant d.

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(2) If n = 1, then the equivalent class of a J-skew-hermitian form h is defined by the determinant det(h). If n = 2, then h is isotropic if and only if det(h) ≡ 1( mod .k∗2 ). If n = 3, then h is anisotropic if and only if det(h) ≡ 1( mod.k∗2 ). (3) J-skew-hermitian forms are equivalent if and only if they have the same dimension and determinant. Proof. The main idea is already in [35, p. 363], but for the convenience of the readers we give it in more details. (1) Let D = (a, b/k). If d ∈ k∗ is given, then there is a local subfield L ⊂ k, such that D, h are defined over L and d ∈ L∗ . Since d ∈ / (−1) × k∗2 , we also have d ∈ / (−1) × L∗2 . Then it is known (see [35, pp. 363–364]), that the equation Nrd(X) = d has solution X ∈ D− . (Setting X = xi +yj+z(ij), x, y, z ∈ k, we show that the equation Nrd(X) = −ax2 − by 2 + abz 2 = d has a solution, or equivalently, the quadratic form −ax2 − by 2 + abz 2 − dt2 is isotropic. Indeed, if the quadratic form −ax2 − by 2 + abz 2 − dt2 were anisotropic, then it is known that it would be equivalent to the anisotropic quadratic norm form of D, since D is the only (up to isomorphism) quaternion division algebra over L. Therefore we have an equivalence of quadratic forms  − d, −a, −b, ab 1, −a, −b, ab and by Witt cancellation (see [35, Corollary 9.2(i), p. 268]), one has  − d 1, a contradiction. One notes that in [35, Theorem 3.6(ii), p. 363], either the condition = 1 should be changed to = −1, or the same, the word “determinant” should be changed to “discriminant”, where the discriminant is defined (as in [45]) disc(h) := (−1)dim(h) det(h).) It is also clear that if n > 1, then for any d ∈ k∗ , there exists a form with determinant d. (2) Consider the case n = 1. Assume that det(α) ≡ det(β)(mod.k∗2 ), where α, β ∈ D− \ {0} Then Nrd(α) = Nrd(β)γ, γ ∈ k∗ . By Theorem 2.4, there is δ ∈ D∗ such that Nrd(δ) = γ, so Nrd(α) = Nrd(β)γ 2 = Nrd(δ J ) Nrd(β) Nrd(δ) = Nrd(δ J βδ). Thus, by replacing β by δ J βδ, we may assume from the very beginning that Nrd(α) = Nrd(β), so α2 = β 2 (∈ k∗ ). It implies that k(α) k(β). By the Skolem–Noether Theorem, one may find x ∈ D∗ such that xαx−1 = β, so xαxJ = βxxJ = Nrd(x)β. By [35, p. 361], one has to show that Nrd(x) is represented by either 1, −a or b, −ab. If not, the following holds true: 1, −a x, −ax and b, −ab x, −ax. Consequently, these relations show that (a) the quaternion algebra (a, d/k) is non-trivial; (b) the quaternion algebras (b, −ab/k) (d, a/k). The uniqueness of quaternion division algebra over k (see Lemma 2.5) leads this to a contradiction. Therefore α β. Consider the case n = 2. Let h = α, β. If h is isotropic, say xJ αx + y J βy = 0, with x = 0, then for z = yx−1 , we have α = −z J βz, so det(h) = Nrd(α) Nrd(β) =

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1(mod.k∗2 ). Conversely, if det(h) ≡ 1(mod.k∗2 ), then Nrd(α) ≡ Nrd(β)(mod.k∗2 ), so the forms α and β have the same determinant. From above we know α β, also β  − β (they have the same dimension and determinant), so α, β β, −β and the last one is isotropic. If n = 3, a form h of determinant 1 must be anisotropic, since other wise, it has a diagonal form h α, −α, β, which shows that Nrd(β) = 1, which is impossible. (3) The rest is the same as in [35, p. 363] (same reference as above). 2 5. Cohomological Hasse principle for simply connected groups. Applications In this section, we prove the cohomological Hasse principle for H1 for semisimple simply connected algebraic groups defined over infinite global fields. As applications, we derive certain cohomological Hasse principles for non-simply connected algebraic groups over such fields, related to the groups of automorphisms of quadratic and hermitian forms, namely the (special) unitary and (special) orthogonal groups. Also, we derive some true extensions of classical results due to Albert–Brauer–Hasse–Noether and Hasse–Maass–Schilling. 5.1. Infinite local (localization) field case First we recall the following well known result due to Kneser, Harder and Chernousov (see [31] and [13]). Theorem 5.1 (Kneser–Harder–Chernousov). Let k be a global field and let G be a semisimple simply connected algebraic group defined over k. Then the Hasse principle holds for Galois cohomology in degree 1 of G. The following theorem, due to Kneser (see [16–18]) in the case of p-adic fields and to Bruhat–Tits (see [5]) in the general case, is indispensable for the proof of Theorem 5.1. Theorem 5.2 (Kneser, Bruhat–Tits). Let k be a local non-archimedean field and G a semisimple simply connected algebraic group defined over k. Then H1flat (k, G) = 1. We have the following analog of Theorem 5.2 in the case of localization fields and their completions. Theorem 5.3. Let k(v) (resp. kv ) be the non-archimedean localization field (resp. completion) of an infinite global field k and let G be a semisimple simply connected algebraic group defined over k(v). Then H1flat (k(v), G) and H1flat (kv , G) are trivial. Proof. Step 1. H1flat (k(v), G) = 1. By definition we have k = ∪i ki , where each ki is a local field. Let P be a G-torsor defined over k. Since G, thus also P , is of finite type, they are defined over some finite extension K of Qp or Fp ((t)), i.e., a local field contained in

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k. By Theorem 5.2, we have H1flat (K, G) = 1, which means P (K) = ∅, and a fortiori, P (k) = ∅, i.e., P is a trivial G-torsor over k. Since P is arbitrary, we have H1flat (k, G) = 1. Step 2. H1flat (kv , G) = 1. For this we need the following very general result regarding the weak approximation property over henselian fields (see [32, Proposition A4, p. A2]). Lemma 5.4 (Rousseau). Let X be a scheme which is locally algebraic over a henselian field k with a place v. Let kv be the completion of k with respect to v. If kv /k is separable or X is smooth, then X(k) has weak approximation with respect to v, i.e., X(k) is v-adically dense in X(kv ). Now from Remark 4.2.1 of [42] (where, instead of considering a number field, we consider the field k(v) (which is henselian by Lemma 2.1) and its completion kv , G being a smooth group, together with Lemma 5.4, we conclude that the localization map λv : H1flat (k(v), G) → H1flat (kv , G) is surjective. Since H1flat (k(v), G) = 1 by Step 1, it implies that H1flat (kv , G) = 1 as well. 2 Remark 5.5. It was noticed in [4], that many results obtained in the theory of Bruhat–Tits remain valid also for henselian discretely valued field. It is natural to ask if H1flat(k, G) = 1 for any simply connected semisimple k-group G, where k is just henselian and has its residue field of cohomological dimension ≤ 1. (It is desirable if one can modify the proof given by Bruhat–Tits in the case of discretely valued complete fields with residue field of cohomogical dimension not exceeding 1 (see [5]) to this more general case. Notice that the valuation v used in Bruhat–Tits theory is required to be discrete and kv is complete with respect to such one.) 5.2. Infinite global field case Now we are able to extend the classical Hasse principle for semisimple simply connected groups over global fields to infinite global fields. First we need some preliminary results. Theorem 5.6. If G is a connected affine algebraic group defined over an infinite global field k and ∞k is the set of all archimedean places of k, then (1) G has weak approximation with respect to ∞k , i.e., for any finite subset S ⊂ ∞k ,  G(k) is dense in the product v∈S G(kv ). (2) For any finite subset S ⊂ ∞k , the localization map λS : H1flat (k, G) →

 v∈S

is surjective.

