PRESCRIBED BEHAVIOR OF CENTRAL SIMPLE ALGEBRAS AFTER SCALAR EXTENSION

ULF REHMANN, SERGEY V. TIKHONOV, AND VYACHESLAV I. YANCHEVSKIĬ

Abstract. 1. Let A_{1}, …, A_{n} be central simple disjoint algebras over a field F. Let also l_{i}|exp(A_{i}), m_{i}|ind(A_{i}), l_{i}|m_{i}, and for each i = 1, …, n, let l_{i} and m_{i} have the same sets of prime divisors. Then there exists a field extension E/F such that exp(A_{i}_{E}) = l_{i} and ind(A_{i}_{E}) = m_{i}, i = 1, …, n.

2. Let A be a central simple algebra over a field K with an involution τ of the second kind. We prove that there exists a regular field extension E/K preserving indices of central simple K-algebras such that A ⊗_{K} E is cyclic and has an involution of the second kind extending τ.

Introduction and motivations

This paper is a continuation of [18] where some properties of central simple algebras after scalar extensions were examined. In [18] we solved two problems.

1. For a given central simple K-algebra A, some K-variety X was constructed such that for a field extension L/K the variety X has an L-rational point iff A ⊗_{K} L has some prescribed properties (e.g., being a symbol-algebra).

2. For a given central simple K-algebra A, a regular field extension E/K was constructed preserving indices of all central simple K-algebras, such that A⊗_{K} E becomes a cyclic algebra. Note that if a field extension E/K preserves indices of all central simple K-algebras then E/K preserves exponents for all such K-algebras, but in some applications one needs to reduce exponents and indices of algebras in a prescribed manner. Below we fix the following notations and conventions. Let A be a finite dimensional central simple algebra over a field F. By Wedderburn’s theorem, there is a unique integer m ≥ 1 and a central division F -algebra D which is unique up to F -isomorphism such that A ≅ M_{m}(D). The degree of A is defined by deg(A) = \sqrt{dim_{F} A}, the index of A is said to be ind(A) = deg(D). Two central simple F -algebras A = M_{m}(D) and A′ = M_{m′}(D′) are said to be Brauer equivalent if D ≅ D′. In this case we write A ∼ A′ and denote the equivalence class of A by [A]. The tensor product of central simple algebras defines an abelian group structure on this set of equivalence classes, called the Brauer group of F and denoted by Br(F). The inverse of the class [A] is induced by the opposed algebra A^{op} of A. A^{m} will denote the central simple algebra A ⊗ ⋯ ⊗ A (m times).

Date: October 9, 2011.

The second and the third authors are grateful to the Department of Mathematics at the University of Bielefeld for the hospitality during the preparation of the paper.

The neutral element is defined by the class A ∼ F, in this case we write A ∼ 1. The exponent exp(A) of A in Br(F) is the order of [A] in Br(F). It is known that exp(A) and ind(A) have the same prime divisors and exp(A)|ind(A) [17, §14.4, Prop. b]. For a field extension K/F, A_{K} will denote the K-algebra A⊗_{F} K. If [K : F] is coprime to ind(A), then ind(A_{K}) = ind(A) [17, §13.4, Prop.]. Let us recall three special types of central simple algebras: Crossed products (L/F, Gal(L/F), f). Let L/F be a Galois field extension, Gal(L/F) its Galois group and f a 2-cocycle of Gal(L/F) with values in L^{∗}. Then the left L-module with L-base {u_{τ}}_{τ∈Gal(L/F)} and multiplication table

u_{s}l = l^{s}u_{s} for l ∈ L, u_{s}u_{t} = f(s, t)u_{st} for any s, t ∈ Gal(L/F)

is a central simple F -algebra and denoted by (L/F, Gal(L/F), f). Cyclic algebras (E/F, σ, a). They are a special form of crossed products. Let E/F be a cyclic field extension of degree n, σ a generator of Gal(E/F) and a ∈ F^{∗}. Then (E/F, σ, a) is a left E-module with E-base {u_{σ}^{i}}_{i=1,...,n} and multiplication table:

u_{σ}^{i}c = c^{σ^{i}}u_{σ}^{i}

and

u_{σ}^{n} = a

for any i = 0, …, n − 1 and c ∈ E. The corresponding cocycle is the following

c_{a}(σ^{i}, σ^{j}) = \{ \begin{matrix}1, & if i + j < [E : F]; \\ a, & if i + j ≥ [E : F]. \end{matrix}

Symbol algebras (a, b)_{n}. These algebras also have a simple set of generators and defining relations. Let ρ_{n} ∈ F be a primitive root of unity of degree n and a, b ∈ F^{∗}. Then (a, b)_{n} is an n^{2}-dimensional vector F -space with an F -base

{A^{i}B^{j}}_{i,j=1,...,n}

and multiplication table

A^{i}B^{j} = ρ_{n}^{ij}B^{j}A^{i}, A^{n} = a, B^{n} = b.

Following some arguments from [12] we prove in this paper, for disjoint algebras (see the Definition 1.1 below), the following

Theorem 1. Let A_{1}, …, A_{n} be central simple disjoint algebras over F. Let also l_{i}|exp(A_{i}), m_{i}|ind(A_{i}), l_{i}|m_{i} such that, for each i = 1, …, n, both numbers l_{i}, m_{i} have the same prime divisors. Then there exists a regular finitely generated field extension E/F such that exp(A_{i}_{E}) = l_{i} and ind(A_{i}_{E}) = m_{i}, i = 1, …, n.

The remaining part of the paper is devoted to algebras with involutions. Using ideas similar to those in [18] we prove the following

Theorem 2. Let A be a central simple algebra over a field K with an involution τ of the second kind. Then there exists a regular field extension E/K preserving indices of central simple K-algebras such that A_{E} is cyclic and has an involution of the second kind extending τ.

