Group action on bimodule categories

GR OUP A CTION ON BIMODULE CA TEGORIES YURIY A. DR OZD T o the memolry of A. V. Ro iter Abstract. W e cons ide r actions of gro ups o n categories and bi- mo dules, the related cross ed group categories and bimodules, a nd prov e for them analog ue s of the res ult know for repres en tations of crossed gro up algebras a nd catego r ies. Sk ew group algebras arise natura lly in lo ts of questions. In partic- ular, the prop erties of t he categories of represen tations of sk ew group algebras and, more generally , sk e w group categories ha v e b een studied in [11, 8]. On the other hand, “ matrix pr oblems ,” esp ecially , bi m o d- ule c ate gories pla y no w a crucial role in the theory of represen tations [5, 6]. The situation, when a group acts on a bimo dule, thus also on the bimo dule category is a lso rather ty pical. Therefore one needs to deal with skew bimo dules and their bimo dule catego ries. In this pap er w e shall study sk ew bimo dules and bimo dule categories and pro v e for them some a nalogues of the results of [11, 8]. In Section 1 w e recall g eneral notions related to bimo dule categories. In Section 2 w e consider actions of groups on bimo dule and bimo d- ule categories and the arising functors. The ma in results are those of Section 3, where w e define sep ar able actions and prov e that in the sep- arable case the bimo dule category o f the sk ew bimo dule is equiv alent to the ske w category of t he original one. W e also consider specially the case of the ab elian groups, since in this case the original category can b e restored from the sk ew one using the group of c haracters. Section 4 is dev oted to the decomp osition of ob j ects in ske w group categories, es- p ecially , to the n um b er of non-isomorphic direct summands in suc h de- comp ositions. W e also consider the r adi c al and almost split mo rp h isms in the sk ew group catego r ies (under the separabilit y condition). 2000 Mathematics S ubje ct Classific ation. Primar y 16 S35, Secondary 16G10 , 16G70. Key wor ds and phr ases. ca tegories, bimo dules, gr o up action, cro s sed g roup categorie s. This resear ch was partially supp orted by the INT AS Gr a n t no. 06 -1000 017-90 93. 1 2 YURIY A. DROZD 1. Bimodule ca tegories W e recall the main definitions related to bimo dule categories [5, 6]. W e fix a comm utativ e ring K . All categories tha t w e cons ider are supp osed t o b e K -c ate g ories , whic h means that all sets of morphisms are K -mo dules, while the m ultiplication is K -bilinear. W e denote the set of morphisms from an ob ject X to an ob ject Y in a category A by A ( X , Y ). A mo dule (more precise, a le f t mo dule ) ov er a category A , or a A -mo dule is, by definition, a K -linear functor M : A → K - Mo d , where K - Mo d denotes the category of K -mo dules. If M is s uc h a mo dule, x ∈ M ( X ) and a ∈ A ( X , Y ), we write, as usually , ax instead of M ( a )( x ). Suc h mo dules ha v e all usual prop erties of mo dules o v er rings. The category of all A -mo dules is denoted by A - Mo d . A bimo dule o v er a category A , o r an A -bimo dule, is, b y definition, a K -bilinear functor B : A op × A → K - Mo d , where A op is the opp osite category to A . If x ∈ B ( X , Y ), a : X ′ → X (i.e. a : X → X ′ in A op ), b : Y → Y ′ , w e write bxa instead of B ( a, b )( x ) (this elemen t b elongs to B ( X ′ , Y ′ )). In particular, xa and bx denote, resp ectiv ely , B ( a, 1 Y )( x ) and B (1 X , b )( x ). If a bimo dule B is fixed, w e of ten write x : X 99 K Y instead of x ∈ B ( X , Y ). A category A is called ful ly add itive if it is additiv e (i.e. has direct sums X ⊕ Y o f any pair of o b j ects X , Y and a zero ob ject 0) and ev ery idemp oten t endomorphism e ∈ A ( X , X ) splits , i.e. there is an ob ject Y and a pair of morphisms ι : Y → X and π : X → Y suc h that π ι = 1 Y and ιπ = e . Cho osing an ob ject Y ′ and morphisms ι ′ : Y ′ → X and π ′ : X → Y ′ suc h that π ′ ι ′ = 1 Y ′ and ι ′ π ′ = 1 − e , w e presen t X as a direct sum Y ⊕ Y ′ , where ι and ι ′ are canonical em b eddings, while π and π ′ are canonical pro jections. F or ev ery K - category A there is the smallest fully additiv e category add A containing A . This category is unique (up to equiv alence). It can b e iden tified either with the category of matrix idemp otents o v er A or with the category o f finitely generated pro jectiv e A - mo dules [9]. W e call it the additive hul l of A . Eac h A - mo dule M (bimo dule B ) extends uniquely (up to isomorphism) to a mo dule (bimo dule) o v er the category add A , whic h w e also denote by M (resp ectiv ely , b y B ) If B is an A -bimo dule, a differ entiation 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 L eibniz rule : ∂ ( ab ) = ( ∂ a ) b + a ( ∂ b ) GROUP A CTION ON BIMOD ULE CA T EGORIES 3 for an y morphisms a, b suc h that the pro duct ab is defined. It implies, in particular, that ∂ 1 X = 0 for an y o b ject X . Again, suc h a differentiation extends t o the a dditiv e h ull of A and w e denote this extension by t he same letter ∂ . A t riple T = ( A , B , ∂ ), where A is a catego ry , B is a A -bimo dule and ∂ is a differentiation from A to B , is called a bimo dule triple . If T ′ = ( A ′ , B ′ , ∂ ′ ) is another bimodule triple, a bifunctor from T to T ′ is defined as a pair F = ( F 0 , F 1 ), where F 0 : A → A ′ is a functor, F 1 : B → B ′ ( F 0 ) is a homomorphism o f A -bimo dule, where B ′ ( F 0 ) is the A - bimo dule o bta ined from B ′ b y t he transfer along F 0 (i.e. F 1 ( x ) : F 0 ( X ) 99K F 0 ( Y ) if x : X 99K Y , and F 1 ( bxa ) = F 0 ( b ) F 1 ( x ) F 0 ( a ) ), suc h that F 1 ( ∂ a ) = ∂ ′ ( F 0 ( a )) fo r all a ∈ Mor A . As a rule, w e write F ( a ) and F ( x ) instead of F 0 ( a ) and F 1 ( x ). Let F = ( F 0 , F 1 ) a nd G = ( G 0 , G 1 ) b e t w o bifunctors from a triple T = ( A , B , ∂ ) to another triple T ′ = ( A ′ , B ′ , ∂ ′ ). A morphism of bi - functors φ : F → G is defined as a morphism of functors φ : F 0 → G 0 suc h that φ ( Y ) F 1 ( x ) = G 1 ( x ) φ ( X ) for eac h x ∈ B ( X , Y ) , ∂ ′ φ ( X ) = 0 for eac h X ∈ Ob A . If φ is an isomorphism of functors, the inv erse morphism is ob viously a morphism of bifunctors to o. Then w e call φ an is o morphism of bi- functors and write φ : F ∼ → G . If suc h an isomorphism exists, w e say that the bifunctors F are G is o morphic and write F ≃ G . W e call a bifunctor F : T → T ′ an e quivalenc e of bi m o dule triples if there is suc h a bifunctor G : T ′ → T that F G ≃ id T ′ and GF ≃ id T , where id T denotes the identit y bifunctor T → T . If suc h a bifunctor exists, w e call the triples T a nd T ′ e quival e n t a nd write T ≃ T ′ . Lemma 1.1. A b i f unctor F = ( F 0 , F 1 ) is an e quivalenc e of bimo d ule triples if and on l y if the fol low i n g c ondi tion s h old: (1) The functor F 0 is fully faithful , i.e. al l induc e d map s A ( X , Y ) → A ′ ( F 0 X , F 0 Y ) ar e bije ctive. (2) This functor is also ∂ - dense , i.e. for every obj e ct X ′ of the c ate gory A ′ ther e ar e a n obje ct X ∈ Ob A and an isomorphi s m α : X ′ → F 0 X such that ∂ α = 0 . (3) The map F 1 ( X , Y ) : B ( X , Y ) → B ′ ( F 0 X , F 0 Y ) is bije ctive for any X , Y ∈ Ob A . Mor e over, if these c o nditions hold, ther e is a bifunctor G : T ′ → T and an isomorphism λ : id T ′ → F G such that GF = id T and λ ( F X ) = 1 F X for al l X ∈ Ob A . 4 YURIY A. DROZD Pr o of. The necessit y of these conditions is eviden t, so w e pro v e t heir sufficiency . Supp ose that these conditions hold. F or eac h ob ject X ′ ∈ A ′ c ho ose an ob ject X a nd an isomorphism α : X ′ → F 0 X suc h that ∂ a = 0, alw a ys setting α = 1 X ′ for X ′ = F 0 X . Se t G 0 X ′ = X and λ ( X ′ ) = α . F or eac h mor phism a : X ′ → Y ′ set G 0 a = F − 1 0 ( X , Y )( λ ( Y ′ ) aλ − 1 ( X ′ )), where X = G 0 X ′ , Y = G 0 Y ′ (then λ ( X ′ ) : X ′ ∼ → F 0 X , λ ( Y ′ ) : Y ′ ∼ → F 0 Y ). Ob viously , the set { λ ( X ′ ) } de- fines an isomorphism of functors λ : id → F 0 G 0 . W e also define a homomorphism of bimo dules G 1 : B ′ → B ( G 0 ) setting G 1 ( x ) = F 1 ( X , Y ) − 1 ( λ ( Y ′ ) xλ − 1 ( X ′ )) if x : X ′ 99K Y ′ , X = G 0 X ′ , Y = G 0 Y ′ . Then G = ( G 0 , G 1 ) is a bifunctor T ′ → T and λ is an isomorphism of bifunctors id T ′ → F G . Moreov er, b y this construction, GF = id T and λ ( F X ) = 1 F X for all X .  Ev ery bimo dule triple T = ( A , B , ∂ ) gives rise to the bimo dule c a te- gory (or the c ate gory of r epr esentations , or the c ate gory of elements ) of this triple [5]. The ob j e cts of t his category a r e elemen ts S X B ( X , X ), where X runs thro ug h ob j ects of the category add A . Morphism s from an o b ject x : X 9 9 K X to an ob ject y : Y 9 9 K Y are suc h morphisms a : X → Y that ax = y a + ∂ ( a ) in B ( X , Y ). It is easy to see that these definitions really define a f ully additive K - category El ( T ). The set o f morphisms x → y in this category is denoted b y Hom T ( x, y ). If ∂ = 0, w e write El ( A , B ) or eve n El ( B ) instead of El ( A , B , ∂ ). Each bifunctor b etw een bimo dule triples F : T → T ′ giv es rise to a functor F ∗ : El ( T ) → El ( T ′ ), whic h maps an ob j ect x to the ob ject F 1 ( x ) and a morphism a : x → y to the morphism F 0 ( a ) : F 1 ( x ) → F 1 ( y ) . As w ell, eac h mor phism of bifunctors φ : F → G induces a morphism of functors φ ∗ : F ∗ → G ∗ , whic h correlate an ob ject x ∈ B ( X , X ) with the morphism φ ( X ) considered as a morphism F ( x ) → G ( x ). Ob viously , if φ is an isomorphism of bifunctors, φ ∗ is an isomorphism of functors. Esp ecially , if F is an equiv alence of bimo dule triples, the functor F ∗ is an equiv alence of their bimo dule categories. If B = A and ∂ = 0, w e say that t he bimo dule triple T = ( A , A , 0 ) is the principle triple f o r the category A . O bviously , a bifunctor b etw een principle triples is just a functor b etw een t he corresponding categories and a morphism of suc h bifunctors is just a morphism of functors. The bimo dule category of the principle triple for a category A is denoted b y El ( A ). If A and A ′ are t w o categories, one can consider A - A ′ -bimo dules , i.e. bilinear functors B : A op × A ′ → K - Mo d . Actually , an y suc h bimodule can b e iden tified with a A × A ′ -bimo dule ˜ B with ˜ B (( X , X ′ ) , ( Y , Y ′ )) = B ( X , Y ′ ) and ( a, a ′ ) x ( b, b ′ ) = axb ′ . Suc h bimo dules are called bip artite . GROUP A CTION ON BIMOD ULE CA T EGORIES 5 In particular, ev ery A - bimo dule B defines a bipartite A - A - bimo dule, whic h w e denote by B (2) and call the double of the A - bimo dule B . Certainly , bimo dules B and B (2) are quite differen t and they define differen t bimo dule categories. If B = A the category El ( A (2) ) coincides with the c ate gory of morphisms of the a dditiv e hull add A . F urther on we often iden tify the catego ries A and add A and sa y ”an ob ject (morphism) o f A ” instead of “an ob ject (morphism) of add A .” W e hop e that this p etty am biguit y will not em barrass the reader. 