HOPF-GALOIS EXTENSIONS AND AN EXACT SEQUENCE FOR H-PICARD GROUPS

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1 HOPF-GALOIS EXTENSIONS AND AN EXACT SEQUENCE FOR H-PICARD GROUPS STEFAAN CAENEPEEL AND ANDREI MARCUS Abstract. Let H be a Hopf algebra, and A an H-Galos extenson. We nvestgate H-Morta autoequvalences of A, ntroduce the concept of H- Pcard group, and we establsh an exact sequence lnng the H-Pcard group of A and the Pcard group of A coh. 1. Introducton The am of ths paper s the followng generalzaton, presented n Secton 7 below, of the man result of M. Beatte and A. del Río [4] see also [14] for an approach based on [13]). Theorem 1.1. Assume that H s a cocommutatve Hopf algebra over the feld. Let A be a fathfully flat H-Galos extenson. There s an exact sequence 1 H 1 H, ZA coh )) g 1 Pc H A) g 2 PcA coh ) H g 3 H 2 H, ZA coh )). Here H H, ZA coh )) are the Sweedler cohomology groups wth respect to the Myashta-Ulbrch acton of H on ZA coh )), PcA coh ) H s the group of H-nvarant elements of PcA coh ) and Pc H A) s the group of somorphsm classes of nvertble relatve Hopf bmodules. We shall gve later more detals about these notatons. Moreover, g 1 and g 2 are group-homomorphsms, whle g 3 s not. We gve a proof of the theorem by usng the deas of [14] and the results of [6] and [15], obtanng n ths way an nterestng nterpretaton of the above theorem n terms of Clfford extendblty to A of A coh -modules. The paper s dvded as follows. In Secton 2 we present our general settng, whch nvolves Hopf-Galos extensons, the Myashta-Ulbrch acton, and most mportantly, the concepts of H-Morta context and H -Morta context ntroduced n [6], and ther relatonshp wth Hopf subalgebras. The man result of Secton 3 says that f H s cocommutatve and A s 2000 Mathematcs Subject Classfcaton. 16W30, 16D90. Key words and phrases. Hopf-Galos extenson, Morta equvalence, Pcard group, cleft extenson, Sweedler cohomology. Ths research was supported by the blateral project BWS04/04 New Technques n Hopf algebras and graded rng theory of the Flemsh and Romanan governments and by the research project G Deformaton quantzaton methods for algebras and categores wth applcatons to quantum mechancs from FWO-Vlaanderen. The second author acnowledges the support of a Bolya Fellowshp of the Hungaran Academy of Scence and of the Romanan PN-II-IDEI-PCE project, code ID 532, contract no. 29/

2 2 S. CAENEPEEL AND A. MARCUS a fathfully flat H-Galos extenson of B := A coh, then the cotensor product A e := A H A op s a fathfully flat Hopf-Galos extenson of the envelopng algebra B e := B B op. In the frst part of Secton 4 we dscuss the partcular case when A s a cleft extenson of the commutatve algebra B := A coh, and especally, the characterzaton of ths stuaton n terms n Sweedler s 1- and 2-cohomology. Ths s needed n the second part of Secton 4, where we revew and adapt to our needs the results of Mltaru and Ştefan [15] on Clfford extendblty of modules. The cleft extenson n dscusson s the subalgebra E := A ENDA B M) op of ratonal elements n A EndA B M) op, where M s an H-nvarant B-module, and E coh B EndM) op s assumed to be commutatve. In Secton 5 we ntroduce the H-Pcard group Pc H A) and the H -Pcard group Pc H A coh ) of A coh. It s a consequence of the results of [6] that the groups Pc H A) and Pc H A coh ) are somorphc. In the stuaton where H s cocommutatve, we can ntroduce the subgroup PcA coh ) H of PcA coh ) consstng of H-stable elements of PcA coh ) Secton 6). The defntons of the maps g 1, g 2 and g 3, as well as the proof of the man theorem are gven n Secton 7. The man ngredent here s the applcaton of the Mltaru-Stefan lftng theorem to an H-stable nvertble B, B)-bmodule M, by consderng the cleft extenson E := A eenda e B e M) op of E coh = ZB). Note that the acton of H on ZB) comng from E s the same as the Myashta-Ulbrch acton comng from A, hence t s ndependent of M. Secton 8 s concerned wth the analyss of the map g 3. It turns out that the acton PcB) on ZB) nduces an acton of PcB) H on H n H, ZB)), and that g 3 s an 1-cocycle of the group PcB) H wth values n H 2 H, ZB)). The exact sequence descrbng Pc H A) gven n Secton 7 holds n the case where H s cocommutatve; n the general case, we can stll gve a descrpton of Pc H A), n the case where the convarants of A concde wth the groundfeld, that s, A s an H-Galos object. Ths s done n Secton 9, and nvolves Schauenburg s theory of bgalos objects. Modules wll be untal and left, unless otherwse stated. For general results on Hopf algebras the reader s referred to [7], [9] or [16]. For group graded versons of the topcs dscussed here we also menton [3] and [11]. 2. Hopf-Galos extensons Throughout ths paper, H s a Hopf algebra, wth bjectve antpode S, over a feld. We use the Sweedler notaton for the comultplcaton on H: h) = h 1) h 2). M H respectvely H M) s the category of rght respectvely left) H-comodules. For a rght H-coacton ρ respectvely a left H-coacton λ) on a -module M, we denote ρm) = m [0] m [1] and λm) = m [ 1] m [0]. The submodule of convarants M coh of a rght respectvely left) H-comodule M conssts of the elements m M satsfyng ρm) = m 1 respectvely λm) = 1 m). Let A be a rght H-comodule algebra. A M H and M H A are the categores of left and rght relatve Hopf modules, and A M H A s the category of relatve Hopf bmodules, see [6]. B = A coh wll be the subalgebra of convarants of