H1flat (k(v), G) =

 v∈S

H1flat (kv , G)

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Proof. (1) We may assume that G is defined over a global field L ⊂ k. Let v be an archimedean place of k. If kv is real, or if v is complex then we may assume (since k is algebraic over L and we can take L sufficiently large) that Lv = kv . If k has no real embeddings, then we may assume also that so does L. Thus we may arrange so that Lv = kv hence we have k(v) = kv , thus G(Lv ) = G(k(v)) = G(kv ) for all v ∈ S ⊂ ∞k . It is well known (by a result basically due to Serre, unpublished), that G has weak  approximation over L with respect to ∞L , that is G(L) is dense in v∈∞L G(Lv ). Since  G(L) ⊂ G(k), it implies that G(k) is dense in the product v∈S G(kv ) for any finite subset S ⊂ ∞k and the assertion (1) follows. (2) For the same field L as in (1), we consider the following commutative diagram λ

S H1flat (L, G) →

 v∈S

↓δ

H1flat (Lv , G)

=

→

 v∈S

γ  ↓=

γ ↓=

H1flat (k, G)

λ

S →

 v∈S

H1flat (Lv , G)

H1flat (k(v), G)

=

→

 v∈S

H1flat (kv , G)

It is a well known result due to Harder, that λS is always surjective (see [12, Satz 5.5.1]). It implies that so is λS . 2 Now we have a true analog of classical Hasse cohomological principle for semsimple simply connected algebraic groups over infinite global fields. Theorem 5.7 (The Hasse principle for simply connected groups). Let k be an infinite algebraic extension of a global field L, k = ∪n Ln , [Ln : L] < ∞, and let G be a connected smooth affine algebraic group defined over L. (1) If the (classical) cohomological Hasse principle holds for G over every finite extension L ⊂ Ln ⊂ k, then the same holds for G over k, i.e., the natural map H1flat (k, G) →



H1flat (k(v), G)

v

is an injective map. (2) If G is one of the following groups defined over k: a semisimple simply connected algebraic group, an absolutely almost simple group, an adjoint group, then we have natural injective map λ : H1flat (k, G) →

 v

H1flat (k(v), G)



H1flat (kv , G).

v

In other words, the cohomological Hasse principle for H1 holds for such a group G both in the classical and in the new setting.

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(3) (True analog of classical Hasse principle) Let G be a semisimple simply connected k-group. The set ∞k := {v ∈ ∞k | H1flat (kv , G) = 0} is finite if and only if H1flat (k, G) is finite. If it is so, then there are bijections λ : H1flat (k, G)



H1flat (k(v), G)

v∈∞k



H1flat (kv , G).

v∈∞k

Proof. (1) The proof, like the one we gave for the proof of the Hasse principle for Brauer group, also uses König’s Lemma. We need the following result, where the connectedness of the group plays an essential role (see [29, Chapter 4, Corollaire, p. 48]). Proposition 5.8 (Oesterlé). Let G be a connected smooth affine algebraic group defined over a global field L, VL the set of places of L. Then the image of the natural map H1flat (L, G) →



H1flat (Lv , G)

v∈VL

 1 actually lies in v∈VL Hflat (Lv , G). In particular, each G-torsor is trivial on Lv for almost all places v. Next we consider a G-torsor P defined over an infinite global field k. Assume that P is not trivial over k, i.e., P (k) = ∅. Consider any field tower L = L0 ⊂ L1 · · · ⊂ k, such that k = ∪n Ln . Since P is not trivial over k, so is it over all Ln . Since the Hasse principle holds over any global field Ln , it implies that P is not trivial at certain place vn of Ln . Also, by Proposition 5.8, P is non-trivial only at a finite set Sn of places. Just as in the proof of the Hasse principle for the Brauer group of k, we consider a the infinite graph ΓS := (VS , AS ), where the set of vertices consists of the fields Ln,w , where w runs over Sn and the set of arrows consists of arrows Lm,w → Ln,u , wherever u ∈ Sn , w ∈ Sm , m < n, and u restricts to w and then we say that Lm,w < Ln,u . It is clear that ΓS is an infinite tree, which satisfies König’s Lemma. Thus we may find an infinite branch Lv → Lv1 → · · · → Lvn → · · · → k  , where vi ∈ Si , k := ∪i Li,vi is a localization of k, which is nothing else than k = k(v). By our choice, P is non-trivial over all Li,vi , i = 1, 2, . . . . So P (Li,vi ) = ∅, for all i. Now we claim that P is non-trivial over k , i.e., P (k ) = ∅. Assume that it is not the case, P (k ) = ∅. Since k = ∪i Li,vi , we have P (k ) = ∪i P (Li,vi ) = ∅, so there exists an index i such that P (Li,vi ) = ∅, which is a contradiction. Thus P must be trivial, and the Hasse principle holds for H1flat (k, G). (2) The statement follows from (1) and the fact that over any global field, the semisimple groups mentioned satisfy the cohomological Hasse principle. In the case of number fields, the statement follows from [33, Section 5, Corollaire 5.4]. In the case of global function field, any semisimple simply connected group satisfies the cohomological Hasse

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N.T. Ngoan, N.Q. Thˇ ańg / Linear Algebra and its Applications ••• (••••) •••–•••

principle by Theorem 5.1 (due to Harder in this case). The case of adjoint groups also ˜ be is due to Harder (see [13, Bemerkung, p. 129]). We recall the following result. Let G ˜ a semisimple simply connected k-covering of semisimple k-group G and let Z(G) denote ˜ The first one is the cohomological Hasse principle for Z(G) ˜ in degree the center of G. 2 2 (see [14, Case (2), p. 173]) which generalizes the Hasse principle for H of n-th roots of unity (which in turn is a consequence of Hasse principle for Brauer groups) and the second one is an analog of Kneser’s Theorem (see [41]). Lemma 5.9. ˜ Let k be a global field and let n be a positive integer. Then (1) (Hasse principle for Z(G)) the natural homomorphism ˜ → H2flat (k, Z(G))



˜ H2flat (kv , Z(G))

v

is injective. ˜ → G is the canonical projection from (2) If k is a local or global function field, π : G the simply connected covering of a semisimple k-group G, F := Ker(π), then the coboundary map H1flat (k, G) → H2flat (k, F ) is a bijection. If the Dynkin type of G is either type Bn , Cn , trialitarian 3 D4 ,6 D4 , or of type E, F, G, then G is either simply connected or adjoint. Since in either cases, the cohomological Hasse principle holds for G and the assertion follows from (1). If G is an absolutely almost simple k-group, which is neither simply connected nor adjoint, then G is of type A or D (not of trialitarian type). Assume that G is of type An . ˜ Then by the same method of proof as in Then F is a subgroup scheme of μn+1 = Z(G). [40, pp. 1060–1061], we are reduced to the case F μm and in this case we invoke again Lemma 5.9. If G is of type Dn , which neither simply connected nor adjoint, then for the fundamental group F of G, we have F μ2 , and by Lemma 5.9, λ is injective. (3) By Theorem 5.3, we have H1flat (k(v), G) = 1, H1flat (kv , G) = 1 for all nonarchimedean places v, hence the rest follows from Theorem 5.6 and Theorem 5.7(2). 2 We say that a global field k is (totally) imaginary if it has no real places. The definition also extends to infinite global fields. It is so, if in particular the characteristic of k is > 0. In such a case, by a theorem of Harder (see [11, Haupsatz], [13, Satz A]), for any semisimple simply connected k-group G, we have H1flat (k, G) = 1. The same also holds in the infinite global field case, as the following shows. Corollary 5.10. Let k be an infinite global field and let G be a semisimple simply connected group defined over k.

