Coactions on Hochschild Homology of Hopf-Galois Extensions and Their Coinvariants

CO A CTIONS ON HOCHSCHILD HOMOLOGY OF HOPF-GALOIS EXTENSION S AND THEIR COINV ARIANT S A. MAKHLOUF AND D. S ¸ TEF AN Abstract. Let B ⊆ A b e an H -Galois extension, wher e H is a Hopf algebra o ver a ﬁeld K . If M is a Hopf bimodule then HH ∗ ( A , M ), the Hochsc hild homology o f A with co eﬃcients in M , is a rig ht co mo dule ov er the coa lgebra C H = H / [ H , H ]. Given an injective left C H -como dule V , our aim is to under stand the relationship betw een HH ∗ ( A , M )  C H V and HH ∗ ( B , M  C H V ). The ro ots o f this problem can be found in [Lo2], where HH ∗ ( A , A ) G and HH ∗ ( B , B ) are shown to be isomorphic for an y cen trally G -Galois extension. T o approa ch the a bove mentioned pr oblem, in t he case when A is a faithfully ﬂat B -mo dule and H satisﬁes so me technical conditions, we cons truct a spectra l sequence T or R H p ( K , HH q ( B , M  C H V )) = ⇒ H H p + q ( A , M )  C H V , where R H denotes the subalgebra of co commutativ e elemen ts in H . W e also ﬁnd conditions on H suc h that the edge ma ps of the ab ov e spe c tr al sequence yie ld isomorphisms K ⊗ R H HH ∗ ( B , M  C H V ) ∼ = HH ∗ ( A , M )  C H V . In the last par t of the pa p er w e deﬁne centrally Hopf-Galois extensions and we show that for such an extension B ⊆ A , the R H -action on HH ∗ ( B , M  C H V ) is trivial. As an applica tio n, we compute the subspace of H -coinv ariant elements in HH ∗ ( A , M ). A similar res ult is derived for HC ∗ ( A ), the cyclic homolog y of A . Intr oduction Chase and Sw eedler [CS], more than 35 y ears a go, deﬁned a sp ecial case of Ho pf - Galois extensions , similar to the theory of Galois group a ctions o n comm utativ e rings that had b een deve lop ed b y Chas e, Harrison and R osen b erg [CHR]. The general deﬁnition, fo r arbitrary Hopf alg ebras, is due to T a k euc hi and Kreimer [K T]. Besides Galois group actions that w e already men t ioned, strongly graded algebras are Hopf- Galois extensions. Other examples come from the theory of aﬃne quotien ts and of en veloping algebras of Lie alg ebras. By deﬁnition, an H -Ga lois extens ion is given b y an algebra map ρ A : A → A ⊗ H that deﬁnes a coaction of a Hopf algebra H on the algebra A suc h that the map β A : A ⊗ B A → A ⊗ H , β A = ( m A ⊗ H ) ◦ ( A ⊗ B ρ A ) is bijectiv e. Here m A : A ⊗ A → A denotes the m ultiplication map in A and B is the subalgebra of coin v arian t elemen ts, i.e. of all a ∈ A suc h that ρ A ( a ) = a ⊗ 1 . F or the remaining par t of the in tro duction w e ﬁx an H -G alois extens ion B ⊆ A . If there is no danger o f confusion, w e shall sa y tha t B ⊆ A is a Hopf-Galo is extension. Key wor ds and phr ases. Hopf-Galois extensions, Ho chsc hild homology and cyclic homology . 2000 Mathematics Subje ct Classiﬁc ation . Primary 16E40 ; Secondar y 16W30. D. S ¸ tefan w as ﬁnancially supp orted by CNCSIS, Con tract 560/2 009 (CNCSIS co de ID 69 ). 1 2 A. MAKHLO UF AN D D. S ¸ TEF A N F or an A -bimo dule M , let HH ∗ ( A , M ) denote Ho c hsc hild homology of A with co eﬃcien ts in M . Since B is a subalgebra o f A , the Ho c hsc hild ho mo lo gy o f B with co eﬃcien ts in M also mak es sense. A striking feature of HH ∗ ( B , M ) is tha t H acts on these linear spaces via the Ulbric h-Miy ashita a ctio n, cf. [S ¸ 1]. By ta king M to b e a Hopf bimo dule, more s tructure can b e deﬁned not only on HH ∗ ( B , M ) but o n HH ∗ ( A , M ) to o. Recall that M is a Hopf bimo dule if M is an A -bimo dule and an H -como dule suc h that the maps that deﬁne the mo dule structures are H - colinear. By deﬁnition of Hopf modules, HH 0 ( B , M ) is a quotien t H -como dule of M . This structure can be ex tended to an H -coaction on HH n ( B , M ) , for eve ry n. On the other hand, HH 0 ( A , M ) is not an H - como dule, in general. Nev ertheless, the quotien t coa lg ebra C H := H / [ H , H ] coacts on HH ∗ ( A , M ) , where [ H , H ] denotes the subspace spanned b y a ll commutators in H , see [S ¸ 2]. These actions and coactions pla y ed an imp ortan t r o le in the study of Ho chsc hild (co)homology of Hopf-Galois extensions, having dee p application in the ﬁeld. Let us brieﬂy discuss some of them, that are related to the presen t work. First, in degree zero, the coin v ariants of Ulbrich-Miy ashita action HH 0 ( B , M ) H := K ⊗ H HH 0 ( B , M ) equal HH 0 ( A , M ). This iden tiﬁcation is one of t he main ingredien ts that are used in [S ¸ 1] to pro v e the existenc e of t he sp ectral sequenc e T or H p ( K , HH q ( B , M )) = ⇒ HH ∗ ( A , M ) . (1) It ge neralizes, in an unifying w a y , Lyndon-Ho c hsch ild-Serre sp ectral sequence for group homolo gy [W e, p. 195], Ho c hsc hild-Serre sp ectral s equence for Lie algebra homology [W e, p. 232] and Lorenz sp ectral sequence for strongly graded algebras [Lo1]. If H is semis imple then (1) collapses and yields the isomorphisms HH n ( B , M ) H ∼ = HH n ( B , M ) H ∼ = HH n ( A , M ) , where HH n ( B , M ) H denotes the space o f H -inv a r ia n t Ho c hsc hild homology classes. Similar isomorphisms in Ho c hsc hild cohomology w ere pro v ed in the same wa y and used, for example, to in ves tigate a lgebraic deformations arising from orbifolds with discrete torsion [CGW ], to c har acterize defo r ma t ions of certain bialgebras [MW] and to study the G -structure o n t he cohomolog y of a Hopf alg ebra [F S]. In [S ¸ 2], for a s ub coalgebra C of C H , a ne w homolog y theory HH C ∗ ( A , − ) with co eﬃcien ts in the category of Ho pf bimo dules is deﬁned. In the case when C is injectiv e a s a left C H -como dule w e hav e HH C ∗ ( A , M ) ∼ = HH ∗ ( A , M )  C H C , for ev ery Hopf bimo dule M (for the deﬁnition of the cotensor pro duct  C H see the preliminaries of this pap er). Th us, HH C ∗ ( A , M ) ma y be regarded as a sort of C - coin v ar ia n t part of HH ∗ ( A , M ) . The main result in lo c. cit. is the sp ectral sequence T or H p ( K , HH q ( B , M  C H C )) = ⇒ HH C ∗ ( A , M ) , (2) that exists for ev ery Hopf bimodule M , provided that H is co comm utativ e and C is injectiv e a s a left C H -como dule. Let G b e a group and let K b e a ﬁeld. If H is the group algebra K G , then C K G := L σ ∈ T ( G ) C σ , where T ( G ) denotes the set of conjugacy classes in G and C σ is CO ACTIONS ON HOCHSCHILD HOMOLOGY OF HOPF-GA LO IS EXTENSIONS 3 a sub coalgebra of dimension o ne, fo r ev ery σ ∈ T ( G ) . Hence, in this particular case, the homogeneous comp onen ts HH σ ∗ ( A , M ) := HH C σ ∗ ( A , M ) completely determine Ho c hsc hild homology of A with co eﬃcien ts in M , cf. [Lo1, S ¸ 2]. F or strongly graded algebras ( i.e. K G -Galois extensions) and C = C σ , the sp ectral sequence (2 ) is due to Lorenz [Lo1]. On the other hand, Burghelea and Nistor deﬁned and studied similar homogeneous comp o nen ts of Ho c hsc hild and cyclic cohomology of group algebras and crosse d pro ducts in [Bu] and [Ni ], respectiv ely . W e ha v e already remark ed that, for an arbitrary H - Galois extension B ⊆ A and ev ery Ho pf bim o dule M , Hochs c hild homology HH ∗ ( B , M ) is a righ t H - como dule and a left H - mo dule. In par ticular, H acts and coacts on A B := HH 0 ( B , A ). No- tably , with resp ect to these structures, A B is a stable-an ti- Y etter-Drinfeld H -mo dule (SA YD H - mo dule, for short). T hese mo dules were indep enden tly disco v ered in [JS ¸ ] and [HKRS], and they can be though t of as co eﬃcien ts for Hopf-cyclic homology . In [JS ¸ ] the authors sho w ed that Hopf-cyclic homolog y of H with co eﬃcien ts in A B equals relativ e cyclic homology HC ∗ ( A / B ) . This iden tiﬁcation is t hen used to com- pute cyclic homolog y of a strongly G -graded algebra with separable component o f degree 1 (e.g. group a lgebras and quan tum tori). It is w orth while to men tion that cyclic homolog y of a group oid was computed in [BS ¸ ], using the t heory of (general- ized) SA YD mo dules. Let G b e a ﬁnite group of automorphisms of an algebra A ov er a ﬁeld K and let A G denote the ring of G -in v ariants in A . Since G is ﬁnite , the dual vec tor space ( K G ) ∗ has a canonical structure of Hopf alg ebra a nd A is a ( K G ) ∗ -como dule alg ebra. Clearly , the coinv ar ia n t subalgebra with resp ect to this coaction equals A G . It is w ell-known that A G ⊆ A is ( K G ) ∗ -Galois if and only if t his exte nsion is G alois in the sense of [CHR]. The cen ter Z of A is G -inv ar ia n t . F o llowing [Lo2], w e sa y that A G ⊆ A is centrally Galois if Z G ⊆ Z is ( K G ) ∗ -Galois. The Galois gro up G acts, of course bo th on Ho chsc hild homology HH ∗ ( A , A ) and cyclic homology HC ∗ ( A ). T o simplify the notation, w e shall write HH ∗ ( A ) for HH ∗ ( A , A ). By [Lo2, § 6], for a cen trally G alois extens ion A G ⊆ A , HH ∗ ( A ) G ∼ = HH ∗ ( A G ) (3) and a similar isomorphism exists in cyclic ho mology , provided that the order of G is in ve rtible in K . Since ( K G ) ∗ is commutativ e, t he coalgebras C ( K G ) ∗ and ( K G ) ∗ are equal. The category of left K G -mo dules is isomorphic to the category of righ t ( K G ) ∗ -como dules and through this iden tiﬁcation X G ∼ = X  ( K G ) ∗ K . In particular, HH ∗ ( A ) G ∼ = HH ∗ ( A )  ( K G ) ∗ K . This isomorphism suggests that the main result in [Lo2] migh t be approac hed in the spirit of [S ¸ 2], i.e. us ing the theory of Hopf-G alois extensions and an appropr ia te sp ectral sequenc e that con verges to HH ∗ ( A )  ( K G ) ∗ K . Since in general ( K G ) ∗ is not co comm utativ e, the sp ectral sequence in (2) cannot b e used directly . The main obstruction to extend it fo r a not necessarily co commutativ e Hopf alg ebra H , is the fact that Ulbric h- Miy ashita action do es not induce a n H - action on HH ∗ ( B , M  C H C ) . 4 A. MAKHLO UF AN D D. S ¸ TEF A N T o o verc ome this diﬃcult y w e deﬁne R H := k er (∆ − τ ◦ ∆) , where τ : H ⊗ H − → H ⊗ H denotes the usual ﬂip map. Since ∆ and τ ◦ ∆ are morphisms of alg ebras, R H is a subalgebra in H . Moreov er, if A is faithfully ﬂat as a left (or rig h t) B -mo dule and the an tip o de of H is an in v olution, then w e pro v e that HH ∗ ( B , M  C H V ) a r e left R H -mo dules, for ev ery injectiv e left C H -como dule V ; see Prop osition 2.14. Th us, under the ab ov e assumptions, for ev ery pair ( p, q ) of natural n umbers, it makes sense to deﬁne the ve ctor spaces E 2 p,q := T or R H p ( K , HH q ( B , M  C H V )) . (4) F urthermore, in Theorem 2.23 w e pro ve that there is a sp ectral seq uence that has E 2 p,q in the ( p, q )-sp ot of the second pag e and conv erges to HH ∗ ( A , M )  H V . This result follows as an application of Prop osition 2.3, where w e indicate a new fo rm of Grothendiec k’s sp ectral sequence. It also relies o n sev eral prop erties of the Ulbrich- Miy ashita action tha t a re pro v ed in the ﬁrst part of pap er. Here, w e just men tion equation (21) that plays a k ey role, as it explains the relationship b et we en the mo dule and como dule structures on HH ∗ ( B , M ). More precisely , using the terminology from [HKRS, JS ¸ ], relation (21) means tha t Ho c hsc hild homology of B with co eﬃcien ts in M is an SA YD H - mo dule. The most restrictiv e conditions that we imp ose in Theorem 2.2 3 are the relations R + H H = H + and T or R H n ( K , H ) = 0 , (5) for ev ery n > 0 , wh ere H + is the ke rnel of the c ounit and R + H := R H T H + . The second relat io n in (5) is easier to handle. F or example, if H is semisimple and cosemisimple ov er a ﬁeld of c hara cteristic zero we sho w t ha t R H is semisimple and C H is cose misimple, cf. Prop o sition 2 .24. The pro of of this result is based on the iden tiﬁcation R H ∗ ∼ = K ⊗ Q C Q ( H ) , where C Q ( H ) denotes the c har a cter algebra of H , a nd on the fact that C Q ( H ) is semisimple if H is so. Thus in this case the second relation in (5) holds true. Obviously b oth relatio ns in (5) a re satisﬁed if H is co comm utative . Notably , b y Prop osition 2.31, they are also v eriﬁed if H is semisimple and comm utativ e. The other tw o assumptions in Theorem 2.23 are not v ery s trong. The an tip o de of H is in v olutiv e for comm utativ e and co comm utative Hopf algebras. In characteris tic zero, b y a result of Larson and Radford, the an- tip o de of a ﬁnite-dimensional Hopf algebra is an in v olution if and only if the Hopf algebra is semisimple and cosemisimple. F aithfully ﬂat Hopf-Galois extensions a r e c har a cterized in [SS , Theorem 4.10]. In view of this theorem, if t he antipo de of H is bijectiv e, then an H -Galo is extension B ⊆ A is faithfully ﬂat if and only if A is injectiv e as a n H -como dule. Hence, ev ery H - Galois extension is faithfully ﬂat if H is cosemisim ple and its an tip o de is bijectiv e. If the algebra R H is semisimple then the sp ectral sequence in Theorem 2.23 collapses. W e ha v e already noticed that R H is semisim ple if H is either semisimple and cosemis imple o v er a ﬁeld of c haracteristic zero, or comm utativ e and semisimple. In these situations, the edge maps induce isomorphisms K ⊗ R H HH n ( B , M  H V ) ∼ = HH n ( A , M )  H V . (6) CO ACTIONS ON HOCHSCHILD HOMOLOGY OF HOPF-GA LO IS EXTENSIONS 5 By sp ecializing t he isomorphism in ( 6 ) to the case H := ( K G ) ∗ , w e deduce the isomorphism in Corollary 2.35 that can b e thought of as a generalization of (3). Our result also explains why the isomorphism (3) works for centrally Ga lo is ex- tensions but no t for arbitrary ones. Namely , the action of R ( K G ) ∗ on HH n ( B ) is trivial for cen trally Ga lois extensions, but not in general. In the last part of the pap er w e sho w that a similar result holds for Hopf- Galois extensions. Let B ⊆ A b e an H - como dule algebra, where H is a comm utative Hopf algebra. If the ce n ter Z of A is an H -sub como dule of A and Z T B ⊆ Z is a faithf ully ﬂat H -Galois extension, then w e say that B ⊆ A is cen trally H -Galo is. W e ﬁx suc h an extension B ⊆ A . In view of Prop osition 3.5, the R H -action on HH ∗ ( B , M  H V ) is trivial. If H is ﬁnite-dimensional and dim H is not zero in K , then b y Theorem 3.6 HH ∗ ( A , M )  H V ≃ HH ∗ ( B , M  H V ) , for ev ery Hopf bimo dule M whic h is symmetric as a Z - bimo dule and ev ery left H -como dule V . Assuming that H := ( K G ) ∗ and that the order of G is not zero in K , and taking M := A and V := K in the ab o ve isomorphism, we obtain the isomorphism in ( 3 ), cf. Corollary 3.9. Another application of Theorem 3.6 is given in Coro llary 3.7. Our approach has also the adv a ntage that one can easily reco ver HH ∗ ( A , M ) from HH ∗ ( B , M co H ) . More precisely , in Theorem 3.1 1 , we prov e the fo llo wing isomorphism of Z -mo dules and H -como dules HH ∗ ( A , M ) ≃ Z ⊗ Z T B HH ∗ ( B , M co H ) , for an y cen tra lly H -G alois extension B ⊆ A and an y Ho pf bimo dule M whic h is symmetric as a Z - bimo dule, pro vided that H is ﬁnite-dimensional and that dim H is not zero in K . Under the same assumptions, we also sho w that HC ∗ ( A ) co H and HC ∗ ( B ) are isomorphic, cf. Theorem 3.13. W e conclude the pap er by indicating a metho d to pro duce examples of cen trally Hopf-G alois extensions of non-commutativ e algebras. 1. Preliminaries In order to state and pro v e our main result w e need sev eral basic facts concerning Ho c hsc hild homology of Hopf-Galois extensions. Those that are w ell-known will b e only stated, for details the reader being referred to [JS ¸ , SS, S ¸ 1, S ¸ 2]. 1.1 . Let H b e a Hopf algebra with com ultiplication ∆ H and counit ε H . T o denote the elemen t ∆ H ( h ) w e shall use the Σ-notation ∆ H ( h ) = P h (1) ⊗ h (2) . Similarly , for a left H -como dule ( N , ρ N ) and a righ t H -como dule ( M , ρ M ) w e shall write ρ N ( n ) := P n h− 1 i ⊗ n h 0 i and ρ M ( m ) = P m h 0 i ⊗ m h 1 i . F or ( M , ρ M ) as a b o v e w e deﬁne the set of coin v ariant elemen ts in M b y M co H := { m ∈ M | ρ M ( m ) = m ⊗ 1 } . 6 A. MAKHLO UF AN D D. S ¸ TEF A N Recall tha t a como dule algebra is an algebra A whic h is a righ t H -como dule via a morphism of algebr as ρ A : A → A ⊗ H . Equiv a lently , ( A , ρ A ) is an H -como dule algebra if and o nly if ρ A (1) = 1 ⊗ 1 and ρ A ( ab ) = P a h 0 i b h 0 i ⊗ a h 1 i b h 1 i , for an y a, b in A . The set A co H is a subalgebra in A . If there is no da ng er of confusion w e shall also denote this subalgebra by B and w e shall sa y that B ⊆ A is an H -como dule algebra. F or an H -como dule algebra A , the catego r y M H A of right Hopf mo dules is deﬁned as follows . An ob ject in M H A is a rig ht A - mo dule M tog ether with a righ t H -coaction ρ M : M → M ⊗ H such that, fo r an y m ∈ M a nd a ∈ A , the following compatibilit y relation is veriﬁed ρ M ( ma ) = P m h 0 i a h 0 i ⊗ m h 1 i a h 1 i . (7) Ob viously , a morphism in M H A is a map whic h is b o t h A -linear and H - colinear. The category A M H is deﬁned similarly . A left A -mo dule and right H -como dule ( M , ρ M ) is a left Hopf mo dule if, for any m ∈ M and a ∈ A , ρ M ( am ) = P a h 0 i m h 0 i ⊗ a h 1 i m h 1 i . (8) By deﬁnition, a Hopf bimo dule is an A -bimo dule M together with a righ t H -coaction ρ M suc h that r elat io ns (7) and (8) are satisﬁed for all m ∈ M and a ∈ A . A morphism betw een t w o Hopf bimo dules is, b y deﬁnition, a map of A -bimo dules and H -como dules. The catego ry of Hopf bimodules will b e denoted by A M H A . F or example, A is a Hopf bimo dule. 1.2 . Let B ⊆ A b e an H -como dule algebra. Recall tha t B ⊆ A is an H -Galo is extension if t he canonical K -linear map β : A ⊗ B A → A ⊗ H , β ( a ⊗ x ) = P ax h 0 i ⊗ x h 1 i is bijectiv e. Note that A ⊗ B A is an ob ject in A M H A with resp ect to the canonical bimo dule structure and the H - coa ction deﬁned b y A ⊗ B ρ A . One can also regard A ⊗ H as an ob ject in A M H A with the A -bimo dule structure a · ( x ⊗ h ) · a ′ = P axa ′ h 0 i ⊗ ha ′ h 1 i , and the H -coaction deﬁned b y A ⊗ ∆ H . With resp ect to these Hopf bimo dule struc- tures β is a morphism in A M H A . Definition 1.3 . F or a K -algebra R a nd an R -bimo dule X w e deﬁne X R := X/ [ R , X ] and X R := { x ∈ X | r x = xr , ∀ r ∈ R} , where [ R , X ] is the K -subspace of X generated b y all comm utators r x − xr , with r ∈ R and x ∈ X . The class of x ∈ X in X R will b e denoted b y [ x ] R . Remark 1.4 . If X is a n R -bimo dule then X R ∼ = R ⊗ R e X , where R e := R ⊗ R op denotes the env eloping algebra of R . The isomorphism is giv en b y [ x ] R 7→ 1 ⊗ R e x . 1.5 . Let now B ⊆ A b e an arbitrary extension of algebras. By [JS ¸ , p. 145] it fo llo ws that ( A ⊗ B A ) B is an asso ciativ e algebra with the m ultiplication giv en by z z ′ = P n i =1 P m j =1 a i a ′ j ⊗ B b ′ j b i , (9) CO ACTIONS ON HOCHSCHILD HOMOLOGY OF HOPF-GA LO IS EXTENSIONS 7 where z = P n i =1 a i ⊗ B b i and z ′ = P m j =1 a ′ j ⊗ B b ′ j are arbitrary elemen ts in ( A ⊗ B A ) B . Moreo v er, if M is an A -bimo dule then M B is a righ t ( A ⊗ B A ) B -mo dule with resp ect to the action that, fo r m in M and z = P n i =1 a i ⊗ B b i in ( A ⊗ B A ) B , is deﬁned b y [ m ] B · z = P n i =1 [ b i ma i ] B . (10) 1.6 . Supp ose now that B ⊆ A is an H -Galois extension and let M b e an A -bimo dule. Let i : H − → A ⊗ H denote the canonical map i ( h ) = 1 ⊗ h . F ollowing [JS ¸ , p. 146] w e deﬁne κ : H − → ( A ⊗ B A ) B , κ := β − 1 ◦ i. F or h ∈ H w e shall use the no tation κ ( h ) = P κ 1 ( h ) ⊗ B κ 2 ( h ). Th us, b y deﬁnition, P κ 1 ( h ) κ 2 ( h ) h 0 i ⊗ κ 2 ( h ) h 1 i = 1 ⊗ h. (11) By [JS ¸ , p. 146], κ is an an ti–mo r phism of a lgebras. Hence h · [ m ] B = P [ κ 2 ( h ) mκ 1 ( h )] B (12) deﬁnes a le ft H -action on M B . Ob viously this structure is functorial in M , so we get a functor ( − ) B : A M A → H M . 1.7 . Let H b e a Hopf algebra with m ultiplication m and com ultiplicatio n ∆. Let τ : H ⊗ H → H ⊗ H denote the usual ﬂip map x ⊗ y 7→ y ⊗ x. One can prov e that C H := cok er ( m − m ◦ τ ) is a quotien t coalgebra of H , as the linear space gene rated by all comm uta t ors in H is a coideal of H . The canonical pro jection on t o C H will b e denoted b y π H . Note that π H is a trace map, that is π H ( hk ) = π H ( k h ) for all h and k in H . Dually , R H := k er(∆ − τ ◦ ∆) = { r ∈ H | X r (1) ⊗ r (2) = X r (2) ⊗ r (1) } is a subalgebra of H . It is clear that R H = { r ∈ H | P r (1) ⊗ r (2) ⊗ r (3) = P r (2) ⊗ r (3) ⊗ r (1) } . (13) 1.8 . If C is a coalgebra a nd R is an algebra w e deﬁne the category R M C as follo ws. The o b jects in R M C are left R -mo dules and right C -como dules suc h that the map ρ M that deﬁnes the coaction on M is R -linear, that is ρ M ( r m ) = P r m h 0 i ⊗ m h 1 i . A map f : M → N is a morphism in R M C if it is R -linear a nd C -colinear. F or a right C -como dule ( M , ρ M ) and a left C -como dule ( V , ρ V ) w e deﬁne their cotensor pro duct b y M  C V := k er ( ρ M ⊗ N − M ⊗ ρ V ) , Recall t ha t V is said to b e c oﬂat if the functor ( − )  C V : M C → K M is exact. By [DNR, Theorem 2.4.1 7 ] V is coﬂat if and only if V is an injectiv e ob ject in the category of left C - como dules. Note that, if M ∈ R M C and V ∈ C M , then M  C V is an R - submo dule of M ⊗ V , as M  C V is the k ernel of an R -linear map. Dually , for a righ t R -mo dule X and an ob ject M in R M C , the tensor pro duct X ⊗ R M is a quotien t C - como dule of X ⊗ M . 8 A. MAKHLO UF AN D D. S ¸ TEF A N In some s p ecial cases the cotensor product and the tensor pro duct “commute ”. F or instance, if X is a righ t R -mo dule, V is a left C -como dule a nd M ∈ R M C then ( X ⊗ R M )  C V ∼ = X ⊗ R ( M  C V ) , (14) pro vided that either X is ﬂat or V is injectiv e. 1.9 . Let A and B b e t w o ab elian cat ego ries and assume that, for eac h n ∈ N , a functor T n : A − → B is giv en. W e sa y that T ∗ is a homolo gic a l δ -functor if , f or ev ery short exact sequence 0 − → X ′ − → X − → X ′′ − → 0 (15) in A and n > 0, there are “connecting” morphism δ n : T n ( X ′′ ) → T n − 1 ( X ′ ) suc h that · · · − → T n ( X ′ ) − → T n ( X ) − → T n ( X ′′ ) δ n − → T n − 1 ( X ′ ) − → · · · is exact and functorial in the sequence in (1 5 ). F urthermore, T ∗ is said to b e ef- fac e able if, for eac h ob ject X in A , there is an ob ject P in A together with an epimorphism f rom P to X suc h that T n ( P ) = 0 fo r any n > 0 . A morphism o f δ -f unctors T ∗ and S ∗ with connecting homomorphisms δ ∗ and ∂ ∗ , resp ectiv ely , is a sequence of natural tra nsformations φ ∗ : T ∗ → S ∗ suc h that, fo r n > 0 , φ n − 1 ◦ δ n = ∂ n ◦ φ n . By Theorem 7.5 in [Br, Chapter I I I], homolog ical and eﬀaceable δ -functors ha v e the follo wing unive rsal prop ert y . If T ∗ and S ∗ are homological and eﬀaceable δ -functors and φ 0 : T 0 → S 0 is a natural tr a nsformation, then there is a un ique morphis m of δ -f unctors φ ∗ : T ∗ → S ∗ that lifts φ 0 . Proposition 1.10 . L et H b e a Hopf alge b r a with antip o de S H . Supp ose that B ⊆ A is an H -Galois extension, ( M , ρ M ) is a Hopf bi m o dule and V is a left H -c omo dule. (1) Th e r e is a K -line a r map ρ 0 ( M ) : M B → M B ⊗ H such that ( M B , ρ 0 ( M )) is a quotient H -c omo dule of ( M , ρ M ) . Mor e over, for m ∈ M a nd h ∈ H we have ρ 0 ( M )( h · [ m ] B ) = P h (2) · [ m h 0 i ] B ⊗ h (3) m h 1 i S H h (1) . (16) (2) I f S H is an involution, then ( M B ⊗ π H ) ◦ ρ 0 ( M ) : M B → M B ⊗ C H is a morphism of left R H -mo