In particular, this theorem has applications to a unitary variant of Suslin’s conjecture. To formulate this conjecture we will recall a few notions. The notion of R-equivalence in the set X(F) of F -points of an algebraic variety defined over a field F was introduced by Manin in [14] and studied firstly for linear algebraic groups by Colliot-Thélène and Sansuc in [7] (See also [5], [8], [16], [22].) It is an important birational invariant of an algebraic variety defined over an arbitrary field F. For an algebraic group G defined over a field F, the subgroup RG(F) of R-trivial elements in the group G(F) of all F points is defined as follows. An element g belongs to RG(F) if there is a rational morphism f : 𝔸_{n}^{1} → G over F, defined at the points 0 and 1 such f(0) = 1 and f(1) = g. In other words, g can be connected with the identity of the group by the image of a rational curve. The subgroup RG(F) is normal in G(F) and the factor group G(F)/RG(F) = G(F)/R is called the group of R-equivalence classes. The group G is called R-trivial, if the group of R-equivalence classes G(L)/R is trivial for any field extension L/F. Let K/F be a quadratic field extension and let A be a central simple algebra over K with an involution τ of the second kind trivial on F. Let U(A, τ) be the unitary group of A. Let also SU(A, τ) be the special unitary group, that is, the set of elements of U(A, τ) with reduced norm 1. It is known that if ind(A) is square-free, then SU(A, τ)/R = 1 ([5], [16], [24], [25], [26], [27]). In the case ind(A) is not square-free, a unitary variant of Suslin’s conjecture states that the group SU(A, τ) is not R-trivial. Since SU(A, τ)_{K} ≅ SL_{1,A}, it follows that this conjecture is true if ind(A) is divisible by 4 ([5, Remark 6.6]). The latter isomorphism says also that Suslin’s conjecture about reduced Whitehead groups implies the conjecture above. Thus Theorem 2 allows to reduce the conjecture about special unitary groups to algebras of a special type. Acknowledgment: The authors are grateful to the referee for careful reading and useful hints. In particular, he pointed out a gap in our first draft of the proof of Thm. 1.

1. Reducing exponent

In this section we show that for disjoint algebras the exponents and indices can be reduced in a prescribed manner over some field extension. We need the following definitions and facts.

Definition 1.1. ([11, Def.2.5]) The central simple F -algebras A_{1}, …, A_{n} are called disjoint if

ind(A_{1}^{j_{1}} ⊗ ⋯ ⊗ A_{n}^{j_{n}}) = ind(A_{1}^{j_{1}}) … ind(A_{n}^{j_{n}})

for all j_{1}, …, j_{n}.

Proposition 1.2. Let A be a central simple algebra over a field F and E the function field of the generalized Severi-Brauer variety SB_{n}(A), n ≤ deg(A). Then

(i) ([2, Th.7]) the relative Brauer group Br(E/F) is generated by the class of A^{n} in Br(F);

(ii) ([2, Th.3]) ind(A ⊗_{F} E) = gcd(n, ind(A)).

Proposition 1.3. ([23, Th.2]) Let A be a central simple algebra over a field F, deg(A) = d and s|d. Let also E be the function field of the generalized Severi-Brauer variety SB_{s}(A). Then for any central simple F -algebra D,

ind(D ⊗_{F} E) = gcd \{\frac{s}{gcd(i, s)} ind(D ⊗_{F} A^{−i}) | 1 ≤ i ≤ d\}

= min\{\frac{s}{gcd(i, s)} ind(D ⊗_{F} A^{−i}) | 1 ≤ i ≤ d\}.

Remark: This was proved in [23, Th.2]), however, the fact that the gcd is actually a min (which is more or less obvious in our examples below) has been pointed out in [15, see (0.3), p. 520 and (5.11), p. 565].

In order to prove Theorem 1 we need the following preliminary

Proposition 1.4. Let A_{1}, …, A_{n} be central simple algebras over F, deg(A_{i}) = d_{i}, i = 1, …, n, and s_{i}|d_{i}. Let also E_{i} be the function field of the generalized Severi-Brauer variety SB_{s_{i}}(A_{i}), i = 1, …, n, and E_{1} ⋯ E_{n} the free composite over F. Then for any central simple F -algebra D,

ind(D ⊗_{F} E_{1} ⋯ E_{n}) =

= gcd \{\frac{s_{1}}{gcd(j_{1}, s_{1})} ⋯ \frac{s_{n}}{gcd(j_{n}, s_{n})} ind(D ⊗_{F} A_{1}^{−j_{1}} ⊗_{F} ⋯ ⊗_{F} A_{n}^{−j_{n}}) | 1 ≤ j_{i} ≤ d_{i}\}.

Proof. We will use induction on n. In the case n = 1 the statement follows from Proposition 1.3. Suppose that the statement of proposition is true for n = n_{0}, i.e., for any field K and central simple K-algebras B, C_{1}, …, C_{n_{0}}, c_{i}|deg(C_{i}), 1 ≤ i ≤ n_{0}, the following holds:

ind(B ⊗_{K} L) =

= gcd \{\frac{c_{1}}{gcd(j_{1}, c_{1})} ⋯ \frac{c_{n_{0}}}{gcd(j_{n_{0}}, c_{n_{0}})} ind(B ⊗_{K} C_{1}^{−j_{1}} ⊗_{K} ⋯ ⊗_{K} C_{n_{0}}^{−j_{n_{0}}}) | 1 ≤ j_{i} ≤ deg(C_{i})\}

where L is the free composite over K of the function fields of generalized Severi-Brauer varieties SB_{c_{i}}(C_{i}), 1 ≤ i ≤ n_{0}. Consider the case n = n_{0} + 1. By Proposition 1.3,

ind(D ⊗_{F} E_{1} ⋯ E_{n_{0}+1}) =

= gcd \{\frac{s_{n_{0}+1}}{gcd(j_{n_{0}+1}, s_{n_{0}+1})} ind(D_{E_{1}···E_{n_{0}}} ⊗_{E_{1}···E_{n_{0}}} A_{n_{0}+1}^{−j_{n_{0}+1}}_{E_{1}···E_{n_{0}}}) | 1 ≤ j_{n_{0}+1} ≤ d_{n_{0}+1}\}

By induction hypothesis, for a fixed j_{n_{0}+1},

ind(D_{E_{1}···E_{n_{0}}} ⊗_{E_{1}···E_{n_{0}}} A_{n_{0}+1}^{−j_{n_{0}+1}}_{E_{1}···E_{n_{0}}} ) =

= gcd \{\frac{s_{1}}{gcd(j_{1}, s_{1})} ⋯ \frac{s_{n_{0}}}{gcd(j_{n}, s_{n_{0}})} ind(D⊗_{F} A_{n_{0}+1}^{−j_{n_{0}+1}} ⊗_{F} A_{1}^{−j_{1}} ⊗_{F} ⋯⊗_{F} A_{n_{0}}^{−j_{n_{0}}} ) | 1 ≤ j_{i} ≤ d_{i}\}.