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(1) If k is totally imaginary (for example if k has positive characteristic), then H1flat (k, G) = 1. (2) There exist an infinite global field and a semisimple simply connected k-group G such that H1flat (k, G) is infinite. Proof. (1) It follows from the Hasse principle for H1 (see Theorem 5.7) that the natural map H1flat (k, G) →



H1flat (k(v), G)

v

 has trivial kernel. Since k has no real places, we have v H1flat (k(v), G) = 1 according to Theorem 5.3. Hence we also have H1flat (k, G) = 1. (Another proof goes as follows. Let P be any G-torsor defined over k. Since k = ∪i ki , where k0 ⊂ k1 ⊂ · · · ⊂ k, where each ki is a global field. We may find n such that P is defined over kn . Since k is totally imaginary, we may take n sufficiently large, so as kn is also imaginary. By Harder’s Theorem, H1flat (kn , G) = 1, i.e., P (kn ) = ∅. A fortiori, we have P (k) = ∅, i.e., P is a trivial G-torsor. Hence H1flat (k, G) = 1.) (2) We start with a semisimple simply connected group G defined over a real global  field L, such that v∈∞L H1flat (Lv , G) = 1. Then we construct a finite extension L1 /L such that |∞L1 | > |∞L |, so one has |

 v∈∞L1

H1flat (L1,v , G)| > |



H1flat (Lv , G)|

v∈∞L

and keep iterating this process. In the end, we obtain an infinite tower L ⊂ L1 ⊂ · · · ⊂ Ln ⊂ · · · , and let k be the union of this tower. Then by Theorem 5.6(2), it is clear that the field k and the group G have the required property. 2 Corollary 5.11. Let k be an infinite global field and let G be a semisimple simply connected group defined over k. Denote the set of real places of k by ∞k,R and assume that it is non-empty and finite and that for some global field L ⊂ k, G is defined over L and k(v) = Lv for all v ∈ ∞k . Then there is a natural bijection H1flat (k, G) H1flat (L, G). Proof. Indeed, we have the following bijections α, β due to the validity of the cohomological Hasse principle

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N.T. Ngoan, N.Q. Thˇ ańg / Linear Algebra and its Applications ••• (••••) •••–•••

H1flat (L, G)

r

→

H1flat (k, G)

↓α  v∈∞L

H1flat (Lv , G)

↓β r

→

 v∈∞k

H1flat (k(v), G)

By assumption, the bottom row is a bijection, hence so is the top and we derive the conclusion. 2 Remark 5.12. As an application of the results just proved, under the assumption of Corollary 5.11, we give another short proof of the validity of the cohomological Hasse principle for simply connected groups over infinite global fields k. We consider the following cases regarding ∞k,R . Case 1. ∞k,R = ∅, i.e., k is imaginary. Then by Corollary 5.10(1) (where the second proof does not depend on Theorem 5.7), H1flat (k, G) = 1 and we are done. Case 2. ∞k,R = ∅. Let x ∈ H1flat (k, G), which has trivial images via all localizationrestriction maps. If P is a k-torsor, whose class is equal to x, then we may assume that there exists a global subfield L ⊂ k such that P is defined over L. This implies that x ∈ Im(q : H1flat (L, G) → H1flat (k, G)), and we may find x ∈ H1flat (L, G) such that x = q(x ). According to Theorem 5.3, we need only consider the archimedean places, and we remove all complex ones, since they do not contribute to the cohomology. Let consider the following commutative diagram H1flat (L, G)

q

→

H1flat (k, G)

α↓ 

↓β q

v∈∞L,R

H1flat (Lv , G) →

 w∈∞k,R

H1flat (k(w), G)

Notice that the map q  is described as follows. If ∞L,R = {v1 , ..., vm }, and xi ∈ H1flat (Lvi , G) = H1flat (R, G), then r (x1 , ..., xm ) = ([x1 ], ..., [xm ]), where for each i,   [xi ] ∈ w∈∞k ,w|vi H1flat (kw , G) = w∈∞k ,w|vi H1flat (R, G), and each of the components of [xi ] is the element xi ∈ H1flat (R, G). Since q  is injective by the description, we have α(x ) = 1, which means x = 1, due to the cohomological Hasse principle for α over L. From this we derive that β has trivial kernel. By twisting, it is also injective. 5.3. Applications to the Hasse norm principle We consider some applications of cohomological treatment above to some classical results of the arithmetic (say the Hasse Norm principle) and to Weak Hasse principles for hermitian and skew-hermitian forms. It is interesting to note that in the classical case of global fields, the cohomological Hasse principle for H1 of simply connected groups of classical types A, B, C, D is proved by using the Weak Hasse principle for forms. However, in the case of infinite global fields, the process is converse. For simplicity, we assume that

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k has characteristic = 2 wherever we consider a quadratic or (skew-)hermitian form over k. First we have true analogs of the Hasse–Maass–Schilling Norm Theorem and the Hasse Norm Theorem in the case of infinite global fields as follows (compare Proposition 3.1). Theorem 5.13 (Analog of Hasse, Hasse–Maass–Schilling Norm Theorem). Let k be an infinite global field, Vk the set of all places of k and let A be a finite dimensional central simple algebra over k. (1) If x ∈ k∗ is such that x >v 0 (x is considered as an element of the real field kv ), for all those real places v such that Av := A ⊗ kv is not split, then it is a norm from A. Equivalently, (2) If x ∈ k ∗ is a reduced norm from A ⊗ kv for all places v, then it is also a norm from A. (3) If L/k is a finite cyclic extension and if for all v ∈ Vk , x is a norm from Lw /kv , where w is an extension of v to L, then it is also a norm from L. Proof. We consider the exact sequence of algebraic k-groups 1 → SLA → GLA → Gm → 1, where for any k-algebra F , GLA (F ) := (A ⊗k F )∗ and SLA (F ) := {x ∈ GLA (F ) | Nrd(x) = 1}. (1) If v is non-archimedean, then by Theorem 5.3 (the analog of Kneser–Bruhat–Tits’ Theorem), H1flat (kv , SLA ) = 1, so from the long exact sequence of Galois cohomology together with Hilbert’s Theorem–90 associated with above sequence, it follows that we have an isomorphism H1flat (k, SLA ) Gm (k)/ Nrd(GLA (k)),

(5.13.1)

and for any v ∈ V , the isomorphisms H1flat (kv , SLA ) Gm (kv )/ Nrd(GLA (kv )).

(5.13.2)

So every element from kv is a reduced norm. If v is real, then xv is automatically a reduced norm, if Av is kv -split. Otherwise, xv is a reduced norm from Av , if and only if xv > 0. Thus (1) is equivalent to (2). (2) From (5.13.1) and (5.13.2), and from the validity of the cohomological Hasse principle for simply connected groups of type A (Theorem 5.7(2)), we deduce that an element from k∗ , which is locally a reduced norm, is also globally a reduced norm. (3) Given a cyclic extension L/k, one may construct an appropriate cyclic algebra over k, associated with L (see for example [35, Chapter 8]). Thus it is clear that (3) is a consequence of (2). 2

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The following gives a true analog of the Albert–Brauer–Hasse–Noether Theorem (the local–global part) (compare Theorem 3.5) for infinite global fields. Notice that in the case of infinite number fields, another method of proof was sketched by Schilling in [36, Theorem 4]. Theorem 5.14 (Hasse principle for Brauer groups). Let k be an infinite global field. Then  (1) The natural homomorphism β : Br(k) → v Br(kv ) is injective. (1) (2) If G is one of the following k-groups: μn , RK/k (Gm ), RK/k (Gm ), then the classical cohomological Hasse principle in degree 2 holds for G, namely the natural map  H2flat (k, G) → v H2flat (kv , G) is injective. Proof. (1) Let x ∈ Ker(β) and let A be a representative from the class x. Thus A is a central simple k-algebra, so it is also defined over a global subfield L ⊂ k. Assume that A is of degree n, thus its class [A] ∈ n Br(L) ⊂ n Br(k). Similarly, for all v ∈ V , we have [A]v ∈ n Br(kv ), and by assumption, [A]v is trivial for all v. Consider the exact sequence of L-groups 1 → μn → SLn → PGLn → 1 and related sequences of flat cohomology Δ

1 → H1flat (L, PGLn ) →L H2flat (L, μn ) = Δ

1 → H1flat (k, PGLn ) → H2flat (k, μn ) =

n Br(L), n Br(k).