3 H-PICARD GROUPS 3 A. We have two pars of adjont functors F 1 = A B, G 1 = ) coh ) and F 2 = B A, G 2 = ) coh ) between the categores B M and A M H, and between M B and M H A. Consder the canoncal maps can : A B A A H, cana B b) = ab [0] b [1] ; can : A B A A H, can a B b) = a [0] b a [1]. We have the followng result, due to H.-J. Schneder [18, Theorem I]. Theorem 2.1. For a rght H-comodule algebra A, the followng statements are equvalent. 1) F 2, G 2 ) s a par of nverse equvalences; 2) F 2, G 2 ) s a par of nverse equvalences and A B M s flat; 3) can s an somorphsm and A B M s fathfully flat; 4) F 1, G 1 ) s a par of nverse equvalences; 5) F 1, G 1 ) s a par of nverse equvalences and A M B s flat; 6) can s an somorphsm and A M B s fathfully flat. If these condtons are satsfed, then we say that A s a fathfully flat H- Galos extenson of B. The Myashta-Ulbrch acton. Let A be a fathfully flat rght H-Galos extenson, and consder the map γ A = can 1 η A H) : H A B A, h l h) B r h). Then the element γ A h) s characterzed by the property 1) l h)r h) [0] r h) [1] = 1 h. For all h, h H and a A, we have see [19, 3.4]): 2) 3) γ A h) A B A) B ; γ A h 1) ) h 2) = l h) B r h) [0] r h) [1] ; 4) 5) 6) 7) γ A h 2) ) Sh 1) ) = l h) [0] B r h) l h) [1] ; l h)r h) = εh)1 A ; a [0] l a [1] ) B r a [1] ) = 1 B a; γ A hh ) = l h )l j h) B r j h)r h ).,j Usng the above formulas, t s straghtforward to show that ZB), the center of B, s a rght H-module algebra under the Myashta-Ulbrch acton: x h = l h)xr h),

4 4 S. CAENEPEEL AND A. MARCUS for all x ZB), h H. H-module algebra va In what follows, we wll vew ZB) as a left 8) h x = x S 1 h) = l S 1 h))xr S 1 h)). We wll need the followng commutaton rule n the sequel. Lemma 2.2. For x ZB) and a A, we have 9) xa = a [0] Sa [1] ) x) and ax = a [1] x)a [0]. Proof. From 6), we now that a [0]l a [1] ) B r a [1] ) = 1 B a B B A A B A, and then we can see that x B a = xa [0] l a [1] ) B r a [1] ) = a [0] l a [1] )x B r a [1] ), hence xa = a [0] l a [1] )xr a [1] ) = a [0] Sa [1] ) x). For all h H, we have that h x ZB). Apply the frst formula of 9) wth x replaced by a [1] x; ths gves the second formula: a [1] x)a [0] = a [0] Sa [1] )a [2] ) x) = ax. Morta equvalences. We recall here some concepts and results from [6]. These are the man ngredents n the defnton of Pc H A) and of the maps g 1 and g 2 n Theorem 1.1. Defnton 2.3. Let A and A be rght H-comodule algebras. An H-Morta context connectng A and A s a Morta context A, A, M, N, α, β) such that M A M H A, N A M H A, α : M A N A s a morphsm n AM H A and β : N A M A s a morphsm n A M H A. Defnton 2.4. Assume that A and A are rght fathfully flat H-Galos extensons of A coh = B and A coh = B. A H -Morta context between B and B s a Morta context B, B, M 1, N 1, α 1, β 1 ) such that M 1 resp. N 1 ) s a left A H A op -module resp. A H A op -module) and α 1 : M 1 B N 1 B s left A H A op -lnear, β 1 : N 1 B M 1 B s left A H A op -lnear. MortaB, B ) s the category wth Morta contexts connectng B and B as objects. A morphsm between the Morta contexts B, B, M 1, N 1, α 1, β 1 ) and B, B, M 2, N 2, α 2, β 2 ) s a couple µ, ν), wth µ : M 1 M 2 and ν : N 1 N 2 bmodule maps such that α 1 = α 2 µ B ν) and β 1 = β 2 ν B µ). In a smlar way see [6]), we ntroduce the categores Morta H B, B ) and Morta H A, A ). We recall the followng result, see [6, Theorems 5.7 and 5.9]. Theorem 2.5. Assume that A and A are rght fathfully flat H-Galos extensons of B and B. 1) The categores Morta H A, A ) and Morta H B, B ) are equvalent. The equvalence functors send strct contexts to strct contexts.