Combining the latter formulas we obtain the statement of the proposition. □

Now we are in a position to prove Theorem 1. Proof of Theorem 1. Let E_{i} be the function field of the generalized Severi-Brauer variety SB_{m_{i}}(A_{i}), and let F_{i} be the function field of the Severi-Brauer variety SB(A_{i}^{l_{i}}) = SB_{1}(A_{i}^{l_{i}}). By Proposition 1.4,

ind(A_{1}^{j}_{E_{2}F_{2}···E_{n}F_{n}}) =

= gcd\{ \frac{m_{2}}{gcd(j_{2}, m_{2})} ⋯ \frac{m_{n}}{gcd(j_{n}, m_{n})} ind(A_{1}^{j} ⊗ A_{2}^{−j′_{2}−l_{2}j_{2}} ⊗ ⋯ ⊗ A_{n}^{−j′_{n}−l_{n}j_{n}}) |

1 ≤ j′_{i} ≤ deg(A_{i}), 1 ≤ j_{i} ≤ deg(A_{i}^{l_{i}}) }

for all j. Now,

ind(A_{1}^{j} ⊗ A_{2}^{−j′_{2}−l_{2}j_{2}} ⊗ ⋯ ⊗ A_{n}^{−j_{n}−l_{n}j_{n}}) = ind(A_{1}^{j})ind(A_{2}^{−j′_{2}−l_{2}j_{2}}) … ind(A_{n}^{−j′_{n}−l_{n}j_{n}})

since the algebras A_{1}, …, A_{n} are disjoint. Hence ind(A_{1}^{j}_{E_{2}F_{2}···E_{n}F_{n}}) = ind(A_{1}^{j}) for every j and therefore exp(A_{1}_{E_{2}F_{2}···E_{n}F_{n}}) = exp(A_{1}). Let ind(A_{1}) have the prime power decomposition ind(A_{1}) = ∏ p^{ν_{p}(ind(A_{1}))}. By [12, Lemma 1.3], we obtain

exp(A_{1}_{F_{1}E_{2}F_{2}···E_{n}F_{n}}) = l_{1}

and

ind(A_{1}_{F_{1}E_{2}F_{2}···E_{n}F_{n}}) = ∏_{p|l_{1}} p^{ν_{p}(ind(A_{1}))} = ∏_{p|m_{1}} p^{ν_{p}(ind(A_{1}))}, (∗)

the latter equation being true because the prime divisors of l_{1} and m_{1} are the same. We define E := E_{1}F_{1}E_{2}F_{2} ⋯ E_{n}F_{n}, E′ := F_{1}E_{2}F_{2} ⋯ E_{n}F_{n}, and apply 1.2 to the extension E/E′, using the variety SB_{m_{1}}(A_{E′}) = SB_{m_{1}}(A_{1}) ×_{F} E′. By 1.2 (i), we get Br(E/E′) = ⟨[A_{1}_{E′}^{m_{1}}]⟩. Since l_{1} = exp(A_{1}_{E′}) | m_{1}, the latter group is trivial and hence the restriction map Br(E′) ⟶ Br(E) is injective. Therefore exp(A_{1}_{E}) = exp(A_{1}_{E′_{2}}) = l_{1}. By 1.2 (ii) and by equation (∗) above, we obtain ind(A_{1}_{E}) = gcd(m_{1}, ind(A_{1}_{E′})) = m_{1}. In view of symmetry we obtain the same results for algebras A_{i}, 2 ≤ i ≤ n. □

2. Algebras after a scalar extension

The main ingredient of the proof of Theorem 2 is the following statement obtained in [18, Th. 2.11]).

Theorem 2.1. Let A be a central simple algebra over a field F. Then there exists a regular field extension M/F such that

(i) A_{M} is cyclic, (ii) for any central simple F -algebra C, ind(C_{M}) = ind(C), (iii) for any central simple F -algebra C, exp(C_{M}) = exp(C), (iv) the restriction map res : Br(F) ⟶ Br(M) is an injection.

For the reader’s convenience, we present a modified proof here. The original proof in [18]) based on a technical construction of a tower of field extensions with certain properties ([18, Lemmas 2.5 and 2.6]). Lemma 2.9 below allows to avoid these difficulties and leads to a slight generalization (see Theorem 2.11 below). In order to prove Theorem 2.1 we need a few preliminary statements.

Proposition 2.2. ([3, Th. 1.3], [20, Th. 13.10]) Let D, E be central division algebras over F of indices m and n respectively. Let SB(E) be the Severi-Brauer variety of E and let K be its function field. Then

ind(D ⊗_{F} K) = gcd{ind(D ⊗_{F} E^{i})}

where i ranges from 1 to exp(E).

Remark 2.3. In the literature the latter formula is called the index reduction formula.

Corollary 2.4. Let D, E be central division algebras over F. Let K be the function field of the Severi-Brauer variety SB(E). Assume that ind(D) is coprime to ind(E). Then ind(D ⊗_{F} K) = ind(D).

Proof. Use the index reduction formula. □

Lemma 2.5. Let A and B be central simple F -algebras. Assume ind(A) = p^{m} and ind(B) = p^{n}. Then ind(A ⊗_{F} B) ≥ p^{|m−n|}.

Proof. Assume for definiteness that m ≥ n. Let E/F be a field extension of degree p^{n} which splits B. Let also ind(A ⊗_{F} B) = p^{s}. Assume p^{s} < p^{m−n}. Then there exists a field extension L/F of degree p^{s} splitting A ⊗_{F} B. Hence

1 ∼ (A ⊗_{F} B)_{EL} ∼ A_{EL} ⊗_{EL} B_{EL} ∼ A_{EL}.

Thus EL is a splitting field of A. Since [EL : F] < p^{m}, then ind(A) < p^{m}. Contradiction. □

Lemma 2.6. Let A be a central simple F -algebra with ind(A) = p^{m}. Then ind(A^{p^{t}}) ≤ p^{m−t} for 0 ≤ t ≤ m.

Proof. Without loss of generality we can assume that there exists a splitting field L of A such that [L : F] = ind(A) and L contains a subfield K with [L : K] = p. (see e.g. [1, Ch.IV, Th.31]). Then ind(A_{K}) = p. Hence 1 = ind(A_{K}^{p}). Thus ind(A^{p}) ≤ [K : F] < [L : F] = ind(A). The formula now follows by induction. □

Lemma 2.7. Let K/F be a cyclic field extension, ⟨σ⟩ = Gal(K(z)/F (z)), and let z be transcendental over F. Also let C be a central division F -algebra such that C_{K} is a division algebra. Then

(K(z)/F (z), σ, z) ⊗ C_{F (z)}

is a division F (z)-algebra.