It is well known that the image of ΔL consists of exactly those elements which have orders dividing n, so it implies that Δ is surjective. (It also follows from a more general result of Kneser [18, Theorem 2 of Chapter V] (for number fields) and Lemma 5.9(2) (for global function fields).) Therefore, by twisting argument, Δ is also bijective. This is also Δ

true when k is replaced by kv and consider the exact sequence 1 → H1flat (kv , PGLn ) →v H2flat (kv , μn ) = n Br(kv ). Thus it suffices to show that if x ∈ H1flat (k, μn ) such that all of its localizations are trivial then x is itself trivial. Since PGLn satisfies the cohomological Hasse principle by Theorem 5.7, it implies that the same also holds for n Br(k) due to the bijections Δ, Δv above. Therefore β is injective as desired. (2) Follows directly from (1). 2 5.4. The weak Hasse principle for forms and cohomological Hasse principle for unitary groups of type A Let k be an infinite global field of characteristic = 2, D a division algebra over its √ center K = k( a), J an involution of the second kind of D, k = K J , h a J-hermitian form with values in D (i.e., a form of type A). There are two k-groups associated to h to

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consider, namely the special unitary group G = SU(h) and the unitary group G1 = U(h). In the global field case, the Weak Hasse principle for hermitian forms with respect to unitary involution was established by Landherr (see [35, p.373]) and we show here that it also holds over infinite global fields. Theorem 5.15 (The Weak Hasse principle for forms of type A). Let g and h be J-hermitian forms with values in D. The forms g and h are equivalent over k if and only if they are so locally everywhere, both in new and classical setting. The original proof of the Weak Hasse principle due to Landherr in the case of global fields combines at the same time the proof of the Strong Hasse principle. The proof we present here reduces to proving the cohomological Hasse principle for G1, which follows from that of G. The following proof also clarifies a minor point in the one given in [40, Lemma 4] for the case of global field. Proposition 5.16. Let k be a (finite or infinite) global field of characteristic = 2. With above notation, G1 satisfies the Cohomological Hasse principle for H1 both in new and classical setting. Proof. It is well-known (see [13, Section 2]) that we have the following exact sequence of algebraic k-groups 1 → G → H → S → 1, which is obtained from the standard exact sequence det

1 → SLn → GLn → Gm → 1, by a suitable twisting. In particular, S is a one-dimensional k-torus, which comes from another exact sequence, namely N

1 → S → RK/k (Gm ) → Gm → 1, √ where K = k( a) and where N stands for the norm. It is well known that H1flat (L, S) = L∗ / NK⊗k L/L ((K ⊗k L)∗ ) for any extension L/k. In particular, H1flat (k, S) = k∗ / NK/k (K ∗ ), H1flat (kv , S) = kv∗ / NKv /kv (Kv∗ ). The Hasse principle for the (quadratic) norm over (infinite) global fields (see Proposition 3.1 and Theorem 5.14 (2)) shows that the natural homomorphism g : H1flat (k, S) →  1 v Hflat (kv , S) is injective.

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(1) k is a global field. If k is an imaginary global field, then by results of Harder (see [13, Satz A]) and Bruhat–Tits (see [5, Theorem 4.7]), we have the following commutative diagram with exact rows

0 →

α

→

H1flat (k, H)

H1flat (k, S)

↓f 0 →

 v

↓g β

→

H1flat (kv , H)

 v

H1flat (kv , S)

Since g is injective, so is f . If k is a number field, we have the following commutative diagram with exact rows

S(k)

δ

→

H1flat (k, G)

↓q  v

S(kv )

β

→

H1flat (k, H)

↓p δ

→



H1flat (k, S)

↓f β

v

α

→

H1flat (kv , G) →

 v

↓g α

H1flat (kv , H) →

 v

H1flat (kv , S)

By the cohomological Hasse principle for G and S shown above, p and g are injective. One notes also, that H1flat (kv , G) = 1 for non-archimedean places v by Theorem 5.3 and  that S(k) → v∈∞k S(kv ) has dense image by a well known result of Serre, so all of these imply that f is injective. (2) k is an infinite global field. We may assume that H is defined over a global field L contained in k. First, we consider the Hasse principle in new setting. By (1), the classical Hasse principle holds for H over any field extension K/L contained in k. Hence by Theorem 5.7(1), the cohomological Hasse principle (in new setting) also holds for H. Second, we prove the cohomological Hasse principle in classical setting. If ∞k,R is finite, then the same argument as above also applies, by considering the diagram similar to the above one, where we replace the set of all places by ∞k,R and the argument as above goes through. In the general case, we may proceed as follows. Let x ∈ Ker(f ). Then by chasing on the diagram and by using the injectivity of g (by Theorem 5.14), there is y ∈ H1flat (k, G) such that β(y) = x. One may assume that y comes from some cocycle class over some global field L ⊂ k, and so does x, where we may assume that G is also defined over L. So y = t(y  ), y  ∈ H1flat (L, G), where t : H1flat (L, G) → H1flat (k, G) is the natural restriction map. From this we derive the following commutative diagram

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S(L)

HH H

α



-

s

v∈∞L,R

Hflat (L, G)

r

H HH j H

p

S(k)

? S(Lv )

1

p p p p p p p p p p p p p p p p αp p p p p p sp p p p p p pp pp p p p p ppp 

HH r H j H

29



pp HH pp pp Ht H pp HH pp j pp β 1 Hflat (k, G) pp pp pp pp ? 

1

Hflat (Lv , G)

β

v∈∞L,R

HH t HH j 





?

w∈∞k,R

S(kw )

p

? 1

Hflat (kw , G)

w∈∞k,R

Here we denote by ∞L,R (resp. ∞k,R ) the set of all real places of L (resp. of k). Here we remove all complex places, since they do not contribute to the cohomology. Notice that the map r (and similarly t ) is described as follows. If ∞L,R = {v1 , ..., vm }, and xi ∈ S(Lvi ) = S(R), then r (x1 , ..., xm ) = ([x1 ], ..., [xm ]), where for each i,   [xi ] ∈ w∈∞k,R ,w|vi S(kw ) = w∈∞k,R ,w|vi S(R), and each of the components of [xi ] is the element xi ∈ S(R). Since β(t(y  )) = t (β  (y  )) ∈ Im(p ), from the description of t it implies that β  (y  ) ∈ Im(s ). It is well known (and easy to prove) that α is surjective, so the bijectivity of β  (the classical Hasse principle for the simply connected group G) implies that y  ∈ Im(s). From this we derive that γ has trivial kernel. By twisting, it is also injective. 2 5.5. The weak Hasse principle for forms and cohomological Hasse principle for unitary groups of type C

Let g, h be hermitian forms of type C with values in a division algebra D, all are defined over an infinite global field k of characteristic = 2. As above, we may assume also that they are defined already over a finite global field L, such that D is a quaternion division algebra over L with the canonical involution J. Theorem 5.17 (The Weak Hasse principle for forms of type C). Let g, h be hermitian forms of type C as above. Then g and h are equivalent over k if and only if they are so locally everywhere both in new and classical setting. Like in the case of hermitian forms with respect to an unitary involution, the proof is reduced to considering the Hasse principle for Galois cohomology. Since the Weak Hasse principle is equivalent to the cohomological Hasse principle for unitary groups of the hermitian forms, the proof of Theorem 5.17 follows from that of Theorem 5.7. 2

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5.6. The cohomological Hasse principle for spin, orthogonal and unitary groups of type Bn , Dn . applications to the weak Hasse principle 5.6.1. The cohomological Hasse principle for spin groups of type Bn , n ≥ 2 or Dn , n ≥ 4 Beside the general proof for all types given in Theorem 5.7, there is another (complicated) proof for the cohomological Hasse principle for Spin groups of type B, D if one imitates the original proof given by Kneser in [18] by using the validity of the cohomological Hasse principle for almost simple simply connected groups of type A1 and A3 . We omit the proof here and just indicate that beside the Hasse principle for quadratic forms (see Theorem 2.6), one needs also the Strong Hasse principle for skew-hermitian forms of type D (see Theorem 6.6) and also to assume that the set of real places of k is finite (in order the arguments related with weak approximation go through).