dules. Henc e, w i th r esp e ct to the ab o ve C H -c omo dule structur e, M B is an obje ct in R H M C H and, for any right R H -mo dule X , t he c o algebr a C H c o acts c anonic al ly o n X ⊗ R H M B . The C H -c o action on M B wil l b e denote d by ρ 0 ( M ) to o. (3) I f V is a le f t C H -c omo dule then M  C H V i s a B e -submo dule of M ⊗ V . Under the additional assumption that V is inje ctive, ( M  C H V ) B and M B  C H V ar e isomorphic line ar sp ac es. I n p articular, the action of R H on the latter ve ctor sp ac e c an b e tr ansp orte d to ( M  C H V ) B . Pr o of. (1 ) Ob viously , ( M B , ρ 0 ( M )) is a quotient H -como dule of M , as [ B , M ] is a sub como dule of M . F or M = A , iden tity (1 6) is prov en in [JS ¸ , Proposition 2.6]. The general case can be handled in a similar manner, rep lacing a ∈ A by m ∈ M ev erywhere in the pro of o f [JS ¸ , Relation (6)]. (2) Recall that S H is an in volution, i.e. S 2 H = Id H , if and only if P r (2) S H r (1) = P S H r (2) r (1) = ε ( r )1 H . (17) CO ACTIONS ON HOCHSCHILD HOMOLOGY OF HOPF-GA LO IS EXTENSIONS 9 Clearly ( M B ⊗ π H ) ◦ ρ 0 ( M ) deﬁnes a C H -como dule structure on M B . F o r brevit y w e shall denote this map by ρ 0 ( M ) to o. F or r ∈ R H and m ∈ M w e get ρ 0 ( M )( r · [ m ] 0 ( M )) = P r (2) · [ m h 0 i ] B ⊗ π H ( r (3) m h 1 i S H r (1) ) = P r (2) · [ m h 0 i ] B ⊗ π H ( m h 1 i S H r (1) r (3) ) = P r (3) · [ m h 0 i ] B ⊗ π H ( m h 1 i r (2) S H r (1) ) = P r · [ m h 0 i ] B ⊗ π H ( m h 1 i ) = r · ρ 0 ( M )([ m ] B ) . Note that the second and the third equalities a r e consequenc es of the fa ct that π H is a trace map and respective ly of relation (13). T o deduc e the penu ltimate iden tity w e use (17). In conclusion, ρ 0 ( M ) is a morphism of R H -mo dules. Hence M is an ob ject in R H M C H and, in view of § 1.8, o ne can regar d M B ⊗ R X as a quotient como dule of M B ⊗ X . (3) Obviously , ρ ′ := ( M ⊗ π H ) ◦ ρ deﬁne s a C H -coaction on M and it is a morphism of B -bimo dules, as ρ is so. Th us ( M , ρ ′ ) is a n ob ject in B e M C H and M  C H V is a B e - submo dule of M ⊗ V , cf. § 1.8. If V is an injectiv e C H -como dule, then ( M  C H V ) B ∼ = B ⊗ B e ( M  C H V ) ∼ = ( B ⊗ B e M )  C H V ∼ = M B  C H V . (18) Since M B is an ob ject in R H M C H , it follows that M B  C H V is an R H -submo dule of M B ⊗ V . In particular, M B  C H V is an R H -mo dule . T o conclude the pro of, w e tak e on ( M  C H V ) B the unique R H -action tha t mak es the comp osition of the isomorphisms in ( 1 8) an R H -linear map.  Remark 1.11 . W e k eep the assumptions in the third part of Prop o sition 1 .10. Let z := P n i =1 m i ⊗ v i b e an elemen t in M  C H V . The comp o sition of the R H -linear isomorphisms in (18) maps [ z ] B to P n i =1 [ m i ] B ⊗ v i . Ob viously , this isomorphism is natural in M ∈ A M H A . It is not har d to see that, for h ∈ R H , h · [ z ] B = P n i =1 [ κ 2 ( h ) m i κ 1 ( h ) ⊗ v i ] B . 2. The spectral s e quence In this section, given an H -G alois extension B ⊆ A , a Hopf bimo dule M and an injectiv e left C H -como dule V , we construct a sp ectral sequence that conv erges to H ∗ ( A , M )  C H V . Our result, Theorem 2.23 , will b e o bt a ined as a direct application of a v ar ian t of G rothendiec k’s sp ectral sequence, whic h will b e deduced from the follo wing t w o lemmas and [W e, Corollary 5.8.4]. Recall that a category A is co complete if and only if an y set of ob jects in A has a direct sum. If X is an ob j ect in a category , then we shall also write X fo r the iden tity map of X . Lemma 2.1 . L et A and B b e c o c om p lete ab elian c ate gori e s and let H , H ′ : A → B b e two right exact functors that c om mute with dir e ct sums. If U is a gener ator in A and ther e is a natur al morph i sm φ : H → H ′ such that φ ( U ) is an isomorphism , then φ ( X ) is an isomorphism, f or every obje ct X in A . 10 A. MAKHLO UF AN D D. S ¸ TEF A N Pr o of. Let X b e an ob ject in A . Since U is a generator in A , there is an exact sequence U ( J ) u − → U ( I ) v − → X − → 0 , where I and J are certain sets. Hence in the follow ing diagram H ( U ( J ) ) φ ( U ( J ) )   H ( u ) / / H ( U ( I ) ) φ ( U ( I ) )   H ( v ) / / H ( X ) φ ( X )   / / 0 H ′ ( U ( J ) ) H ′ ( u ) / / H ′ ( U ( I ) ) H ′ ( v ) / / H ′ ( X ) / / 0 the squares are comm utativ e and the lines are exact. Recall that H commute s with direct sum s if the canonical map α : ⊕ i ∈ I H ( X i ) → H ( ⊕ i ∈ I X i ) is an isomorphism for eac h family of ob jects ( X i ) i ∈ I in A . Now one can see easily that φ ( U ( I ) ) and φ ( U ( J ) ) are isomorphisms, as H and H ′ comm ute with direct sums and φ ( U ) is a n isomorphism. Th us φ ( X ) is an isomorphism to o.  Lemma 2.2 . L et A , B and C b e c o c omplete ab elian c ate gories with en o ugh pr oje ctive obje cts. L et F : B → C and G : A → B b e right exact functors that c ommute with dir e ct sums. If U is a gener ator in A such that G ( U ) is F -acyclic, then G ( P ) is F -acyclic f o r any pr oje ctive o bje ct P in A . Pr o of. R ecall tha t G ( U ) is F -a cyclic if L n F ( G ( U )) = 0 fo r a n y n > 0 . Let P b e a pro jectiv e ob ject in A . There is a set I suc h that P is a direct summand of U ( I ) . Hence G ( P ) is a direct summand of G ( U ( I ) ) . On the ot her hand, the pro of of [W e , Corollary 2.6.11] w orks fo r any functor tha t comm utes with direct sums. Th us L n F comm utes with direct sums, so G ( U ( I ) ) ∼ = G ( U ) ( I ) is F -acyclic. Then G ( P ) is also F -acyclic.  Proposition 2.3 . L et G : A → B , F : B → C and H : A → C b e right exa ct functors that c ommute with dir e ct sums, wher e A , B and C ar e c o c omplete ab elian c ate gories with enough pr oje ctive obje cts. Assume that U is a gener ator in A and that φ : F ◦ G → H is a natur al tr ansformation. If φ ( U ) is an isomorphism and G ( U ) is F -ac yclic then, for every obje ct X in A , ther e is a functorial sp e ctr al se quenc e L p F (L q G ( X )) = ⇒ L p + q H ( X ) . (19) Pr o of. By Lemma 2.1, for eve ry ob ject X the morphism φ ( X ) is an isomorphism. On the other hand, if X is pro jectiv e in A then L n F ( G ( X )) = 0 for ev ery n ∈ N ∗ . Hence, w e obtain (19) as a particular case of [W e, Corollary 5.8.4].  W e tak e B ⊆ A to b e a f aithfully ﬂat Galois extension. F or proving Theorem 2.23, one o f our main results, w e shall apply Prop osition 2.3. In order to do that w e need some prop erties of the category A M H A . W e start with the follow ing. Proposition 2.4 . L et H b e a Hopf algebr a with b i j e ctive antip o de. L et B ⊆ A b e a faithful ly ﬂat H -Galois extension. Then A ⊗ A is a p r oje ctive g e ner ator in the c ate gory of Hopf bimo dules. It is als o pr oje ctive as a B -bimo dule. CO ACTIONS ON HOCHSCHILD HOMOLOGY OF HOPF-GA LO IS EXTENSIONS 11 Pr o of. By [SS, Theorem 4.10] the induction functor ( − ) ⊗ B A : M B → M H A is an equiv alence of categor ies and its in verse is ( − ) co H . W e deduce that, for an arbitra r y righ t Hopf mo dule X , the canonical map X co H ⊗ B A → X induced b y the mo dule structure of X is an isomorphism of righ t Hopf mo dules. Let M b e a Hopf bimo dule. Hence, the A -bimo dule struc ture on M deﬁnes an epim orphism A ⊗ M co H ⊗ A → M of Hopf bimo dules. Thus A ⊗ A is a generator in the category A M H A . Let p : X → Y b e an epimorphism of Hopf bimo dules and f : A ⊗ A → Y b e an arbitr a ry morphism in A M H A . W e w a nt to show t ha t there is a morphism g : A ⊗ A → X of Hopf bimo dules suc h tha t p ◦ g = f . Indeed, if y := f (1 A ⊗ 1 A ) then y ∈ Y co H . Since ( − ) co H : M H A → M B is an equiv alence of categories it follo ws that ( − ) co H is exact. Hence p ( X co H ) = Y co H . Let x ∈ X co H b e an elemen t suc h that p ( x ) = y . There is a unique morphism of A -bimo dules g : A ⊗ A → X suc h that g ( a ′ ⊗ a ′′ ) = a ′ xa ′′ . Since x is an H -coinv ariant elemen t in X , one can c hec k easily that g is a map of H - como dules to o. Ob viously , p ◦ g = f . By [SS, Theorems 4.9 and 4.10] A is pro jectiv e as a left and righ t B -mo dule. Th us A e is pro jectiv e as a left B e -mo dule, that is A ⊗ A is a pro jectiv e B -bimo dule.  Cor ollar y 2.5 . L et H b e a Hopf algebr a with bije ctive antip o de. If B ⊆ A is a faithful ly ﬂat H -Galois extension then A M H A has enough pr oje c tive obje cts. Pr o of. Eve ry category with a pro j ectiv e generator has enough pro jectiv e ob jects.  Definition 2.6 . F or a K -algebra R and an R -bimodule X , let ( C ∗ ( R , X ) , b ∗ ) b e the c hain complex give n by C n ( R , X ) = X ⊗ R ⊗ n and b n ( x ⊗ r 1 ⊗ · · · ⊗ r n ) = xr 1 ⊗ · · · ⊗ r n + P n − 1 i =1 ( − 1) i x ⊗ r 1 ⊗ · · · ⊗ r i r i +1 ⊗ · · · ⊗ r n + ( − 1) n r n x ⊗ r 1 ⊗ · · · ⊗ r n − 1 . Ho chschild homo lo gy of R with co eﬃcien ts in X is, b y deﬁnition, the homology of ( C ∗ ( R , X ) , b ∗ ). It will b e denoted b y HH ∗ ( R , X ) . 2.7 . Let R and X b e as in the abov e deﬁnition. It is w ell-kno wn that Ho chsc hild homology of R with co eﬃcien ts in X may be deﬁned in an equiv alen t w ay b y HH ∗ ( R , X ) = T or R e ∗ ( R , X ) . Since X R ∼ = R ⊗ R e X , it also follow s that HH ∗ ( R , − ) are the left derive d functor s of ( − ) R : R M R → M K . 2.8 . Let B ⊆ A b e an H - como dule algebra and M b e a Hopf bimo dule. Ob viously , ρ n ( M ) : C n ( B , M ) → C n ( B , M ) ⊗ H giv en by ρ n ( M ) ( m ⊗ b 1 ⊗ · · · ⊗ b n ) = P  m h 0 i ⊗ b 1 ⊗ · · · ⊗ b n  ⊗ m h 1 i . deﬁnes a como dule structure on C n ( B , M ) suc h that C ∗ ( B , M ) is a complex of right H -como dules. Note that, if Z is the cen ter of A then C ∗ ( B , M ) is a complex of left Z 0 -mo dules, where Z 0 := Z T B . Indeed, Z 0 -acts on M ⊗ B ⊗ n b y z · ( m ⊗ b 1 ⊗ · · · ⊗ b n ) = ( z · m ) ⊗ b 1 ⊗ · · · ⊗ b n , and the diﬀeren tial maps b ∗ are morphisms of Z 0 -mo dules. Clearly , ρ n ( M ) is a mor- phism of Z 0 -mo dules, so C n ( B , − ) can b e seen as a functor fro m A M H A to the catego ry of c hain complexes in Z 0 M H . Therefore, a fortior i, the functors HH ∗ ( B , − ) map a 12 A. MAKHLO UF AN D D. S ¸ TEF A N Hopf bimodule to an ob ject in Z 0 M H . T he H -coaction on HH ∗ ( B , M ) will still b e denoted b y ρ ∗ ( M ). Remark 2.9 . By deﬁnition, Ho c hsc hild homolog y of B with co eﬃcien t s in M in degree zero equals M B . Th us in Prop osition 1.10 (1 ) and § 2.8, w e constructed t w o H -coactions on M B , b oth o f them b eing denoted b y ρ 0 ( M ). The notation w e ha v e used is consisten t, as these coactions a re iden tical. Lemma 2.10 . L et H b e a Hopf algebr a with bije ctive antip o de. If B ⊆ A is a faithful ly ﬂat H -Galo i s extension then HH ∗ ( B , − ) : A M H A → Z 0 M H is a homolo gic al and eﬀac e able δ -functor, wher e Z is the c enter of A and Z 0 := Z T B . Pr o of. W e take a short exact sequence of Hopf bimodules 0 − → M ′ − → M − → M ′′ − → 0 . (20) W e ha v e to prov e that there are connecting ma ps δ n : HH n ( B , M ′′ ) → HH n − 1 ( B , M ′ ) , whic h are homomorphisms of Z 0 -mo dules and H -como dules, making · · · − → HH n ( B , M ′ ) − → HH n ( B , M ) − → HH n ( B , M ′′ ) δ n − → HH n − 1 ( B , M ′ ) − → · · · a functorial exact sequen ce. In our setting, the connecting maps are obtained b y applying the long exact sequence in homolog y to the following short exact sequence of complexes in Z 0 M H 0 − → C ∗ ( B , M ′ ) − → C ∗ ( B , M ) − → C ∗ ( B , M ′′ ) − → 0 . Let us prov e that HH ∗ ( B , − ) is eﬀace able to o. Let M b e a giv en Hopf bimodule. By Propo sition 2.4, there exists a certain set I such that M is the quotien t of P := ( A ⊗ A ) ( I ) as a Hopf bimo dule. In view of the same proposition, A ⊗ A is pro jectiv e as a B - bimo dule. Th us, for n > 0, HH n ( B , P ) ∼ = T or B e n ( B , P ) = 0 . Hence the lemma is completely prov en.  