On linear algebraic groups over pseudoglobal fields
Some properties of R–equivalence and weak approximation in linear algebraic group over global field are generalized to the case of linear algebraic group over an algebraic function field in one variable with pseudofinite constant field.
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irk-123456789-1523792019-06-11T01:25:28Z On linear algebraic groups over pseudoglobal fields Andriychuk, V. Some properties of R–equivalence and weak approximation in linear algebraic group over global field are generalized to the case of linear algebraic group over an algebraic function field in one variable with pseudofinite constant field. 2007 Article On linear algebraic groups over pseudoglobal fields / V. Andriychuk // Algebra and Discrete Mathematics. — 2007. — Vol. 6, № 4. — С. 11–22. — Бібліогр.: 19 назв. — англ. 1726-3255 http://dspace.nbuv.gov.ua/handle/123456789/152379 2000 Mathematics Subject Classification:11E72, 14G27, 20G35. en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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Some properties of R–equivalence and weak approximation in linear algebraic group over global field are generalized to the case of linear algebraic group over an algebraic function field in one variable with pseudofinite constant field. |
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Andriychuk, V. On linear algebraic groups over pseudoglobal fields Algebra and Discrete Mathematics |
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Andriychuk, V. |
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On linear algebraic groups over pseudoglobal fields |
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On linear algebraic groups over pseudoglobal fields |
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On linear algebraic groups over pseudoglobal fields |
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On linear algebraic groups over pseudoglobal fields |
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On linear algebraic groups over pseudoglobal fields |
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on linear algebraic groups over pseudoglobal fields |
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Інститут прикладної математики і механіки НАН України |
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On linear algebraic groups over pseudoglobal fields / V. Andriychuk // Algebra and Discrete Mathematics. — 2007. — Vol. 6, № 4. — С. 11–22. — Бібліогр.: 19 назв. — англ. |
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Algebra and Discrete Mathematics |
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AT andriychukv onlinearalgebraicgroupsoverpseudoglobalfields |
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2025-07-13T02:57:22Z |
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2025-07-13T02:57:22Z |
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1837498837742125056 |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Number 4. (2007). pp. 11 – 22
c© Journal “Algebra and Discrete Mathematics”
On linear algebraic groups
over pseudoglobal fields
Vasyl Andriychuk
Communicated by M. Ya. Komarnytskyj
Dedicated to Professor V. V. Kirichenko
on the occasion of his 65th birthday
Abstract. Some properties of R–equivalence and weak ap-
proximation in linear algebraic group over global field are general-
ized to the case of linear algebraic group over an algebraic function
field in one variable with pseudofinite constant field.
Let X be a smooth algebraic variety defined over a field K. Recall
that two points x, y ∈ X(K) are R-equivalent if there is a sequence of
points zi ∈ X(K), x = z1, . . . , y = zn, such that for each pair zi, zi+1
there exists a K-rational map fi : P1 → X, regular at 0 and 1, with
fi(0) = zi, fi(1) = zi+1, 1 ≤ i ≤ n − 1). We shall denote the set of R-
equivalence classes on X(K) by X(K)/R. If G is a linear algebraic group
defined over a field K, the set G(K)/R can be endowed with a natural
group structure.
Let V K be the set of all valuations of a field K, S be a finite subset of
V K , and G be a connected linear algebraic group defined over K. Denote
by G(K) the closure of G(K) in the product topology, where G(K) is
embedded diagonally into the direct product
∏
v∈S G(Kv), and G(Kv) is
endowed with the v-adic topology induced from that of Kv. One says that
G has weak approximation with respect to S if G(K) =
∏
v∈S G(Kv), and
G has weak approximation over K if it is so for any finite subset S ⊂ V K .
2000 Mathematics Subject Classification: 11E72, 14G27, 20G35.
Key words and phrases: linear algebraic group, Hasse principle, weak approx-
imation, Tate-Shafarevich group, algebraic function field.