5.6.2. The weak Hasse principle for forms and cohomological Hasse principle for (special) orthogonal groups of type Bn , n ≥ 2 and Dn , n ≥ 4 We have the following Theorem 5.18. (1) The cohomological Hasse principle holds over any infinite global field k of characteristic = 2 for special orthogonal groups of the type indicated both in new and classical setting. (2) There are infinite global fields k and quadratic forms q of any dimension over k, for which the cohomological Hasse principle does not hold for O(q). In particular, the Weak Hasse principle does not hold. Proof. (1) First, the assertion regarding the cohomological Hasse principle in new setting follows from Theorem 5.7(1). Next we consider the Hasse principle in classical setting. Let G = SO(f ), where f is a quadratic form over k. Then the usual method of proving the cohomological Hasse principle for H1flat (k, SO(f )) (by using the Strong or Weak Hasse principle for quadratic forms in three or more variables and also the validity of the cohomological Hasse principle for Spin groups proved above) combined with an argument from the proof of Theorem 5.16 also applies. Since the argument is short, we briefly recall it. We have the following exact sequence of algebraic groups

1 → μ2 → Spin(f ) → SO(f ) → 1, and we derive the following commutative diagram with exact rows

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→

H1flat (k, Spin(f )) ↓β 

H2flat (k, μ2 )

↓γ 

q

v∈Vk

Δ

→

H1flat (k, SO(f ))

H1flat (kv , Spin(f )) →

↓ζ Δ

v∈Vk

31

H1flat (kv , SO(f )) →

 v∈Vk

H2flat (kv , μ2 )

Since H1flat (k(v), Spin(f )) = 1, H1flat (kv , Spin(f )) = 1 for v ∈ / ∞k (see Theorem 5.3), so if ∞k = ∅, then we also have the following commutative diagram with exact rows q

→

H1flat (k, Spin(f )) ↓β 

H2flat (k, μ2 )

↓γ 

q

v∈∞k

Δ

→

H1flat (k, SO(f ))

H1flat (kv , Spin(f )) →

v∈Vk

↓ζ Δ

H1flat (kv , SO(f )) →

 v∈Vk

H2flat (kv , μ2 )

Let x ∈ Ker(γ). Then by chasing on the diagram and by using the injectivity of ζ (by Theorem 5.14), there is y ∈ H1flat (k, Spin(f )) such that q(y) = x. One may assume that y comes from some cocycle class over some global field L ⊂ k, and so does x. So y = t(y  ), y  ∈ H1flat (L, Spin(f )). From this we derive the following commutative diagram, where we set G := Spin(f ) (to make the diagram compact) HH r HH HH j

α



-

s

1

Hflat (L, μ2 )

1

Hflat (L, G)

p

1

Hflat (k, μ2 )

? 1

Hflat (Lv , μ2 )

v∈∞L,R

H

p p p p p p p p p p p p p αp p p p psp p p p p p p p p p ppp





HH r H j H



1

Hflat (Lv , G)

β

v∈∞L,R

HH t HH j H  

?





pp HH pp t pp HH pp HH pp j pp β - H1flat (k, G) pp pp pp pp ?

1

Hflat (kw , μ2 )

p

w∈∞k,R

? 1

Hflat (kw , G)

w∈∞k,R

(We remove all complex places, since they do not contribute to the cohomology.) Notice that the map r (and similarly t ) is described as follows. If ∞L,R = {v1 , ..., vm }, and xi ∈ H1flat (Lvi , μ2 ) = H1flat (R, μ2 ), then r (x1 , ..., xm ) = ([x1 ], ..., [xm ]), where for each i, [xi ] ∈

 w∈∞k,R ,w|vi

H1flat (kw , μ2 ) =



H1flat (R, μ2 ),

w∈∞k,R ,w|vi

and each of the components of [xi ] is the element xi ∈ H1flat (R, μ2 ). Since β(t(y  )) = t (β  (y  )) ∈ Im(p ), from the description of t it implies that β  (y  ) ∈ Im(s ). It is well known (and easy to prove) that α is surjective, so the bijectivity of β  (the classical

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Hasse principle) implies that y  ∈ Im(s). From this we derive that γ has trivial kernel. By twisting, it is also injective. If k is imaginary, then by Theorem 5.7 we have that H1flat (k, Spin(f )) = 1, H1flat (k(v), Spin(f )) = 1, and H1flat (kv , Spin(f )) = 1, so the assertion follows from similar commutative diagram as above and we are done. (2) The assertion follows from Theorem 2.6, since the validity of the Weak Hasse principle is equivalent to that of the cohomological Hasse principle of the corresponding orthogonal group. 2 5.6.3. The weak Hasse principle for forms and cohomological Hasse principle for (special) unitary groups of type Dn , n ≥ 4 We have the following Theorem 5.19. The cohomological Hasse principle both in new and classical setting holds over any infinite global field k of characteristic = 2 for special unitary groups of type Dn , n ≥ 4. In particular, if two skew-hermitian forms of type Dn , n ≥ 4 over a division algebra having the same dimension and determinant are equivalent locally everywhere (over k(v) or kv ), then so are they globally over k. Proof. Let G be an almost simple group of type Dn , n ≥ 4 defined over an infinite global field k of characteristic = 2. Assume that G is associated with a skew-hermitian form h. Then the same proof of the cohomological Hasse principle as in the case of special orthogonal groups (associated with quadratic forms) above can be carried over to this case. 2 Note that if n is even, the fundamental group of Spin(h) is of the form μ2 × μ2 . Therefore, besides the special orthogonal (resp. unitary) group, there are two other groups obtained from Spin(h) by taking a quotient by a finite subgroup scheme of order 2 of μ2 ×μ2 . The same proof as above shows that they too, also satisfy the cohomological Hasse principle for H1 . It is well known that in the case n = 2, the Strong (and Weak) Hasse principle may fail for skew-hermitian forms over global fields. Namely, if D is a quaternion division algebra over a global field k, which is not split at s ≥ 2 places of k, then for any skew-element λ ∈ D∗ with respect to the canonical involution J of D, there are exactly 2s−2 elements α ∈ k∗ /k∗2 such that xJ λx − αy J λy represents zero over kv non-trivially locally everywhere, but does not represent zero non-trivially over k (see [18, pp. 136–138, p. 148], or [35, Chapter 10, 4.5, 4.6]). It appears that the failure of the Strong (and also Weak) Hasse principle for skew-hermitian forms with respect to the canonical involution J of D also happens in the case of infinite global fields. We have

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Proposition 5.20 (Failure of the Weak Hasse principle). (1) There are an infinite global field k of characteristic = 2 and a quaternion division algebra D = (a, b/k) with the properties that D is defined over a global subfield L ⊂ k (i.e., a, b ∈ L) and D ⊗L Kv is not trivial for at least 4 places v of K for any finite extension K/L contained in k. (2) For a quaternion division algebra D satisfying (1), for any natural number n, there are J-skew-hermitian forms g and h of dimension n with values in D, which are locally everywhere equivalent, but not globally equivalent. In particular, the Weak Hasse principle does not hold. (3) The cohomological Hasse principle does not hold for U(h), if n ≥ 4. Proof. (1) First we choose a global field L of characteristic = 2 and a quaternion division algebra D = (a, b/L). Let S be the (finite) set of all places of L, such that D ⊗ Lv is not trivial. We may choose D such that |S| ≥ 4. For each v ∈ S, we choose an odd number nv > 1. Then the least common multiple n of nv s is also odd. By global class field theory (see [3, Theorem 5, p. 105]), we may choose a finite cyclic extension L1 /L of degree n, such that for all v ∈ S, and for each extension v1 of v to L1 , the local degree [L1,v1 : Lv ] is exactly nv . It is clear that since nv is odd, and D is not trivial over Lv , so is D over L1,v1 . Indeed, by a theorem of Springer (see [35, Ch. II, Theorem 5.3]), the norm form of D, being anisotropic over Lv , remains so over L1,v1 . Thus the finite set S1 of all places of L1 such that D ⊗ L1 is non-trivial has at least four places. Keep iterating ¯ this process, we obtain an infinite tower of finite field extensions L = L0 ⊂ L1 ⊂ · · · ⊂ L. We set k = ∪n Ln . From our construction, each Ln has at least four places where D ⊗ Ln remains non-split. Then it is clear that k(v) = ∪n Ln,vn . If D ⊗ k(v) is trivial then there is a splitting field of D inside k(v) of finite degree over Lv , which is impossible. Thus D ⊗ k(v) is not trivial and it is clear this is so for at least four places of k. If K is any finite extension of L, L ⊂ K ⊂ k then from above we can choose n such that L ⊂ K ⊂ Ln , Lv ⊂ Kw ⊂ Ln,vn . Since D ⊗ Ln,vn is not trivial, so is D ⊗ Kw . Therefore (1) is proven. (2) We consider a J-skew-element λ ∈ D∗ . We choose λ2 = a, where D = (a, b/k). To prove the assertion, it suffices to use the construction of 2-dimensional forms h = xJ λx − αy J λy for which the Strong Hasse principle fails over L (see [35, p. 368]). According to [35, p. 361], the equation h = 0 has a solution α ∈ L∗ (resp. α ∈ k∗ ) (equivalently, the forms λ and αλ are equivalent), if and only if α is represented globally over L (resp. over k) either by the form 1, −a or by the form b, −ab. Then the elements α ∈ L∗ (resp. α ∈ k∗ ) for which the Strong Hasse principle fails for h satisfy the following local conditions for Hilbert symbol (., .)v over Lv (resp. over kv ) (∗) For every place v of S, either (a, α)v = 1 or (a, α)v = (a, b)v = −1, but any of such conditions does not hold globally.