5 H-PICARD GROUPS 5 2) Let B, B, M 1, N 1, α 1, β 1 ) be strct Morta context. If M 1 has a left A H A op -module structure, then there s a unque left A H A op - module structure on N 1 such that B, B, M 1, N 1, α 1, β 1 ) s a strct H -Morta context. The correspondng strct H-Morta context A, B, M, N, α, β) s gven by the followng data M = A A op ) AA op M 1 A M H A ; N = A A op ) A A op N 1 A M H A ; α = A A op ) AA op β 1 ; β = A A op ) A A op β 1. Hopf subalgebras. Now let K be a Hopf subalgebra of H. We assume that the antpode of K s bjectve, and that H s fathfully flat as a left K-module. Let K + = Ker ε K ). It s well-nown, and easy to prove see [21, Sec. 1]) that H = H/HK + = H K s a left H-module coalgebra, wth operatons h l = hl, H h) = h 1) h 2), ε H h) = εh). The class n H represented by h H s denoted by h. 1 s a grouple element of H, and we consder convarants wth respect to ths element. A rght H- comodule M s also a rght H-comodule, by corestrcton of coscalars: ρ H m) = m [0] m [1]. The H-convarants of M M H are then M coh = {m M m [0] m [1] = m 1} = {m M ρm) M K} = M H K. If A s a rght H-comodule algebra, then A coh s a rght K-comodule algebra, and A coh ) cok = A coh. In [6, Cor. 7.3], we have seen the followng result, based on [19, Remar 1.8]. Proposton 2.6. Let H, K and A be as above, and assume that A s a fathfully flat H-Galos extenson of B. Then A coh s a fathfully flat K- Galos extenson of B. Let : A coh A and j : K H be the ncluson maps. Then we have a commutatve dagram A coh B A coh can A coh A coh K B A B A can A j A H The map j s njectve here we use the fact that we wor over a feld ). From the fact that can A coh s an somorphsm, t follows that B s also njectve. For K, we then have can A B ))γ A coh ) = j) can A coh )γ A coh ) = 1 j), hence 10) B )γ A coh ) = γ A j)).

6 6 S. CAENEPEEL AND A. MARCUS 3. Cotensor product of Hopf-Galos extensons Troughout ths Secton, we assume that H s cocommutatve. : H H H s a Hopf algebra map, so we can consder H as a Hopf subalgebra of H H. Then H H s a left H-module by restrcton of scalars. Lemma 3.1. H H s fathfully flat as a left H-module. Proof. Let H H be the vector space H H, but wth left H-acton h l) = h l. Then H H and H H are somorphc as left H- modules, and we have the followng natural somorphsms of functors: H H H) = H H H ) = H, an the result follows from the fact that H s fathfully flat as a -vector space. In a smlar way, we have an somorphsm H H) H M = H M, for every left H-module M. In partcular, s a left H-module va the count ε, so we have an somorphsm f : H H) H H, fh ) = hs) of H-module coalgebras, wth left H-acton on H gven by h = εh). Lemma 3.2. Let A and A be fathfully flat H-Galos extensons of B and B. Then the followng statements hold. 1) A A s a fathfully flat H H-Galos extenson of B B. 2) A A ) coh H = A H A. 3) A H A ) coh = B B. Proof. 1) We frst show that A A ) coh H) = B B. We have a map f : B B A A ) coh H), fb b ) = b b. B B = A B ) B A ) and A A ) coh H) are both subspaces of A A, so t suffces to show that f s surjectve. Tae a a A A ) coh H). Then a [0] a [0] a [1] a [1] = a a 1 1. Applyng ε to the fourth tensor factor, we fnd a [0] a a [1] = a a 1. Ths means that a a B A. In a smlar way, we fnd that a a A B. It s easy to show that can A A s bjectve. Fnally A A s fathfully flat as a rght B B -module: B A s fathfully flat as a rght B B -module because for every left B B -module M there s a natural somorphsm B A ) B B M = A B M. Smlarly, A A s fathfully flat as a rght B A -module. Then apply the followng general property: f f : A B and g : B C are algebra morphsms, and B/A and C/B are fathfully flat, then C/A s fathfully flat.