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.12 On linear algebraic groups . . .
Let
AS(G) =
∏
v∈S
G(Kv)/G(K), A(G) =
∏
v∈V K
G(Kv)/G(K),
be the defect of weak approximation in S and over K respectively.
J.-L. Colliot-Thélène, P. Gille and R. Parimala [8] showed that many
arithmetical properties of linear algebraic groups and of their homoge-
neous spaces over totally imaginary number fields have counterparts over
the fields of one of the following types:
(gl) a function field K in two variables over an algebraically closed
field k of characteristic zero, i.e. the function field of a smooth, projective,
connected surface over K;
(ll) the field of fractions K of a henselian, excellent, two-dimensional
local domain A with an algebraically closed residue field k of characteristic
zero;
(sl) the Laurent series field l((t)) over a field l of characteristic zero
and cohomological dimension 1.
All these fields K have the folloving properties:
(I) their cohomological dimension is two;
(II) index and exponent of central simple algebras over K coincide;
(III) H1(K, G) = 1 for any semisimple simply connected group G over
K.
J.-L. Colliot-Thélène, P. Gille and R. Parimala also proved in [8] that
properties (I) and (II) imply (III) for groups without E8-factors. More-
over, they proved that for any semisimple simply connected linear alge-
braic group G defined over a field K satisfying properties (I) and (II)
(and cohomological dimension cd(Kab) of maximal abelian extension of
K satisfies cd(Kab) ≤ 1 if factors of type E8 are allowed), the group
G(K)/R is trivial.
In this paper we consider the linear algebraic groups over an algebraic
function field K in one variable with a pseudofinite [5] constant field k.
We call such a field K pseudoglobal. Recall that a perfect field k is called
pseudofinite if it has exactly one extension of degree n for every natural
number n, and if every absolutely irreducible affine variety defined over k
has a k-rational point. For example, the following fields are pseudofinite
[12]:
i) the infinite extension of a finite field having exactly one extension
of each degree n;
ii) the fixed field Q(σ) in the algebraic closure Q of Q for almost all (in
the sense of the canonical Haar measure on Gal(Q/Q)) automorphisms
σ ∈ Gal(Q/Q);
iii) the infinite models of the first-order theory of finite fields.
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.V. Andriychuk 13
In particular, these examples show that there exist the pseudoglobal
fields of characteristic zero as well as of positive characteristic.
The important feature of pseudoglobal fields is that for these fields
there is an analogue (cf. [15], [14], [2]) of the classical global class
field theory (see also [11] for such an analogue for more wide class of
fields, namely for algebraic function fields in one variable over pseudo-
algebraically closed constant fields). In particular, the Hasse principle
for Brauer group holds for a pseudoglobal field K, i.e. the canonical map
BrK →
∏
v∈V K BrKv is injective, where BrK (resp. BrKv) is the Brauer
group of K (resp. the Brauer group of Kv, Kv being the completion of
K at a valuation v), v runs over the set V K of all (trivial on the constant
field) valuations of K. Also, the Tate-Nakayama theorems for algebraic
tori hold for a pseudoglobal field and all its completions.
In order to adapt the results of J.-L. Colliot-Thélène, P. Gille and
R. Parimala [8] to the case of pseudoglobal ground field we need some
relevant facts about pseudoglobal fields. Let us enumerate them in the
following theorem.
Theorem 1. A pseudoglobal field field K has the following properties.
(a) K has the C2 property: for any r ≥ 1 and any system of homoge-
neous forms fj(X1, . . . , Xn) ∈ K[X1, . . . , Xn], j = 1, . . . , r, if n >
∑
j d2
j ,
where dj is the degree of fj, then there is a nontrivial common zero in
Kn for these r forms.
(b) The cohomological dimension cd(K) of K is 2.
(c) The reduced norm map Nrd : D∗ → L∗ is surjective for any finite
field extension L/K and any central simple algebra D/L.
