Group action on bimodule categories

arXiv:0802.4266v2 [math.RT] 15 Oct 2008 GROUP ACTION ON BIMODULE CATEGORIES YURIY A. DROZD To the memolry of A. V. Roiter Abstract. We consider actions of groups on categories and bimodules, the related crossed group categories and bimodules, and prove for them analogues of the result know for representations of crossed group algebras and categories. Skew group algebras arise naturally in lots of questions. In particular, the properties of the categories of representations of skew group algebras and, more generally, skew group categories have been studied in [11, 8]. On the other hand, “matrix problems,” especially, bimodule categories play now a crucial role in the theory of representations [5, 6]. The situation, when a group acts on a bimodule, thus also on the bimodule category is also rather typical. Therefore one needs to deal with skew bimodules and their bimodule categories. In this paper we shall study skew bimodules and bimodule categories and prove for them some analogues of the results of [11, 8]. In Section 1 we recall general notions related to bimodule categories. In Section 2 we consider actions of groups on bimodule and bimodule categories and the arising functors. The main results are those of Section 3, where we define separable actions and prove that in the separable case the bimodule category of the skew bimodule is equivalent to the skew category of the original one. We also consider specially the case of the abelian groups, since in this case the original category can be restored from the skew one using the group of characters. Section 4 is devoted to the decomposition of objects in skew group categories, especially, to the number of non-isomorphic direct summands in such decompositions. We also consider the radical and almost split morphisms in the skew group categories (under the separability condition). 2000 Mathematics Subject Classification. Primary 16S35, Secondary 16G10, 16G70. Key words and phrases. categories, bimodules, group action, crossed group categories. This research was partially supported by the INTAS Grant no. 06-1000017-9093. 1 2 YURIY A. DROZD 1. Bimodule categories We recall the main definitions related to bimodule categories [5, 6]. We fix a commutative ring K. All categories that we consider are supposed to be K-categories, which means that all sets of morphisms are K-modules, while the multiplication is K-bilinear. We denote the set of morphisms from an object X to an object Y in a category A by A(X, Y ). A module (more precise, a left module) over a category A, or a A-module is, by definition, a K-linear functor M : A → K-Mod, where K-Mod denotes the category of K-modules. If M is such a module, x ∈ M(X) and a ∈ A(X, Y ), we write, as usually, ax instead of M(a)(x). Such modules have all usual properties of modules over rings. The category of all A-modules is denoted by A-Mod. A bimodule over a category A, or an A-bimodule, is, by definition, a K-bilinear functor B : Aop × A → K-Mod, where Aop is the opposite category to A. If x ∈ B(X, Y ), a : X ′ → X (i.e. a : X → X ′ in Aop ), b : Y → Y ′ , we write bxa instead of B(a, b)(x) (this element belongs to B(X ′ , Y ′ )). In particular, xa and bx denote, respectively, B(a, 1Y )(x) and B(1X , b)(x). If a bimodule B is fixed, we often write x : X 99K Y instead of x ∈ B(X, Y ). A category A is called fully additive if it is additive (i.e. has direct sums X ⊕ Y of any pair of objects X, Y and a zero object 0) and every idempotent endomorphism e ∈ A(X, X) splits, i.e. there is an object Y and a pair of morphisms ι : Y → X and π : X → Y such that πι = 1Y and ιπ = e. Choosing an object Y ′ and morphisms ι′ : Y ′ → X and π ′ : X → Y ′ such that π ′ ι′ = 1Y ′ and ι′ π ′ = 1 − e, we present X as a direct sum Y ⊕ Y ′ , where ι and ι′ are canonical embeddings, while π and π ′ are canonical projections. For every K-category A there is the smallest fully additive category add A containing A. This category is unique (up to equivalence). It can be identified either with the category of matrix idempotents over A or with the category of finitely generated projective A-modules [9]. We call it the additive hull of A. Each Amodule M (bimodule B) extends uniquely (up to isomorphism) to a module (bimodule) over the category add A, which we also denote by M (respectively, by B) If B is an A-bimodule, a differentiation from A to B is, by definition, a set of K-linear maps ∂ = { ∂(X, Y ) : A(X, Y ) → B(X, Y ) | X, Y ∈ Ob A } , satisfying the Leibniz rule: ∂(ab) = (∂a)b + a(∂b) GROUP ACTION ON BIMODULE CATEGORIES 3 for any morphisms a, b such that the product ab is defined. It implies, in particular, that ∂1X = 0 for any object X. Again, such a differentiation extends to the additive hull of A and we denote this extension by the same letter ∂. A triple T = (A, B, ∂), where A is a category, B is a A-bimodule and ∂ is a differentiation from A to B, is called a bimodule triple. If T′ = (A′ , B′ , ∂ ′ ) is another bimodule triple, a bifunctor from T to T′ is defined as a pair F = (F0 , F1 ), where F0 : A → A′ is a functor, F1 : B → B′ (F0 ) is a homomorphism of A-bimodule, where B′ (F0 ) is the A-bimodule obtained from B′ by the