Proof. See [17, §19.6, Prop.] □

In the notations of the previous lemma we have immediately the following

Corollary 2.8. (i) For any j ≥ 1, the algebra (K(z)/F (z), σ, z^{j}) is Brauer-equivalent to (K′(z)/F (z), τ, z) for some K′ ⊂ K and some generator τ of Gal(K′/F). (ii) Let A be a central simple F -algebra such that ind(A_{K}) = ind(A). Then for any j ≥ 1,

ind ( (K(z)/F (z), σ, z^{j}) ⊗ A_{F (z)}) = ind((K(z)/F (z), σ, z^{j})) · ind(A).

Proof.

(i) Let n := [K : F], d := gcd(j, n), n′ =: n/d, and j′ := j/d. As j′ is relatively prime to n′, there is a natural number j′′ such that j′j′′ ≡ 1 mod n′. Moreover, let K′/F be the subfield of K such that [K′ : F] = n′. We obtain

(K(z)/F (z), σ, z^{j}) = (K(z)/F (z), σ, z^{j′d}))
≅ (K′(z)/F (z), σ|_{K′(z)}, z^{j′})  by [17, §15.1, Cor. b]
∼ (K′(z)/F (z), σ^{j′′}|_{K′(z)}, z)  by [17, §15.1, Cor. a (i)],

and of course τ = σ^{j′′}|_{K′(z)} generates Gal(K′(z)/F (z)). (ii) Since ind(A_{K}) = ind(A), then ind(A_{K′}) = ind(A) and we may apply Lemma 2.7 to the algebra (K′(z)/F (z), τ, z) ⊗ A_{F (z)} obtained from (i). □

Lemma 2.9. Let F be a field and G a finite group. Then there exists a tower of field extensions

F ⊂ K ⊂ E

such that (i) E/F is a finitely generated purely transcendental extension; (ii) E/K is Galois with the group G; (iii) for any central simple F -algebra C, ind(C_{E}) = ind(C).

Proof. Let E/F be a purely transcendental extension of degree |G| with algebraically independent variables x_{g}, g ∈ G. Define an action of G on E as follows. For h ∈ G, h(x_{g}) = x_{hg} and h is trivial on F. Let K = E^{G} be the subfield of fixed elements. Then E/K is Galois with the group G. Moreover, since E/F is purely transcendental, then E preserves indices of central simple F -algebras. □

Remark 2.10. Our proof for this Lemma in [18] was very technical and did work only for finite cyclic groups. We have to thank J.-L. Colliot-Thélène, who provided us with a much simpler and more elegant proof which works for arbitrary finite groups. Our argument above is a further simplification of his suggestion.

Proof of Theorem 2.1. Let deg(A) = n. It follows from Lemma 2.9 that there exists a tower of field extensions F ⊂ K ⊂ E such that E/F is a finitely generated purely transcendental extension, E/K is cyclic of degree n and E preserves indices of central simple F -algebras. Consider the cyclic algebra

D = (E(z)/K(z), σ, z),

where ⟨σ⟩ = Gal(E(z)/K(z)) and z is a transcendental variable. The algebra D is of exponent and index n with a maximal subfield E(z). One has

D ∼ D ⊗_{K(z)} A_{K(z)}^{op} ⊗_{K(z)} A_{K(z)}.

Let M be the function field of the Severi-Brauer variety SB(D ⊗_{K(z)} A_{K(z)}^{op}). Then A_{M} ∼ D_{M}. Since deg(A_{M}) = deg(D_{M}), then A_{M} ≅ D_{M}. Let C be a central simple F -algebra and C = ⊗_{i=1}^{m}C_{i} the decomposition of C as a tensor product of algebras of relatively prime primary indices. Since ind(C_{M}) = ∏_{i=1}^{m} ind(C_{i}_{M}), then to prove the statement about preserving indices it is enough to consider the case where ind(C) = p^{m} for a power of some prime p. Using the index reduction formula we obtain

ind(C_{M}) = gcd{ind(D^{j} ⊗_{K(z)} A_{K(z)}^{op}^{j} ⊗_{K(z)} C_{K(z)})}

where j ranges from 1 to n. Consider the algebra B_{j} = D_{p}^{j} ⊗_{K(z)} A_{p}_{K(z)}^{op}^{j} ⊗_{K(z)} C_{K(z)}, where D_{p} and A_{p} are p-primary parts of algebras D and A. Let p^{k} = ind(D_{p}) and p^{l} = ind(A_{p}). Note that k ≥ l. Since ind(C) is a power of p, then ind(C_{M}) = min_{j=1}^{n}{ind(B_{j})}. Fix some j. Let j = p^{t}j_{1}, where p does not divide j_{1}. Then exp(D_{p}^{j}) = p^{k−t}. Hence ind(D_{p}^{j}) = p^{k−t} in view of Lemma 2.6. Let ind(A_{p}^{op} ^{j}) = p^{s}. Then s ≤ l − t by Lemma 2.6. Note that by Corollary 2.8,

ind(B_{j}) = ind(D_{p}^{j})ind(A_{p}_{K}^{op} ^{j} ⊗_{K} C_{K}).

In view of Lemma 2.5,

ind(B_{j}) ≥ p^{k−t}p^{|s−m|} = p^{k−t+|s−m|}.

Finally consider two cases.

(i) s ≥ m. Then k − t ≥ l − t ≥ s ≥ m and k − t + |s − m| ≥ m. (ii) s < m. Then k − t + |s − m| = k − t − s + m ≥ l − t − s + m ≥ m. Therefore, ind(B_{j}) ≥ p^{m} = ind(C) for any j. Thus ind(C_{M}) = ind(C). Note that, for a field extension M/F, preserving indices for all central simple F -algebras implies also preserving exponents of central simple F -algebras. Indeed, assume C_{M}^{m} ∼ 1 for some central simple F -algebra C. Since

1 = ind(C_{M}^{m}) = ind(C^{m}),

then C^{m} ∼ 1. Thus exp(C_{M}) = exp(C). Moreover, preserving exponents implies, in turn, that the restriction homomorphism

res : Br(F) ⟶ Br(M)

is an embedding. □

We have also the following kind of generalization of Theorem 2.1 to the case of abelian groups.