Remark 2.11 . Both HH ∗ ( B , − ) and HH ∗ ( B , − ) ⊗ H can b e seen as homological and eﬀaceable δ - functors that map a Hopf bimo dule to an ob ject in Z 0 M H . The natural transformations ρ ∗ ( − ) in § 2.8 deﬁne a morphisms of δ -functors that lifts ρ 0 ( − ) : HH 0 ( B , − ) − → HH 0 ( B , − ) ⊗ H . Proposition 2.12 . L et H b e a Hopf algebr a with bije ctive a n tip o de. We assume that B ⊆ A is a fa ithful ly ﬂat H -Galois extension and that M is a Hop f bimo dule. (1) Th e r e is an H -action on HH n ( B , M ) that extends the mo dule structu r e de- ﬁne d in (12). Mor e over, for any h ∈ H and ω ∈ HH n ( B , M ) , ρ n ( M )( h · ω ) = P h (2) · ω h 0 i ⊗ h (3) ω h 1 i S H h (1) . (21) (2) I f the antip o de of H is involutive then HH ∗ ( B , − ) is a homolo gic a l and ef- fac e able δ -functor that take s values in R H ⊗Z 0 M C H . Pr o of. (1 ) W e ﬁx h ∈ H . The mo dule structure constructed in formula (12) deﬁnes a natural map µ h 0 ( M ) : HH 0 ( B , M ) → HH 0 ( B , M ) , µ h 0 ( M )([ m ] B ) = h · [ m ] B . CO ACTIONS ON HOCHSCHILD HOMOLOGY OF HOPF-GA LO IS EXTENSIONS 13 In view of Lemma 2.10 the δ -functor HH ∗ ( B , − ) : A M H A → Z 0 M H is homological and eﬀaceable. Hence, b y the univ ersal prop ert y o f these functors (se e § 1.9) there is a unique morphism of δ -functors µ h ∗ ( − ) : HH ∗ ( B , − ) → HH ∗ ( B , − ) that lifts µ h 0 ( − ) . Note that, by deﬁnition, µ h ∗ ( − ) and the connecting morphisms δ ∗ are morphisms o f Z 0 -mo dules and H -como dules. F or ω ∈ HH n ( B , M ) , w e set h · ω := µ h n ( M )( ω ) . Pro ceeding as in the pro of of [S ¸ 1, Prop osition 2.4], one can easily see that the a b o v e form ula deﬁnes a natural action o f H on HH n ( B , M ). By construction, it lifts the action in (12). Note that, fo r an y n, the connecting maps δ n are morphisms o f H -mo dules, since µ h ∗ ( − ) is a morphism of δ -functors. T o conclude the pro of o f this part, it remains t o pro ve relation (21). W e pro ceed b y induction. In degree zero the required iden tity holds by (16). Let us assume that (21) ho lds in degree n for an y Hopf bimodule. Let M b e a g iv en Hopf bimo dule. W e tak e an exact sequence 0 − → K − → P − → M − → 0 of Hopf bimo dules suc h that P := ( A ⊗ A ) ( I ) . Since δ n +1 is a homomorphism of H -mo dules and H -como dules a nd using the induction h yp othesis, for h ∈ H and ω ∈ HH n +1 ( B , M ) , w e get ( δ n +1 ⊗ H ) ( ρ n +1 ( M )( h · ω )) = ρ n ( K ) ( δ n +1 ( h · ω )) = ρ n ( K ) ( h · δ n +1 ( ω ) ) = P h (2) · δ n +1 ( ω h 0 i ) ⊗ h (3) ω h 1 i S H h (1) = ( δ n +1 ⊗ H )  P h (2) · ω h 0 i ⊗ h (3) ω h 1 i S H h (1)  . As HH n +1 ( B , P ) = 0 it fo llo ws that δ n +1 is injectiv e. Consequen tly , δ n +1 ⊗ H is also injectiv e. Thus t he for ego ing computation implies relation (21). (2) Let M b e a giv en Ho pf bimo dule. F or z ∈ Z 0 w e deﬁne ν z n ( M ) : HH n ( B , M ) → HH n ( B , M ) , ν z n ( M ) ( ω ) = z · ω . W e claim that µ h ∗ ( − ) and ν z ∗ ( − ) comm ute for a ll h ∈ H , i.e. µ h ∗ ( − ) ◦ ν z ∗ ( − ) = ν z ∗ ( − ) ◦ µ h ∗ ( − ) . (22) In degree zero t his iden tit y follows f rom the computatio n b elow, where for brevit y w e write µ h 0 and ν z 0 instead of µ h 0 ( M ) and ν z 0 ( M ) . Indeed,  µ h 0 ◦ ν z 0  ([ m ] B ) = P [ κ 2 ( h ) z mκ 1 ( h )] B = z · P [ κ 2 ( h ) mκ 1 ( h )] B =  ν z 0 ◦ µ h 0  ([ m ] B ) , where f or the second equalit y w e used that z is a cen tral elemen t. F urthermore, the natural transformat io ns that a pp ear in the left and right hand sides of (22) are morphisms of δ -functors that lift resp ectiv ely µ h 0 ( − ) ◦ ν z 0 ( − ) and ν z 0 ( − ) ◦ µ h 0 ( − ). Hence ( 2 2) fo llo ws by the univ ersal prop ert y of homolo gical and δ -functors. In view of the relation (2 2) it follo ws that HH n ( B , M ) is an H ⊗ Z 0 -mo dule with resp ect to ( h ⊗ z ) · ω =  µ h n ( M ) ◦ ν z n ( M )  ( ω ) . 14 A. MAKHLO UF AN D D. S ¸ TEF A N W e can now pro v e that HH n ( B , M ) is an ob ject in R H ⊗Z 0 M C H . As R H ⊗ Z 0 is subalgebra of H ⊗ Z 0 , it acts on HH n ( B , M ). The coalgebra C H coacts on the Ho c hsc hild homology of B with co eﬃcien ts in M via ¯ ρ ∗ ( M ) :=  HH ∗ ( B , M ) ⊗ π H  ⊗ ρ ( M ) . T o simplify the notation, w e shall write ρ ∗ ( M ) instead of ¯ ρ ∗ ( M ). W e ha v e to sho w that ρ n ( M ) is a morphism of R H ⊗ Z 0 -mo dules. By Lemma 2.10 we already know that ρ n ( M ) is a morphism of Z 0 -mo dules. Th us, it remains to che c k tha t ρ n ( M ) is a morphism of R H -mo dules to o. F or the c ase n = 0 see the pro of of Prop osition 1.10 (2). In fact, the same pro of w orks f or an arbitrary n, just replacing [ m ] B b y an elemen t ω ∈ HH n ( B , M ) a nd using (21) instead o f (16). W e still ha v e to pro v e t ha t HH ∗ ( B , − ) : A M H A → R H ⊗Z 0 M C H is a homolog ical and eﬀaceable δ -functor, i.e. for ev ery short exact o f Hopf bimo dules the cor r espo nding connecting maps δ ∗ are morphisms of R H ⊗ Z 0 -mo dules and H -como dules. By Lemma 2.10, it follows t ha t δ ∗ are morphisms of Z 0 -mo dules and H -como dules. By the pro of of the ﬁrst part o f the pro p osition, δ ∗ are a lso morphisms o f H -mo dules. Hence, a fortiori, they are morphisms of R H -mo dules.  2.13 . T he natural transformations t hat deﬁne the R H -mo dule a nd the C H -como dule structures o f HH ∗ ( B , − ) , as in the a b o v e prop osition, will b e denoted by µ ∗ ( − ) a nd ρ ∗ ( − ) , respective ly . Let us tak e an injective left C H -como dule V . By § 1.8, for a Hopf bimo dule M , the cotensor pro duct HH ∗ ( B , M )  C H V is a left R H ⊗ Z 0 -mo dule. It fo llows that HH ∗ ( B , − )  C H V is a homo lo gical and eﬀaceable functor from the category of Hopf bimo dules to the category of left R H ⊗ Z 0 -mo dules. Of course, its connecting maps are δ ∗  C H V , where δ ∗ are the connecting homomorphisms of the δ -functor HH ∗ ( B , − ) . T o simplify the notation, w e shall denote HH 0 ( B , − )  C H V b y G V . By the fore- going observ ations G V maps a Hopf bimo dule to a left R H ⊗ Z 0 -mo dule. Our aim no w is to describ e the left deriv ed functors of G V . Proposition 2.14 . L et H b e a Hopf a lg ebr a such that S 2 H = Id H . If B ⊆ A is a faithful ly ﬂat H -Galois ex tension and V is an inje ctive C H -c omo dule, then HH ∗ ( B , −  C H V ) : A M H A − → R H ⊗Z 0 M is a homolo gic al and e ﬀ ac e able δ -functor. As δ -functors fr om A M H A to R H ⊗Z 0 M , L ∗ G V ∼ = HH ∗ ( B , − )  C H V ∼ = HH ∗ ( B , −  C H V ) . (23) Pr o of. By Prop osition 1.10 (3) the cotensor pro duct M  C H V is a B -bimo dule, for ev ery Hopf bimo dule M . Hence Ho c hsc hild ho mo lo gy of B with co eﬃcien ts in M  C H V make s sense. W e set T ∗ := HH ∗ ( B , −  C H V ) and tak e a short exact sequence of Hopf bimo dules a s in (20). Since V is an injectiv e como dule, 0 − → C ∗ ( B , M ′  C H V ) − → C ∗ ( B , M  C H V ) − → C ∗ ( B , M ′′  C H V ) − → 0 is exact. By the deﬁnition of T ∗ and the long exact sequence in homolo gy w e deduce that T ∗ is homolo g ical, rega rded as a δ -functor to the category of v ector spaces. Recall that U := A ⊗ A is a generator in the category of Hopf bimodules. Therefore, to pro v e tha t T ∗ is eﬀaceable, it is enough to sho w that T n ( X ) = 0 , where X is an arbitrary direct sum of copies of U and n > 0 . In fact, as Ho chsc hild homology and CO ACTIONS ON HOCHSCHILD HOMOLOGY OF HOPF-GA LO IS EXTENSIONS 15 the cotensor pro duct comm ute with direct sums, w e ma y assume that X = U . W e claim that U  C H V is ﬂat as a B -bimo dule. By (14), for an arbitrary B -bimo dule N , N ⊗ B e ( U  C H V ) ∼ = ( N ⊗ B e A e )  C H V ∼ = ( A ⊗ B N ⊗ B A )  C H V . Hence, the functors ( − ) ⊗ B e ( U  C H V ) and ( −  C H V ) ◦ ( A ⊗ B − ⊗ B A ) are isomorphic. Since the antipo de of H is bijectiv e, by [SS, Theorems 4.9 and 4.10], A is faithfully ﬂat as a left a nd a right B -mo dule. Therefore, A ⊗ B − ⊗ B A is an exact functor. As V is injectiv e, the functor ( − )  C H V is also exact, so U  C H V is ﬂat. Thus T n ( U ) ∼ = T or B e n ( B , U  C H V ) = 0 . Summarizing, T ∗ : A M H A − → K M is a homological a nd eﬀaceable δ -functor. F or eac h Hopf bimo dule M , our aim now is to endo w T n ( M ) with a left mo dule structure o v er R H ⊗ Z 0 . Let us ﬁrst consider the case n = 0 . By Prop o sition 1.10 (3), there is a canonical left R H -action on T 0 ( M ) suc h that T 0 ( M ) ∼ = HH 0 ( B , M )  C H V , (24) the natural K -linear isomorphism c onstructed in (18), is a homomorphism of R H - mo dules. As M is an ob ject in Z 0 M C H , b y § 1.8 , it follows that M  C H V is a left Z 0 -submo dule of M ⊗ V . In particular t his cotensor pro duct is a left Z 0 -mo dule. F urthermore, T 0 ( M ) is a quotien t Z 0 -mo dule o f M  C H V , as the commutator space [ B , M  C H V ] is a Z 0 -submo dule of M  C H V . Ob viously , with resp ect to this mo dule structure, T 0 ( M ) b ecomes a mo dule ov er R H ⊗ Z 0 and the isomorphism in (2 4) is a map of R H ⊗ Z 0 -mo dules. Since T ∗ is a homological and eﬀaceable functor, one can pro ceed as in the pro of of Prop osition 2.12 to lift the R H ⊗ Z 0 -action on T 0 ( M ) to a natural R H ⊗ Z 0 -mo dule structure on T ∗ ( M ) , fo r ev ery Hopf bimo dule M . Again as in the pro o f of the ab o v e men tioned result, w e can show that T ∗ : A M H A → R H ⊗Z 0 M is homological and eﬀaceable. It remains to pro v e the isomorphisms in (23 ) . Note that the left deriv ed functors of a right exact functor deﬁne a homological a nd eﬀaceable δ -functor. Th us L ∗ G V is a homological and eﬀaceable δ -functor. Clearly , L 0 G V = HH 0 ( B , − )  C H V . Hence, b y the univ ersal prop erty of homological and eﬀaceable δ -functors, this iden tit y ma y b e lifted to giv e the ﬁrst isomorphism in (23). The second isomorphism is obtained in a similar manner, by lifting t he natura l transformation in (24).  2.15 . Let B ⊆A b e an H -como dule alg ebra and let M b e a Hopf bimo dule. F ollo wing [S ¸ 2 , Theorem 1.3] w e regard C ∗ ( A , M ) as a complex in the category M C H with resp ect to the coaction that in degree n is giv en b y  n ( M )( m ⊗ a 1 ⊗ · · · ⊗ a n ) = X m h 0 i ⊗ a 1 h 0 i ⊗ · · · ⊗ a n h 0 i ⊗ π H  m h 1 i a 1 h 1 i · · · a n h 1 i  . Recall t ha t π H denotes the pro j ection of H onto C H and that Z 0 = Z T B . It is not diﬃcult to see that C ∗ ( A , M ) is a comple x of left Z 0 -mo dules with resp ect to the action that in degree n is deﬁned by z · ( m ⊗ a 1 ⊗ · · · ⊗ a n ) = z m ⊗ a 1 ⊗ · · · ⊗ a n . In fa ct, since Z 0 con t a ins only coinv aria nt elemen t s, it follo ws that C ∗ ( A , M ) is a complex in Z 0 M C H . Therefore, HH n ( A , M ) is an ob ject in the same category , for 16 A. MAKHLO UF AN D D. S ¸ TEF A N ev ery n . In view of § 1.8 it follo ws that HH ∗ ( A , M )  C H V is a left Z 0 -mo dule, f o r an y injectiv e left C H -como dule V . Therefore H V : A M H A → Z 0 M , H V ( M ) := M A  C H V . is a w ell deﬁne d functor, as b y the foregoing remarks M A = HH 0 ( A , M ) is a righ t C H -mo dule and H V ( M ) is a Z 0 -mo dule. Proposition 2.16 . L et B ⊆ A b e a faithful ly ﬂat H -Galois extension, w her e H is a Hopf algeb r a with bije ctive antip o de. If V is an inje ctive C H -c omo dule, then ther e is an isomorphi s m of δ -functors L ∗ H V ∼ = HH ∗ ( A , − )  C H V . (25) Pr o of. F irst, let us show tha t T ∗ := HH ∗ ( A , − )  C H V is a homolo gical and eﬀaceable δ -f unctor to the category of left Z 0 -mo dules. F or a short exact sequence as in (20), 0 − → C ∗ ( A , M ′ ) − → C ∗ ( A , M ) − → C ∗ ( A , M ′′ ) − → 0 is an exact sequence of complexes in Z 0 M C H . Therefore, the corresp onding long exact sequence in homology liv es in the same category . In particular, its connecting maps δ ∗ are morphisms of Z 0 -mo dules and C H -como dules, so are Z 0 -linear. Since V is injectiv e, the functor ( − )  C H V is exact. Th us T ∗ is a homological functor with connecting maps δ ∗  C H V . By Prop osition 2.4, t he Hopf bimo dule U := A ⊗ A is a generator . W e also hav e HH n ( A , U )  C H V ∼ = T or A e n ( A , A e )  C H V = 0 . In conclusion T ∗ is eﬀaceable, as Ho c hsc hild homology and the cotensor pro duct comm ute with direct sums. The isomorphisms in (25) are obtained b y lifting the iden tity L 0 H V = T 0 , as in the pro of of Prop osition 2.14.  