transfer along F0 (i.e. F1 (x) : F0 (X) 99K F0 (Y ) if x : X 99K Y , and F1 (bxa) = F0 (b)F1 (x)F0 (a)), such that F1 (∂a) = ∂ ′ (F0 (a)) for all a ∈ Mor A. As a rule, we write F (a) and F (x) instead of F0 (a) and F1 (x). Let F = (F0 , F1 ) and G = (G0 , G1 ) be two bifunctors from a triple T = (A, B, ∂) to another triple T′ = (A′ , B′ , ∂ ′ ). A morphism of bifunctors φ : F → G is defined as a morphism of functors φ : F0 → G0 such that φ(Y )F1 (x) = G1 (x)φ(X) for each x ∈ B(X, Y ), ∂ ′ φ(X) = 0 for each X ∈ Ob A. If φ is an isomorphism of functors, the inverse morphism is obviously a morphism of bifunctors too. Then we call φ an isomorphism of bi∼ functors and write φ : F → G. If such an isomorphism exists, we say that the bifunctors F are G isomorphic and write F ≃ G. We call a bifunctor F : T → T′ an equivalence of bimodule triples if there is such a bifunctor G : T′ → T that F G ≃ idT′ and GF ≃ idT, where idT denotes the identity bifunctor T → T. If such a bifunctor exists, we call the triples T and T′ equivalent and write T ≃ T′ . Lemma 1.1. A bifunctor F = (F0 , F1 ) is an equivalence of bimodule triples if and only if the following conditions hold: (1) The functor F0 is fully faithful, i.e. all induced maps A(X, Y ) → A′ (F0 X, F0 Y ) are bijective. (2) This functor is also ∂-dense, i.e. for every object X ′ of the category A′ there are an object X ∈ Ob A and an isomorphism α : X ′ → F0 X such that ∂α = 0 . (3) The map F1 (X, Y ) : B(X, Y ) → B′ (F0 X, F0 Y ) is bijective for any X, Y ∈ Ob A. Moreover, if these conditions hold, there is a bifunctor G : T′ → T and an isomorphism λ : idT′ → F G such that GF = idT and λ(F X) = 1F X for all X ∈ Ob A. 4 YURIY A. DROZD Proof. The necessity of these conditions is evident, so we prove their sufficiency. Suppose that these conditions hold. For each object X ′ ∈ A′ choose an object X and an isomorphism α : X ′ → F0 X such that ∂a = 0, always setting α = 1X ′ for X ′ = F0 X. Set G0 X ′ = X and λ(X ′ ) = α. For each morphism a : X ′ → Y ′ set G0 a = F0−1 (X, Y )(λ(Y ′ )aλ−1 (X ′ )), where X = G0 X ′ , Y = G0 Y ′ (then λ(X ′ ) : ∼ ∼ X ′ → F0 X, λ(Y ′ ) : Y ′ → F0 Y ). Obviously, the set {λ(X ′)} defines an isomorphism of functors λ : id → F0 G0 . We also define a homomorphism of bimodules G1 : B′ → B(G0 ) setting G1 (x) = F1 (X, Y )−1 (λ(Y ′ )xλ−1 (X ′ )) if x : X ′ 99K Y ′ , X = G0 X ′ , Y = G0 Y ′ . Then G = (G0 , G1 ) is a bifunctor T′ → T and λ is an isomorphism of bifunctors idT′ → F G. Moreover, by this construction, GF = idT and λ(F X) = 1F X for all X.  Every bimodule triple T = (A, B, ∂) gives rise to the bimodule category (or the category of representations, or the category ofSelements) of this triple [5]. The objects of this category are elements X B(X, X), where X runs through objects of the category add A. Morphisms from an object x : X 99K X to an object y : Y 99K Y are such morphisms a : X → Y that ax = ya + ∂(a) in B(X, Y ). It is easy to see that these definitions really define a fully additive K-category El(T). The set of morphisms x → y in this category is denoted by HomT(x, y). If ∂ = 0, we write El(A, B) or even El(B) instead of El(A, B, ∂). Each bifunctor between bimodule triples F : T → T′ gives rise to a functor F∗ : El(T) → El(T′ ), which maps an object x to the object F1 (x) and a morphism a : x → y to the morphism F0 (a) : F1 (x) → F1 (y). As well, each morphism of bifunctors φ : F → G induces a morphism of functors φ∗ : F∗ → G∗ , which correlate an object x ∈ B(X, X) with the morphism φ(X) considered as a morphism F (x) → G(x). Obviously, if φ is an isomorphism of bifunctors, φ∗ is an isomorphism of functors. Especially, if F is an equivalence of bimodule triples, the functor F∗ is an equivalence of their bimodule categories. If B = A and ∂ = 0, we say that the bimodule triple T = (A, A, 0) is the principle triple for the category A. Obviously, a bifunctor between principle triples is just a functor between the corresponding categories and a morphism of such bifunctors is just a morphism of functors. The bimodule category of the principle triple for a category A is denoted by El(A). If A and A′ are two categories, one can consider A-A′ -bimodules, i.e. bilinear functors B : Aop × A′ → K-Mod. Actually, any such bimodule can be identified with a A × A′ -bimodule B̃ with B̃((X, X ′ ), (Y, Y ′ )) = B(X, Y ′ ) and (a, a′ )x(b, b′ ) = axb′ . Such bimodules are called bipartite. GROUP ACTION ON BIMODULE CATEGORIES 5 In particular, every A-bimodule B defines a bipartite A-A-bimodule, which we denote by B(2) and call the double of the A-bimodule B. Certainly, bimodules B and B(2) are quite different and they define different bimodule categories. If B = A the category El(A(2) ) coincides with the category of morphisms of the additive hull add A. Further on we often identify the categories A and add A and say ”an object (morphism) of A” instead of “an object (morphism) of add A.” We hope that this petty ambiguity will not embarrass the reader. 