Theorem 2.11. Let A be a central simple algebra of degree n over a field F and G an abelian group of order n. Then there exists a regular finitely generated field extension L/F such that

(i) A_{L} is a crossed product with the group G, (ii) ind(A_{L}) = ind(A).

Before proving this theorem, we introduce some notations and prove a preliminary lemma. Let F ⊂ K ⊂ E be a tower of field extension such that E/K is Galois with the group G and E preserves indices of F -algebras. Let

G = H_{1} ⊕ ⋯ ⊕ H_{m},

where H_{1}, …, H_{m} are cyclic with generators respectively σ_{1}, …, σ_{m}. Let E_{i} be the subfield of E fixed by

Ĥ_{i} := ⊕_{{j=1 \atop j≠i}}^{m} H_{j}.

Then the extension E_{i}/K is cyclic with the Galois group H_{i}. Denote the canonical surjective homomorphism from G to H_{i} by τ_{i}. Let y_{1}, …, y_{m} be transcendental variables over K. Consider the cyclic algebras

D_{i} = (E_{i}(y_{1}, …, y_{m})/K(y_{1}, …, y_{m}), σ_{i}, y_{i}).

Note that

D_{i} ∼ (E(y_{1}, …, y_{m})/K(y_{1}, …, y_{m}), G, c_{i}),

where

c_{i}(g, h) = c_{y_{i}}(τ_{i}(g), τ_{i}(h))

and the cocycle c_{y_{i}} is defined by

c_{y_{i}}(σ_{i}^{k}, σ_{i}^{j}) = \{ \begin{matrix}1, & if k + j < [E_{i} : F]; \\ y_{i}, & if k + j ≥ [E_{i} : F]. \end{matrix}

(see e.g. [10, Th. 2.13.8] or [17, §14.5]). Let

D = D_{1} ⊗ ⋯ ⊗ D_{m}.

Then

D ≅ (E(y_{1}, …, y_{m})/K(y_{1}, …, y_{m}), G, c_{1} … c_{m})

is a crossed product with the group G and deg(D) = n ([17, §14.3]). In the notations above, we have the following

Lemma 2.12. For any central division F -algebra C, D ⊗ C_{K(y_{1},...,y_{m})} is a division algebra. Moreover, for any j ≥ 1,

ind(D^{j} ⊗ C_{K(y_{1},...,y_{m})}) = ind(D^{j}) · ind(C).

Proof: We will prove the lemma using induction on m. If m = 1, then the statement is true in view of Lemma 2.7 and of Corollary 2.8. Assume that the statement is true for m = m_{0}. That is, for any tower of field extensions F ⊂ K′ ⊂ E′ such that E′ preserves indices of F -algebras and E′/K′ is Galois with the group H′_{1} ⊕ ⋯ ⊕ H′_{m_{0}}, (H′_{i}, 1 ≤ i ≤ m_{0}, is cyclic with generator σ′_{i}) and any central division F -algebra C, the algebra

(E′_{1}(y′_{1}, …, y′_{m_{0}})/K′(y′_{1}, …, y′_{m_{0}}), σ′_{1}, y′_{1}) ⊗ ⋯ ⋯ ⊗ (E′_{m_{0}}(y′_{1}, …, y′_{m_{0}})/K′(y′_{1}, …, y′_{m_{0}}), σ′_{m_{0}}, y′_{m_{0}}) ⊗ C_{K′(y′_{1},...,y′_{m_{0}})}

is division, where E′_{i} is a subfield of E′ fixed by the group ⊕_{j=1,j≠i}^{m_{0}}H′_{j} and y′_{1}, …, y′_{m_{0}} are transcendental variables over K′.

Let m = m_{0} + 1. We have the following diagram of field extensions.

[figure: E_{1} ⋯ E_{m_{0}} · E_{m}; E_{1} ⋯ E_{m_{0}}; H_{m}; E_{i} · E_{m}; G_{i}; E_{i}; H_{m}; G_{i}; E_{m}; H_{i}; K; H_{m}; H_{i}]

for any 1 ≤ i ≤ m_{0}, where

G_{i} = ⊕_{{j=1 \atop j≠i}}^{m_{0}} H_{j}.

Denote B = (E_{1}(y_{1}, …, y_{m_{0}})/K(y_{1}, …, y_{m_{0}}), σ_{1}, y_{1}) ⊗ ⋯

⋯ ⊗ (E_{m_{0}}(y_{1}, …, y_{m_{0}})/K(y_{1}, …, y_{m_{0}}), σ_{m_{0}}, y_{m_{0}}) ⊗ C_{K(y_{1},...,y_{m_{0}})}.

Then D ⊗ C_{K(y_{1},...,y_{m})} = (E_{1}(y_{1}, …, y_{m})/K(y_{1}, …, y_{m}), σ_{1}, y_{1}) ⊗ …

⋯ ⊗ (E_{m}(y_{1}, …, y_{m})/K(y_{1}, …, y_{m}), σ_{m}, y_{m}) ⊗ C_{K(y_{1},...,y_{m})} ≅

≅ B_{K(y_{1},...,y_{m_{0}})(y_{m})} ⊗ (E_{m}(y_{1}, …, y_{m_{0}})(y_{m})/K(y_{1}, …, y_{m_{0}})(y_{m}), σ_{m}, y_{m}).

To prove that the latter algebra is division it is enough by Lemma 2.7 to show that the algebra B_{E_{m}(y_{1},...,y_{m_{0}})} is division. But this follows from the induction hypothesis (take K′ = E_{m}, E′ = E_{1} ⋯ E_{m_{0}} · E_{m}, E′_{i} = E_{i} · E_{m}) since B_{E_{m}(y_{1},...,y_{m_{0}})} ≅ (E_{1} · E_{m}(y_{1}, …, y_{m_{0}})/E_{m}(y_{1}, …, y_{m_{0}}), σ_{1}, y_{1}) ⊗ ⋯

⋯ ⊗ (E_{m_{0}} · E_{m}(y_{1}, …, y_{m_{0}})/E_{m}(y_{1}, …, y_{m_{0}}), σ_{m_{0}}, y_{m_{0}}) ⊗ C_{E_{m}(y_{1},...,y_{m_{0}})}.