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Now let D, S be as in (1). Denote by Sk the set of all places of k extending those places belonging to S. Take any subset S1 ⊆ S with an even number of elements and at least 4. Take any element α ∈ L∗ such that (∗) holds. In particular, we require that (a, α)v = 1 for all v ∈ / S, (a, α)v = −1 for all v ∈ S1 , and (a, α)v = 1 for all v ∈ S \ S1 . Then for any place w of k, if w ∈ S1,k , then (a, α)w = −1, and (a, α)w = 1 otherwise. Therefore, the element α ∈ k∗ satisfies the condition (∗) and the forms λ and αλ are equivalent (equivalent) over kv for all v, but not so over k. The number of such globally non-equivalent but locally everywhere equivalent 1-forms is finite (resp. infinite), if Sk is finite (resp. infinite). The case of arbitrary dimension follows from the one-dimensional case. (3) Follows from (2). 2 Remark 5.21. (1) In the situation of Theorem 5.7(1), one can say also that if a connected smooth affine algebraic group defined over a global subfield L of an infinite global field k satisfies the cohomological Hasse principle “locally” everywhere, that is, over each sufficiently large global subfield L ⊂ K contained in k, then it does so over k. (2) The remark above suggests that one may consider the problem of investigating the validity of a fibration method for proving the Hasse principle. (3) As it was mentioned in Remark 2.7, the classical cohomological Hasse principle for H1 may fail for certain non-connected algebraic groups over infinite global fields, though such groups do satisfy the principle over any global fields. So the connectedness of the groups plays an essential role here. (4) From the Hasse principle for an arbitrary simply connected semisimple group G over a global field k it implies that for such group G, the set H1 (k, G) is finite. The situation is different in the case of infinite global fields, as Corollary 5.10(3) shows. (5) The Hasse–Maass–Schilling Norm Theorem and the Hasse Norm Theorem in the case of infinite global fields do not seem to be deduced in a straight forward way from the classical theorem. Notice that in the classical case, we deduce the Hasse principle for H1 for groups of type 1 An by using the Hasse–Maass–Schilling Norm Theorem. (6) Let k be a non-archimedean local field or a global field. For any simply connected ˜ with its center F˜ let F ⊂ F˜ be a k-subgroup of F˜ . Set G = semisimple k-group G ˜ ¯ ˜ ˜ G/F , G = G/F , the adjoint group of G. It is a classical result due to Kneser (see [18, Chapter IV, Theorem 2 and Chapter V, Theorem 2]) that if k is of characteristic 0, then the natural map Δ : H1flat (k, G) → H2flat (k, F ) is surjective (and bijective if k is a p-adic field). Later on, Douai in his thesis (see [8, Chapter VI, Corollaire 1.16 and Chapter VIII, Corollaire 1.5]) proved that the surjectivity result also holds for k either a local or a global function field (see another elementary proof that was given in [41]). ˜ : H1 (k, G) ¯ → H2 (k, F˜ ) is surjective, then some people (see [10]) name such a If Δ flat flat field k as of Douai type. We think that it is more appropriate to consider all subgroup schemes F ⊂ F˜ , and the corresponding coboundary maps Δ : H1flat (k, G) → H2flat (k, F ). We propose to call fields of Kneser type those fields k for which the coboundary map

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Δ is surjective for any such G and F . Of course, the class of fields of Kneser type is a subclass of those of Douai type, and it is not clear whether they are the same. Also, it is natural to ask if infinite local or global fields are also fields of Douai type or of Kneser type. 6. The strong Hasse principle and global classification 6.1. The strong Hasse principle for forms of type A The following well known Hasse principle for forms of type A was proved by W. Landherr in 1938 (see [35, 6.1, 6.2, p. 373]). Let k be a global field of characteristic = 2, Vk the set of all its places and let h be a hermitian form with respect to an involution J √ of the second kind over a division algebra D of center K = k( a), k = K J . Theorem 6.1 (Landherr). If h represents zero non-trivially locally everywhere over all kv , v ∈ Vk then it does so over globally over k. We have a complete analogue in the infinite global field case. Theorem 6.2 (The Strong Hasse principle for forms of type A). Let k be an infinite global field of characteristic = 2 and keep other notation as above. If h represents zero non-trivially locally everywhere over all k(v), v ∈ Vk then it does so globally over k. First we need the following result, which is folklore Lemma 6.3. Let k be a global field of characteristic = 2 and keep other notation as above. Let h be a J-(skew-)hermitian form with values in D. If dim(h) ≥ 2, then there is a finite set S of places of k, such that if v ∈ / S, then h is isotropic over kv . Proof. For simplicity, assume that h is a hermitian form of dimension n ≥ 2. It is well known (see Theorem 4.4) that over non-archimedean fields kv h becomes a hermitian form with values in Kv . The trace form qh corresponding to h has dimension 4, and we assume that qh = a1 , a2 , a3 , a4 . For some finite set S of places of k, the entries ai are v-adic units for v ∈ / S and by Springer’s Theorem, qh ⊗ kv is isotropic over kv , hence so is h. 2 Proof of Theorem 6.2. The proof is the same as in the case of quadratic forms, by using König’s Lemma and the Hasse principle for hermitian forms over global fields (see Theorem 6.1 and Lemma 6.3). 2

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6.2. Strong Hasse principle for forms of type C Let k be an infinite global field of characteristic = 2, Vk the set of all places of k. Let h be a hermitian form of dimension n with respect to an involution J of symplectic type over a division algebra D over k. We have the following Proposition 6.4 (The Strong Hasse principle for forms of type C). If h represents zero non-trivially locally everywhere, then it does so over k globally. Proof. Since h, D, J are given by a finite datum, it follows that they are defined over a global field L. Thus we may assume from the very beginning that it is so, and therefore by Proposition 3.8, we may assume that D is a quaternion division algebra with the canonical involution J, for which the set of fixed points is L. We assume also that h is given in a diagonal form h = a1 x1 xJ1 + · · · + an xn xJn , n ≥ 2, where ai ∈ L, xj ∈ D. Let D = (a, b/L) with a basis {1, i, j, ij} such that i2 = a, j 2 = b, ij = −ji, where a, b ∈ L. The norm form of D is given by N (x) = x21 − ax22 − bx23 + abx24 , x = x1 + x2 i + x3 j + x4 , xi ∈ k. The trace form associated to h is given by qh = a1 N (x1 ) + · · · + an N (xn ). For any field extension K/L, it is clear that h ⊗ K is isotropic if and only if qh ⊗ K is so over K. Thus qh locally represents zero non-trivially over all completions of L. By the Hasse principle for quadratic forms, it follows that qh is isotropic over L, and therefore, so is h. 2 6.3. The strong Hasse principle for forms of type D Let k be an infinite global field of characteristic = 2, D a division algebra D with the center k and an involution J. Let h be a J-skew-hermitian form of (orthogonal) type D with values in D. We know that D is a quaternion division algebra with the center k and that h is associated with the canonical involution J. Over global fields, we have the following the strong Hasse principle for such forms, due to Kneser (see [18, p. 128], [35, p. 366]). Theorem 6.5 (Kneser). (1) With above notation, if k is a global field of characteristic = 2 and if dim(h) ≥ 3, then the Strong Hasse principle holds for h. Equivalently, we have (2) If dim(h) ≥ 2, x ∈ D∗ is a J-skew-symmetric element, such that h represents x locally everywhere, then so does h over k. We have the following analogue of Kneser result in the infinite case.