2.17 . Since H is a Hopf algebra, the category of right H - como dules is monoidal, with respect to t he tensor product of v ector spaces, on which w e put the diago na l coaction. More precisely , if V and W a re righ t H -como dules then H coa cts on V ⊗ W via the map ρ V ⊗ W ( v ⊗ w ) = P v h 0 i ⊗ w h 0 i ⊗ v h 1 i w h 1 i , where v ∈ V , w ∈ W. Let us denote the right adjoin t coaction of H on itself by H ad . Recall that the ma p ρ H : H → H ⊗ H that deﬁne this coaction is giv en b y ρ H ( h ) = P h (2) ⊗ S h (1) h (3) . Hence A ⊗ H ad is a righ t H -como dule. Consequen tly , it is a C H -como dule via ρ A⊗H ( a ⊗ h ) = P a h 0 i ⊗ h (2) ⊗ π H  a h 1 i S h (1) h (3)  . (26) 2.18 . F o r a Hopf alg ebra H let H + := k er ε H and R + H := H + ∩ R H . If X is a n R H ⊗ Z 0 -mo dule, then R + H X is a Z 0 -submo dule of X , as the corresponding actions of R H and Z 0 on X comm ute. F or the same reason, K ⊗ R H X is a Z 0 -mo dule. Ob viously , with resp ect to these mo dule structures, the canonical isomorphism K ⊗ R H X ∼ = X/  R + H X  CO ACTIONS ON HOCHSCHILD HOMOLOGY OF HOPF-GA LO IS EXTENSIONS 17 is Z 0 -linear. F or example, Z 0 acts on K ⊗ R H G V ( M ) suc h that, for ζ = P n i =1 [ m i ] B ⊗ v i (27) in G V ( M ) and z in Z 0 w e ha ve z · (1 ⊗ R H ζ ) = P n i =1 1 ⊗ R H [ z m i ] B ⊗ v i . In view of this observ ation, w e shall regard K ⊗ R H G V as a functor from the category of Hopf bimo dules to the catego r y of left Z 0 -mo dules. Lemma 2 .1 9 . L et R and C den ote a n algebr a and a c o algebr a, r e s p e ctively, over a ﬁeld K . If ( M , ρ M ) ∈ R M C then ther e is a uniq ue morphis m of δ -functors ρ ∗ ( − ) : T or R ∗ ( − , M ) → T or R ∗ ( − , M ) ⊗ C that lifts ρ 0 ( − ) := ( − ) ⊗ R ρ M and deﬁn es a C -c o ac tion on T or R ∗ ( − , M ) . Pr o of. O bviously , T or R ∗ ( − , M ) and T or R ∗ ( − , M ) ⊗ C are homological and eﬀaceable δ -f unctors, whic h are deﬁned on the category of righ t R -modules. In de gree zero, ρ 0 ( − ) := ( − ) ⊗ R ρ M deﬁnes a C - como dule structure on T o r R 0 ( − , M ) = ( − ) ⊗ R C . By the univ ersal prop ert y , there is a morphism of δ - functors ρ ∗ ( − ) : T or R n ( − , M ) → T or R n ( − , M ) ⊗ C that lifts ρ 0 . W e w an t to prov e that ρ ∗ ( − ) deﬁnes a coaction on T o r R ∗ ( − , M ) . W e ha v e already remark ed that this prop erty holds in degree zero, so  ρ 0 ( − ) ⊗ C  ◦ ρ 0 ( − ) = ( − ⊗ ∆ C ) ◦ ρ 0 ( − ) . (28) W e need a similar iden tity for ρ n ( − ). No t e that T or R ∗ ( − , M ) ⊗ C ⊗ C is also a homological and eﬀaceable δ - f unctor. Clearly ,  ρ ∗ ( − ) ⊗ C  ◦ ρ ∗ ( − ) and ( − ⊗ ∆ C ) ◦ ρ ∗ ( − ) lift the natural tr ansformations in the left and respectiv ely righ t hand sides of (28). Hence, by the uniqueness of the lifting, these morphisms of δ - f unctors are equal (see the univ ersal prop erty in § 1.9).  Remark 2 .20 . In view of the previous lemma, ρ ∗ ( − ) is a homomorphism of δ - functors. Hence, for an exact sequence of rig h t R -mo dules as in (1 5 ) and an ob ject M in R M C , the connecting maps δ ∗ : T o r R ∗ ( X ′′ , M ) → T or R ∗− 1 ( X ′ , M ) are morphisms o f C -como dules. Lemma 2.2 1 . L e t B ⊆ A b e a faithful ly ﬂat H -Galois extension ove r a Hopf algebr a such that S 2 H = Id H . L et V b e an inje ctive C H -c omo dule and set U := A ⊗ A . (1) L et G V and H V b e the functors deﬁne d r esp e ctively in § 2.13 and § 2.1 5 . F or every Hop f bimo dule M an d ζ as in (27), the formula φ ( M )(1 ⊗ ζ ) = P n i =1 [ m i ] A ⊗ v i deﬁnes a natur al tr ansformation φ ( − ) : K ⊗ R H G V ( − ) − → H V ( − ) . (2) Th e algebr a R H acts on A ⊗ H via the multiplic ation in H so that A ⊗ H is an obje ct in R H M C H with r esp e ct to the c o action (2 6 ). (3) Th e C H -c o action on A ⊗ H induc e s a c omo dule structur e on T o r R H ∗ ( K , A ⊗ H ) . 18 A. MAKHLO UF AN D D. S ¸ TEF A N (4) We assume, in addition, t hat R + H H = H + and T or R H n ( K , H ) = 0 , for every n > 0 . T hen φ ( U ) is a n isomorphism and T o r R H n ( K , G V ( U )) = 0 , for n > 0 . Pr o of. (1 ) Let M b e a Hopf bimo dule and let p ( M ) : M B → M A denote t he canonical pro jection. Clearly , p ( M ) is a natural morphism of C H -mo dules. T o simplify the no- tation, set p := p ( M ) . By the foregoing, f := p  C H V is w ell-deﬁned. F urthermore, b y [JS ¸ , Prop osition 2.6 ], P κ 1 ( r ) κ 2 ( r ) = ε ( r ) . Let ζ b e an elemen t in M B  C H V satisfying relation (27) . Since [ am ] A = [ ma ] A , for ev ery r ∈ R H , a straigh tfo rw ar d computatio n yields f ( r · ζ ) = P n i =1 [ κ 2 ( r ) m i κ 1 ( r )] A ⊗ v i = ε ( r ) f ( ζ ) . Th us, t here exists a natural map φ ( M ) : K ⊗ R H G V ( M ) → H V ( M ) of Z 0 -mo dules, whic h is uniquely deﬁned suc h that φ ( M )(1 K ⊗ R H ζ ) = f ( ζ ) . (2) W e regard A ⊗ H as a le ft R H -mo dule via the m ultiplication in H . Let us pro v e tha t ρ A⊗H is a morphism of R H -mo dules. W e pic k up a ∈ A , h ∈ H and r ∈ R H . Th us ρ A⊗H ( a ⊗ r h ) = P a h 0 i ⊗ r (2) h (2) ⊗ π H  a h 1 i S H ( r (1) h (1) ) r (3) h (3)  = P a h 0 i ⊗ r (2) h (2) ⊗ π H  a h 1 i S H h (1) S H r (1) r (3) h (3)  = P a h 0 i ⊗ r (3) h (2) ⊗ π H  a h 1 i S H h (1) S H r (2) r (1) h (3)  = P a h 0 i ⊗ r h (2) ⊗ π H  a h 1 i S H h (1) h (3)  = r · ρ A⊗H ( a ⊗ h ) . Note t hat the third equalit y follo ws b y (13), while in the fourth one w e used (17). (3) This part is a direct application of Lemma 2.19. (4) Let λ := β ◦ η , where β is the canonical map in the deﬁnition of Hopf-G a lois extensions and η is the follo wing K -linear isomorphism η : ( A ⊗ A ) B → A ⊗ B A , η ([ a ⊗ x ] B ) = x ⊗ B a. By the deﬁnition o f β and η w e can easily sho w that λ ([ a ⊗ x ] B ) = P xa h 0 i ⊗ a h 1 i . W e claim that λ is an isomorphism in R H M C H . Ob viously , λ is bijective as β and η are so. If r ∈ R H and a, x ∈ A then λ ( r · [ a ⊗ x ] B ) = λ ( P [ κ 2 ( r ) a ⊗ xκ 1 ( r )] B ) = P xκ 1 ( r ) κ 2 ( r ) h 0 i a h 0 i ⊗ κ 2 ( r ) h 1 i a h 1 i = P xa h 0 i ⊗ r a h 1 i , where for the la st equalit y w e used (1 1). Th us λ is a morphism of R H -mo dules. Let ρ denote the coaction of C H on ( A ⊗ A ) B . Hence, by the deﬁnition of ρ and the fact that π H is a trace map, we g et ( λ ⊗ H ) ◦ ρ ([ a ⊗ x ] B ) = P λ  [ a h 0 i ⊗ x h 0 i ] B  ⊗ π H ( a h 1 i x h 1 i ) = P x h 0 i a h 0 i ⊗ a h 1 i ⊗ π H ( x h 1 i a h 2 i ) . CO ACTIONS ON HOCHSCHILD HOMOLOGY OF HOPF-GA LO IS EXTENSIONS 19 On the o t her hand, by (26) , it follows ρ A⊗H ◦ λ ([ a ⊗ x ] B ) = ρ A⊗H  P xa h 0 i ⊗ a h 1 i  = P ( xa h 0 i ) h 0 i ⊗ ( a h 1 i ) (2) ⊗ π H  ( xa h 0 i ) h 1 i S H ( a h 1 i (1) ) a h 1 i (3)  = P x h 0 i a h 0 i ⊗ a h 3 i ⊗ π H ( x h 1 i a h 1 i S H a h 2 i a h 4 i ) = P x h 0 i a h 0 i ⊗ a h 1 i ⊗ π H ( x h 1 i a h 2 i ) . Summarizing, the computation ab o ve sho ws us tha t λ is a morphism of C H -como dules to o. F urthermore, fo r a righ t R H -mo dule N , w e get N ⊗ R H G V ( U ) ∼ = N ⊗ R H [( A ⊗ H )  C H V ] ∼ = [ N ⊗ R H ( A ⊗ H ) ]  C H V , where the ﬁrst isomorphism is deﬁned by N ⊗ R H ( λ  C H V ) and the second one comes from the comm utation of the tensor pro duct and the cotensor pro duct. W e ha v e obtained a natural isomorphism ν ( N ) : N ⊗ R H G V ( U ) → [ A ⊗ ( N ⊗ R H H )]  C H V giv en, for z := P n i =1 [ a i ⊗ b i ] B ⊗ v i in G V ( U ) and x ∈ N , b y ν ( N )( x ⊗ R H z ) = P k i =1 P b i a i h 0 i ⊗ ( x ⊗ R H a i h 1 i ) ⊗ v i . Let us pro ve that φ ( U ) is an isomorphism. Since H / R + H H ∼ = K , K ⊗ R H H ∼ = R H / R + H ⊗ R H H ∼ = H / R + H H ∼ = K . Note t hat this isomorphism maps 1 ⊗ R H h t o ε ( h ) . Let γ b e the comp osition of the isomorphism [ A ⊗ ( K ⊗ R H H )]  C H V ∼ = A  C H V a nd ν ( K ) . Then γ (1 ⊗ R H z ) = P k i =1 P b i a i ⊗ v i , where z ∈ G V ( U ) is give n by the s ame formula as ab ov e. F urthermore, the m ulti- plication in A induces an isomorphism of C H -como dules µ : U A − → A , µ ([ a ′ ⊗ a ′′ ] A ) = a ′′ a ′ . It is easy to see that φ ( U ) = ( µ  C H V ) ◦ γ , so φ ( U ) is an isomorphism. Let P ∗ b e a resolution of K in M R H . The natural transformation ν yields iso- morphisms P ∗ ⊗ R H G V ( U ) ∼ = [ A ⊗ ( P ∗ ⊗ R H H )]  C H V . Since A ⊗ ( − ) and ( − )  C H V a r e exact functors it follo ws T or R H n ( K , G V ( U )) ≃ [ A ⊗ T o r R H n ( K , H )]  C H V . Hence the lemma is completely prov en as T or R H n ( K , H ) = 0, for ev ery n > 0 .  Definition 2 .22 . W e say that a Hopf algebra H ha s enough c o c omm utative elements if R + H H = H + . Theorem 2.23 . L et H b e a Hopf alge br a such that S 2 H = Id H . We assume that H has enough c o c ommutative elements and T or R H ∗ ( K , H ) = 0 . If B ⊆ A is a faithful ly ﬂat H -Galois extension an d V is a n inje ctive left C H -c omo dule then, for ev e ry Hopf bimo dule M , ther e is a sp e ctr al se quenc e in the c ate gory Z 0 M T or R H p ( K , HH q ( B , M  C H V )) = ⇒ HH p + q ( A , M )  C H V . (29) 20 A. MAKHLO UF AN D D. S ¸ TEF A N Pr o of. W e kno w that U := A ⊗ A is a generator in A M H A . In view of Lemma 2.21, one can apply Prop osition 2.3 to the f ollo wing categories: A := A M H A , B := R H ⊗Z 0 M , C := Z 0 M . The functors F , G V and H are giv en b y F := K ⊗ R H ( − ) , G V := ( − ) B  C H V , H V := ( − ) A  C H V and the natural transformation φ : F ◦ G V → H V is deﬁned in Lemma 2.21 (1). T o compute the left deriv ed functors of G V and H V w e use Prop ositions 2.14 and 2.16. Since any pro jectiv e R H ⊗ Z 0 -mo dule is also pro j ectiv e as an R H -mo dule, it follows that L n F ∼ = T or R H n ( K , − ) .  