Thus for any central division F -algebra C, D ⊗ C_{K(y_{1},...,y_{m})} is a division algebra. Hence

ind(D ⊗ C_{K(y_{1},...,y_{m})}) = deg(D ⊗ C_{K(y_{1},...,y_{m})}) = deg(D) · deg(C) = ind(D) · ind(C). Since D^{j} ∼ D_{1}^{j}⊗⋯⊗D_{m}^{j}, then using properties of cyclic algebras (see proof of Corollary 2.8) we obtain the formula

ind(D^{j} ⊗ C_{K(y_{1},...,y_{m})}) = ind(D^{j}) · ind(C).

□

Now we are in a position to prove Theorem 2.11.

Proof of Theorem 2.11: Given a field F and a finite group G of order n, it follows from Lemma 2.9 that there exists a tower of field extensions F ⊂ K ⊂ E such that E/F is a finitely generated purely transcendental extension and E/K is Galois with the group G. Let

G = H_{1} ⊕ ⋯ ⊕ H_{m},

where H_{1}, …, H_{m} are cyclic. One can construct a corresponding division algebra D (see the text before Lemma 2.12). Then deg(D) = ind(D) = n ≥ ind(A). One has

D ∼ D ⊗ A_{K(y_{1},...,y_{m})}^{op} ⊗ A_{K(y_{1},...,y_{m})}.

Let L be the function field of the Severi-Brauer variety SB(D ⊗ A_{K(y_{1},...,y_{m})}^{op}). Then A_{L} ≅ D_{L}, i.e., A_{L} is a crossed product with the group G. To finish the proof we need to show that ind(A_{L}) = ind(A). Let A_{p} and D_{p} be the p-primary parts of A and D. It is enough to prove that ind(A_{p}_{L}) = ind(A_{p}). By the index reduction formula,

ind(A_{p L}) = gcd{ind(A_{p}_{K(y_{1},...,y_{m})} ⊗ D_{p}^{j} ⊗ A_{p}^{op} _{K(y_{1},...,y_{m})}^{j})},

where j ranges from 1 to n. By Lemma 2.12, for any 1 ≤ j ≤ n,

ind(A_{p}_{K(y_{1},...,y_{m})} ⊗ D_{p}^{j} ⊗ A_{p}^{op} _{K(y_{1},...,y_{m})}^{j}) = ind(D_{p}^{j}) · ind(A_{p}^{op} ^{j−1}).

If p does not divide j, then ind(D_{p}^{j}) = ind(D_{p}) ≥ ind(A_{p}). If p divides j, then ind(A_{p}^{op} ^{j−1}) = ind(A_{p}). Hence obtain that ind(A_{p}_{L}) = ind(A_{p}).

□

3. Algebras with involutions after a scalar extension

Proof of Theorem 2. Let F ⊂ K be the subfield of fixed elements of τ and τ|K = σ. By Theorem 2.1, there exists a regular field extension M/K preserving indices of central simple K-algebras such that A_{M} is cyclic. For the following constructions, in particular for the construction of the transfer of a regular field extension, we refer to [19, p. 220]). The automorphism σ : K ⟶ K can be extended to an isomorphism of M and another regular extension of K denoted by M_{σ}. That is, the following diagram commutes:

\xymatrix{K \ar@{^{(}->}[r] \ar[d]^{σ} & M \ar[d]^{σ} \\ K \ar@{^{(}->}[r] & M_{σ} .}

Let E = MM_{σ} be the free composite over K of M and M_{σ} (see definition in [9, p. 203]). Then E is a regular extension of K. The automorphism σ can be extended to an automorphism σ̄ of E. Let T = T_{K/F} (M) be the transfer of M with respect to the ground field descent F ⊂ K, i.e., the subfield of E of elements fixed under the action of σ̄. Thus T is a subfield of E of degree 2. Then the composite T K coincides with E.

The algebra A_{E} is cyclic. Moreover, the latter algebra has an involution of the second kind defined by the formula

\overline{τ}(a ⊗ e) = τ(a) ⊗ \overline{σ}(e),

where a ∈ A, e ∈ E and \overline{σ} is an automorphism of E extending σ. As for preserving indices note that the field E can be constructed using the same procedure as for the field M (see proof of Theorem 2.1). We just replace the ground field K by M_{σ}. Hence E preserves indices of central simple M_{σ}-algebras, but M_{σ} preserves indices of all central simple K-algebras. □

Using above results we prove immediately the following.

Theorem 3.1. Suslin’s conjecture about special unitary groups is true iff it is true for all cyclic division algebras.

We can also prove the following

Theorem 3.2. Let A be a central simple algebra over a field K with an involution τ of the second kind. Assume that p^{2} divides ind(A) for some prime number p. Then there exists a regular field extension M/K such that A_{M} is an algebra of index p^{2}, which is Brauer equivalent to a bicyclic algebra of degree p^{2} and has an involution of the second kind extending τ.

Proof: Let F ⊂ K be the subfield of fixed elements of τ and τ|K = σ. Denote by G the group ℤ/p⊕ℤ/p. Let ϕ_{1} be a generator of the first summand, and ϕ_{2} of the second one. Then by Lemma 2.9, there exists a tower of field extensions F ⊂ F_{0} ⊂ E such that E/F_{0} is Galois with the group G and E is a purely transcendental extension of F. Let E_{2} be the subfield of E fixed by the first summand of G and E_{1} the subfield fixed by the second summand. Then the extensions E_{i}/F_{0}, i=1,2, are cyclic of degree p. Let F_{0}(y_{1}, y_{2}) be a purely transcendental extension of F_{0} of degree 2. Consider the cyclic algebras

D_{i} = (E_{i}(y_{1}, y_{2})/F_{0}(y_{1}, y_{2}), ϕ_{i}, y_{i}), i = 1, 2.

Set D = D_{1} ⊗ D_{2}. By Lemma 2.12, D is a division algebra. Let L be the function field of the Severi-Brauer variety SB(D_{KF_{0}(y_{1},y_{2})} ⊗ A_{KF_{0}(y_{1},y_{2})}^{op}). Then A_{L} ∼ D_{L}. Thus ind(A_{L})|p^{2}. By means of the index reduction formula we obtain that ind(A_{L}) = p^{2}. Indeed,

ind(A_{L}) = gcd{ind(A_{KF_{0}(y_{1},y_{2})} ⊗ D_{KF_{0}(y_{1},y_{2})}^{j} ⊗ A^{op} _{KF_{0}(y_{1},y_{2})}^{j}},

where j ranges from 1 to ind(A). By Lemma 2.12,

ind(A_{KF_{0}(y_{1},y_{2})} ⊗ D_{KF_{0}(y_{1},y_{2})}^{j} ⊗ A^{op} _{KF_{0}(y_{1},y_{2})}^{j}) = ind(D_{KF_{0}(y_{1},y_{2})}^{j}) · ind(A^{op} _{KF_{0}(y_{1},y_{2})}^{j−1}).