2. Group actions Let T = (A, B, ∂) be a bimodule triple and G be a group. One says that the group G acts on the triple T if a bifunctor Tσ : T → T is defined for each element σ ∈ G so that T1 = idT and Tστ ≃ Tσ Tτ for any σ, τ ∈ G. It implies, in particular, that all Tσ are equivalences. Further on we write X σ instead of Tσ (X). We only note that according to this notation X στ ≃ (X τ )σ . A system of factors λ for such an action is ∼ defined as a set of isomorphisms of bifunctors λσ,τ : Tστ → Tσ Tτ , which satisfy the relations: (2.1) λρσ,τ λρ,στ = λρ,σ λρσ,τ for any triple of elements ρ, σ, τ ∈ G, and λσ,1 = λ1,σ = 1 for any σ ∈ G. We omit the arguments (objects of A) in these formulae (and later on in analogous cases), since their values can easily be restored. Since λσ,τ is a morphism of bifunctors, one has λσ,τ : X στ → (X τ )σ and (2.2) λσ,τ xστ = (xτ )σ λσ,τ for every morphism from A and every element from B, and also ∂λσ,τ = 0 for all σ, τ . Note also that the relations (2.1) and (2.2) imply, in particular, that −1 λσσ−1 ,σ = λσ,σ−1 and λσ,σ−1 x = (xσ )σ λσ,σ−1 . Given an action T = { Tσ } of a group G on a bimodule triple T = (A, B, ∂) and a system of factors λ for this action, we define the crossed group triple TG = T(G, T, λ). Namely, we consider the crossed group category AG = A(G, T, λ) [11, 8]. Its objects coincide with those of A, but morphisms X → Y P in the category AG are defined as formal (finite) linear combinations σ∈G aσ [σ], where aσ ∈ A(X σ , Y ), and the multiplication of such morphisms is defined by bilinearity and the rule (2.3) aσ [σ]bτ [τ ] = aσ bστ λσ,τ [στ ]. 6 YURIY A. DROZD The condition (2.1) for a system of factors is equivalent to the associativity of this multiplication. The AG-bimodule BG = B(G, T, λ) is constructed in an analogous P way: elements of BG(X, Y ) are formal (finite) linear combinations σ∈G xσ [σ], where xσ ∈ B(X σ , Y ), and their products with morphisms from AG are defined by the same formula (2.3), with the only difference that one of the elements aσ , bτ is a morphism from A, while the second onePis an element Pfrom B. The differentiation ∂ extends to AG if we set ∂( σ aσ [σ]) = σ ∂aσ [σ]. We identify every morphism a ∈ A(X, Y ) with the morphism a[1] ∈ AG(X, Y ) and every element x ∈ B(X, Y ) with the element x[1] ∈ BG(X, Y ) getting the embedding bifunctor T → TG. An action T of a group G on a bimodule triple T induces its action T∗ on the bimodule category El(T): an element σ ∈ G defines the functor (Tσ )∗ : x 7→ xσ . Moreover, if λ is a system of factors for the action T , it induces the system of factors λ∗ for the action T∗ : one has to set (λ∗ )σ,τ (x) = λσ,τ (X) if x ∈ B(X, X). Thus the crossed group category El(T)G = El(T)(G, T∗ , λ∗ ) is defined, as well as the embedding El(T) → El(T)G. One can also define the natural functor Φ : El(T)G → El(TG) as follows. PFor an object x ∈ B(X, X), set Φ(x) = x[1] ∈ BG(X, X). Let α = σ aσ [σ] be a morphism from x to y ∈ B(Y, Y ) in the category El(T)G. It means that aσ : xσ → y in the category El(T), i.e. aσ ∈ A(X σ , Y ) and aσ xσ = yaσ + ∂aσ . Then one can consider X → YP in the category AG(X, Y ), and P α as a morphism P αx[1] = σ aσ [σ]x[1] = σ aσ xσ [σ] = σ (yaσ + ∂aσ )[σ] = y[1]α + ∂α, so α is a morphism x[1] → y[1] in the category El(TG) and one can set Φ(α) = α. Proposition 2.1. The functor Φ is fully faithful, i.e. for any objects x, y from El(T)G it induces the bijective map HomTG(x, y) → HomTG (x, y), where HomTG denotes the morphisms in the category El(T)G. P Proof. Obviously, = σ aσ [σ] : x[1] → y[1], P thisσmap is injective. Let αP i.e. αx[1] = σ aσ x [σ] = y[1]α + ∂α = σ (yaσ + ∂aσ )[σ]. Then aσ xσ = yaσ + ∂aσ for all σ, so aσ : xσ → y in the category El(T), thus α : x → y in the category El(T)G. Therefore, this map is also surjective.  If the group G is finite, one can also construct a functor Ψ : El(TG) → L El(T). For every object X ∈ Ob A, set X̃ = σ∈G X σ and for every P element ξ = σ xσ [σ] ∈ BG(X, X), where xσ : X σ 99K X, denote by ξ˜ L the element from B(X̃, X̃) = σ,τ B(X τ , X σ ) such that its component −1 ξ˜σ,τ ∈ B(X τ , X σ ) equals xσσ−1 τ λσ,σ−1 τ . Note that xσ−1 τ : X σ τ 99K Y , GROUP ACTION ON BIMODULE CATEGORIES 7 −1 σ τ σ hence xσσP ) 99K Y σ , thus xσσ−1 τ λσ,σ−1 τ : X τ 99K Y σPindeed. −1 τ : (X Let η = σ yσ [σ] ∈ BG(Y, Y ), where yσ ∈ B(Y σ , Y ) and α = σ aσ [σ] be a morphism from ξ to η, where aσ ∈ A(X σ , Y ). Since XX XX αξ = aρ [ρ]xσ [σ] = aρ xρσ λρ,σ [ρσ] = ρ = σ ρ X X τ ρ σ
aρ xρρ−1 τ λρ,ρ−1 τ [τ ], and ηα = XX ρ = yρ [ρ]aσ [σ] = τ ρ σ X X XX yρaρρ−1 τ λρ,ρ−1 τ ρ yρ aρσ λρ,σ [ρσ] = σ
[τ ], it means that, for each τ , X X (2.4) aρ xρρ−1 τ λρ,ρ−1 τ = yρ aρρ−1 τ λρ,ρ−1 τ + ∂aτ . ρ ρ Consider the morphism α̃ : X̃ → Ỹ such that α̃σ,τ = aσσ−1 τ λσ,σ−1 τ : X τ → Y σ . Then the (σ, τ )-component of the product α̃ξ˜ equals X X −1
σ I= aσσ−1 ρ λσ,σ−1 ρ xρρ−1 τ λρ,ρ−1 τ = aσσ−1 ρ (xσρ−1 τρ λσ,σ−1 ρ λρ,ρ−1 τ , ρ ρ while the (σ, τ )-component of the product η̃ α̃ equals X X −1
σ II = yσσ−1 ρ λσ,σ−1 ρ aρρ−1 τ λρ,ρ−1 τ = yσσ−1 ρ (aσρ−1 τρ λσ,σ−1 ρ λρ,ρ−1 τ . ρ ρ (In both cases we used the relation (2.2) replacing τ by σ −1 ρ). Since, by the condition (2.1) for the system of factors, λσ,σ−1 ρ λρ,ρ−1 τ = λσσ−1 ρ,ρ−1 τ λσ,σ−1 τ , and ∂λσ,σ−1 τ = 0, we get from the relation (2.4) that I = II + ∂ α̃σ,τ (we just replace ρ by σ −1 ρ, τ by σ −1 τ , then apply the functor Tσ to both sides). Therefore, α̃ is a morphism ξ˜ → η̃ and one can define the functor Ψ setting Ψ(ξ) = ξ˜ and Ψ(α) = α̃. Proposition 2.2. The functors Φ and Ψ form an adjoint pair, i.e. there is a natural isomorphism HomTG(Φx, η) ≃ HomT(x, Ψη) for each objects x ∈ El(T) and η ∈ El(TG). 8 YURIY A. DROZD P Proof. Let x ∈ B(X, X), η ∈ BG(Y, Y ), η = σ yσ [σ], where y : Y σ 99K Y , andPα : Φ(x) = x[1] → η in the category El(TG). By definition, α = σ aσ [σ], where aσ : X σ → Y , and X X X