(d) Every central simple algebra A over K is cyclic, and the index
and exponent of A coincide.
(e) For a given prime p, over any finite field extension L of K, the
tensor product of two central simple algebras of index p has index at most
p.
(f) Over any finite field extension L/K, any quadratic form in at least
5 variables has a nontrivial zero.
(g) The cohomological dimension cd(Kab) of the maximal abelian ex-
tension of K is 1.
Proof. There are some relations between properties stated in Theorem 1.
In particular (cf. [8], Theorem 1.1) for a field K of characteristic zero
the following implications hold: (i) (a) implies (b) and (f); (ii) (b) is
equivalent to (c); (iii) (d) is equivalent to (e). So it is sufficient to prove
(a), (d), (f) and (g).
The properties (a) and (d) were proved in [3], and (f) was proved in
[4]. Recall briefly the argument from [3]. In [5] J. Ax showed that every
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.14 On linear algebraic groups . . .
pseudo algebraically closed field with abelian absolute Galois group is C1.
Next, it is known that if k is a Ci-field, and K is an extension of k of
transcendence degree n, then K is a Ci+n-field. Hence a pseudoglobal
field is a C2-field.
To prove (d) we apply the argument which essentially follows the
classical case of algebras over global fields. Let A be a finite-dimensional
central simple algebra over a pseudoglobal field K. Let v1, . . . , vr be all
the valuations of K at which A has nontrivial local invariants. Set ni =
indAvi
, Avi
= A⊗KKvi
, where Kvi
is the completion of K at vi. Let m be
a smallest common multiple of n1, . . . , nr. For all i, 1 ≤ i ≤ r, ni|n, where
n = degA = [A : K]
1
2 , thus we have m|n. By Saltman’s theorem ([16],
Theorem 5.10) for any abelian group G if Li/Kvi
are G Galois extension,
then there is a G Galois extension L/K such that L⊗K Kvi
= Li. Thus
we may suppose that there are cyclic extensions L/K and M/K of degree
m and n respectively, such that L/K and Li/Kvi
are cyclic of degree m,
and M/K and Mi/Kvi
are cyclic of degree n. We can take Li and Mi to
be the unramified extension of Kvi
of degrees m and n respectively. Then
taking into account the class field theory for general local field [18] one
sees that ni is the invariant of the algebra Avi
, 1 ≤ i ≤ r. In view of class
field theory for pseudoglobal field (see [3] for more details) A splits over
L and M , and the field M is isomorphic to a strongly maximal subfield
of A, hence the algebra A is cyclic.
It remains to prove that indA = expA. Since expA|indA, it is enough
to prove that indA ≤ expA. Since the field L splits A, indA ≤ m.
Further, if e = expA, then i(e·[A]) = e·(i[A]) = 0, where [A] ∈ BrK is the
corresponding element of the Brauer group of K, and i : BrK →֒ ⊕vBrKv
is the monomorphism [A] 7→
∑
v[A ⊗K Kv] [3]. Denoting by invvA the
local invariant of A at v, we have e · invvA = 0 for all valuation v of V K .
Hence ni|e, 1 ≤ i ≤ r, and m|e. Thus indA ≤ m ≤ e = expA, and this
completes the proof of (d).
As for (g), we note that since all finite extensions of the constant
field k of K are abelian (in fact cyclic), the maximal abelian extension
of K contains the subfield ksK (ks is the separable closure of k in Kab),
cd(ksK) = 1 by Tsen’s theorem. Thus Kab being an algebraic extension
of ksK has cohomological dimension 1.
Taking into account these results, we get the following results about
linear algebraic groups over a pseudoglobal field stated as Corollaries 1 –
3 below.
Corollary 1. Let K be a pseudoglobal field of characteristic zero. Let G
be a connected linear algebraic group defined over K.
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.V. Andriychuk 15
(i) If G is a simply connected group, then H1(K, G) = 1.