If p does not divide j, then ind(D_{KF_{0}(y_{1},y_{2})}^{j}) = p^{2}. If p|j, then p^{2}|ind(A^{op} _{KF_{0}(y_{1},y_{2})}^{j−1}). So, in both cases ind(A_{KF_{0}(y_{1},y_{2})} ⊗ D_{KF_{0}(y_{1},y_{2})}^{j} ⊗ A^{op} _{KF_{0}(y_{1},y_{2})}^{j}) is divisible by p^{2}. Thus, ind(A_{L}) = p^{2}. Note also that L preserves the index of the F_{0}(y_{1}, y_{2})-algebra D. The automorphism σ : K ⟶ K can be extended to an isomorphism of L and another regular extension of K denoted by L_{σ}. Let T = T_{K/F} (L) be the transfer of L with respect to the ground field descent F ⊂ K. Then the composite T K over F coincides with the free composite LL_{σ} over K. Thus A_{LL_{σ}} has an involution of the second kind extending τ. To finish the proof we need to show that ind(A_{LL_{σ}}) = p^{2}. The isomorphism σ : L ⟶ L_{σ} can be extended in such a way that the following diagram commutes

\xymatrix{K \ar@{^{(}->}[r] \ar[d]^{σ} & KF_{0}(y_{1}, y_{2}) \ar@{^{(}->}[r] \ar[d]^{σ} & L \ar[d]^{σ} \\ K \ar@{^{(}->}[r] & KF_{0}(y_{1}, y_{2}) \ar@{^{(}->}[r] & L_{σ}}

where the middle arrow σ acts as σ on K and trivially on F_{0}(y_{1}, y_{2}). We have also the following commutative diagram.

\xymatrix{ & KF_{0}(y_{1}, y_{2}) \ar@{^{(}->}[r] \ar[dd]^{σ} & L \ar[dd]^{σ} \\ F_{0}(y_{1}, y_{2}) \ar@{^{(}->}[ur] \ar@{^{(}->}[dr] \\  & KF_{0}(y_{1}, y_{2}) \ar@{^{(}->}[r] & L_{σ}.}

As we noted before, L preserves the index of the F_{0}(y_{1}, y_{2})-algebra D and σ is trivial on F_{0}(y_{1}, y_{2}). Hence L_{σ} also preserves the index of D. Now consider the free composite LL_{σ}. It can be constructed using the same procedure as for the field L. We just replace the ground field K by L_{σ}. Thus LL_{σ} can be constructed as follows. Instead of the tower of fields extensions K ⊂ KF_{0} ⊂ KE and the algebra D_{KF_{0}(y_{1},y_{2})} in the same way we construct the tower of field extensions L_{σ} ⊂ L_{σ}(KF_{0}) ⊂ L_{σ}(KE) and the algebra D̄ over K̄(z_{1}, z_{2}) where K̄ = L_{σ}(KF_{0}) and K̄(z_{1}, z_{2}) is a purely transcendental extension of K̄ of degree 2. Then LL_{σ} is the function field of the Severi-Brauer variety

SB(D̄ ⊗ A_{K̄(z_{1},z_{2})}^{op}).

Note that L_{σ}(KE) is a purely transcendental extension of L_{σ}. Hence K̄ as a subfield of L_{σ}(KE) preserves the index of D. Then K̄(z_{1}, z_{2}) also preserves this index. Moreover, we conclude that ind(A_{L_{σ}}) = p^{2}. Indeed, assume that ind(A_{L_{σ}}) < p^{2}. Then the p-primary component of A_{L_{σ}}^{p} is trivial. By the index reduction formula,

ind(D_{LL_{σ}}) = gcd{ind((D̄^{j} ⊗ A^{op} _{K̄(z_{1},z_{2})}^{j}) ⊗ D_{K̄(z_{1},z_{2})})}

where j ranges from 1 to ind(D̄ ⊗ A_{K̄(z_{1},z_{2})}^{op}). Note that since L_{σ}(KE) is purely transcendental extension of L_{σ}, then it preserves indices of central simple L_{σ}-algebras. By Lemma 2.12,

ind((D̄^{j} ⊗ A^{op} _{K̄(z_{1},z_{2})}^{j}) ⊗ D_{K̄(z_{1},z_{2})}) = ind(D̄^{j}) · ind(A^{op} _{K̄(z_{1},z_{2})}^{j} ⊗ D_{K̄(z_{1},z_{2})}).

If p does not divide j, then ind(D̄_{K̄(z_{1},z_{2})}^{j}) = p^{2}. If p divides j, then the p-primary part A_{p}^{op} _{K̄(z_{1},z_{2})}^{j} of A^{op} _{K̄(z_{1},z_{2})}^{j} is trivial. Hence

ind(A_{p}^{op} _{K̄(z_{1},z_{2})}^{j} ⊗ D_{K̄(z_{1},z_{2})}) = ind(D_{K̄(z_{1},z_{2})}) = ind(D_{L_{σ}}) = p^{2}.

Thus ind(D_{LL_{σ}}) = ind(A_{LL_{σ}}) > ind(A_{L_{σ}}) and we have a contradiction. Hence ind(A_{L_{σ}}) = p^{2}, then ind(A_{K̄(z_{1},z_{2})}) = p^{2}. Using the index reduction formula we obtain that ind(A_{LL_{σ}}) = p^{2}. The proof of the latter equality is the same as for the index of A_{L}. □

Corollary 3.3. Let A be a central simple algebra over a field K with an involution τ of the second kind. Assume that p^{2} divides ind(A) for some prime number p and the primitive p-th root of unity belongs to K. Then there exists a regular field extension M/K such that A_{M} is an algebra of index p^{2}, which is Brauer equivalent to a tensor product of two symbol algebras and has an involution of the second kind extending τ.

Combining Theorems 3.1 and 3.2 one can prove:

Theorem 3.4. Suslin’s conjecture about special unitary groups is true iff it is true for all cyclic division algebras which are bicyclic algebras of degree p^{2} for any prime p.