αx[1] = aσ xσ [σ] = ηα + ∂α = yρ aρρ−1 σ λρ,ρ−1 σ + ∂aσ [σ], σ σ ρ i.e. aσ xσ = (2.5) X yρ aρρ−1 σ λρ,ρ−1 σ + ∂aσ ρ L σ for every σ. Consider the morphism f (α) = β : X τ → Ỹ = σY σ σ such that its component βσ : X → Y equals aσ−1 λσ,σ−1 . Compute the σ-components of the products βx and η̃β, where η̃ = Ψη. They equal, respectively, −1 βσ xτ = aσσ−1 λσ,σ−1 x = aσσ−1 (xσ )σ λσ,σ−1 and X yσσ−1 ρ λσ,σ−1 ρ aρρ−1 λρ,ρ−1 = −1 yσσ−1 ρ (aσρ−1 ρ )σ λσ,σ−1 ρ λρ,ρ−1 = ρ ρ = X X −1 yσσ−1 ρ (aσρ−1 ρ )σ λσσ−1 ρ,ρ−1 λσ,σ−1 . ρ The relation (2.5), where σ is replaced by σ −1 and ρ by σ −1 ρ, these two expressions differ exactly by ∂βσ = ∂aσσ−1 λσ,σ−1 , hence β = f (α) is a morphism x → η̃ in the category El(T). Obviously, if α 6= α′ , then f (α) 6= f (α′ ) as well. Moreover, one easily checks that the correspondence α 7→ f (α) is functorial in x and η, i.e. f (α)b = f (αΦb) and f (γα) = (Ψγ)f (α) for any morphisms b : x′ → x and γ : η → η ′ . On the contrary, let β : x → η̃ be a morphism in the category σ El(T). Denote by βσ : X → YP the corresponding component of β and consider the morphism α = σ aσ [σ] : X → Y in the category AG, σ σ where aσ = λ−1 σ,σ−1 βσ−1 : X → Y . Comparing the σ-components in the equality βx = η̃β, we get X (2.6) βσ x = yσσ−1 ρ λσ,σ−1 ρ βρ + ∂βσ . ρ The coefficients near [σ] in the products α(Φx) = αx[1] and ηα equal, respectively, σ σ aσ xσ = λ−1 σ,σ−1 βσ−1 x GROUP ACTION ON BIMODULE CATEGORIES 9 and X yρ aρ λρ−1 σ = X ρ ρ σ −1 yρ (λ−1 ρ−1 σ,σ−1 ρ ) βσ−1 ρ λρ,ρ σ = = X ρ σ −1 yρ (λ−1 ρ−1 σ,σ−1 ρ ) λρ,ρ σ βσ−1 ρ . ρ −1 ρ ρ The relation (2.6), with σ replaced by σ −1 , implies that X σ−1 σ σ σ aσ xσ − ∂aσ = λ−1 σ,σ−1 (yσρ ) λσ−1 ,ρ βσ−1 ρ = ρ = X = X σ σ yσρ λ−1 σ,σ−1 λσ−1 ,ρ βσ−1 ρ = ρ X σ σ yρ λ−1 σ,σ−1 λσ−1 ,σρ βρ = ρ σ yρλ−1 σ,σ−1 ρ βσ−1 ρ = ρ X ρ σ −1 yρ (λ−1 ρ−1 σ,σ−1 ρ ) λρ,ρ σ βσ−1 ρ . ρ (Passing from the second row to the third, we used the relation (2.1) for the triple σ, σ −1 , ρ, while in the third row we used the same relation for the triple ρ, ρ−1 σ, σ −1 ρ.) Therefore, αx[1] = ηα + ∂α, thus α is a morphism Φx → η. Moreover, the σ-component of f (α) equals −1 σ σ −1 σ σ −1 = (λ −1 ) λσ,σ −1 βσ = βσ . aσσ−1 λσ,σ−1 = (λ−1 σ−1 ,σ ) (βσ ) λσ,σ σ ,σ Hence f (α) = β and the map α 7→ f (α) is bijective.  3. Separable actions We call the center Z(T) of a bimodule triple T = (A, B, ∂) the endomorphism ring of the identity bifunctor idT. In other words, the elements of this center are the sets of morphisms α = { αX : X → X | X ∈ Ob A } , such that αY a = aαX for every morphism a : X → Y , αY x = xαX for every element x : X 99K Y and ∂αX = 0 for all X. In particular, the element αX belongs to the center of the algebra A(X, X). One easily sees that if α = { αX } and β = { βX } are two such sets, then the sets α + β = { αX + βX } and αβ = { αX βX } also belong to Z(T). Hence, this center is a ring (even a K-algebra), commutative, since αX βX = βX αX . If F = (F0 , F1 ) is an equivalence of bimodule triples ∼ T → T′ = (A′ , B′ , ∂ ′ ), it induces an isomorphism FZ : Z(T) → Z(T′ ). Namely, for any X ′ ∈ Ob A′ , choose an isomorphism λ : X ′ → F0 X for some X ∈ Ob A, and, for each element α = { αX } ∈ Z(T), set ∼ (FZ α)X ′ = λ−1 (F0 αX )λ. Let Y ′ be another object from A, µ : Y ′ → 10 YURIY A. DROZD F0 Y and (FZ α)Y ′ = µ−1 (F0 αY )µ. If a′ ∈ A′ (X ′ , Y ′ ), the morphism µa′ λ−1 : F0 X → F0 Y is of the form F0 a for some a : X → Y . It gives (FZ α)Y ′ a′ = µ−1 (F0 αY )µ · µ−1 (F0 a)λ = (3.1) = µ−1 (F0 αY )(F0 a)λ = µ−1 (F0 (αY a))λ = = µ−1 F0 (aαX )λ = µ−1 (F0 a)(F0 αX )λ = = a′ λ−1 (F0 αX )λ = a′ (FZ α)X ′ . Especially, if Y ′ = X ′ and a′ = 1X ′ , we see that FZ (α)X ′ does not depend on the choice of X and λ. Just in the same way one checks that (FZ α)Y ′ x′ = x′ (FZ α)X ′ for every x′ ∈ B′ (X ′ , Y ′ ). Note that an isomorphism λ can always be chosen such that ∂λ = 0: for instance, one can use the isomorphism of bifunctors φ : idT′ → F G for some bifunctor G and set X = G0 X ′ , λ = φ(X ′). Therefore ∂ ′ (FZ α)X ′ = 0, so the set FZ α = { (FZ α)X ′ } belongs to Z(T′ ). Obviously, FZ (α + β) = FZ α + FZ β and FZ (αβ) = (FZ α)(FZ β), and if F ′ : T′ → T′′ is another equivalence, then (F ′ F )Z = FZ′ FZ . Moreover, similarly to the equalities (3.1), one easily verifies that if F ≃ F ′ , then FZ = FZ′ . In particular, if G : T′ → T is such a bifunctor that F G ≃ idT′ and GF ≃ idT, then GZ = FZ−1 , thus FZ is an isomorphism. These considerations imply that every action T of a group G on a triple T induces an action of the same group on the center of this triple with the trivial system of factors: if λ is a system of factors for the σ −1 for every α ∈ Z(T). Espeaction T , then (ασ )X = λ−1 σ,σ−1 αX σ −1 λσ,σ cially, if the group G is finite, any element α P from Z(T) its trace is P for σ σ −1 