2. Group a ctions Let T = ( A , B , ∂ ) b e a bimo dule triple and G b e a g roup. One sa ys that the group G acts on the triple T if a bifunctor T σ : T → T is defined fo r eac h ele men t σ ∈ G so that T 1 = id T and T στ ≃ T σ T τ for an y σ , τ ∈ G . It implies, in particular, that all T σ are equiv alences. F urther on we write X σ instead of T σ ( X ). W e only note that according to this notation X στ ≃ ( X τ ) σ . A system of factors λ for suc h an action is defined a s a set of isomorphisms of bifunctors λ σ ,τ : T στ ∼ → T σ T τ , whic h satisfy the relations: (2.1) λ ρ σ ,τ λ ρ,στ = λ ρ,σ λ ρσ ,τ for an y triple o f elemen ts ρ, σ, τ ∈ G , a nd λ σ , 1 = λ 1 ,σ = 1 for any σ ∈ G . W e omit the argumen ts (ob jects of A ) in these form ulae (a nd later on in analogo us cases), since their v alues can easily b e restored. Since λ σ ,τ is a mor phism of bifunctors, one has λ σ ,τ : X στ → ( X τ ) σ and (2.2) λ σ ,τ x στ = ( x τ ) σ λ σ ,τ for ev ery morphism from A and eve ry elemen t from B , and also ∂ λ σ ,τ = 0 for all σ, τ . Note also that the relations (2.1) and (2.2) imply , in particular, t ha t λ σ σ − 1 ,σ = λ σ ,σ − 1 and λ σ ,σ − 1 x = ( x σ − 1 ) σ λ σ ,σ − 1 . Giv en an action T = { T σ } of a group G on a bimo dule triple T = ( A , B , ∂ ) and a system of factors λ for this a ction, we define the cr osse d gr oup triple T G = T ( G , T , λ ). Namely , w e consider the cr o s s e d gr oup c ate gory A G = A ( G , T , λ ) [11 , 8]. Its ob j ects coincide with those of A , but morphisms X → Y in t he category A G are defined as formal (finite) linear com binations P σ ∈ G a σ [ σ ], where a σ ∈ A ( X σ , Y ), and the m ultiplication of suc h morphisms is defined b y bilinearity and the rule (2.3) a σ [ σ ] b τ [ τ ] = a σ b σ τ λ σ ,τ [ σ τ ] . 6 YURIY A. DROZD The condition (2.1) for a system of factors is equiv alen t to the asso cia- tivit y of this m ultiplication. The A G - bimo dule B G = B ( G , T , λ ) is con- structed in a n a na logous w a y: elemen ts of B G ( X , Y ) are formal (finite) linear com binations P σ ∈ G x σ [ σ ], where x σ ∈ B ( X σ , Y ), and their pro d- ucts with morphisms from A G are defined b y the same formula ( 2.3), with the only difference that one of the elemen ts a σ , b τ is a morphism from A , while the second one is an elemen t f rom B . The differen tiation ∂ extends to A G if w e set ∂ ( P σ a σ [ σ ]) = P σ ∂ a σ [ σ ]. W e identify ev ery morphism a ∈ A ( X, Y ) with the morphism a [1] ∈ A G ( X , Y ) and ev ery elemen t x ∈ B ( X , Y ) with the elemen t x [1] ∈ B G ( X , Y ) getting the em bedding bifunctor T → T G . An action T of a group G on a bimo dule triple T induces its action T ∗ on the bimodule category El ( T ): an elemen t σ ∈ G defines the functor ( T σ ) ∗ : x 7→ x σ . Moreo v er, if λ is a system of fa cto r s for the action T , it induces the system of factors λ ∗ for the a ction T ∗ : one has to set ( λ ∗ ) σ ,τ ( x ) = λ σ ,τ ( X ) if x ∈ B ( X , X ). Th us the crossed group category El ( T ) G = El ( T )( G , T ∗ , λ ∗ ) is defined, as w ell as t he em bedding El ( T ) → El ( T ) G . One can also define the natura l functor Φ : El ( T ) G → El ( T G ) as follo ws. F or an ob ject x ∈ B ( X , X ), set Φ( x ) = x [1] ∈ B G ( X , X ). Let α = P σ a σ [ σ ] b e a morphism from x to y ∈ B ( Y , Y ) in the category El ( T ) G . It means that a σ : x σ → y in the category El ( T ), i.e. a σ ∈ A ( X σ , Y ) and a σ x σ = y a σ + ∂ a σ . Then one can consider α as a morphism X → Y in the category A G ( X , Y ), and αx [1] = P σ a σ [ σ ] x [1] = P σ a σ x σ [ σ ] = P σ ( y a σ + ∂ a σ )[ σ ] = y [1 ] α + ∂ α , so α is a morphism x [1] → y [1] in the category El ( T G ) and one can set Φ( α ) = α . Prop osition 2.1. The functor Φ is ful ly faithful, i.e. for any ob- je cts x, y fr om El ( T ) G it induc es the bi j e ctive map Ho m T G ( x, y ) → Hom T G ( x, y ) , wher e Hom T G denotes the morphi s ms in the c ate g o ry El ( T ) G . Pr o of. Obvious ly , this map is injectiv e. Let α = P σ a σ [ σ ] : x [1] → y [1], i.e. αx [1] = P σ a σ x σ [ σ ] = y [1] α + ∂ α = P σ ( y a σ + ∂ a σ )[ σ ]. Then a σ x σ = y a σ + ∂ a σ for all σ , so a σ : x σ → y in the category El ( T ), th us α : x → y in the category El ( T ) G . Therefore, this map is also surjectiv e.  If the group G is finite, one can also construct a functor Ψ : El ( T G ) → El ( T ). F or ev ery ob ject X ∈ Ob A , set ˜ X = L σ ∈ G X σ and for ev ery elemen t ξ = P σ x σ [ σ ] ∈ B G ( X , X ), where x σ : X σ 99K X , denote b y ˜ ξ the elemen t from B ( ˜ X , ˜ X ) = L σ ,τ B ( X τ , X σ ) such tha t its comp onen t ˜ ξ σ ,τ ∈ B ( X τ , X σ ) equals x σ σ − 1 τ λ σ ,σ − 1 τ . Note that x σ − 1 τ : X σ − 1 τ 99K Y , GROUP A CTION ON BIMOD ULE CA T EGORIES 7 hence x σ σ − 1 τ : ( X σ − 1 τ ) σ 99K Y σ , thu s x σ σ − 1 τ λ σ ,σ − 1 τ : X τ 99K Y σ indeed. Let η = P σ y σ [ σ ] ∈ B G ( Y , Y ), where y σ ∈ B ( Y σ , Y ) and α = P σ a σ [ σ ] b e a morphism from ξ to η , where a σ ∈ A ( X σ , Y ). Since αξ = X ρ X σ a ρ [ ρ ] x σ [ σ ] = X ρ X σ a ρ x ρ σ λ ρ,σ [ ρσ ] = = X τ  X ρ a ρ x ρ ρ − 1 τ λ ρ,ρ − 1 τ  [ τ ] , and η α = X ρ X σ y ρ [ ρ ] a σ [ σ ] = X ρ X σ y ρ a ρ σ λ ρ,σ [ ρσ ] = = X τ  X ρ y ρ a ρ ρ − 1 τ λ ρ,ρ − 1 τ  [ τ ] , it means that, for each τ , (2.4) X ρ a ρ x ρ ρ − 1 τ λ ρ,ρ − 1 τ = X ρ y ρ a ρ ρ − 1 τ λ ρ,ρ − 1 τ + ∂ a τ . Consider the morphism ˜ α : ˜ X → ˜ Y suc h that ˜ α σ ,τ = a σ σ − 1 τ λ σ ,σ − 1 τ : X τ → Y σ . Then the ( σ, τ )- comp onen t o f the pro duct ˜ α ˜ ξ equals I = X ρ a σ σ − 1 ρ λ σ ,σ − 1 ρ x ρ ρ − 1 τ λ ρ,ρ − 1 τ = X ρ a σ σ − 1 ρ ( x σ − 1 ρ ρ − 1 τ  σ λ σ ,σ − 1 ρ λ ρ,ρ − 1 τ , while the ( σ, τ )- comp onen t o f the pro duct ˜ η ˜ α equals I I = X ρ y σ σ − 1 ρ λ σ ,σ − 1 ρ a ρ ρ − 1 τ λ ρ,ρ − 1 τ = X ρ y σ σ − 1 ρ ( a σ − 1 ρ ρ − 1 τ  σ λ σ ,σ − 1 ρ λ ρ,ρ − 1 τ . (In b oth cases we used the relation ( 2 .2) replacing τ by σ − 1 ρ ). Since, b y the conditio n (2.1) for the system of factors, λ σ ,σ − 1 ρ λ ρ,ρ − 1 τ = λ σ σ − 1 ρ,ρ − 1 τ λ σ ,σ − 1 τ , and ∂ λ σ ,σ − 1 τ = 0 , w e get from the relation (2.4) that I = I I + ∂ ˜ α σ ,τ (w e just replace ρ b y σ − 1 ρ , τ b y σ − 1 τ , then apply the functor T σ to both sides). Therefore, ˜ α is a morphism ˜ ξ → ˜ η and one can define the functor Ψ setting Ψ( ξ ) = ˜ ξ and Ψ( α ) = ˜ α . Prop osition 2.2. The functors Φ and Ψ form an adjoint pa ir , i.e. ther e is a natur al isom orphism Hom T G (Φ x, η ) ≃ Ho m T ( x, Ψ η ) for e ach obje cts x ∈ El ( T ) and η ∈ El ( T G ) . 