(ii) Let G be a semisimple simply connected group, and let 1 → µ →
G → Gad → 1 be the central isogeny associated to the center µ of G.
Then:
(a) the boundary map δ : H1(K, Gad)→ H2(K, µ) is a bijection;
(b) if the group G is not purely of type A, then it is isotropic.
Proof. Taking into account Theorem 1 (d) and (f), the property (i) follows
from Theorem 1.2 (v) in [8] which says that if index and exponent coincide
for 2-primary and 3-primary algebras over finite field extension of K and
the cohomological dimension of Kab is at most one, then H1(K, G) =
0. Further, Theorem 2.1 in [8] asserts that if G is a semisimple simply
connected group defined over a field K of characteristic zero, cd(K) ≤ 2,
and indA = expA for every central simple algebra A over any finite field
extension of K, then G has the properties stated in (ii). Hence, applying
Theorem 1 (b) and (d) completes the proof.
Let K be a field, T be a K-torus, and let
1 −→ F −→ P −→ T −→ 1 (1)
be a flasque resolution of T , where P is a quasitrivial torus and F is a
flasque torus.
The next two corollaries concern the linear algebraic groups G admit-
ting a special covering, i.e. there exists an exact sequence
1 −→ µ −→ G′ −→ G −→ 1, (2)
where G′ is the product of a semisimple simply connected group and a
quasitrivial torus, and G′ → G is an isogeny with kernel µ. It is known
[9], that the group µ has a flasque resolution
1 −→ µ −→ F −→ P −→ 1, (3)
where the torus F is flasque and the torus P is quasitrivial.
Corollary 2. Let G be a connected linear algebraic group defined over a
pseudoglobal field K of characteristic zero. Then
(i) The quotient G(K)/R is a finite abelian group.
(ii) If G has a special covering (2), then G(K)/R ≃ H1(K, F ), where
F is the flasque torus from (3).
Proof. Since the cohomological dimension of a pseudoglobal K field is
2, index and exponent coincide for central simple algebras over K, and
cd(Kab) = 1 we may apply Corollary 4.10 in [8] which asserts that
G(K)/R ≃ H1(K, F ). Thus G(K)/R is a finite abelian group because
H1(K, F ) is a finite abelian group [3].
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.16 On linear algebraic groups . . .
Corollary 3. (i) Let G be a connected linear group defined over a pseu-
doglobal field K of characteristic zero. Let S ⊂ V K be a finite set. Then
the closure G(K) of the image G(K) under the diagonal map G(K) →∏
v∈S G(Kv) is a normal subgroup, and the quotient
AS(G) =
∏
v∈S
G(Kv)/G(K)
is a finite abelian group.
(ii) Suppose that G has a special covering (2). The composite maps
G(K) → H1(K, µ) → H1(K, F ) and G(Kv) → H1(Kv, µ) → H1(Kv, F )
induce isomorphisms of finite abelian groups
AS(G) ≃ Coker
[
H1(K, F ) −→
∏
v∈S
H1(Kv, F )
]
and
AS(G) ≃ Coker[G(K)/R −→
∏
v∈S
G(Kv)/R
]
.
Proof. The examination of the proof of Theorem 4.13 in [8] shows that
the stated properties hold whenever the field K satisfies the conditions:
cd(K) ≤ 2, cd(Kab) ≤ 1, index and exponent coincide for 2-primary
and 3-primary central simple algebras All these conditions hold for a
pseudoglobal field by Theorem 1. The Corollary follows.
As in the case of a number ground field, for a connected reductive
group G defined over a pseudoglobal field K the defect A(G) of weak
approximation and the Tate-Shafarevich group
Ш(G) = Ker
(
H1(K, G) −→
∏
v∈V K
H1(Kv, G)
)
can be inserted in a short exact sequence. In order to state and prove the
corresponding result we will need some more properties of pseudoglobal
fields. In particular, we will need the following result which can be re-
garded as an analogue of Čebotarev density theorem for pseudoglobal
fields. Its proof presented in [V. Andriychuk, An analogue of Tcheb-
otarev’s density theorem for pseudoglobal fields // Visnyk Kyiv. un-tu,
Seriya phiz.-mat. (2000), 4, p. 11-16] is shaped from argument used by
M. Fried [13] for algebraic function field with finite constant field.
Theorem 2. Let L/K be a finite Galois extension of a pseudoglobal field
K, G = Gal(L/K) and τ ∈ G. There exists an infinite set of non equiv-
alent valuation of the field L, whose decomposition group is the cyclic
subgroup generated by τ .
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.V. Andriychuk 17
Let M be a finite Gal(Ksep/K)-module over a pseudoglobal field K,
M̂ = Hom(M, K∗
sep) be its dual module, S ⊂ V K be a finite subset of
valuations of the field K,
Ш1
S(M) = Ker(H1(K, M) −→
∏
v/∈S
H1(Kv, M)),
Ч1
S(M) = Coker(H1(K, M) −→
∏
v∈S
H1(Kv, M)),
Ш1
ω(M) = lim
−→
S
Ш1
S(M), Ч1
ω(M) = lim
←−
S
Ч1
S(M), Ш1(M) = Ш1
∅(M).
The following theorem 3 enumerates some properties of these groups
which are counterparts of corresponding properties of finite modules over
a global field.
Theorem 3. Let M be a finite module over a pseudoglobal field K,
(|M |, charK) = 1.
1) The finite group Ч1
S(M̂) is isomorphic to the dual group of the
group Ш1
S(M)/Ш1(M).
2) The finite groups Ч1
ω(M̂) and Ш1
ω(M)/Ш1(M) are dual each an-
other.
3) There exists the exact sequence of finite groups
0 −→ Ч1
ω(M̂) −→Ш1
ω(M)∗ −→Ш1(M)∗ −→ 0,
where A∗ = Hom(A, Q/Z).
4) Ч1
ω(M) = Ч1
S0
(M), where S0 is the finite subset of valuations of
the field K consisting of ramified valuations with non cyclic decomposition
groups in a finite Galois extension, over which the module M̂ becomes
trivial.
5) If S consists of the valuations having cyclic decomposition groups
in a finite Galois extension, over which the module M̂ becomes trivial,
then ЧS(M) = 0.
To prove this theorem we need the following fact about valuations of
a pseudoglobal field.
Lemma 1. Let L/K is a finite Galois extension, and L 6= K. Then there
exists an infinite set of valuations v of K which does not split completely
in L. In other words, if Lw = Kv for almost all valuations of K (w is
the extension of v to L), then L = K.
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.18 On linear algebraic groups . . .
Proof. For a global field this is Theorem 2 in [6], see also [1], Corollary 8.8.
The proof reduces to the case where L/K is abelian. Let G = Gal (L/K).
Denote by CK and CL the idele class groups of K and L respectively.
Since the idele classes of a pseudoglobal field form a class formation, ([2],
Theorem 1), we have CK/NL/KCL ≃ G. Like the case of global ground
field ([1], p. 275) it follows that for every finite subset S of valuations
of K containing all valuations ramified in L, the group G is generated
by elements σv ∈ Gal (Lw/Kv) ⊂ G for v /∈ S, w is an extension of
v to L. Here σv is the image of σ[k(v):k]|k(w) ∈ Gal (k(w)/k(v)) under
isomorphism Gal (k(w)/k(v)) ≡ Gal (Lw/Kv) ⊂ G, σ is the generator
of the absolute Galois group of constant field k, and k(w)/k(v) is the
corresponding extension of residue field of Kv. Suppose, contrary to our
assertion, that there exist only finitely many valuations of K which not
split completely in L, then adding them to S, we would have that all σv
are trivial for v /∈ S, so they cannot generate G.