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Theorem 6.6 (The Strong Hasse principle for skew-hermitian forms of type D). If k is an infinite global field of characteristic = 2, then the same assertion as in Theorem 6.5 holds. Proof. There are two classical proofs of this theorem in the global field case; one is due to Kneser and another one to Springer. Both are a bit long and complicated. In our new local-global setting, first we give our first proof by using the arguments given in the case of quadratic forms (see also the proof of Theorem 3.5) and some arguments used in the proof given by Springer (see [35, Chapter 10, Section 4]). One needs the following statement (see [35, p. 369] and also Remark 6.8(2) below) Lemma 6.7. Let k be a global field of characteristic = 2 and let D be a quaternion division algebra with the center k and the canonical involution J. Let h be a J-(skew-)hermitian form with values in D. If dim(h) ≥ 3, then there is a finite set S of places of k, such that if v ∈ / S, then h is isotropic over kv . Proof. Consider h in its diagonal form: h = a1 , ..., an , ai ∈ D∗ . For each place v ∈ Vk denote by the same symbol the place on D extending v. Let A be a maximal order in D. Consider the set S of all places v of k, where either v is dyadic (i.e., the residue field has characteristic 2), or Dv is a division algebra, or ai is not invertible in Av for some 1 ≤ i ≤ n. Then it is clear that S is finite, and for any non-archimedean place v ∈ / S, iv : Dv M2 (kv ). Also, iv (Av ) ⊂ M2 (Ov ), where Ov is the ring of v-integers of kv . Denote by qh the quadratic form corresponding to h. Then dim(qh ) = 2 dim(h), so dim(qh ) ≥ 6. Also, due to the choice of S, the images i(a1 ), ..., i(an ) of ai in M2 (Ov ) are invertible, thus qh has a diagonal form over Ov with invertible diagonal elements (i.e., v-units). It follows that the Witt-index of qh is ≥ 2 and therefore h is isotropic over kv . If v is archimedean and v ∈ / S, then it is well known that h is isotropic if dim(h) ≥ 2 (see [35, Chapter 10, Section 3]). Therefore S is a set we need. 2 Proof of Theorem 6.6. We may apply the same argument in the proof of Theorem 4 in [21] to our situation (see also the proof of Theorem 3.5). 2 Remark 6.8. (1) A second (cohomological) proof of Theorem 6.6, can also be obtained if one follows closely the (cohomological) proof given by Kneser (see [18, pp. 128–132]). Here one shows that if dim(h) ≥ 2, x ∈ D∗ is a J-skew-symmetric element, such that h represents x locally everywhere, then so does h over k. One first starts with the case n = dim(h) = 2 and n = 3, and then for the case n > 3. There are several induction steps. The point of this method is that we have to assume that the set of real places of k is finite (in order the arguments related with weak approximation go through) and we have to use the validity of the cohomological Hasse principle for tori, which are split over a quadratic extension, and also the one for simply connected almost simple groups of type A1 and A3 . (They are consequences of Theorem 5.7, which is a more general result.)

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Regarding the tori which are split over quadratic extensions, one may use Corollary 3.6, and for anisotropic k-groups G of type A1 , where G(k) = SL1 (D), where D = (a, b/k) is a quaternion division algebra over k, one may prove the Hasse principle for G by using the Hasse principle for the quadric Nrd(X) = c, c ∈ k∗ , which follows from the Hasse principle for the quadratic form in five variables x2 − ay 2 − bz 2 + abt2 − cu2 over k. (2) Lemma 6.3 and Lemma 6.7 are particular cases of the following general well known result, the proof of which we refer to [33, p.68]: Let G be a semisimple algebraic group defined over a global field k. Then there is a finite (possibly empty) finite set S of places of k, such that for all v ∈ / S, G becomes quasi-split over kv . In particular, it means that if G is the semisimple group associated to a quadratic (resp. (skew-)hermitian) form h, then h has maximal possible Witt index over kv , for v∈ / S, thus h is also isotropic over kv . 6.4. Global classification By combining the previously proved assertions, we have the following global classification result for forms over infinite global fields of characteristic = 2. Theorem 6.9 (Global classification). Let k be an infinite global field of characteristic = 2, ∞k,R the set of its real places, D a division algebra of center K with an involution J, √ where K = k (resp. K = k( a) and k = K J ), if J is of the first (resp. the second) kind. Let g, h be quadratic or J-(skew-)hermitian forms of dimension n and of the same type over D. (1) Assume that n ≥ 3 if h is of type D. Then the form h represents zero non trivially over k if and only if it does so locally everywhere. (2) Assume that the forms g and h are neither of type D, nor quadratic forms. Then they are equivalent over k if and only if they are so locally everywhere. (3) Assume that the forms g and h have the same determinant. Then they are equivalent over k if and only if they are so locally everywhere. (4) (The Classical Strong Hasse principle) Assume that n ≥ 5. Then the form h represents zero non trivially over k if and only if it does so locally everywhere over kv when v runs over the set of all real places ∞k,R . Proof. Only (4) needs a proof. Since dim(h) ≥ 5, from above we see that h represents 0 locally everywhere on non-archimedean localization fields. The fact that h represents 0 on kv for all v ∈ ∞k,R implies that h represents 0 on k(v), v ∈ ∞k,R , since these fields are the same. Now from (1) we deduce (4). 2 Remark 6.10. It is an open question to the see if the Strong Hasse principle for (skew-)hermitian forms of all type holds in the classical setting for other values of n.

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7. Injectivity of restriction maps and counter-examples to the Hasse principle for connected groups We have seen that the cohomological Hasse may fail for finite group schemes over infinite algebraic extensions of global fields (see Remark 2.7). It is natural to investigate the analogous phenomenon for connected algebraic groups over infinite global fields k. 7.1. Injectivity of the restriction map In order to tackle the problem mentioned above, we need some results regarding restriction maps for Galois cohomology. Let k be a field and let G be a connected reductive k-group. We say (after Sansuc [33, p. 14]) that another reductive k-group H is a special covering of G if H is a direct ˜ with an induced k-torus T and if product of a semisimple simply connected k-group G there exists a central k-isogeny π : H → G. The following technical result is very useful in dealing with isogenies and coverings, which generalizes a result due to T. Ono to any connected reductive group (see [33, Lemma 1.10]). Lemma 7.1 (Sansuc). Let k be a field, G a connected reductive k-group and let K/k be a Galois extension of k over which G is split. Then there are a positive integer m > 0, an induced k-torus Q which is split over K, another induced k-torus P and a semisimple simply connected k-group H such that there is a central k-isogeny H ×k P → Gm ×k Q, i.e., H ×k P is a special covering of the latter. In [33, Corollaire 4.6], there was shown a generalization of the Springer Theorem (on the anisotropicity of anisotropic quadratic forms via odd degree extensions) to a wide class of algebraic groups over number fields. The following result extends a result by Sansuc in the number field case to function field case and also the case of infinite algebraic extensions of global fields. We consider a class of fields F which consists of (possibly infinite) local and global fields k such that for any given natural number n, there exist finite extensions K/k of odd degree [K : k] which is co-prime with n. We say that k is of type F if k ∈ F. From the local and global class field theory (see [3, Theorem 5, p. 105], [9, Chapter II], [37, Chapter II]), we know that any non-archimedean local and global field belongs to F. The following extends [33, Corollary 4.6] to infinite local or global fields. Proposition 7.2. Let k be either a (possibly infinite) non-archimedean local or global field of type F and let G be a connected reductive k-group. Then there exists a natural number N , such that if K/k is any finite Galois extension of odd degree prime to N then the restriction map H1flat (k, G) → H1flat (K, G) is injective.