For abelian crossed products, we have the following

Theorem 3.5. Let A be a central simple algebra over a field K of degree n with an involution τ of the second kind, and let G be an abelian group of order n. Then there exists a regular field extension E/K preserving the index of A such that A_{E} is a crossed product with the group G and A_{E} has an involution of the second kind extending τ.

Proof: Let F ⊂ K be the subfield of fixed elements of τ and τ|K = σ. It follows from Lemma 2.9 that there exists a tower of field extensions F ⊂ M ⊂ N such that N/F is a finitely generated purely transcendental extension and N/M is Galois with the group G. Let G = H_{1}⊕⋯⊕H_{r} be the decomposition of G as a sum of cyclic subgroups. As in the proof of Theorem 2.11 we can construct an algebra D over a purely transcendental extension M(y_{1}, …, y_{r}) of degree r of M which is a crossed product with the group G. Let L be the function field of the Severi-Brauer variety SB(A_{KM(y_{1},...,y_{r})}^{op}⊗D_{KM(y_{1},...,y_{r})}). Then L preserves the index of A (the proof is analogous to that of Theorem 2.1) and A_{L} is a crossed product with the group G. Further, as in the proof of Theorem 3.2 we construct a free composite LL_{σ} and prove that A_{LL_{σ}} has prescribed properties. □

References

[1] A. A. Albert, Structure of algebras, rev. printing, Amer. Math. Soc., Providence, 1961.

[2] A. Blanchet, Function fields of generalized Brauer-Severi varieties, Comm. Algebra 19 (1991), no. 1, 97–118.

[3] M. Van den Berg, A. Schofield, The index of a Brauer class on a Brauer-Severi variety, Trans. AMS. 341 (1994), 505–517.

[4] N. Bourbaki, Éléments de mathématiques / Livre 2, Algèbre, ch 8, Modules et anneaux semisimples, Actualités Sci. Indust. No. 1261, Hermann, Paris, 1958.

[5] V.I. Chernousov, A.S. Merkurjev, R-equivalence and special unitary groups, J. Algebra 209 (1998), no. 1, 175-198.

[6] P. M. Cohn. Free rings and their relations, London Mathematical Society Monographs, No. 2. Academic Press, London-New York, 1971.

[7] J.-L. Colliot-Thélène, J.-J. Sansuc, La R-équivalence sur les tores, Ann. Sci. École Norm. Sup. (4) 10 (1977), no. 2, 175–229.

[8] P. Gille Le problème de Kneser-Tits, Séminaire Bourbaki. Vol. 2007/2008. Astérisque No. 326 (2009), Exp. No. 983, vii, 39-81 (2010).

[9] N. Jacobson, Lectures in abstract algebra. Vol III: Theory of fields and Galois theory Inc., Princeton, N.J.-Toronto, Ont.-London-New York 1964.

[10] N. Jacobson, Finite-dimensional division algebras over fields, New York-Heidelberg-Berlin: Springer-Verlag, 1996.

[11] O.T. Izhboldin, N.A. Karpenko Generic splitting fields of generic simple algebras: Galois cohomology and nonexcellence , Algebras and Representation Theory 2 (1999), 19–59.

[12] I. Kersten, U. Rehmann, Excellent algebraic groups. I, J. Algebra 200 (1998), no. 1, 334–346.

[13] B. È. Kunyavskiĭ, L. H. Rowen, S. V. Tikhonov, and V. I. Yanchevskiĭ. Bicyclic algebras of prime exponent over function fields, Trans. Amer. Math. Soc. 358 (2006), no. 6, 2579-2610.

[14] Yu. I. Manin, Cubic forms, North-Holland, Amsterdam, 1974.

[15] A. S. Merkurjev, I. A. Panin and A. R. Wadsworth, Index Reduction Formulas for Twisted Flag Varieties, I, K-Theory 10 (1996), 517-596.

[16] A. P. Monastyrnyĭ, V. I. Yanchevskiĭ, Whitehead groups of spinor groups, Izv. Akad. Nauk SSSR Ser. Mat. 54 (1990), no. 1, 60–96, 221; translation in Math. USSR-Izv. 36 (1991), no. 1, 61–100.

[17] R. S. Pierce, Associative algebras, Springer, New York, 1982.

[18] U. Rehmann, S.V. Tikhonov, V.I. Yanchevskiĭ, Symbols and cyclicity of algebras after a scalar extension, Journal of Mathematical Sciences 164, n^{∘} 1, 131-142.

[19] P. Roquette, Isomorphisms of generic splitting fields of simple algebras, J. Reine Angew. Math. 214/215, (1964), 207-226.

[20] D. J. Saltman, Lectures on division algebras, Amer. Math. Soc., Providence, RI, 1999.

[21] A. A. Suslin, SK_{1} of division algebras and Galois cohomology, Adv. Soviet Math. 4, Amer. Math. Soc., Providence, RI (1991), 75-100.

[22] V. E. Voskresenskiĭ, The reduced Whitehead group of a simple algebra, Uspehi Mat. Nauk 32 (1977), no. 6(198), 247-248.

[23] A. R. Wadsworth, The index reduction formula for generic partial splitting varieties, Comm. Algebra 21 (1993), 1063–1070.

[24] V. I. Yanchevskiĭ, Simple algebras with involutions, and unitary groups, Mat. Sb. 22 (1974), 372–385.

[25] V. I. Yanchevskiĭ, Reduced unitary K-theory and division algebras over Henselian discretely valued fields, Izv. Akad. Nauk SSSR Ser. Mat. 42 (1978), no. 4, 879–918; translation in Math. USSR-Izv. 13 (1979), 175–213.

[26] V. I. Yanchevskiĭ, The inverse problem of reduced K-theory, Mat. Zametki 26 (1979), no. 3, 475–482; translation in Math. Notes. 26 (1979), 728–731.

[27] V. I. Yanchevskiĭ, Reduced unitary K-theory: applications to algebraic groups., Mat. Sb. 38 (1981), 533–561.

Rehmann: Universität Bielefeld, POB 10 01 31, 33501 Bielefeld, GERMANY

E-mail address: rehmann@math.uni-bielefeld.de

Tikhonov, Yanchevskiĭ: Institute of Mathematics of the National Academy of Sciences of Belarus, ul. Surganova 11, 220072 Minsk, BELARUS

E-mail address: tsv@im.bas-net.by , yanch@im.bas-net.by