Proof of Theorem 3. Note, that the proof of this theorem is carried out
by the argument analogous to used in the case of a number ground field.
First we show, that all the group indicated in the statement of Theorem 3
are finite. For any valuation v ∈ V K the completion Kv is a general local
field. Hence the group H1(Kv, M) and H1(Kv, M̂) are finite by II.5.2 in
[18], and the finiteness of groups Ч1
S(M) and Ч1
S(M̂) follows immediately
from their definitions.
Consider the group Ш1
S(M). If M is a trivial Gal(Ksep/K)-module,
then Ш1
S(M) = 0. Indeed, since the groups H1(K, M) and H1(Kv, M)
are inductive limits relative to the inflation homomorphisms of groups
H1(G, M) and H1(Gv, M̂), where G (respectively Gv) is the Galois group
of finite Galois subextensions L/K (respectively Lw/Kv, w is an exten-
sion of valuation v to the field L), it suffices to prove that the groups
Ш(L/K, Z/nZ) = Ker(H1(G, Z/nZ) −→
∏
v/∈S
H1(Gv, Z/nZ)) =
= Ker(Hom(G, Z/nZ) −→
∏
v/∈S
Hom(Gv, Z/nZ))
are trivial for all finite Galois extensions L/K. If a homomorphism f lies
in Ш1(L/K, Z/nZ), then f defines a cyclic subextension K ′ = LKerf of
the field L. The restrictions fv of the homomorphism f to the groups
Gv are trivial. This means that K ′
w′ = Kv, where w′ is an extension of
the valuation v to K ′. Lemma 1 implies that K ′ = K, so Kerf = G
and f = 0. Now let M be an arbitrary finite Gal(Ksep/K)-module.
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.V. Andriychuk 19
The module M becomes trivial over some finite Galois extension L/K.
Consider the following commutative diagram
H1(L/K, M)
�
�
//
α
��
H1(K, M) //
��
H1(L, M)
��∏
v 6∈S
H1(Lw/Kv, M) �
�
//
∏
v 6∈S
H1(Lw/Kv, M) res
//
∏
v 6∈S
H1(Lw, M)
with exact rows (here w is an extension of valuation v to the field L).
In this diagram the left vertical homomorphism has trivial kernel by the
preceding argument, and therefore Ш1
S(M) = Kerα = Ш1
S(L/K, M) is a
finite group.
Theorem 2 implies that after eliminating from the set S a finite subset
of valuations with cyclic decomposition group, for the obtained set of
valuations S′ we will have Ш1
S′(M) = Ш1
S(L/K, M) = Ш1
S(M), and
therefore Ш1
ω(M) = Ш1
S0
(M), where S0 is the finite set of valuations
ramified in L, and with non cyclic decomposition group. In particular,
Ш1
ω(M) is a finite group.
Denote by
∏′
v∈V K H1(Kv, M) the restricted topological product of
the groups H1(Kv, M) relative to the subgroups H1
un(Kv, M). It is proved
in [10] that when the constant field of algebraic function field K is quasi-
finite field of formal power series k0((t)) over an algebraically closed field
k0 of characteristic 0, then there exists an exact sequence
H1(K, M) −→
∏
v∈V K
′
H1(Kv, M) −→ H1(K, M̂)∗,
where H1(K, M̂)∗ = Homcont(H
1(K, M̂), Q/Z). By using the argument,
analogous to that of [10] we get the exact sequence
H1(K, M)
β
−→
∏
v/∈S
′
H1(Kv, M)×
∏
v∈S
H1(K, M)
γ
−→ H1(K, M̂)∗
for finite modules M over a pseudoglobal field K.