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Proof. We use the idea of the proof of [33, Corollary 4.6]. By Lemma 7.1, there are a natural number m, induced k-tori P, Q, and a semisimple simply connected k-group H and a central k-isogeny π : H ×k P → Gm ×k Q. It is clear that if the assertion of the proposition holds for Gm ×k Q, then the same is true for G. Thus without loss of any generality, we may assume that G has a special covering H ×k P . Let B := Ker(π). Then B is a finite commutative k-group scheme of multiplicative type and it becomes split over a finite Galois extension L/k; in particular, B L μn1 × · · · × μns . Let N := n1 · · · ns and consider the following exact sequence of k-groups 1 → B → H ×k P → G → 1. We need to consider the following cases. (a) k is a non-algebraically closed algebraic extension of a local field L. Since L is non-archimedean, according to a result due to Kneser and Bruhat–Tits and its extension (see Theorems 5.2, 5.3) we have H1flat (L, H ×k P ) = 1, H1flat (k, H ×k P ) = 1, thus we obtain an exact sequence Δ

1 → H1flat (k, G) →k H2flat (k, B). By using twisting we check that α is also injective. The same also holds if k is replaced by its finite extension K/k. We take any finite Galois extension K/k of odd degree n such that H2flat (k, B) has no n-torsion, for example (n, N ) = 1. Then the assertion is trivial in this case, by looking at the following commutative diagram 1 →

H1flat (k, G)

Δk

→

H2flat (k, B)

↓ rG 1 → H1flat (K, G)

↓ rB ΔK

→

H2flat (K, B)

due to the facts that rB is injective (by our choice of K and n) and that Δk and ΔK are injective (see above). (b) k is a non-algebraically closed algebraic extension of a global field. Step 1. The natural map ξ : H1flat (k, G) → H2flat (k, B) ×

 v∈∞k

is injective.

H1flat (kv , G)

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For, we have the following commutative diagram with exact rows q

H1flat (k, H ×k P )

→

H1flat (k, G)

↓β  v∈∞k

Δk

→

H2flat (k, B)

↓γ q

H1flat (kv , H ×k P )

→

 v∈∞k

↓ζ

H1flat (kv , G)



Δ

→

v∈∞k

H2flat (kv , B)

First we assume that k is an imaginary global field. Then by Harder’s Theorem (see [11, Haupsatz], [13, Satz A]), or by Theorem 5.8, we have H1flat (k, H ×k P ) = 1, since P is induced. Thus Δk has trivial kernel, and by a twisting argument, Δk is injective as well. Now we assume that ∞k,R = ∅ and let x be an element such that Δk (x) = γ(x) = 1. For simplicity, we assume that the set of real places ∞k,R of k is finite. (The general case follows from an argument used in the proof of Theorem 5.16 (or 5.18), namely by descending the k-torsor representing x to the one defined over a global field L contained in k.) We consider the following commutative diagram with exact rows p

→

H1flat (k, B)

H1flat (k, H ×k P )

↓α  v∈∞k

q

→

H1flat (k, G)

↓β

H1flat (kv , B)

p

→



↓γ q

v∈∞k

H1flat (kv , H ×k P ) →

 v∈∞k

H1flat (kv , G)

Then x = q(y), y ∈ H1flat (k, H ×k P ), so β(y) ∈ Ker(q  ) = Im(p ), say β(y) = p (b1 ),  b1 ∈ v∈∞k H1flat (kv , B). Since B is a finite algebraic group of multiplicative type, B can be embedded into an induced k-torus I. Consider the following exact sequence 1 → B → I → I/B → 1 and the following associated commutative diagram with exact rows, where S is any finite subset of ∞k I(k)

r

→

(I/B)(k)

↓ α  v∈S

I(kv )

s

→

H1flat (k, B)

↓ β r

→



v∈S (I/B)(kv )

δ

→

H1flat (k, I) = 1 ↓ ζ

↓α s

→

 v∈S

H1flat (kv , B)

δ

→

 v∈S

H1flat (kv , I) = 1

By a result of Serre (see [33, Corollaire 3.5(ii)]) I and I/B have the weak approximation property with respect to any finite subset of ∞k , so this implies that (I/B)(k) is dense in  the product v∈S (I/B)(kv ) via the diagonal mapping. Since the map r is an open map

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 with respect to S-topology, there is a natural topology on v∈S H1flat (kv , B), namely the discrete one, since each H1flat (kv , B) is finite. The fact that the image of β  is dense implies that so is the image of α. Together with the discreteness just discussed, it implies that α is surjective. Now we take S = ∞k,R , which is finite by assumption. Let b0 ∈ H1flat (k, B) such that α(b0 ) = b1 . Then β(y) = β(p(b0 )). Since H satisfies the cohomological Hasse principle (where k is either a global field (by the Kneser–Harder–Chernousov Theorem 5.1) or an infinite global field (by Theorem 5.7), β is an injection. Hence in the previous diagram, we have y = p(b0 ), so x = 1, i.e, ξ has trivial kernel. The usual argument of twisting shows that ξ is injective. Step 2. We take a finite Galois extension K/k of odd degree (n, N ) = 1 so that the finite algebraic group B has no n-torsion and consider the following commutative diagram H1flat (k, G)

ξ

→

×

H2flat (k, B)

 v∈∞k

H1flat (kv , G)

↓r

↓ rG ξ

H1flat (K, G) →

×

H2flat (K, B)

 w∈∞k

H1flat (Kw , G)

where r = (rB , r∞ ). Then for such K and n, the set ∞K,R is also finite and rB and r∞ are all injective. Now since ξ, ξ  are injective by Step 1, all of this implies that so is rG . 2 As an immediate consequence, we have the following result extending [33, Corollary 4.7, Corollary 4.8] to infinite global fields. Corollary 7.3. Let k be a (possibly infinite) non-algebraically closed non-archimedean local or global field of type F and let G be a connected reductive k-group. Let n1 , ..., ns > 1 be co-prime natural numbers and let ki /k be a finite extension of degree ni . Then the natural map H1flat (k, G) →



H1flat (ki , G)

i

is injective. In particular, every G-torsor which is defined over k having a rational point (resp. rational points) in K (in case (1)) (resp. in each of ki , in case (2)), has already a k-point. Proof. The proof is the same as in [33, Proof of Corollary 4.8] by using Proposition 7.2. 2

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Remark 7.4. In Proposition 7.2 and Corollary 7.3, if k is algebraically closed, then H1flat (k, G) = 1, so the assertion is “formally true”, though there are no non-trivial finite extensions K/k. 7.2. Construction of counter-examples to the cohomological Hasse principle By Theorem 5.7, if G is a connected affine algebraic group defined over an infinite global field k, for which the cohomological Hasse principle does not hold, then there exists some global subfield L ⊂ k, such that G is defined over L and does not satisfy the (classical) cohomological Hasse principle over L. There are many examples of connected groups over global fields for which the cohomological Hasse principle fails, cf. for example [6], [33], [38] etc. Here we prove the converse statement. Proposition 7.5. Let L be a global field, G a connected reductive L-group, such that the cohomological Hasse principle for H1 fails for G over L. Then there exists an infinite algebraic extension k/L such that G ×L k does not satisfy the cohomological Hasse principle for H1 over k. Proof. We apply Proposition 7.2 to construct such an infinite global field k and a connected k-group G such that the cohomological Hasse principle fails for H1 . Namely, let L be a global field over which a connected reductive L-group G fails to satisfy the cohomological Hasse principle. Set L = L0 . By Proposition 7.2, we may find an extension L1 /L0 of odd degree such that H1flat (L0 , G) → H1flat (L1 , G) is injective, and keep iterating this process. Then we obtain an infinite tower of extensions L = L0 ⊂ L1 ⊂ · · · ⊂ Ln ⊂ · · · inside a separable closure of L. Set k := ∪n Ln . Then we claim that for the k-group G, the cohomological Hasse principle fails over k. Indeed, by assumption, there is some G-torsor P defined over L such that P is trivial locally everywhere, but P is not trivial. We show that P is also non-trivial as a G-torsor defined over k. If not, let P (k) = ∅. By our construction, k = ∪n Ln , so for some m we have P (Lm ) = ∅, i.e., P is a trivial G-torsor, considered as a torsor over Lm , by base change. But the class [P ]m of P in H1flat (Lm , G) is just the image of the class [P ] of P in H1flat (L, G) via the restriction map 1 rLm /L : H1flat (L, G) → Hflat (Lm , G).

By the construction in the proof of Proposition 7.2, all these maps are injective, hence [P ]m = 0, since [P ] = 0, a contradiction. 2

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