defined as tr α = trG α = σ α , i.e. (tr α)X = σ λ−1 σ,σ−1 αX σ −1 λσ,σ . Obviously, the center of the triple TG is a subalgebra of the center of T. Proposition 3.1. The center Z(TG) coincides with the subalgebra Z(T)G of elements of the center Z(T) that are invariant under the action of G. In particular, if this group is finite, the trace of each element α ∈ Z(T) belongs to Z(TG). Proof. Let α = { αX } be an element of the center Z(T). Since αY a[σ] = σ [σ] for each morphism a : X σ → Y , this aαX σ [σ] and a[σ]αX = aαX σ element belongs to the center of the triple TG if and only if αX σ = αX for every X and every σ. But then −1 −1 σ σ σ −1 = λ −1 = αX , (ασ )X = λ−1 σ,σ−1 (αX ) λσ,σ σ,σ−1 αX σ −1 λσ,σ so α is invariant under the action of G. Just in the same way one verifies that every invariant element from Z(T) belongs to Z(TG). The last GROUP ACTION ON BIMODULE CATEGORIES 11 statement follows from the fact that tr α is always invariant under the action of the group.  Definition. We call an action of a finite group G on a bimodule triple T separable, if there is an element of the center α ∈ Z(T) such that tr α = 1. Certainly, it is enough tr α to be invertible. For instance, if the order of the group G is invertible in the ring K, any action of this group is separable. Another important case is when the center of the triple T contains a subring R such that it is G-invariant, the group G acts effectively (i.e. for any σ 6= 1 there is r ∈ R such that r σ 6= r) and R is a separable extension of its subring of invariants RG [4]. If R is a field and G acts effectively on R, the last condition always holds. In general case it is necessary and sufficient that every element σ 6= 1 induce a non-identity automorphism of the residue field R/m for each maximal ideal m ⊂ R such that mσ = m [4, Theorem 1.3]. For an action of a group on a category (that is, on a principle triple) the notion of separability was introduced in [8]. Obviously, if an action of a group on a bimodule triple is separable, so is also its induced action on the corresponding bimodule category. We also note that if an action of a group G is separable, P so is the action of every subgroup H ⊆ G: if trG α = 1 and β = σ∈R ασ , where R is a set of representatives of right cosets H\G, then trH β = 1. Recall that a ring homomorphism A → A′ is called separable if the ′ ′ ′ natural homomorphism of A′ -bimodules a⊗b P A ⊗A A →′ A sending ′ to ab splits, i.e. there is an element b ⊗ c in A ⊗ A such that i i A i P P P ′ b c = 1 and ab ⊗ c = b c a for all a ∈ A . i i i i i i i i i Lemma 3.2. An action of a finite group G on a triple T is separable if and only if so is the ring homomorphism Z → ZG, where Z = Z(T). Proof. Suppose that the action is separable, α = an P{ ασX } is such −1 element of the center σ α [σ] ⊗ [σ ] ∈ P that tr α = 1. Let t = ZG⊗Z ZG. Then σ ασ [σ][σ −1 ] = tr α = 1 and, for any β ∈ Z, τ ∈ G, X X β[τ ] · t = βατ σ [τ σ] ⊗ [σ −1 ] = ατ σ β[τ σ] ⊗ [σ −1 ] = σ = X σ σ σ −1 α β[σ] ⊗ [σ τ ] = X ασ [σ] ⊗ [σ −1 ]β[τ ] = t · β[τ ], σ so the homomorphism Z → ZG is separable. Now let the homomorphism Z → ZG be P separable. Note that every element from ZG ⊗Z ZG is of the form ⊗ [τ ] for some σ,τ zσ,τ [σ] P zσ,τ ∈ Z. Hence there are elements zσ,τ such that σ,τ zσ,τ [στ ] = 12 YURIY A. DROZD
P P zσ,σ−1 τ [τ ] = 1, i.e. σ zσ,σ−1 = 1, and σ zσ,σ−1 τ = 0 if τ 6= 1, moreover, for every ρ ∈ G we have: X X ρ
X ρ [ρ] zσ,τ [σ] ⊗ [τ ] = zσ,τ [ρσ] ⊗ [τ ] = zρ−1 σ,τ [σ] ⊗ [τ ] = P P τ σ σ,τ = X σ,τ Thus zσ,σ−1 σ,τ σ,τ X X
zσ,τ [σ] ⊗ [τ ] [ρ] = zσ,τ [σ] ⊗ [τ ρ] = zσ,τ ρ−1 [σ] ⊗ [τ ]. zρρ−1 σ,τ σ = z1,1 . σ,τ σ,τ = zσ,τ ρ−1 for ρ, σ, τ . Especially, for σ = ρ, τ = 1 we get Therefore, tr z1,1 = 1 and the action is separable.  Corollary 3.3. If an action of a group G on a triple T = (A, B, ∂) is separable, so is also the embedding functor A → AG, i.e. the homomorphism of AG-bimodules φ : AG ⊗A AG → AG splits, or, the same, for every object X ∈ Ob A there is an element tX ∈ (AG⊗A AG)(X, X) such that φ(tX ) = 1X and atX = tY a for each a ∈ AG(X, Y ). In particular, the action of a group G on a category A is separable if and only if so is the embedding functor A → AG. Theorem 3.4. If an action of a finite group G on a bimodule triple T = (A, B, ∂) is separable, the functor Φ : El(T)G → El(TG) induces an equivalence of the categories add El(T)G → El(TG). Proof. First we prove a lemma about fully additive categories. Lemma 3.5. Let C be a fully additive category, F : C → C ′ be a fully faithful functor. F is an equivalence of categories if and only if every object X ′ ∈ C ′ is isomorphic to a direct summand of an object of the form F Y , where Y ∈ Ob C. Proof. The necessity of this condition is obvious, so we only have to prove the sufficiency. If X ′ is a direct summand of F Y , there are morphisms ι′ : X ′ → F Y and π ′ : F Y → X ′ such that π ′ ι′ = 1X ′ . Then e′ = ι′ π ′ is an idempotent endomorphism of the object F Y . Since the functor F is fully faithful, e′ = F e for an idempotent endomorphism e : Y → Y . Since the category C is fully additive, there are an object X and morphisms ι : X → Y and π : Y → X such that e = ιπ and πι = 1X . Then (F ι)(F π) = e′ and (F π)(F ι) = 1F X . Let u = π ′ F (ι), v = (F π)ι′ ; then we immediately get that uv = 1X ′ and vu = 1F X , i.e. X ′ ≃ F X, the functor F is also dense, so it is an equivalence of categories.  