8 YURIY A. DROZD Pr o of. Let x ∈ B ( X , X ) , η ∈ B G ( Y , Y ) , η = P σ y σ [ σ ], where y : Y σ 99K Y , a nd α : Φ( x ) = x [1] → η in the category El ( T G ). By definition, α = P σ a σ [ σ ], where a σ : X σ → Y , and αx [1] = X σ a σ x σ [ σ ] = η α + ∂ α = X σ  X ρ y ρ a ρ ρ − 1 σ λ ρ,ρ − 1 σ + ∂ a σ  [ σ ] , i.e. (2.5) a σ x σ = X ρ y ρ a ρ ρ − 1 σ λ ρ,ρ − 1 σ + ∂ a σ for ev ery σ . Consider the mo r phism f ( α ) = β : X τ → ˜ Y = L σ Y σ suc h that its comp o nen t β σ : X → Y σ equals a σ σ − 1 λ σ ,σ − 1 . Compute the σ -comp onents of the pr o ducts β x and ˜ η β , where ˜ η = Ψ η . They equal, resp ectiv ely , β σ x τ = a σ σ − 1 λ σ ,σ − 1 x = a σ σ − 1 ( x σ − 1 ) σ λ σ ,σ − 1 and X ρ y σ σ − 1 ρ λ σ ,σ − 1 ρ a ρ ρ − 1 λ ρ,ρ − 1 = X ρ y σ σ − 1 ρ ( a σ − 1 ρ ρ − 1 ) σ λ σ ,σ − 1 ρ λ ρ,ρ − 1 = = X ρ y σ σ − 1 ρ ( a σ − 1 ρ ρ − 1 ) σ λ σ σ − 1 ρ,ρ − 1 λ σ ,σ − 1 . The relation (2 .5), where σ is replaced b y σ − 1 and ρ b y σ − 1 ρ , these t w o expre ssions differ exactly b y ∂ β σ = ∂ a σ σ − 1 λ σ ,σ − 1 , hence β = f ( α ) is a morphism x → ˜ η in the category El ( T ). Obviously , if α 6 = α ′ , then f ( α ) 6 = f ( α ′ ) as w ell. Moreov er, one easily ch ec ks that the corresp on- dence α 7→ f ( α ) is f unctorial in x a nd η , i.e. f ( α ) b = f ( α Φ b ) and f ( γ α ) = (Ψ γ ) f ( α ) for any morphisms b : x ′ → x and γ : η → η ′ . On the con trary , let β : x → ˜ η b e a morphism in the category El ( T ). Denote b y β σ : X → Y σ the corresponding comp onen t of β and consider the morphism α = P σ a σ [ σ ] : X → Y in the category A G , where a σ = λ − 1 σ ,σ − 1 β σ σ − 1 : X σ → Y . Comparing the σ - comp onen ts in the equalit y β x = ˜ η β , w e get (2.6) β σ x = X ρ y σ σ − 1 ρ λ σ ,σ − 1 ρ β ρ + ∂ β σ . The co efficien t s near [ σ ] in the pro ducts α ( Φ x ) = α x [1] and η α equal, resp ectiv ely , a σ x σ = λ − 1 σ ,σ − 1 β σ σ − 1 x σ GROUP A CTION ON BIMOD ULE CA T EGORIES 9 and X ρ y ρ a ρ λ ρ − 1 σ = X ρ y ρ ( λ − 1 ρ − 1 σ ,σ − 1 ρ ) ρ β ρ − 1 σ σ − 1 ρ λ ρ,ρ − 1 σ = = X ρ y ρ ( λ − 1 ρ − 1 σ ,σ − 1 ρ ) ρ λ ρ,ρ − 1 σ β σ σ − 1 ρ . The relation (2.6), with σ replaced b y σ − 1 , implies that a σ x σ − ∂ a σ = X ρ λ − 1 σ ,σ − 1 ( y σ − 1 σρ ) σ λ σ σ − 1 ,ρ β σ σ − 1 ρ = = X ρ y σρ λ − 1 σ ,σ − 1 λ σ σ − 1 ,ρ β σ σ − 1 ρ = X ρ y ρ λ − 1 σ ,σ − 1 λ σ σ − 1 ,σ ρ β σ ρ = = X ρ y ρ λ − 1 σ ,σ − 1 ρ β σ σ − 1 ρ = X ρ y ρ ( λ − 1 ρ − 1 σ ,σ − 1 ρ ) ρ λ ρ,ρ − 1 σ β σ σ − 1 ρ . (P assing f r o m the second r ow to the third, w e used the relation (2.1) for the triple σ, σ − 1 , ρ , while in the third row w e used the same relation for the triple ρ, ρ − 1 σ , σ − 1 ρ .) Therefore, αx [1] = η α + ∂ α , thus α is a morphism Φ x → η . Moreo v er, the σ -comp onen t of f ( α ) equals a σ σ − 1 λ σ ,σ − 1 = ( λ − 1 σ − 1 ,σ ) σ ( β σ − 1 σ ) σ λ σ ,σ − 1 = ( λ − 1 σ − 1 ,σ ) σ λ σ ,σ − 1 β σ = β σ . Hence f ( α ) = β and the map α 7→ f ( α ) is bijectiv e.  3. Sep arable a ctions W e call the c enter Z ( T ) of a bimodule t r iple T = ( A , B , ∂ ) the endomorphism ring of the iden tit y bifunctor id T . In other w ords, the elemen ts of this cen ter are the sets of morphisms α = { α X : X → X | X ∈ Ob A } , suc h that α Y a = aα X for ev ery morphism a : X → Y , α Y x = xα X for ev ery elemen t x : X 99K Y and ∂ α X = 0 for all X . In particular, the elemen t α X b elongs to the cen ter of the alg ebra A ( X , X ). One easily sees that if α = { α X } and β = { β X } are t w o suc h sets, then the sets α + β = { α X + β X } and αβ = { α X β X } also b elong to Z ( T ). Hence, this cen ter is a ring (ev en a K -algebra) , comm utativ e, since α X β X = β X α X . If F = ( F 0 , F 1 ) is an equiv alence o f bimo dule triples T → T ′ = ( A ′ , B ′ , ∂ ′ ), it induces an isomorphism F Z : Z ( T ) ∼ → Z ( T ′ ). Namely , for an y X ′ ∈ Ob A ′ , choose an isomorphism λ : X ′ → F 0 X for some X ∈ Ob A , and, for each elemen t α = { α X } ∈ Z ( T ), set ( F Z α ) X ′ = λ − 1 ( F 0 α X ) λ . Let Y ′ b e another ob j ect f rom A , µ : Y ′ ∼ → 10 YURIY A. DROZD F 0 Y and ( F Z α ) Y ′ = µ − 1 ( F 0 α Y ) µ . If a ′ ∈ A ′ ( X ′ , Y ′ ), t he morphism µa ′ λ − 1 : F 0 X → F 0 Y is o f the fo rm F 0 a for some a : X → Y . It giv es ( F Z α ) Y ′ a ′ = µ − 1 ( F 0 α Y ) µ · µ − 1 ( F 0 a ) λ = = µ − 1 ( F 0 α Y )( F 0 a ) λ = µ − 1 ( F 0 ( α Y a )) λ = = µ − 1 F 0 ( aα X ) λ = µ − 1 ( F 0 a )( F 0 α X ) λ = = a ′ λ − 1 ( F 0 α X ) λ = a ′ ( F Z α ) X ′ . (3.1) Esp ecially , if Y ′ = X ′ and a ′ = 1 X ′ , we see that F Z ( α ) X ′ do es not dep end on the c hoice of X and λ . Just in the same wa y one c hec ks that ( F Z α ) Y ′ x ′ = x ′ ( F Z α ) X ′ for ev ery x ′ ∈ B ′ ( X ′ , Y ′ ). Note that an isomorphism λ can alw a ys b e chos en suc h that ∂ λ = 0 : for instance, one can use the isomorphism of bifunctors φ : id T ′ → F G for some bifunctor G and set X = G 0 X ′ , λ = φ ( X ′ ). Therefore ∂ ′ ( F Z α ) X ′ = 0, so the set F Z α = { ( F Z α ) X ′ } b elongs to Z ( T ′ ). Obviously , F Z ( α + β ) = F Z α + F Z β and F Z ( αβ ) = ( F Z α )( F Z β ), and if F ′ : T ′ → T ′′ is another equiv alence, then ( F ′ F ) Z = F ′ Z F Z . Moreo v er, similarly to the equalities (3 .1), o ne easily ve rifies that if F ≃ F ′ , then F Z = F ′ Z . In particular, if G : T ′ → T is suc h a bifunctor that F G ≃ id T ′ and GF ≃ id T , then G Z = F − 1 Z , th us F Z is an isomorphism. These considerations imply t ha t ev ery action T o f a group G on a triple T induces an action of the same g roup on the cen ter of this triple with the trivial system of factors: if λ is a system of fa cto r s for the action T , then ( α σ ) X = λ − 1 σ ,σ − 1 α σ X σ − 1 λ σ ,σ − 1 for ev ery α ∈ Z ( T ). Esp e- cially , if the g r oup G is finite, for any elemen t α from Z ( T ) its tr ac e is defined as tr α = tr G α = P σ α σ , i.e. (tr α ) X = P σ λ − 1 σ ,σ − 1 α σ X σ − 1 λ σ ,σ − 1 . Ob viously , the cen ter of the triple T G is a subalgebra of the cen ter of T . Prop osition 3.1. The c enter Z ( T G ) c o i ncides with the sub algebr a Z ( T ) G of elements of the c enter Z ( T ) that ar e invariant under the action of G . In p a rticular, if this gr oup is fin i te, the tr ac e of e ach element α ∈ Z ( T ) b elongs to Z ( T G ) . Pr o of. Let α = { α X } b e an elemen t of the cen ter Z ( T ). Since α Y a [ σ ] = aα X σ [ σ ] and a [ σ ] α X = aα σ X [ σ ] fo r eac h morphism a : X σ → Y , this elemen t b elongs to the cen ter of the triple T G if and only if α X σ = α σ X for ev ery X and eve ry σ . But then ( α σ ) X = λ − 1 σ ,σ − 1 α σ X σ − 1 λ σ ,σ − 1 = λ − 1 σ ,σ − 1 ( α σ − 1 X ) σ λ σ ,σ − 1 = α X , so α is in v aria n t unde r the action o f G . Just in the same wa y one v erifies that ev ery inv ariant elemen t from Z ( T ) b elong s to Z ( T G ). The last GROUP A CTION ON BIMOD ULE CA T EGORIES 11 statemen t follow s fro m the fact that tr α is a lw a ys in v ariant under the action of the group.  Definition. W e call an action of a finite group G on a bimo dule t r iple T sep a r able , if there is an elemen t o f the cen ter α ∈ Z ( T ) suc h that tr α = 1. Certainly , it is enough tr α to b e inv ertible. F or instance, if the o rder of the gro up G is in v ertible in the ring K , an y a ction of this gro up is separable. Another importa n t case is when the cente r of the triple T con tains a subring R suc h that it is G -in v ariant, the group G acts effe c tively (i.e. for any σ 6 = 1 there is r ∈ R suc h that r σ 6 = r ) and R is a sep ar able ex tension of its subring of inv a r ia n ts R G [4]. If R is a field and G acts effectiv ely o n R , the last condition a lw a ys ho lds. In general case it is necess ary and sufficien t that ev ery elemen t σ 6 = 1 induce a non-iden tit y automorphism of the residue field R / m fo r eac h maximal ideal m ⊂ R such that m σ = m [4, Theorem 1.3]. F or an action of a g r oup on a category (that is, on a principle triple) the notion of separabilit y was in tro duced in [8]. Obv iously , if an action of a group on a bimo dule triple is separable, so is a lso its induced action on the corresp onding bimo dule category . W e also note that if an action of a gr o up G is separable, so is the action of ev ery subgroup H ⊆ G : if tr G α = 1 and β = P σ ∈ R α σ , where R is a set of r epresen tativ es of righ t cosets H \ G , then tr H β = 1. Recall that a r ing homomorphism A → A ′ is called sep ar able if the natural homomorphism of A ′ -bimo dules A ′ ⊗ A A ′ → A ′ sending a ⊗ b to ab s plits , i.e. there is an elemen t P i b i ⊗ c i in A ′ ⊗ A A ′ suc h that P i b i c i = 1 and P i ab i ⊗ c i = P i b i c i a for all a ∈ A ′ . Lemma 3.2. A n action of a finite gr oup G on a triple T is sep ar able if and only if so is the ring homomorphis m Z → Z G , w h er e Z = Z ( T ) . Pr o of. Supp ose that the action is separable, α = { α X } is suc h an elemen t of the cen ter t hat tr α = 1 . Let t = P σ α σ [ σ ] ⊗ [ σ − 1 ] ∈ Z G ⊗ Z Z G . Then P σ α σ [ σ ][ σ − 1 ] = tr α = 1 and, for any β ∈ Z , τ ∈ G , β [ τ ] · t = X σ β α τ σ [ τ σ ] ⊗ [ σ − 1 ] = X σ α τ σ β [ τ σ ] ⊗ [ σ − 1 ] = = X σ α σ β [ σ ] ⊗ [ σ − 1 τ ] = X σ α σ [ σ ] ⊗ [ σ − 1 ] β [ τ ] = t · β [ τ ] , so the homomor phism Z → Z G is separable. No w let the ho mo mo r phism Z → Z G be separable. Note that every elemen t from Z G ⊗ Z Z G is o f the form P σ ,τ z σ ,τ [ σ ] ⊗ [ τ ] for some z σ ,τ ∈ Z . Hence there are elemen ts z σ ,τ suc h that P σ ,τ z σ ,τ [ σ τ ] = 12 YURIY A. DROZD P τ  P σ z σ ,σ − 1 τ  [ τ ] = 1, i.e. P σ z σ ,σ − 1 = 1, and P σ z σ ,σ − 1 τ = 0 if τ 6 = 1, moreo v er, fo r ev ery ρ ∈ G we hav e: [ ρ ]  X σ ,τ z σ ,τ [ σ ] ⊗ [ τ ]  = X σ ,τ z ρ σ ,τ [ ρσ ] ⊗ [ τ ] = X σ ,τ z ρ ρ − 1 σ ,τ [ σ ] ⊗ [ τ ] = =  X σ ,τ z σ ,τ [ σ ] ⊗ [ τ ]  [ ρ ] = X σ ,τ z σ ,τ [ σ ] ⊗ [ τ ρ ] = X σ ,τ z σ ,τ ρ − 1 [ σ ] ⊗ [ τ ] . Th us z ρ ρ − 1 σ ,τ = z σ ,τ ρ − 1 for ρ, σ , τ . Esp ecially , for σ = ρ, τ = 1 w e get z σ ,σ − 1 = z σ 1 , 1 . Therefore, tr z 1 , 1 = 1 a nd the action is separable.  Corollary 3.3. If an ac tion of a gr oup G on a triple T = ( A , B , ∂ ) is sep a r able, so is also the emb e dding functor A → A G , i.e . the ho m o- morphism of A G -b i mo dules φ : A G ⊗ A A G → A G splits, or, the same , for every obje ct X ∈ Ob A ther e is an element t X ∈ ( A G ⊗ A A G )( X , X ) such that φ ( t X ) = 1 X and at X = t Y a for e ach a ∈ A G ( X , Y ) . In p ar- ticular, the action of a gr oup G on a c ate g o ry A is sep ar able if and only if so is the emb e dding functor A → A G . Theorem 3.4. If an action of a finite gr oup G o n a bim o dule triple T = ( A , B , ∂ ) is sep ar able, the functor Φ : El ( T ) G → El ( T G ) induc es an e q uiva l e nc e of the c ate gories add El ( T ) G → El ( T G ) . Pr o of. First w e prov e a lemma ab out fully additive categories. Lemma 3.5. L et C b e a ful ly additive c ate g ory, F : C → C ′ b e a ful ly faithful functor. F is an e quivalenc e of c ate gories if and only i f every obje ct X ′ ∈ C ′ is isomorphic to a dir e ct summand of an obje ct of the form F Y , wher e Y ∈ Ob C . Pr o of. The neces sit y of this condition is o bvious, so w e only ha v e to pro v e the sufficiency . If X ′ is a dir ect summand o f F Y , there are morphisms ι ′ : X ′ → F Y and π ′ : F Y → X ′ suc h that π ′ ι ′ = 1 X ′ . Then e ′ = ι ′ π ′ is an idempot en t endomorphism of the ob ject F Y . Since the functor F is f ully fa it hf ul, e ′ = F e for an idemp oten t endomorphism e : Y → Y . Since t he category C is fully additiv e, there are an ob ject X a nd morphisms ι : X → Y and π : Y → X suc h that e = ιπ and π ι = 1 X . Then ( F ι )( F π ) = e ′ and ( F π )( F ι ) = 1 F X . Let u = π ′ F ( ι ) , v = ( F π ) ι ′ ; then we immediately get that uv = 1 X ′ and v u = 1 F X , i.e. X ′ ≃ F X , the functor F is also dense, so it is an equiv alence of categories.  