Proposition 2.24 . L et H b e a ﬁnite-dimensio n al Hopf al g ebr a over a ﬁ eld K of ch a r acteristic zer o such that S 2 H = Id H . Then R H is semisimple and C H is c osemisimple . Pr o of. W e ﬁrst pro v e that R H is semisimple in the case when K is algebraically closed. By Larson-Ra df o rd Theorem [DNR, Theorem 7.4.6], it follows that H is semisimple and cosemisimple. W e claim that, in this particular case, R H ∗ equals the K -subalgebra C K ( H ) of H ∗ , whic h is generated by the set of c haracters of H . F or the deﬁnition and prop erties of characters of a semisimple Hopf algebra, the reader is referred to [D NR, Section 7.5]. Recall that an elemen t α ∈ H ∗ is said to b e a tra ce map on H if and only if α v anishes on the space o f comm utators [ H , H ] . Let us sho w that R H ∗ equals the space of all tra ce maps on H . By the deﬁnition of com ultiplicatio n of H ∗ , ∆( α ) = X α (1) ⊗ α (2) if and only if α ( xy ) = P α (1) ( x ) α (2) ( y ) , f or all x, y ∈ H . On the o ther hand, α ∈ R H ∗ if and only if ∆( α ) = P α (2) ⊗ α (1) . There fore, for α ∈ R H ∗ , w e get α ( xy ) = X α (2) ( x ) α (1) ( y ) = X α (1) ( y ) α (2) ( x ) = α ( y x ) , so α is a trace map. The other implicatio n can b e pro ved s imilarly . W e can no w sho w that R H ∗ = C K ( H ) . By deﬁnition, a c haracter is a trace map, so C K ( H ) is a subspace of R H ∗ . Therefore, it is enough to sho w that dim R H ∗ ≤ dim C K ( H ). As the base ﬁeld is algebraically closed, H ∼ = Q n i =1 M d i ( K ) . F or ev ery i = 1 , . . . , n, let V i b e a simple left H -mo dule asso ciat ed to the blo c k M d i ( K ) and let χ i denote the irreducible c haracter corres p onding to V i . By [DNR, Proposition 7.5.7], χ 1 , . . . , χ n are linearly independen t o v er K , as eleme n ts in H ∗ . On the other hand, using the canonical basis { E ip i q i | i = 1 , . . . , n, p i , q i = 1 , . . . , d i } on Q n i =1 M d i ( K ), one can sho w that α is a trace map if and only if there a re a 1 , . . . , a n in K suc h that α ( E ip i q i ) =  0 , if p i 6 = q i , a i , if p i = q i . Hence, dim R H ∗ = n ≤ dim C K ( H ). T o deduce that R H ∗ is semisim ple, w e no w use [DNR, Theorem 7.5 .12] and the fact that C K ( H ) = K ⊗ Q C Q ( H ). W e hav e a lready remark ed that H is cosemisim ple to o. Thus , R H ∼ = R H ∗∗ is also semisimple. W e no w assume tha t K is an a r bit r a ry ﬁeld of c haracteristic ze ro. Let K b e an algebraic closure o f K and let H := K ⊗ K H . W e claim that K ⊗ K R H = R H . Indeed, CO ACTIONS ON HOCHSCHILD HOMOLOGY OF HOPF-GA LO IS EXTENSIONS 21 let { α i | i ∈ I } b e a basis of K as a K -v ector space and z = P i ∈ I α i ⊗ h i ∈ H . By the deﬁnition of the comultiplic ation of H , z b elongs to R H if and only if P i ∈ I ( α i ⊗ h i (1) ) ⊗ K (1 ⊗ h i (2) ) = P i ∈ I (1 ⊗ h i (2) ) ⊗ K ( α i ⊗ h i (1) ) . Th us P i ∈ I α i ⊗ h i (1) ⊗ h i (2) = P i ∈ I α i ⊗ h i (2) ⊗ h i (1) . Since the elemen t s α i are linearly indep enden t ov er K it results that z ∈ R H if and only if eac h h i is an elemen t in R H . Consequen tly , the claimed equalit y is pro ven. Ob viously , S 2 H = Id H . Since H is a Hopf algebra ov er a n algebraically closed ﬁeld, it follow s that R H is semisimple. Let J b e the Jacobson radical of R H whic h is a ﬁnite-dimensional algebra. Th us J is a nilpo ten t ideal. Clearly K ⊗ K J is a nilp ot en t ideal in K ⊗ K R H ∼ = R H , so it is con tained in the Jacobson radical of R H . W e deduce that K ⊗ K J = 0 . Th us J = 0 , so R H is semisim ple, b eing ﬁnite-dimensional. It remains to prov e that C H is cosemisimple. The dual algebra C ∗ H is isomorphic to the su balgebra of trace maps on H , i.e. C ∗ H ∼ = R H ∗ . As H is cose misimple, we ha v e already seen that C ∗ H ∼ = R H ∗ is semisim ple. Hence C H is cosemisim ple.  Theorem 2.25 . L et B ⊆ A b e an H -Galois extension, wh e r e H is a Hopf al g ebr a of ﬁnite dimen sion over a ﬁeld of char acteristic zer o. If H has enough c o c o mmutative elements and S 2 H = Id H then, for e very Hopf bimo dule M and every le f t C H -c omo dule V , ther e is isomorphisms of Z 0 -mo dules K ⊗ R H HH n ( B , M  C H V ) ∼ = HH n ( A , M )  C H V . Pr o of. By the pro of of the previous prop osition, H is cosemisimple. Thus A is injec- tiv e as an H -como dule, s o the extension B ⊆ A is faithfully ﬂat, cf. [SS, Theorem 4.10]. In view of the same pr o p osition V is injectiv e, as C H is cosemisimple, and K is pro jective as a rig ht R H -mo dule. Therefore, under the assumptions of the theo- rem, the sp ectral sequence (29) exists and collapses. The edge maps of this sp ectral sequence yields the r equired isomorphisms.  F or another application of Theorem 2.23, let us ta ke the Hopf algebra H to b e co comm utativ e. In this case R H = H , so the assumptions on H a r e trivially satisﬁed. W e obtain the sp ectral sequence from the following corollary . Note that a related result can b e found in [S ¸ 2, Theorem 3 .1], where the extension B ⊆ A is not necessarily fait hfully ﬂat but V is just a s ub coalgebra o f C H whic h is injectiv e in C H M . Cor ollar y 2.26 . L et B ⊆ A b e a f a ithful ly ﬂat H -Galois extension, with H a c o c ommutative Hopf algeb r a. If V is a n inje ctive right C H -c omo dule and M is a Hopf bimo d ule then ther e exists a sp e ctr al se quenc e in the c ate gory Z 0 M T or H p ( K , HH q ( B , M  C H V )) = ⇒ HH p + q ( A , M )  C H V . Remark 2.27 . By taking V := C H in the ab ov e corollary w e obtain (only in the case of co comm utative Hopf algebras) the sp ectral sequ ence [S ¸ 1, Theorem 4.5]. Let now consider the case when the Hopf alg ebra H is the group algebra K G o f an arbitrary group G . By [Mo, Theorem 8.1.7], an extension B ⊆ A is K G -Galois if 22 A. MAKHLO UF AN D D. S ¸ TEF A N and o nly if A is G -strongly graded and B is its homogeneous comp onen t of degree one, i.e. A is a direct sum of linear subspaces A = L g ∈ G A g suc h that A 1 = B a nd A g A h = A g h , for an y g , h in G . A strongly graded algebra, i.e. a K G - G alois ex tension B ⊆ A , is alwa ys faithfully ﬂat, as K G is cosemisimple . F urthermore, the coalgebra C K G is cosemisimple a nd p ointed, cf. [S ¸ 2, Exemple 1 .2 (a)]. A direct a pplicatio n o f the preceding corollary , for V := C K G , yields [Lo 1 , Theorem 2.5 (a)]. F urthermore, a K G -como dule is a ve ctor space M together with a decomp osition as a direct sum of subspaces M = L g ∈ G M g . Hence, an A - bimo dule M is a Hopf bimo dule if and only if t he a b o v e decomp osition satisﬁes, for an y g and h in G , the follo wing relations A h M g ⊆ M hg and M g A h ⊆ M g h . Th us, t o give a Hopf bimo dule is equiv alen t to giv e a G -g r a ded A - bimo dule. The coalgebra C K G is cosemisimple and p ointed, cf. [S ¸ 2, Exemple 1.2 (a )]. Recall that on C K G there is a canonical basis { e σ | σ ∈ T ( G ) } , where T ( G ) is the set of conjugacy classes in G and eac h e σ is a group-lik e elemen t. A left (or righ t) C K G como dule structure ρ V : V − → V ⊗ K G on a giv en v ector space V is uniqu ely deﬁned b y a decomp osition of V as a direct sum V = L σ ∈ T ( G ) V σ . Note t hat the subspace V σ is giv en b y V σ = { v ∈ V | ρ V ( v ) = e σ ⊗ v } . W e shall sa y that V σ is the homogeneous comp onen t o f V of degree σ . W e no w ﬁx a conjugacy class σ in G and we put V := K e σ . Since e σ is a group- lik e elemen t, V is a left C K G -sub como dule o f C K G and V σ = V . Let M be a G -graded bimo dule and M σ := L g ∈ σ M g . Th us M  C H V = M σ . Therefore, if G V and H V are the functors in the pr o of of Theorem 2.23 then G V ( M ) ∼ = B ⊗ B e M σ , H V ( M ) := ( M A ) σ . Let us notice that M A is a right C K G -como dule, so it mak es sense to sp eak ab o ut ( M A ) σ . Mor e generally , the coaction of C K G on the Ho c hsc hild homology of A with co eﬃcien ts in M induces a decomp osition of HH ∗ ( A , M ) as a direct sum of its homogeneous comp o nen ts HH ∗ ( A , M ) σ , for σ a rbitrary in T ( G ). As an a pplication of Theorem 2.23 w e now get the following result, that was also pro v ed in [Lo1, Theorem 2.5 (b)] b y a diﬀeren t metho d. Cor ollar y 2.28 . If B ⊆ A is a str ong l y G -gr ade d algebr a then , for any gr ade d A -bimo dule M and σ ∈ T ( G ) , ther e is a natur al sp e ctr al se quenc e in Z 0 M H p ( G, HH q ( B , M σ )) = ⇒ HH p + q ( A , M ) σ . (30) Pr o of. G roup homolog y H ∗ ( G, X ) and T o r K G ∗ ( K , X ) are equal for an y K G -module X , cf. [W e, Theorem 3.6 .2 ].  CO ACTIONS ON HOCHSCHILD HOMOLOGY OF HOPF-GA LO IS EXTENSIONS 23 Remark 2.29 . W e k eep the notation fro m the previous corollary . Let us pic k up an elemen t g in σ and denote the cen tra lizer of g in G b y C G ( g ). One can sho w that HH q ( B , M σ )) ∼ = K G ⊗ K C G ( g ) HH q ( B , M g ) . Th us, b y Shapiro’s Lemma, the terms in the second page of the spectral sequence (30) are isomorphic to E 2 p,q = H p ( C G ( g ) , HH q ( B , M g )) . F or details the reader is referred to [Lo1, p. 504]. No w w e are going to in v estigate the case when H is comm utativ e but not neces- sarily co comm utativ e. Thu s C H = H . Our ﬁrst aim is to sho w t hat w e can drop the assumptions on R H in Theorem 2 .23. T o this end, we need the following. Lemma 2.3 0 . L et G b e a ﬁnite gr oup. If H := ( K G ) ∗ then H has enough c o c ommu- tative ele ments and R H is semisi mple. Pr o of. Let { p x | x ∈ G } b e the dual basis of the canonical basis on K G. By deﬁnition, the coalgebra structure on H is g iv en b y ∆( p x ) = P g ∈ G p xg − 1 ⊗ p g and ε ( p x ) = δ x, 1 . Th us, a n elemen t z = P x ∈ G a x p x b elongs to R H if and only if P x,g ∈ G a x p xg − 1 ⊗ p g = P x,g ∈ G a x p g ⊗ p xg − 1 . Since { p x ⊗ p y | x, y ∈ G } is a basis on H ⊗ H , we deduce that a xg = a g x . Therefore, if σ ∈ T ( G ) then there is a σ in K suc h that a g = a σ , for all g ∈ σ . It follow s z = P σ ∈ T ( G ) a σ p σ , where p σ := P x ∈ σ p x . In conclusion, R H is the K -linear subspace g enerated b y all p σ , with σ ∈ T ( G ) . On the other hand, p x p y = δ x,y p x , for arbitr a ry x, y ∈ G. Thus , if x ∈ G and x 6 = 1, then p x = p σ p x , where σ denotes the conjugacy class of x. Hence the relation R + H H = H + is prov en. T o conclude the pr o of w e remark that p σ p τ = δ σ ,τ p σ and P σ ∈ T ( G ) p σ = 1 R H , so R H is semisim ple. In fa ct, the ab ov e relations sho ws us that R H ≃ K # T ( G ) .  Proposition 2.31 . L et H b e a c ommutative Hopf algebr a. I f H is semisimple then H has enough c o c ommutative elements and R H is a semisimpl e K -algebr a. Pr o of. Let us assume ﬁrst that H is semisimple. Then H is ﬁnite-dimensional b y [S ¸ 1 , Remark 3.8(b)]. Let K b e a n alg ebraic closure of K and let H := K ⊗ K H . Hence H is a ﬁnite-dimensional comm utativ e Hopf a lg ebra ov er K . Since H is semisimple it follow s that H is semisimple to o . Th us, the dual Hopf a lg ebra H ∗ is cosemisimple and co comm utative. Since K is algebraically closed, there is a ﬁnite g roup G suc h that H = ( K G ) ∗ . By the previous lemma, R + H H = H + and R H is semisim ple. As K ⊗ K ( H + / R + H H ) ∼ = ( K ⊗ K H + ) / ( K ⊗ K R + H H ) ∼ = H + / R + H H = 0 24 A. MAKHLO UF AN D D. S ¸ TEF A N w e get R + H H = H + . T o prov e that R H is semisimple, w e can pro ceed as in the pro of of Prop osition 2.24. If H is ﬁnite-dimensional o v er a ﬁeld of c haracteristic zero then it is semisimple (and cosemis imple). Hence we can apply the ﬁrst part of the prop osition.  Remark 2.32 . F or a comm utativ e Hopf algebra H o ve r a ﬁeld K , w e ha v e S 2 H = Id H . If H is ﬁnite-dimensional then the trace of S 2 H equals ( dim H )1 K . The refore, by [DNR, Theorem 7.4.1 ], H is semisimple and cosemisim ple if and only if dim H is not zero in K . In this case, H has enough co commutativ e elemen ts. Theorem 2.33 . L et B ⊆ A b e an H -Galois e x tension, wher e H is a c ommutative Hopf algebr a of ﬁnite dimension over a ﬁeld K s uch that dim H is not zer o in K . If V is a left H -c om o dule and M is a Hopf b i m o dule then ther e is an isomo rp hisms of Z 0 -mo dules K ⊗ R H HH n ( B , M  H V ) ∼ = HH n ( A , M )  H V . (31) Pr o of. In view of the ab o ve remark, H has enough co comm utativ e elemen ts a nd H is semisimple and cosemisimple. Th us V is injectiv e and B ⊆ A is a faithfully ﬂat extension. Since R H is semisimple , T or R H p ( K , H ) = 0 for p > 0, so w e can apply Theorem 2.23. F urthermore, for p > 0 T or R H p ( K , HH q ( B , M  H V )) = 0 , as an y R H -mo dule is pro jectiv e. It fo llows that the sp ectral seq uence in Theorem 2.23 collapses, its edge maps g iving the isomorphism in (31). Ob viously these maps are Z 0 -linear, as the sp ectral sequence liv es in Z 0 M b y construction.  