We prove now that every object ξ of the category El(TG) is isomorphic to a direct summand of ΦΨξ. Since Φ is fully faithful (Proposition P 2.1), Theorem 3.4 follows then from Lemma 3.5. Let ξ = σ xσ [σ] ∈ BG(X, X), where xσ ∈ B(X σ , X). Then Ψξ = ξ˜ ∈ B(X̃, X̃), where GROUP ACTION ON BIMODULE CATEGORIES 13 L ˜ X̃ = σ X σ and ξ˜σ,τ = xσσ−1 τ λσ,σ−1 τ , and ΦΨξ = ξ[1]. Choose an element α ∈ Z(T) such that tr α = 1. Consider the morphism π : X̃ → X −1 σ such that its σ-component equals πσ = λ−1 σ−1 ,σ [σ ] : X → X. Then the σ-component of the element ξπ equals X X −1 xρ λ−1 xρ (λρσ−1 ,σ )−1 λρ,σ−1 [ρσ −1 ] = ρσ−1 ,σ [ρσ ] ρ ρ (we use the relation (2.1) for the triple ρ, σ −1 , σ), while the σ-component ˜ equals of the element π ξ[1] X X ρ ρ−1 ρ−1 ρ−1 −1 −1 λ−1 (x ) λ [ρ ] = xρ−1 σ λ−1 −1 −1 −1 ρ ,ρ ρ σ ρ,ρ−1 λρ,ρ−1 σ [ρ ] = ρ,ρ σ ρ = ρ X −1 xρ−1 σ λ−1 ρ−1 ,σ [ρ ] ρ = X −1 xρ λ−1 ρσ−1 ,σ [ρσ ]. ρ Here we used first the relation (2.1) for the triple ρ−1 , ρ, ρ−1 σ and then ˜ and, since ∂π = 0, π is a morphism replaced ρ by σρ−1 . So ξπ = π ξ[1] ˜ → ξ. Now consider the morphism ι : X → X̃ such that its σξ[1] component equals αX σ [σ]. The σ-component of the element ιξ equals X X αX σ xσσ−1 ρ λσ,σ−1 ρ [ρ], αX σ xσρ λσ,ρ [σρ] = ρ ρ ˜ and the σ-component of the element ξ[1]ι equals X X xσσ−1 ρ λσ,σ−1 ρ αX ρ [ρ] = αX σ xσσ−1 ρ λσ,σ−1 ρ [ρ], ρ ρ ˜ ˜ since α ∈ Z(T). Therefore, ξ[1]ι = ιξ, thus ι is a morphism ξ → ξ[1]. P −1 −1 σ But πι = σ λσ−1 ,σ αX σ λσ −1 ,σ = (tr α)X = 1X = 1ξ , which just means ˜ that the element ξ is a direct summand of the element ξ[1].  One can get more information if the group G is finite abelian and the ring K is a field containing a primitive n-th root of unit, where n = #(G), i.e. such an element ζ that ζ n = 1 and ζ k 6= 1 for 0 < k < n. Then certainly char K ∤ n, so any action of the group G on a bimodule triple T = (A, B, ∂) is separable. Let Ĝ be the group of characters of the group G, i.e. the group of its homomorphisms to the multiplicative group K× of the field K. This group acts on the triple TG (with the trivial system of factors) by the 14 YURIY A. DROZD rules: X χ = X for every X ∈ Ob A, X
χ X xσ [σ] = χ(σ)xσ [σ], σ σ P where χ ∈ Ĝ and σ xσ [σ] is a morphism from AG or an element from BG. Recall that also #(Ĝ) = n, so this action is separable as well. We denote by χ0 the unit character, i.e. such that χ0 (σ) = 1 for all σ ∈ G. By definition, morphisms from AGĜ and elements of BGĜ are of the P form σ,χ xσ,χ [σ][χ]. We write [χ] instead of [1][χ] and σ instead of [σ][χ0 ]. In particular an element x[1][χ0 ] is denoted by x. Theorem 3.6. The bimodule triples add T and add TGĜ are equivalent. 1X χ(σ)[χ] from the endomorProof. Consider the elements eσ = n χ phism ring AGĜ(X, X). The formulae of orthogonality for characters [7, Theorem 3.5] immediately imply that eσ are mutually orthogonal P idempotents and σ eσ = 1. Moreover, eσ [τ ] = [τ ]eστ , so all these idempotents are conjugate, thus define isomorphic directLsummands Xσ of the object X in the category add AGĜ, and X = σ Xσ . We define the bifunctor Θ : add T → add TGĜ setting ΘX = X1 and Θx = xe1 = e1 x, where x is a morphism X → Y or an element from B(X, Y ). Obviously, the functor Θ0 : add A → add AGĜ satisfies the conditions of Lemma 3.5, so it defines an equivalence of categories. Since every map Θ1 (X, Y ) is also bijective, the bifunctor Θ is an equivalence by Lemma 1.1.  Corollary 3.7. The categories El(T) and add El(T)GĜ are equivalent. Proof. Indeed, add El(T)GĜ ≃ El(TGĜ) by Theorem 3.4.  4. Radical and decomposition In this section we suppose that the ring K is noetherian, local and henselian [3] (for instance, complete). We denote by m its maximal ideal and by k = K/m its residue field. We call a K-category A piecewise finite if all K-modules A(X, Y ) are finitely generated. Then its additive hull add A is piecewise finite as well. Moreover, each endomorphism ring A P = A(X, X) is semiperfect, i.e. possesses a unit decomposition 1 = ni=1 ei , where ei are mutually orthogonal idempotents and all rings ei Aei are local. Hence the category add A is local, i.e. every object in it decomposes into a finite direct sum of objects with GROUP ACTION ON BIMODULE CATEGORIES 15 local endomorphism rings. Therefore this category is a Krull–Schmidt category, i.e. every objectL X in it decomposes into a finite direct sum m of indecomposables: such a decomposition is unique, Ln X ′= i=1 Xi and i.e. if also X = i=1 Xi , where all Xi′ are indecomposable, then m = n and there is a permutation ε of the set { 1, 2, . . . , m } such that Xi ≃ Xεi′ for all i [2, Theorem I.3.6]. Recall that the radical of a local category A is the ideal rad A consisting of all such morphisms a : X → Y that all components of a with respect to some (then any) decompositions of X and Y into a direct sum of indecomposables are non-invertible. We denote A = A/ rad A. In particular, rad A(X, X) is the radical of the ring A(X, X) and A(X, X) is a semisimple artinian ring [9]. In the case of a piecewise finite category