W e pro v e now that eve ry ob ject ξ o f the category El ( T G ) is isomor- phic to a direct summand of ΦΨ ξ . Since Φ is fully faithful ( Prop osition 2.1), Theorem 3 .4 follows then from Lemma 3.5. Let ξ = P σ x σ [ σ ] ∈ B G ( X , X ), where x σ ∈ B ( X σ , X ). Then Ψ ξ = ˜ ξ ∈ B ( ˜ X , ˜ X ), where GROUP A CTION ON BIMOD ULE CA T EGORIES 13 ˜ X = L σ X σ and ˜ ξ σ ,τ = x σ σ − 1 τ λ σ ,σ − 1 τ , and ΦΨ ξ = ˜ ξ [1 ]. Cho ose an ele- men t α ∈ Z ( T ) suc h that tr α = 1. Consider the morphism π : ˜ X → X suc h that it s σ -comp o nen t equals π σ = λ − 1 σ − 1 ,σ [ σ − 1 ] : X σ → X . Then the σ -comp onen t of the elemen t ξ π equals X ρ x ρ ( λ ρ σ − 1 ,σ ) − 1 λ ρ,σ − 1 [ ρσ − 1 ] = X ρ x ρ λ − 1 ρσ − 1 ,σ [ ρσ − 1 ] (w e use the relation (2.1) f or the triple ρ, σ − 1 , σ ), while the σ -comp onen t of the elemen t π ˜ ξ [1 ] equals X ρ λ − 1 ρ − 1 ,ρ ( x ρ ρ − 1 σ ) ρ − 1 λ ρ − 1 ρ,ρ − 1 σ [ ρ − 1 ] = X ρ x ρ − 1 σ λ − 1 ρ,ρ − 1 λ ρ − 1 ρ,ρ − 1 σ [ ρ − 1 ] = = X ρ x ρ − 1 σ λ − 1 ρ − 1 ,σ [ ρ − 1 ] = X ρ x ρ λ − 1 ρσ − 1 ,σ [ ρσ − 1 ] . Here w e used first the relatio n (2.1) for the triple ρ − 1 , ρ, ρ − 1 σ and then replaced ρ by σ ρ − 1 . So ξ π = π ˜ ξ [1] and, since ∂ π = 0, π is a morphism ˜ ξ [1 ] → ξ . No w consider the morphism ι : X → ˜ X suc h that its σ - comp onen t equals α X σ [ σ ]. The σ -comp onen t o f the elemen t ιξ equals X ρ α X σ x σ ρ λ σ ,ρ [ σ ρ ] = X ρ α X σ x σ σ − 1 ρ λ σ ,σ − 1 ρ [ ρ ] , and the σ -comp onent of the elemen t ˜ ξ [1 ] ι equals X ρ x σ σ − 1 ρ λ σ ,σ − 1 ρ α X ρ [ ρ ] = X ρ α X σ x σ σ − 1 ρ λ σ ,σ − 1 ρ [ ρ ] , since α ∈ Z ( T ). Therefore, ˜ ξ [1 ] ι = ιξ , thus ι is a morphism ξ → ˜ ξ [1 ]. But π ι = P σ λ − 1 σ − 1 ,σ α σ − 1 X σ λ σ − 1 ,σ = ( tr α ) X = 1 X = 1 ξ , whic h just means that the elemen t ξ is a direct summand of the elemen t ˜ ξ [1].  One can get more information if the gr o up G is finite ab elian and the ring K is a field containing a primi tive n -th r o ot of unit , where n = #( G ), i.e. suc h an elemen t ζ that ζ n = 1 and ζ k 6 = 1 for 0 < k < n . Then certainly c har K ∤ n , so any action o f the g roup G on a bimo dule triple T = ( A , B , ∂ ) is separable. Let ˆ G b e the gr oup of char acters of the group G , i.e. the group of its homomorphisms to the multiplicativ e group K × of the field K . This group acts on the triple T G (with the trivial system of fa ctors) by the 14 YURIY A. DROZD rules: X χ = X for ev ery X ∈ Ob A ,  X σ x σ [ σ ]  χ = X σ χ ( σ ) x σ [ σ ] , where χ ∈ ˆ G a nd P σ x σ [ σ ] is a morphism from A G or an elemen t from B G . Recall that also #( ˆ G ) = n , so this action is separable as w ell. W e denote b y χ 0 the unit char acter , i.e. suc h that χ 0 ( σ ) = 1 for all σ ∈ G . By definition, morphisms from A G ˆ G and elemen ts of B G ˆ G are of the form P σ ,χ x σ ,χ [ σ ][ χ ]. W e write [ χ ] instead of [1][ χ ] and σ instead of [ σ ][ χ 0 ]. In particular an elemen t x [1][ χ 0 ] is denoted by x . Theorem 3.6. The bimo dule triples add T and add T G ˆ G ar e e quivalen t. Pr o of. Consider the elemen ts e σ = 1 n X χ χ ( σ )[ χ ] f rom the endomor- phism r ing A G ˆ G ( X, X ). The formulae of orthogonality for characters [7, Theorem 3.5 ] immediately imply that e σ are m utually orthogona l idemp oten ts a nd P σ e σ = 1. Moreo v er, e σ [ τ ] = [ τ ] e στ , so all these idemp oten ts are conjugate, th us define isomorphic direct summands X σ of the ob ject X in the category a dd A G ˆ G , and X = L σ X σ . W e define the bif unctor Θ : add T → add T G ˆ G setting Θ X = X 1 and Θ x = xe 1 = e 1 x , where x is a morphism X → Y or an elemen t from B ( X , Y ). Obvious ly , the functor Θ 0 : add A → add A G ˆ G satisfies the conditions of Lemma 3.5, so it defines an equiv alence of catego ries. Since ev ery map Θ 1 ( X , Y ) is also bijectiv e, the bifunctor Θ is an equiv- alence b y Lemma 1.1.  Corollary 3.7. The c ate gories El ( T ) and a dd El ( T ) G ˆ G ar e e quival e nt. Pr o of. Indeed, add El ( T ) G ˆ G ≃ El ( T G ˆ G ) b y Theorem 3.4.  4. Radical and decomposition In this section w e supp ose t ha t the ring K is no etherian, lo c al an d henselian [3] (for instance, c omplete ). W e denote b y m its maximal ideal and b y k = K / m its residue field. W e call a K -category A pie c e- wise finite if all K -mo dules A ( X , Y ) a r e finitely generated. Then its additiv e h ull add A is piecewise finite as w ell. Moreo v er, eac h endomor- phism ring A = A ( X , X ) is semip erfe ct , i.e. p ossesses a unit decom- p osition 1 = P n i =1 e i , where e i are mutually orthogonal idempo t en ts and all rings e i Ae i are lo cal. Hence the cat ego ry add A is lo c al , i.e. ev ery ob ject in it decomp oses in to a finite dir ect sum of ob jects with GROUP A CTION ON BIMOD ULE CA T EGORIES 15 lo cal endomorphism rings. Therefore this category is a Krul l–Schmidt c ate gory , i.e. ev ery ob ject X in it decomp oses into a finite direct sum of indecomposables: X = L m i =1 X i and suc h a decomp osition is unique, i.e. if also X = L n i =1 X ′ i , where all X ′ i are indecomp osable, then m = n and t here is a p erm utation ε o f the se t { 1 , 2 , . . . , m } suc h that X i ≃ X ′ εi for all i [2, Theorem I.3 .6 ]. Recall that the r adic al of a lo cal category A is the ideal rad A consisting of all suc h mor phisms a : X → Y t ha t all comp onen ts o f a with respect to some (t hen any ) decomp ositions of X and Y into a direct