2.34 . Recall that if G is a ﬁnite g roup of algebra automorphisms of A and B = A G then A is a ( K G ) ∗ -como dule alg ebra and B := A co( K G ) ∗ . Note tha t the corresp onding coaction ρ : A − → A ⊗ ( K G ) ∗ satisﬁes the relation ρ ( a ) = P x ∈ G x ( a ) ⊗ p x , where { p x | x ∈ G } is the dual basis of { x | x ∈ G } ⊆ K G. It is not diﬃcult to see that B ⊆ A is ( K G ) ∗ -Galois if and only if there are elemen ts a ′ 1 , . . . , a ′ n and a ′′ 1 , . . . , a ′′ n in A suc h that P n i =1 a ′ i g ( a ′′ i ) = δ g , 1 , for all g ∈ G. Th us ( K G ) ∗ -Galois extensions generalize Galo is extensions o f com- m uta tiv e rings. F or the deﬁnition of G alois extension of comm uta tiv e rings, the reader is referred to [DeMI, Chapter I I I]. More particularly , a ﬁnite ﬁeld extension is ( K G ) ∗ -Galois if and only if it is separable and normal. In this case , t he Galois group of the extension is G, cf. [DNR, Example 6 .4 .3 (1)]. F or t his reason, in this pap er, ( K G ) ∗ -Galois extensions will b e called (classical) G -Galois extensions. Note that an o b ject in A M ( K G ) ∗ A is an A -bimo dule M together with a G - action on M such t hat, for g ∈ G, a ∈ A and m ∈ M g · ( am ) = g ( a ) [ g · m ] and g · ( ma ) = [ g · m ] g ( a ) . W e shall sa y that suc h an M is a ( G, A )-bimo dule. CO ACTIONS ON HOCHSCHILD HOMOLOGY OF HOPF-GA LO IS EXTENSIONS 25 Cor ollar y 2.35 . L et B ⊆ A b e a G -Galois extension o v er a ﬁeld K such that the or der of G is not zer o in K . If M is a ( G, A ) -bim o dule then HH n ( A , M ) G ∼ = p 1 · HH n ( B , M G ) as Z 0 -mo dules, wher e { p x | x ∈ G } is the dual b asis of the c a n onic al b asis on K G . Pr o of. By Theorem 2.33, K ⊗ R ( K G ) ∗ HH n ( B , M  ( K G ) ∗ K ) ∼ = HH n ( A , M )  ( K G ) ∗ K . On the o t her hand, by the pro o f of Lemma 2.30 , w e get R ( K G ) ∗ ∼ = K # T ( G ) , as S := { p σ | σ ∈ T ( G ) } is a basis on ( K G ) ∗ and a complete set of orthogonal idempo ten ts. There fore, f or a ( K G ) ∗ -mo dule W , w e ha ve W = L σ ∈ T ( G ) p σ · W . Clearly , K ⊗ R ( K G ) ∗ W ∼ =  R ( K G ) ∗ / R + ( K G ) ∗  ⊗ R ( K G ) ∗ W ∼ = W /  R + ( K G ) ∗ W  ∼ = p 1 · W . Note t hat for the last isomorphism w e used that R + ( K G ) ∗ is spanned b y S \ { p 1 } . W e conclude the pro o f in view o f the foregoing remarks and of the isomorphisms X  ( K G ) ∗ V ∼ = X co( K G ) ∗ ∼ = X G . In the ab o ve iden tiﬁcations, for a righ t ( K G ) ∗ -como dule X , the G -in v arian ts are tak en with r esp ect to the left G -action on X t hat corresp onds to the ( K G ) ∗ -como dule structure on X via the isomorphism of catego r ies M ( K G ) ∗ ∼ = K G M .  3. Centrall y Hop f- Galois extensions Throughout this section we ﬁx a comm utativ e Hopf alg ebra H . In the case when H is a ﬁnite-dimensional Hopf a lg ebra a nd B ⊆ A is an H -como dule algebra we shall pro v e that Z , the cen ter of A , is an H -sub como dule. F or a giv en Hopf bimo dule M , our main purp ose is to sho w that, under some assumptions on H and Z co H ⊆ Z , the homology groups HH ∗ ( A , M ) co H and HH ∗ ( B , M co H ) are isomorphic. A similar result will b e pro ved for cyclic homology . Proposition 3.1 . L et B ⊆ A b e an H -c omo dule al g e br a. L et Z denote the c enter of A and set Z ′ := Z T B . (1) I f H is c ommutative and ﬁn itely gen er ate d as an alge b r a then Z is an H - sub c omo dule of A . (2) I f Z is an H -sub c omo dule of A an d Z ′ ⊆ Z is an H -Galois extension then H is c ommutative. L et us assume, in addition, that Z ′ ⊆ Z is a faithful ly ﬂat extension. Then A co H ⊆ A is a faithful ly ﬂat H -Galois extension. Pr o of. (1 ) As A is an H -como dule, ( A , · ) is a left H ∗ -mo dule, where for α in H ∗ and a in A α · a = P α ( a h 1 i ) a h 0 i . (32) T o prov e that Z is a n H -sub como dule we m ust chec k that Z is an H ∗ -submo dule. Let H ◦ denote the ﬁnite dual of H (for the deﬁnition of the ﬁnite dual of a n algebra see [D NR, Section 1.5]). It is w ell-kno wn that H ◦ is an S H -in v arian t subalgebra o f 26 A. MAKHLO UF AN D D. S ¸ TEF A N H ∗ , so it has a canonical structure of Hopf algebra. The com ultiplication of H ◦ is uniquely deﬁned suc h that ∆( α ) := P n i =1 α ′ i ⊗ α ′′ i if and only if α ( xy ) = X n i =1 α ′ i ( x ) α ′′ i ( y ) , for all x, y ∈ H . Clearly , A is an H ◦ -mo dule. In fact A is an H ◦ -mo dule algebra, that is α · ( a ′ a ′′ ) = P ( α (1) · a ′ )( α (2) · a ′′ ) , (33) for α ∈ H ◦ and a ′ , a ′′ ∈ A . W e no w w an t to show that Z is a n H ◦ -submo dule. F or α ∈ H ◦ and a ∈ Z , w e get ( α · a ) x = P ( α (1) · a )[ α (2) · ( S H ◦ α (3) · x )] = P α (1) · [ a ( S H ◦ α (2) · x )] = P α (1) · [( S H ◦ α (2) · x ) a ] , where for the second equalit y w e used (33). By [Ab, Corollary 2.3.17 (ii)], H ◦ is co comm utativ e. Th us P α (1) · [( S H ◦ α (2) · x ) a ] = P [( α (1) S H ◦ α (3) ) · x ]( α (2) · a ) = P [( α (1) S H ◦ α (2) ) · x ]( α (3) · a ) = x ( α · a ) . By the foregoing computation, we conclude that α · a ∈ Z . Since H is ﬁnitely generated as an algebra it follo ws that H ◦ is dense in H ∗ , with resp ect t o the ﬁnite top ology , cf. [Ab, Theorems 2.2.17 and 2.3.19 ]. This means tha t , for ev ery α ∈ H ∗ and every ﬁnite set X ⊆ H there is β ∈ H ◦ suc h that α = β on X. W e can no w pro v e that Z is an H ∗ -submo dule of A . Let α ∈ H ∗ and a ∈ A . If ρ ( a ) = P n i =1 a i ⊗ h i , then there is β ∈ H ◦ suc h that α ( h i ) = β ( h i ) , for eve ry i = 1 , . . . , n. Thus α · a = X n i =1 α ( h i ) a i = X n i =1 β ( h i ) a i = β · a. It f ollo ws that α · a ∈ Z , as β ∈ H ◦ and a ∈ Z . (2) Since Z is a sub como dule of A , it f ollo ws that Z co H = Z ′ . The canonical map β Z : Z ⊗ Z ′ Z → Z ⊗ H , that correspo nds to the H -como dule algebra Z ′ ⊆ Z , is bijectiv e b y assumption. As Z is a comm utativ e a lg ebra, β Z is a morphism of algebras and Z ⊗ Z ′ Z is comm utativ e. W e conclude t ha t H is commutativ e b y remarking that H is a subalgebra of Z ⊗ H , whic h is comm utative. W e now assume that Z ′ ⊆ Z is a faithfully ﬂat H - Galois extension. F or each h ∈ H there are a ′ 1 , . . . , a ′ r and a ′′ 1 , . . . , a ′′ r in Z suc h t ha t β Z ( P r i =1 a ′ i ⊗ Z co H a ′′ i ) = 1 ⊗ h. (34) Ob viously , β A ( P r i =1 a ′ i ⊗ B a ′′ i ) = β Z ( P r i =1 a ′ i ⊗ Z co H a ′′ i ) = 1 ⊗ h and β A is a morphism of left A -mo dules. Th us β A is surjectiv e to o. Since Z ′ ⊆ Z is faithfully ﬂat it follo ws that Z is injectiv e as an H - como dule. By [SS, Lemma 4.1.] there is an H -como dule map φ : H → Z such that φ (1) = 1 . W e ma y regard φ as an H - colinear map from H to A , so A is injectiv e as an H -como dule. Hence B ⊆ A is a faithf ully ﬂat H -Galois extension, cf. [SS, Theorem 4.10 ].  CO ACTIONS ON HOCHSCHILD HOMOLOGY OF HOPF-GA LO IS EXTENSIONS 27 Definition 3 .2 . Let H b e a comm utativ e Hopf algebra. W e sa y tha t an H - como dule algebra B ⊆ A is a c entr al ly H - Galois extension if the cen ter Z of A is a sub como d- ule and Z ′ ⊆ Z is a faithfully ﬂat H -Galois extension, where Z ′ := Z co H . Remark 3 .3 . In the case when H is cosemisimple and ﬁnitely generated as an algebra, an H -como dule algebra A is cen trally H -Galo is if and only if Z ′ ⊆ Z is H -Ga lois. 3.4 . Throughout the remaining part of this section w e ﬁx a comm utative Hopf algebra H and a cen trally H -Ga lois extension B ⊆ A . W e also ﬁx a Hopf bimo dule M and a left H -como dule V . W e ha v e see n that B ⊆ A is H -Galo is, so HH n ( B , M ) is a left H -mo dule. Our aim now is to giv e a n equiv alen t description of this action. W e ﬁx h ∈ H and we pic k up a ′ 1 , . . . , a ′ r and a ′′ 1 , . . . , a ′′ r in Z suc h that (34) holds tr ue. W e now deﬁne λ h n ( M ) : C n ( B , M ) → C n ( B , M ) b y λ h n ( M )( m ⊗ b 1 ⊗ · · · ⊗ b n ) = P r i =1 a ′′ i ma ′ i ⊗ b 1 ⊗ · · · ⊗ b n . It is easy to se e that λ h ∗ ( M ) is a morphism of complexes, as a ′ i and a ′′ i are in the cen ter of A for all i = 1 , . . . , r . Le t ¯ λ h n ( M ) b e the endomorphism o f HH n ( B , M ) induced b y λ h n ( M ). Clearly , b ot h λ h n and ¯ λ h n are natural transformations. Proposition 3.5 . L et h ∈ H . F or a Hopf bimo dule M and ω ∈ HH n ( B , M ) h · ω = ¯ λ h n ( M )( ω ) . If in a ddition M is a symmetric Z -bim o dule then the ab ove action is trivial. In this c ase, for an inje ctive left H -c om o dule V , the action of R H on HH n ( B , M  H V ) is trivial to o. Pr o of. Let µ h ∗ b e the natura l transformat io ns that lift the H - action on M B , as in the pro of of Pro p osition 2.12. Thus , fo r ω in HH n ( B , M ) µ h n ( M )( ω ) = h · ω . W e shall pro ve b y induction on n that ¯ λ h n ( M ) = µ h n ( M ). F or n = 0 that is obv ious, b y construction of ¯ λ h 0 and t he deﬁnition of the H -mo dule structure in (12). Let us assume that ¯ λ h n ( K ) = µ h n ( K ) , for an y Hopf bimo dule K . Since B ⊆ A is a faithfully ﬂat H -Galois extension, U := A ⊗ A is a pro jectiv e generator in A M H A . Th us, there is an exact sequence 0 − → K 0 − → L − → M − → 0 in A M H A suc h that L ∼ = U ( I ) , where I is a certain set. On the other hand, A is pro jectiv e as a left and righ t B -mo dule, so U is pro jectiv e as a B -bimo dule. Hence HH n ( B , L ) = 0, for n > 0 . Consequen tly , δ n +1 : HH n +1 ( B , M ) → HH n ( B , K 0 ) is injectiv e. On the other hand, by construction, µ h ∗ is a morphism o f δ -functors. Th us δ n +1 ◦ µ h n +1 ( M ) = µ h n ( K 0 ) ◦ δ n +1 . (35) Since the long exact sequence in homolog y is natural and λ h ∗ is a natural morphism of complexes δ n +1 ◦ ¯ λ h n +1 ( M ) = ¯ λ h n ( K 0 ) ◦ δ n +1 . ( 3 6) 28 A. MAKHLO UF AN D D. S ¸ TEF A N Using relatio ns (35 ) and (36), the induction h yp othesis and t he fa ct tha t δ n +1 is injectiv e o ne gets µ h n +1 ( M ) = ¯ λ h n +1 ( M ) . Let us assume that M is symmetric as a Z - bimo dule, i.e. z · m = m · z , for any z ∈ Z and m ∈ M . Th us λ h n ( M )( m ⊗ b 1 ⊗ · · · ⊗ b n ) = P k i =1 ma ′ i a ′′ i ⊗ b 1 ⊗ · · · ⊗ b n = ε ( h ) m ⊗ b 1 ⊗ · · · ⊗ b n , where for the second equality w e used [JS ¸ , Relation (5)]. Th us ¯ λ h n ( M )( ω ) = ε ( h ) ω . By the ﬁrst part of the prop o sition w e deduce that t he action of H on HH ∗ ( B , M ) is trivial. Finally , if V is an injectiv e left H -como dule, then there is an isomorphism HH ∗ ( B , M  H V ) ∼ = HH ∗ ( B , M )  H V of R H -mo dules. Note that HH ∗ ( B , M )  H V is a R H -submo dule of HH ∗ ( B , M ) ⊗ V . Hence, the action of R H on HH ∗ ( B , M )  H V is also trivial.  Theorem 3.6 . L et B ⊆ A b e a c entr al ly H -Galois extension, wher e H is a ﬁnite- dimensional Hopf algeb r a over a ﬁeld K such that dim H is not zer o in K . L et M b e a Hopf bi m o dule which is symmetric a s a Z -bimo d ule. I f V is a left H -c omo dule then ther e ar e isomorphisms o f Z ′ -mo dules HH ∗ ( A , M )  H V ≃ HH ∗ ( B , M  H V ) . (37) Pr o of. W e ha v e already noticed that B ⊆ A is a fa ithfully ﬂat H -G alois exten- sion and t hat H ∗ ( B , M  H V ) is a trivial R H -mo dule. The isomorphism of left Z ′ - mo dules(37) follow s b y applying Theorem 2.33.  Cor ollar y 3.7 . Ke ep ing the notation and the assump tion s fr om the pr e c e ding the- or em, ther e ar e isomorphisms of Z ′ -mo dules HH ∗ ( A , M ) co H ≃ HH ∗ ( B , M co H ) . Pr o of. T a k e V = K in Theorem 3.6 and note that ( − ) co H ∼ = ( − )  H K .  Remark 3.8 . A f aithfully ﬂat H -Galois extension of comm utative alg ebras is cen- trally Hopf-G alois. Th us the isomor phisms in the preceding corollary exist for suc h an extens ion, pro vided that H is ﬁnite-dimensional and dim H is not zero in K . Cor ollar y 3.9 . L et G b e a ﬁnite gr oup of automorphisms of an algebr a A over a ﬁeld K such that the or der of G is not zer o in K . L et Z denote the c en ter of A and let M b e an ( G, A ) -Hopf bimo dule. If Z G ⊆ Z is a G -Galois extension then ther e ar e isom o rphisms of Z G -mo dules HH ∗ ( A , M ) G ≃ HH ∗ ( A G , M G ) . Pr o of. Apply Corollary 3.7 for H := ( K G ) ∗ . Note that, for a ( K G ) ∗ -como dule X , w e hav e X co( K G ) ∗ = X G , cf. § 2.3 4.  