As in the classical case of a number field [17], it follows that the image
of the group Ш1
S(M) under homomorphism β, is isomorphic to the group
Ш1
S(M)/Ш1(M) which is dual to the kernel of homomorphism
H1(K, M̂)
γ̃
−→
∏
v∈S
H1(Kv, M)
Here we use the duality of groups H1(Kv, M) and H1(Kv, M̂) (see [19],
p. 113, exercise 2). Therefore, the group Cokerγ̃ ≃ Ч1
S(M̂) is dual to
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.20 On linear algebraic groups . . .
the kernel of the restriction homomorphism γ to the group
∏
v∈S Kv, M),
which is isomorphic to the group Ш1
S(M)/Ш1(M), and assertion 1) of
Theorem 3 is proved. Now we have the following exact sequence
0 −→ Ч1
S(M̂) −→Ш1
S(M)∗ −→Ш1(M)∗ −→ 0.
Passing in this sequence to the projective limit we obtain assertions 2) and
3) of Theorem 3. The equality Ш1
ω(M) = Ш1
S0
(M) follows from Theorem
2, and assertion 2), and the change of M by M̂ implies assertion 4).
Finally, it follows from Theorem 2 that for every v ∈ V K unramified in a
Galois extension L/K there exist infinitely many valuations of K having
the same decomposition group as v, which proves assertion 5).
Corollary 4. Let S be a finite subset of the set of all valuations of a
pseudoglobal field K. Then
1) Ш1
S(L/K, M) is a finite group;
2) H1(KS , M) is a finite group;
3) Ш1
S(K, Z/nZ) = Ш1
S(K, µn) = Ш1(K, µn) = 0, where Z/nZ is a
trivial GK-module, and µn is the group of n-th roots of 1 in the field K,
(n, charK) = 1.
Proof. 1) The finiteness of the group Ш1
S(L/K, M) was proved in the
course of the proof of Theorem 3.
2) By [ [14], Lemma 4.8, p. 68-69] the kernel of the group H1(KS , M)
under localization homomorphism is contained in the group∏
v∈S H1(Kv, M). Since the set S is finite, H1(Kv, M) is a finite group
and the kernel of Ш1
S(KS/K, M) of localization homomorphism is finite,
thus the group H1(KS/K, M) is finite as well.
3) The equality Ш1
S(K, Z/nZ) = 0 was proved in the course of the
proof of Theorem 3. The triviality of the group Ш1
S(K, µn) follows from
assertion 5) of Theorem 3. Finally, the triviality of the group Ш2(K, µn)
follows from the monomorphism 0→ BrK →
∏
v BrKv (see [4]).
Theorem 4. Let G be a connected reductive group defined over a pseu-
doglobal field K. Suppose that G is endowed with a special covering
G′ → G with kernel µ, for example a semisimple connected group with
fundamental group µ. The covering G′ → G define an exact sequence of
finite group
0 −→ A(G) −→Ш1(k, µ̂)∗ −→Ш(G) −→ 0. (4)
As in the classical case (see [17]) the proof of Theorem 4 is based on
the following three facts:
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.V. Andriychuk 21
1. A(G) = Ч1
ω(µ) for connected reductive group defined over a pseu-
doglobal field K and endowed with a special covering G′ → G with
kernel µ
2. The map Ш(K, G)
∂
→ Ш2(K, µ), defined by special k-covering
G′ → G with kernel µ, is a bijection.
3. There is an exact sequence
0 −→Ш1
ω(K, µ̂) −→Ш1
ω(K, µ)∗ −→Ш1(k, µ)∗ −→ 0. (5)
Proof. The equality A(G) = Ч1
ω(µ) follows from Corollary 3 taking into
account the exact sequences (2) and (3). The bijection Ш(K, G)
∂
→
Ш2(K, µ) follows from Corollary 2, and the exact sequence (5) follows
from Theorem 3.
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Contact information
V. Andriychuk Department of Mathematics and Mechanics,
Lviv Ivan Franko University, Lviv, Ukraine
E-Mail: v−andriychuk@mail.ru
Received by the editors: 18.10.2007
and in final form 10.04.2008.
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