sum o f indecomp osables are non-in v ertible. W e de- note 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 se misimple a rtinian ring [9]. In the case of a piecewise finite category alw a ys rad A ⊇ m A , in particular, A ( X, X ) is a finite dimensional k -a lg ebra. The category A is semisim ple , i.e. ev ery ob ject in it decomp oses in to a finite direct sum of indecomp osables and A ( X, Y ) = 0 if X and Y are non-isomorphic indecomposables, while A ( X, X ) is a sk ewfie ld fo r eve ry indecomp osable ob ject X . (Note that an ob ject X is indecomp osable in the category A if and only if it is so in the category A ). Moreo v er, rad A is the biggest a mo ng the I ⊂ A suc h that the factor- cat ego ry A / I is semisimple . If a finite gr o up G acts on a piecew ise finite category A with a sy stem of factors λ , the category A G is piecewise finite as w ell. Moreov er, the radical is a G -in v aria nt ideal, i.e. ( rad A ) σ = rad A for all σ ∈ G , and the ideal (rad A ) G is contained in the radical o f the category A G . Prop osition 4.1. If the action of a gr oup G on a c ate gory A is sep- ar abl e , so is also its induc e d action on the c ate gory A G . In this c a se rad( A G ) = (rad A ) G an d the c ate gory A G is sem i s i m ple. Pr o of is eviden t.  F rom now on, we suppose that A is a piecewise finite lo cal K - category , R = rad A , X ∈ Ob A is an indecomp osable ob ject from A , A = A ( X , X ) a nd G is a finite group acting on A with a system of factors λ so that its action is separable. W e are in terested in the decomp osition of the ob ject X in t he categor y A G in to a direct sum of indecomposables, especially , the num b er ν G ( X ) of non- isomorphic summands in such a decomp osition. Recall that suc h decomp osition comes from a decomp osition of the ring A G ( X , X ) or, equiv alen tly , of the ring A G ( X , X ) in to a direct sum of indecomp osable mo dules. Prop osition 4.2. L et H = { σ ∈ G | X σ ≃ X } . Then A G ( X , X ) / R G ( X, X ) ≃ A H ( X , X ) / R H ( X, X ) , in p articular, ν G ( X ) = ν H ( X ) . 16 YURIY A. DROZD Pr o of is eviden t, since a σ ∈ R for ev ery morphism a σ : X σ → X if σ / ∈ H .  Corollary 4.3. If X σ 6≃ X for al l σ ∈ G , the obje ct X r emains inde- c om p osable in the c ate gory A G . Therefore, dealing with the decompo sition of X , w e can only con- sider the action of the subgroup H . F or ev ery σ ∈ H w e fix an iso- morphism φ σ : X σ → X and consider the action T ′ of the group H on the ring A giv en by the rule T ′ σ ( a ) = φ σ a σ φ − 1 σ . One easily v eri- fies that t he elemen ts λ ′ σ ,τ = φ σ φ σ τ λ σ ,τ φ − 1 στ form a system of factors for this action, moreov er, the map a [ σ ] 7→ aφ σ [ σ ] establishes an isomor- phism A ( H , T ′ , λ ′ ) ≃ A H ( X , X ). Th us, in what follows , w e in v estigate the alg ebras A ( H , T ′ , λ ′ ) a nd D ( H , T ′ , ¯ λ ), where D = A / rad A and ¯ λ σ ,τ denotes the image of λ ′ σ ,τ in the sk ewfield D . The latter f actor- ring is finite dimensional sk ewfie ld (division a lgebra) o v er the field k . W e denote by F the cen ter if this algebra (it is a field). Let N b e the su bgroup of H consisting o f all elemen ts σ suc h that the auto- morphism T ′ σ induces an inner automorphism of the sk ewfield D , or, equiv alen tly , the ide n tit y automorphism of the field F [7 , Corollary IV.4.3]. It is a no rmal subgroup in H . F or ev ery elemen t ρ ∈ N w e c ho ose an elemen t d ρ ∈ D suc h that T ′ ρ ( a ) = d ρ ad − 1 ρ for all a ∈ D . W e also choose a set S of represen tat iv es of cosets H / N a nd, for ev- ery σ ∈ H , denote b y ¯ σ the elemen t from S suc h that σ N = ¯ σ N , and b y ρ ( σ ) the elemen t from N suc h that σ = ρ ( σ ) ¯ σ . Now we set D σ ( a ) = d − 1 ρ ( σ ) T ′ σ ( a ) d ρ ( σ ) . An immediate v erification sho ws that w e get in this w ay an action of the group H on the sk ewfield D with the system of fa cto r s µ σ ,τ = d − 1 ρ ( σ ) ( d σ ρ ( τ ) ) − 1 ¯ λ σ ,τ d ρ ( στ ) and, b esides, the map [ σ ] 7→ d ρ ( σ ) [ σ ] induces an isomorphism D ( H , T ′ , ¯ λ ) ≃ D ( H , D , µ ). Note that now N = { σ ∈ H | D σ = id } = { σ ∈ H | D σ | F = id } . Moreo v er, one easily sees that µ σ ,τ ∈ F if σ, τ ∈ H . F urther on w e denote D H = D ( H , D , µ ) . The n um b er of non- isomorphic indecomp osable summands in the decomposition of D H equals the n um b er of simple comp onen ts of this algebra [7, Theorem I I.6.2], or, the same, the num b er of simple comp onents of its cen ter. Prop osition 4.4. The c enter of the algeb r a D H c oincid es with the set ( F N ) H = { α ∈ F H | ∀ τ [ τ ] α = α [ τ ] } = = n X σ ∈ N a σ [ σ ]    ∀ σ  a σ ∈ F & ∀ τ ( τ ∈ H ⇒ a τ σ µ τ ,σ = a τ στ − 1 µ τ στ − 1 ,τ )  o . GROUP A CTION ON BIMOD ULE CA T EGORIES 17 Esp e cial ly, if N = { 1 } , then D H is a c entr a l simple algebr a ove r the field of invariants F H , henc e, ν G ( X ) = 1 . 1 Pr o of. If an elemen t α = P σ a σ [ σ ] b elongs to the cen ter of D H , then P σ ba σ [ σ ] = P σ a σ [ σ ] b = P σ a σ b σ [ σ ], so if a σ 6 = 0, then b σ = a − 1 σ ba σ , hence, σ ∈ N , b σ = b and a σ ∈ F . Fina lly , the e qualities [ τ ] α = P σ a τ σ µ τ ,σ [ τ σ ] = α [ τ ] = P σ a σ µ σ ,τ [ σ τ ] = P σ a τ στ − 1 µ τ στ − 1 ,τ [ τ σ ] com- plete the pro of.  Corollary 4.5. I f F = k (for instanc e, the r esidue field k is alge- br aic al ly close d ) an d the gr oup H is a b elian , the c enter of the al g ebr a D H c o incides with k H 0 , wher e H 0 is the sub gr oup of H c onsisting of al l elements σ such that µ σ ,τ = µ τ ,σ for al l τ ∈ H . I n p articular, ν G ( X ) = #( H 0 ) . Pr o of. In this case N = H , so the cen ter of D H coincides with k H 0 (one easily c hec ks that H 0 is indeed a subgroup). Since the latter algebra is comm utativ e and semisimple, it is isomorphic to k m , where m = #( H 0 ), therefore, the n um ber of its simple comp onen ts equals m .  