Remark 3.10 . Note that the pro o f of the previous corollary works only if the order of G is not in v ertible in K , as K mus t b e injectiv e a s ( K G ) ∗ -como dule in order t o apply Theorem 3.6. On the o ther hand the isomorphisms in [Lo2, § 6] hold true without an y assumption on the characteristic of K . CO ACTIONS ON HOCHSCHILD HOMOLOGY OF HOPF-GA LO IS EXTENSIONS 29 Theorem 3.11 . L et B ⊆ A b e a c entr al ly Galois extension over a ﬁnite-dime n sional Hopf algebr a H such that dim H is not zer o in K . L et M b e a Hopf bimo dule which is symmetric as a Z -bimo dule. Then , ther e ar e isomorphisms of Z -mo dules and H -c omo dules HH ∗ ( A , M ) ≃ Z ⊗ Z ′ HH ∗ ( B , M co H ) . Pr o of. Since Z ′ ⊆ Z is a f aithfully ﬂat H - Galois extension the categories Z M H and Z ′ M are equiv alent, cf. [SS, Theorems 4.9 and 4.10]. More precisely , Z ⊗ Z ′ ( − ) : Z ′ M − → Z M H is an equiv alence of categories, whose in v erse is the functor X 7→ X co H . Th us, H ∗ ( A , M ) ∼ = Z ⊗ Z ′ H ∗ ( A , M ) co H . W e conclude de pro of in view of Corollary 3.7.  3.12 . Recall that B ⊆ A is a cen t r ally H -Galois extension. In particular, H is comm utat ive. Therefore, the map t n : C n ( A , A ) − → C n ( A , A ) giv en b y t n ( a 0 ⊗ a 1 ⊗ · · · ⊗ a n ) = a n ⊗ a 0 ⊗ · · · ⊗ a n − 1 . is a morphism of righ t H -como dules, where H coacts on A ⊗ n +1 as in (2 .15). Cyclic homology of A , denoted b y HC ∗ ( A ) , is deﬁne d a s the homology of the total complex of the bicomplex CC ∗∗ ( A ), see [W e, D eﬁnition 9 .6.6]. As the op era t o r t n is H - colinear f or ev ery n, it f ollo ws that CC ∗∗ ( A ) is a bicomplex in the category of righ t H -como dules. Th us HC n ( A ) is an H -como dule to o . W e can no w prov e the following. Theorem 3.13 . L et B ⊆ A b e a c entr al ly H -Galois extension, wher e H is a ﬁnite- dimensional Hopf algebr a such that dim H is not zer o in K . Then HC n ( A ) co H ∼ = HC n ( B ) . (38) Pr o of. The case n = 0 is obvious , in view of Corolla ry 3.7 and of the f a ct tha t cyclic homolog y and Ho chs c hild homology are equal in degree zero. F or eac h r ig h t H -como dule X the natural transformation ν ( X ) : X co H − → X  H K , ν ( X )( x ) := x ⊗ 1 is an isomorphism. Since H is comm utat iv e and the characteris tic of K do es not divide the dimension of H , we deduce that H is cosemisimple. Hence K is an injectiv e como dule. Th us the functor that maps a right H -como dule X to X  H K is exact. Consequen tly , by applying the functor ( − ) co H to Connes’ exact sequence [W e , Prop osition 9.6.11], w e get the exact seq uence on the top of the follo wing diagram. f HC n ( A ) e B / / g HH n +1 ( A ) e I / / f HC n +1 ( A ) e S / / f HC n − 1 ( A ) e B / / g HH n ( A ) HC n ( B ) B / / ∼ = O O HH n +1 ( B ) I / / ∼ = O O HC n +1 ( B ) S / / O O HC n − 1 ( B ) B / / ∼ = O O HH n ( B ) ∼ = O O Here, f HC ∗ ( A ) a nd g HH ∗ ( A ) denote HC ∗ ( A ) co H and HH ∗ ( A ) co H , res p ectiv ely . Note that b y induction h yp othesis, the ﬁrst a nd the fo urth v ertical a r r ows are isomor- phisms. F urthermore, b y taking M = A in Corollary 3.7 , w e get that the second 30 A. MAKHLO UF AN D D. S ¸ TEF A N and the ﬁfth v ertical maps are isomorphisms. Th us, b y 5- Lemma [W e, p.13] the v ertical map in the middle is also an isomorphism.  W e conclude this pap er showing that, under some extra assumptions, Or e exten- sions provide non-tr ivial examples of cen trally Hopf- Galois extensions. T o deﬁne a n Ore exten sion of a K -alg ebra A , w e need an algebra automorphism σ : A → A and a σ -deriv ation δ : A → A . Recall that δ is a σ -deriv ation if, for a and b in A , δ ( ab ) = σ ( a ) δ ( b ) + δ ( a ) b. F or σ a nd δ as ab o v e o ne deﬁnes a new algebra A [ X , σ, δ ], the Ore extension of A . As a left A - mo dule, A [ X, σ, δ ] is free with basis { 1 , X , X 2 , . . . } and its multiplication is the unique left A -linear morphism suc h that X n X m = X n + m and X a = σ ( a ) X + δ ( a ) . (39) F or simplicit y , w e shall denote the Ore extension A [ X , σ, δ ] b y T . W e now assume, in addition, that A is an H -como dule algebra a nd that σ and δ a re morphisms of como dules. Set B := A co H . Since σ and δ are morphisms of H -como dules they map B in to B . W e still denote the restrictions of these maps to B b y σ a nd δ. Clearly , δ can b e regarded as a σ -deriv ation of B , so w e can construct the Ore extension S := B [ X , σ, δ ]. Lemma 3 .1 4 . Th e c om o dule structur e map ρ A : A − → A ⊗ H c an b e extende d in a unique way to a n H -c o action ρ T on T such that, for a ∈ A a n d n ∈ N , ρ T ( aX n ) = P a h 0 i X n ⊗ a h 1 i . (40) With r esp e ct to this c o action the sub algebr a of c oinvariant elements in T is S . Pr o of. F or n ∈ N and 0 ≤ k ≤ n let f ( n ) k b e the non-commutativ e p olynomial in σ and δ with co eﬃcien ts in the prime subﬁeld of K suc h that X n a = P n k =0 f ( n ) k ( a ) X k . (41) Let us put f ( n ) − 1 = f ( n ) n +1 = 0. Th us, b y m ultiplying to the left b oth sides of (41) b y X and using (39), for 0 ≤ k ≤ n + 1, w e get f ( n +1) k = σ f ( n ) k − 1 + δ f ( n ) k , F or a 0 , . . . , a n in A w e now deﬁne ρ T ( P n i =0 a i X i ) = P n i =0 a i h 0 i X i ⊗ a i h 1 i . Clearly , ρ T deﬁnes a coaction of H on T and v eriﬁes the iden tit y (4 0). W e ha ve to pro v e that ρ T is a morphism of alg ebras, i.e. ρ T ( f g ) = ρ T ( f ) ρ T ( g ) for an y f , g ∈ T . In fa ct, it is enough to pro v e this equalit y for f = X n and g = a, with a ∈ A and n ∈ N ∗ . Since f ( n ) k are non-commutativ e p o lynomials in σ and δ and these maps are morphism of H -mo dules, it follo ws that f ( n ) k are also H -colinear. Hence, ρ T ( X n a ) = ρ T ( P n k =0 f ( n ) k ( a ) X k ) = P n k =0 f ( n ) k ( a ) h 0 i X k ⊗ f k ( a ) h 1 i = P n k =0 f ( n ) k ( a h 0 i ) X k ⊗ a h 1 i . CO ACTIONS ON HOCHSCHILD HOMOLOGY OF HOPF-GA LO IS EXTENSIONS 31 On the o t her hand, ρ T ( X n ) ρ A ( a ) = P X n a h 0 i ⊗ a h 1 i = P n k =0 f ( n ) k ( a h 0 i ) X k ⊗ a h 1 i = ρ T ( X n a ) . Ob viously ρ T is unital. Th us T is an H -como dule algebra. It remains to pro v e that T co H = S . F or this, w e ﬁx a basis { h j | j ∈ J } on H . W e ma y assume that there is j 0 ∈ J suc h that h j 0 = 1 . Let us take f = P n i =0 a i X i in T and write ρ ( a i ) = P j ∈ J a i j ⊗ h j . There fore, ρ ( f ) = P n i =0 P j ∈ J a i j X i ⊗ h j . It follows that f ∈ T co H if and only if P n i =0 a i j X i = δ j,j 0 P n i =0 a i X i . Th us, f is H -coinv aria nt if and only if ρ ( a i ) = P j ∈ J δ j,j 0 a i ⊗ h j = a i ⊗ 1 , for all i = 0 , . . . , n. W e deduce that f ∈ T co H if and only if f ∈ S .  Lemma 3.15 . L et f = P n i =0 a i X i b e an elemen t in T a n d a − 1 = a n +1 = 0 . Th e n f is in Z ( T ) , the c enter of T , if an d o nly if P n k = i a k f ( k ) i ( a ) = aa i , for i = 0 , . . . , n an d a ∈ A , (42) σ ( a i ) + δ ( a i +1 ) = a i , for i = − 1 , 0 , . . . , n + 1 . (43) Pr o of. As an alg ebra, T is generated b y A and X . Hence, f is central if and only if X f = f X and af = f a, for all a ∈ A . W e get f a = P n k =0 a k X k a = P n k =0 P k i =0 a k f ( k ) i ( a ) X i = P n i =0  P n k = i a k f ( k ) i ( a )  X i . W e deduce that f a = af and (42) are equiv alen t. On the other hand, f X = X f is equiv alen t t o P n i =0 σ ( a i ) X i +1 + P n i =0 δ ( a i ) X i = P n i =0 a i X i +1 . In conclusion, f X = X f a nd (4 3) are equiv alent.  Cor ollar y 3.16 . L et A σ = { a | σ ( a ) = a } and A δ = { a | δ ( a ) = 0 } . If Z is the c enter of A then Z ( T ) ∩ A = A σ ∩ A δ ∩ Z . Pr o of. W e rega r d A as a subalgebra of T . Th us, a 0 ∈ A is in the cen ter o f T if and only if f or an y a ∈ A w e ha v e δ ( a 0 ) = 0 , σ ( a 0 ) = a 0 , aa 0 = a 0 f (0) 0 ( a ) . Since f (0) 0 = Id A , w e get Z ( T ) ∩ A = Z ∩ A σ ∩ A δ .  Theorem 3.17 . L et B ⊆ A b e an H -c omo dule algebr a, wh e r e H is a c ommutative ﬁnite-dimensiona l Hopf algebr a over a ﬁeld of char acteristic zer o. L et σ : A → A b e an algebr a map and δ : A → A b e a σ -derivation. Assume that b oth σ and δ ar e morphisms of H -c omo dules. L et T := A [ X , σ, δ ] and S := B [ X , σ, δ ] . (1) Th e c enter Z o f A and A σ ∩ A δ ∩ Z ar e H -c omo dule sub a l g ebr as of A . The algebr a o f c oinvariant e l e m ents in A σ ∩ A δ ∩ Z is B σ ∩ B δ ∩ Z . (2) I f B σ ∩ B δ ∩ Z ⊆ A σ ∩ A δ ∩ Z is an H -Galoi s extension then the extension S ⊆ T is a c entr al ly H -Galo i s e x tension. 32 A. MAKHLO UF AN D D. S ¸ TEF A N Pr o of. (1 ) By Prop osition 3.1 (2), Z is an H -sub como dule o f A as H is ﬁnite- dimensional and comm utativ e. Since σ and δ are morphisms of H -como dules it follo ws that A σ = k er( σ − Id A ) and A δ = k er δ are H - sub como dules of A . W e deduce that A σ ∩ A δ ∩ Z is an H -como dule algebra. Its subalgebra of coinv ariant elemen ts is [ A σ ∩ A δ ∩ Z ] co H = A σ ∩ A δ ∩ Z ∩ B = B σ ∩ B δ ∩ Z . (2) Again by Prop osition 3 .1 (2), the cen ter Z ( T ) o f T is an H -sub como dule of T . Since H is commutativ e and ﬁnite-dimensional ov er a ﬁeld of characteris tic zero, w e deduce t hat H is cosemisimple. Hence, Z ( T ) is injective as an H -como dule. In view of [SS, Theorem 4.10 ], to prov e that Z ( T ) ∩ S ⊆ Z ( T ) is H - Galois and faithfully ﬂat, w e hav e to sho w that the canonical map β Z ( T ) : Z ( T ) ⊗ Z ( T ) ∩S Z ( T ) − → Z ( T ) ⊗ H is surjectiv e. Proceeding as in the pro of of Prop osition 3.1 (3) it is enough to sho w that 1 ⊗ h is in the image of β Z ( T ) , for ev ery h ∈ H . Let Z ′ := A σ ∩ A δ ∩ Z . By assumption, the canonical map β Z ′ : Z ′ ⊗ Z ′ ∩B Z ′ − → Z ′ ⊗ H is bijectiv e. Th us, there ar e a ′ 1 , . . . , a ′ r and a ′′ 1 , . . . , a ′′ r in Z ′ suc h that β Z ′ ( P r i =1 a ′ i ⊗ Z ′ ∩B a ′′ i ) = 1 ⊗ h. By the previous corollary , Z ′ is an H -submo dule of Z ( T ). Therefore, β Z ( T ) ( P n i =1 a ′ i ⊗ Z ( T ) ∩S a ′′ i ) = β Z ′ ( P n i =1 a ′ i ⊗ Z ′ ∩B a ′′ i ) = 1 ⊗ h. Hence, the theorem is completely pro ven.  A more concrete example can b e obta ined as follo ws. Let K ⊆ K ⊆ A b e ﬁeld extensions suc h that K ⊆ A is ﬁnite, separable and normal of Galois group G. W e assume tha t G = N H , where H and N are subgroups in G suc h that N T H = { 1 } and N is generated b y a cen tral elemen t σ in G. W e set B := A H . W e wish to prov e that this setting fulﬁls the conditions in the preceding theorem, to get the follow ing. Cor ollar y 3 .1 8 . With the ab ove notation, B [ X , σ , 0] ⊆ A [ X, σ, 0] is a c entr al ly ( K H ) ∗ -Galois extensio ns. Pr o of. In order to apply Theorem 3.17, we hav e to c heck tha t σ is a mor phism of ( K H ) ∗ -como dules and that B σ ⊆ A σ is a ( K H ) ∗ -Galois extens ion. The former condition is equiv alen t to the fact that σ is a morphism of K H - mo dules, whic h in our case means that σ h = hσ for any h in H . T rivially this equalit y is satisﬁed as, b y assumption, σ is cen tral in G . F urthermore, B σ =  A H  σ =  A H  N = A H N = K . A similar computation yields us ( A σ ) H = A N H = B σ . On the o ther hand, since N T H = { 1 } one can em b ed H in to the group of ﬁeld automorphisms o f A σ via the restriction map u 7→ u | A σ . By Ar t in’s Lemma, B σ ⊆ A σ is separable and normal of Galois g roup H . W e hav e noticed in § 2.3 4 that a ﬁnite ﬁeld extension is ( K H ) ∗ - Galois if and only if it is separable and normal of Galois group H . In conclusion, the second requiremen t is a lso satisﬁed.  CO ACTIONS ON HOCHSCHILD HOMOLOGY OF HOPF-GA LO IS EXTENSIONS 33 Ac kno wledgemen ts. 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Coactions on Hochschild Homology of Hopf-Galois Extensions and Their Coinvariants

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