always rad A ⊇ mA, in particular, A(X, X) is a finite dimensional k-algebra. The category A is semisimple, i.e. every object in it decomposes into a finite direct sum of indecomposables and A(X, Y ) = 0 if X and Y are non-isomorphic indecomposables, while A(X, X) is a skewfield for every indecomposable object X. (Note that an object X is indecomposable in the category A if and only if it is so in the category A). Moreover, rad A is the biggest among the I ⊂ A such that the factor-category A/I is semisimple. If a finite group G acts on a piecewise finite category A with a system of factors λ, the category AG is piecewise finite as well. Moreover, the radical is a G-invariant ideal, i.e. (rad A)σ = rad A for all σ ∈ G, and the ideal (rad A)G is contained in the radical of the category AG. Proposition 4.1. If the action of a group G on a category A is separable, so is also its induced action on the category AG. In this case rad(AG) = (rad A)G and the category AG is semisimple. Proof is evident.  From now on, we suppose that A is a piecewise finite local Kcategory, R = rad A, X ∈ Ob A is an indecomposable object from A, A = A(X, X) and G is a finite group acting on A with a system of factors λ so that its action is separable. We are interested in the decomposition of the object X in the category AG into a direct sum of indecomposables, especially, the number νG (X) of non-isomorphic summands in such a decomposition. Recall that such decomposition comes from a decomposition of the ring AG(X, X) or, equivalently, of the ring AG(X, X) into a direct sum of indecomposable modules. Proposition 4.2. Let H = { σ ∈ G | X σ ≃ X }. Then AG(X, X)/RG(X, X) ≃ AH(X, X)/RH(X, X), in particular, νG (X) = νH (X). 16 YURIY A. DROZD Proof is evident, since aσ ∈ R for every morphism aσ : X σ → X if σ∈ / H.  Corollary 4.3. If X σ 6≃ X for all σ ∈ G, the object X remains indecomposable in the category AG. Therefore, dealing with the decomposition of X, we can only consider the action of the subgroup H. For every σ ∈ H we fix an isomorphism φσ : X σ → X and consider the action T ′ of the group H on the ring A given by the rule Tσ′ (a) = φσ aσ φ−1 σ . One easily veri′ σ −1 fies that the elements λσ,τ = φσ φτ λσ,τ φστ form a system of factors for this action, moreover, the map a[σ] 7→ aφσ [σ] establishes an isomorphism A(H, T ′ , λ′ ) ≃ AH(X, X). Thus, in what follows, we investigate the algebras A(H, T ′ , λ′) and D(H, T ′ , λ̄), where D = A/ rad A and λ̄σ,τ denotes the image of λ′σ,τ in the skewfield D. The latter factorring is finite dimensional skewfield (division algebra) over the field k. We denote by F the center if this algebra (it is a field). Let N be the subgroup of H consisting of all elements σ such that the automorphism Tσ′ induces an inner automorphism of the skewfield D, or, equivalently, the identity automorphism of the field F [7, Corollary IV.4.3]. It is a normal subgroup in H. For every element ρ ∈ N we choose an element dρ ∈ D such that Tρ′ (a) = dρ ad−1 for all a ∈ D. ρ We also choose a set S of representatives of cosets H/N and, for every σ ∈ H, denote by σ̄ the element from S such that σN = σ̄N, and by ρ(σ) the element from N such that σ = ρ(σ)σ̄. Now we set ′ Dσ (a) = d−1 ρ(σ) Tσ (a)dρ(σ) . An immediate verification shows that we get in this way an action of the group H on the skewfield D with the −1 σ system of factors µσ,τ = d−1 ρ(σ) (dρ(τ ) ) λ̄σ,τ dρ(στ ) and, besides, the map [σ] 7→ dρ(σ) [σ] induces an isomorphism D(H, T ′ , λ̄) ≃ D(H, D, µ). Note that now N = { σ ∈ H | Dσ = id } = { σ ∈ H | Dσ |F = id } . Moreover, one easily sees that µσ,τ ∈ F if σ, τ ∈ H. Further on we denote DH = D(H, D, µ). The number of nonisomorphic indecomposable summands in the decomposition of DH equals the number of simple components of this algebra [7, Theorem II.6.2], or, the same, the number of simple components of its center. Proposition 4.4. The center of the algebra DH coincides with the set (FN)H = { α ∈ FH | ∀τ [τ ]α = α[τ ] } = nX
o = aσ [σ] ∀σ aσ ∈ F & ∀τ (τ ∈ H ⇒ aτσ µτ,σ = aτ στ −1 µτ στ −1 ,τ ) . σ∈N GROUP ACTION ON BIMODULE CATEGORIES 17 Especially, if N = { 1 }, then DH is a central simple algebra over the field of invariants FH , hence, νG (X) = 1.1 P Proof. If an element α = Pσ aσ [σ] belongs to the center of DH, then P P σ σ −1 σ baσ [σ] = σ aσ [σ]b = σ aσ b [σ], so if aσ 6= 0, then b = aσ baσ , σ hence, P τ σ ∈ N, b = b and P aσ ∈ F. Finally, P the equalities [τ ]α = σ aσ µτ,σ [τ σ] = α[τ ] = σ aσ µσ,τ [στ ] = σ aτ στ −1 µτ στ −1 ,τ [τ σ] complete the proof.  Corollary 4.5. If F = k (for instance, the residue field k is algebraically closed) and the group H is abelian, the center of the algebra DH coincides with kH0 , where H0 is the subgroup of H consisting of all elements σ such that µσ,τ = µτ,σ for all τ ∈ H. In particular, νG (X) = #(H0 ). Proof. In this case N = H, so the center of DH coincides with kH0 (one easily checks that H0 is indeed a subgroup). Since the latter algebra is commutative and semisimple, it is isomorphic to km , where m = #(H0 ), therefore, the number of its simple components equals m.  Corollary 4.6. If F = k and the group H is cyclic, the center of the algebra DH coincides with kH and νG (X) = #(H). Proof. Actually, in this case it is well-known that µσ,τ = µτ,σ for all σ, τ ∈ H.  