Corollary 4.6. If F = k and the gr o up H is cyclic, the c e n ter of the algebr a D H c oincides with k H and ν G ( X ) = #( H ) . Pr o of. Actually , in this case it is w ell-kno wn that µ σ ,τ = µ τ ,σ for all σ , τ ∈ H .  Note that all these coro llaries hold if the group G itself is ab elian or cyclic. If K -categor y A is piecewise finite, s o is ev ery bimo dule category El ( T ) as w ell, where T = ( A , B , ∂ ). If a group G a cts separably on the triple T , it acts separably on the category El ( T ) as w ell, and, according to Theorem 3.4, add El ( T ) G ≃ El ( T G ), this equiv alence b eing induced b y the functor Φ : x 7→ x [1]. Therefore, a ll the results abov e can b e applied to the s tudy of the decomp osition of an ele men t x [1] in the category El ( T G ). W e only quote explicitly the reformulations of Corollaries 4.5 and 4.6 for this case. Corollary 4.7. L et the r esidue field k b e algebr aic al ly close d an d the gr oup H = { σ | x σ ≃ x } b e ab elian. C ho ose isom orphisms φ σ : x σ → x for every eleme n t σ ∈ H a nd denote by µ σ ,τ the image of a morph i s m φ σ φ τ λ σ ,τ φ − 1 στ in k ≃ Hom T ( x, x ) / rad T ( x, x ) . Then the numb er of non- isomorphic inde c omp osable dir e ct summands in the de c omp osition of the obje ct x [1] in the c ate g o ry El ( T G ) e quals the o r der of the gr oup H 0 = { σ | ∀ τ µ σ ,τ = µ τ ,σ } . Esp e cial ly, if the gr oup H is cyclic, this numb er e quals the or der of H . 1 The last sta tement is well-known, see [10, Theorem 4 .5 0]. 18 YURIY A. DROZD Remark 4.8. It is eviden t that all thes e statemen ts also hold if separa- ble is the action of t he group H on the sk ewfield D , or, equiv alen tly , on its cen ter F . It is know n [10, Section 4.18] that one only has to v erify that separable is the a ction of the subgroup N , i.e. that c har k ∤ #( N ), since the action of N on F is trivial. Prop osition 4.1 eviden tly implies some more coro lla ries concerning the structure of the radical of the catego ry A G (for instance, bimo dule category El ( T G ) ). Corollary 4.9. L et the action of the gr oup G is sep ar able. If a set of morphisms { a i } is a set of gener ators of the A -mo dule (rad A )( X , ) (or A op -mo dule (rad A )( , X ) ), its ima g e { a i [1] } in A G is a set of gener ators o f the A G -mo dule (rad A G )( X , ) ( r esp e ctively, A op -mo dule (rad A G )( , X ) ). W e call a morphism a : Y → X left almost split (resp ectiv ely , right almost split ) if it generates the A - mo dule (rad A )( , X ) ( r espectiv ely , A op -mo dule (rad A )( Y , ) ), and an equality a = bf implies tha t the morphism f is left in v ertible, or, the same, is a split epimorphism (resp ectiv ely , the equalit y a = f b implies that g is right in v ertible, or, the same, is a split monomorphism). 2 Corollary 4.10. L et the action of G is sep ar able. If a morphism a : Y → X is left (right) almost split, so is a [1] as wel l. A sequence X a − → Y b − → X ′ is called almost split if the mo r phism a is left almost split, the mo r phism b is righ t almost split a nd, b esides, a = Ker b and b = Cok a , i.e., for ev ery ob ject 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. Corollary 4.11. L et the action of G is sep ar able. If a se quenc e X a − → Y b − → X ′ is almost split in the c ate gory A , the se quenc e X a [1] − − → Y b [1] − − → X ′ is almost spli t in the c ate gory A G . 2 In the b o o k [1] one only use s these notions in the ca se when X (resp ectively , Y ) is indecomposa ble. How ever, o ne ca n ea sily see that a left (right) almost split morphism in our sense is just a dire ct sum of those in the sense o f [1]. The sa me also conce r ns the no tion of the almost split se quenc es used b elow. GROUP A CTION ON BIMOD ULE CA T EGORIES 19 Since, under the separabilit y condition, ev ery ob ject fro m add A G is a direct summand of an ob ject that has come from the category A , Corol- laries 4.10 and 4.11 describe almost split morphisms a nd seque nces in the category add A G as so on as they are kno wn in the category A . In particular, these results can b e applied to the bimo dule categories El ( T G ) due to Theorem 3.4. Reference s [1] Auslander M., Reiten I. and Smalø S.O. Repr esentation Theory of Artin Al- gebras. Ca m bridge Universit y Press, 1 9 95. [2] Bass H. Algebra ic K - theo ry . New Y or k, Benjamin Inc. 19 6 8. [3] Bourbaki N. Commutativ e algebra. Chapters 1– 7. Berlin, Spr ing er–V erlag, 1989. [4] Chase S.U., Har rison D.K. a nd Rosen be rg A. Galois Theory and Galois Coho- mology of Co mmutative Rings. Mem. Amer. Math. So c. 5 2 (196 5), 1–19. [5] Crawley-Boevey W.W. Matrix problems and Drozd’s theo r em. Ba nach Cent. Publ. 26, Part 1 (199 0), 199-2 22. [6] Drozd Y.A. Reductio n algorithm a nd r epresentations of boxes and alg ebras. Comtes Rendue Math. Acad. Sci. Canada 23 (2 001), 9 7-125 . [7] Drozd Y.A. and Kirichenk o V.V. Finite Dimeniona l Algebr as. Ber lin, Springer– V erla g, 19 94. [8] Drozd Y.A., Ov sienko S.A. and F ur ch in B.Y. Categ orical constructions in the theory of re pr esentations. Algebra ic Structures and their Applications. Kiev, UMK VO, 1988, 17 –43. [9] Gabriel P . a nd Roiter A.V. Representations of Finite-Dimensional Alg ebras. Algebra VII I, Ency clop e dia of Math. Sci. Berlin: Springer– V erla g, 1992. [10] Ja cobson N. The Theory of Rings. AMS Math. Surveys, vol. 1. 194 3. [11] Reiten I. and Riedtmann C. Skew group algebr a s in the repres en tation theory of Artin a lgebras. J. Algebra , 92 (1 9 85), 224 –282. Institute o f Ma thema tics, Na tional A cademy of Sciences of Ukraine, Tereschenkivska 3, 01601 Kiev, Ukraine E-mail add r ess : drozd@ imath. kiev.ua