Note that all these corollaries hold if the group G itself is abelian or cyclic. If K-category A is piecewise finite, so is every bimodule category El(T) as well, where T = (A, B, ∂). If a group G acts separably on the triple T, it acts separably on the category El(T) as well, and, according to Theorem 3.4, add El(T)G ≃ El(TG), this equivalence being induced by the functor Φ : x 7→ x[1]. Therefore, all the results above can be applied to the study of the decomposition of an element x[1] in the category El(TG). We only quote explicitly the reformulations of Corollaries 4.5 and 4.6 for this case. Corollary 4.7. Let the residue field k be algebraically closed and the group H = { σ | xσ ≃ x } be abelian. Choose isomorphisms φσ : xσ → x for every element σ ∈ H and denote by µσ,τ the image of a morphism φσ φτ λσ,τ φ−1 στ in k ≃ HomT(x, x)/ radT(x, x). Then the number of nonisomorphic indecomposable direct summands in the decomposition of the object x[1] in the category El(TG) equals the order of the group H0 = { σ | ∀τ µσ,τ = µτ,σ }. Especially, if the group H is cyclic, this number equals the order of H. 1 The last statement is well-known, see [10, Theorem 4.50]. 18 YURIY A. DROZD Remark 4.8. It is evident that all these statements also hold if separable is the action of the group H on the skewfield D, or, equivalently, on its center F. It is known [10, Section 4.18] that one only has to verify that separable is the action of the subgroup N, i.e. that char k ∤ #(N), since the action of N on F is trivial. Proposition 4.1 evidently implies some more corollaries concerning the structure of the radical of the category AG (for instance, bimodule category El(TG)). Corollary 4.9. Let the action of the group G is separable. If a set of morphisms { ai } is a set of generators of the A-module (rad A)(X, ) (or Aop -module (rad A)( , X) ), its image { ai [1] } in AG is a set of generators of the AG-module (rad AG)(X, ) (respectively, Aop -module (rad AG)( , X)). We call a morphism a : Y → X left almost split (respectively, right almost split) if it generates the A-module (rad A)( , X) (respectively, Aop -module (rad A)(Y, ) ), and an equality a = bf implies that the morphism f is left invertible, or, the same, is a split epimorphism (respectively, the equality a = f b implies that g is right invertible, or, the same, is a split monomorphism).2 Corollary 4.10. Let the action of G is separable. If a morphism a : Y → X is left (right) almost split, so is a[1] as well. a b A sequence X − →Y − → X ′ is called almost split if the morphism a is left almost split, the morphism b is right almost split and, besides, a = Ker b and b = Cok a, i.e., for every object Z, the induced sequences of groups 0 → A(Z, X) → A(Z, Y ) → A(Z, X ′), 0 → A(X ′ , Z) → A(Y, Z) → A(X, Z) are exact. a Corollary 4.11. Let the action of G is separable. If a sequence X − → b a[1] b[1] Y − → X ′ is almost split in the category A, the sequence X −−→ Y −−→ ′ X is almost split in the category AG. 2 In the book [1] one only uses these notions in the case when X (respectively, Y ) is indecomposable. However, one can easily see that a left (right) almost split morphism in our sense is just a direct sum of those in the sense of [1]. The same also concerns the notion of the almost split sequences used below. GROUP ACTION ON BIMODULE CATEGORIES 19 Since, under the separability condition, every object from add AG is a direct summand of an object that has come from the category A, Corollaries 4.10 and 4.11 describe almost split morphisms and sequences in the category add AG as soon as they are known in the category A. In particular, these results can be applied to the bimodule categories El(TG) due to Theorem 3.4. References [1] Auslander M., Reiten I. and Smalø S.O. Representation Theory of Artin Algebras. Cambridge University Press, 1995. [2] Bass H. Algebraic K-theory. New York, Benjamin Inc. 1968. [3] Bourbaki N. Commutative algebra. Chapters 1–7. Berlin, Springer–Verlag, 1989. [4] Chase S.U., Harrison D.K. and Rosenberg A. Galois Theory and Galois Cohomology of Commutative Rings. Mem. Amer. Math. Soc. 52 (1965), 1–19. [5] Crawley-Boevey W.W. Matrix problems and Drozd’s theorem. Banach Cent. Publ. 26, Part 1 (1990), 199-222. [6] Drozd Y.A. Reduction algorithm and representations of boxes and algebras. Comtes Rendue Math. Acad. Sci. Canada 23 (2001), 97-125. [7] Drozd Y.A. and Kirichenko V.V. Finite Dimenional Algebras. Berlin, Springer– Verlag, 1994. [8] Drozd Y.A., Ovsienko S.A. and Furchin B.Y. Categorical constructions in the theory of representations. Algebraic Structures and their Applications. Kiev, UMK VO, 1988, 17–43. [9] Gabriel P. and Roiter A.V. Representations of Finite-Dimensional Algebras. Algebra VIII, Encyclopedia of Math. Sci. Berlin: Springer–Verlag, 1992. [10] Jacobson N. The Theory of Rings. AMS Math. Surveys, vol. 1. 1943. [11] Reiten I. and Riedtmann C. Skew group algebras in the representation theory of Artin algebras. J. Algebra, 92 (1985), 224–282. Institute of Mathematics, National Academy of Sciences of Ukraine, Tereschenkivska 3, 01601 Kiev, Ukraine E-mail address: drozd@imath.kiev.ua