7 H-PICARD GROUPS 7 2) We can apply Proposton 2.6, wth H replaced by H H, K by H and A by A A. Note that a a A A ) coh H f and only f a [0] a [0] a [1]Sa [1] ) = a a 1, or a [0] a a [1] = a a [0] a [1], whch means precsely that a a A HA. 3) We now that A H A s a rght H-comodule algebra wth structure map ρ gven by 11) ρ a a ) = a [0] a a [1] = a a [0] a [1]. Tae x = a a A HA ) coh. It follows from 11) that x B A ) A B ) = B B. Combnng these observatons wth Proposton 2.6, we obtan the followng result, whch s well-nown n the stuaton where B = B =. Theorem 3.3. Let A and A be fathfully flat H-Galos extensons of B and B. Then A H A s a fathfully flat H-Galos extenson of B B. We want to apply ths theorem n the case when A s the opposte algebra A op. Snce H s cocommutatve, A op s a rght H-comodule algebra, wth coacton ρ gven by ρa) = a [0] Sa [1] ). Lemma 3.4. If A s a fathfully flat H-Galos extenson of B, then A op s a fathfully flat H-Galos extenson of B op. Proof. The map can A op : A op B op A op A op H s gven by can A opa a ) = a [0] a Sa [1] ) = Aop S) can A. Then can A op s bjectve snce can A and S are bjectve. We now from Theorem 2.1 that A M B s fathfully flat, and ths mples that A op B opm s fathfully flat. It then follows from Theorem 2.1 that Aop s also a fathfully flat H-Galos extenson. Proposton 3.5. Let A be a fathfully flat H-Galos extenson of B. Then A e := A H A op s a fathfully flat H-Galos extenson of the envelopng algebra B e := B B op. Moreover, the element 12) γ A eh) :=,j l h 1) ) r j h 2) )) B B op r h 1) ) l j h 2) )) belongs to A e B e A e. Proof. Frst observe that can A e : A e B e A e A e H H s gven by can A A opa b) a b )) = aa [0] b [0] b a [1] Sb [1] ).

8 8 S. CAENEPEEL AND A. MARCUS Recall the notaton γ A h) := l h) B r h). Then we compute that ) can A e l h 1) ) r j h 2) )) B e r h 1) ) l j h 2) )),j = l h 1) )r h 1) ) [0] l j h 2) ) [0] r j h 2) ) r h 1) ) [1] Sl j h 2) ) [1] ),j 1,4) = j 1 l j h 3) )r j h 3) ) h 1) SSh 2) )) 5) =1 1 h). Let : A e A e be the canoncal njecton. It follows from 10) that B B op )γ A eh)) = γ A e h)) = l h 1) ) r j h 2) )) B e r h 1) ) l j h 2) )),,j and the statement s proved. 4. Cleft extensons and the lftng Theorem In ths Secton, we adapt and revew the results from [15], gong bac to older results from graded Clfford theory, see [8]. Cleft extensons. Proposton 4.1. Let H be a Hopf algebra, A a rght H-comodule algebra, and B = A coh. We have a category C A, wth two objects 1 and 2, and morphsms C A 1, 1) = HomH, B) ; C A 1, 2) = Hom H H, A); C A 2, 1) = {u : H A ρuh)) = uh 2) ) Sh 1) ), for all h H}; C A 2, 2) = {w : H A ρwh)) = wh 2) ) Sh 1) )h 3), for all h H}. The composton of morphsms s gven by the convoluton product. Recall that A s called H-cleft f there exsts a convoluton nvertble t Hom H H, A), or, equvalently, f 1 and 2 are somorphc n C A. Then t1) 1 = u1), and t = u1)t Hom H H, A) has convoluton nverse ut1), and t 1) = 1. So f A s H-cleft, then there exsts a convoluton nvertble t Hom H H, A) wth t1) = 1. If H s cocommutatve, then C A 1, 1) = C A 2, 2). If t Hom H H, A) s an algebra map, then t s convoluton nvertble wth convoluton nverse t S), so A s H-cleft. Consder the space Ω A = {t Hom H H, A) t s an algebra map}. We have the followng equvalence relaton on Ω A : t 1 t 2 f and only f there exsts b UB) such that bt 1 h) = t 2 h)b, for all h H. We denote Ω A = Ω A /. Tae t Hom H H, A) wth convoluton nverse u such that t1 H ) = 1 A, and consder the map ω t : H B B, ω t h b) = th 1) )buh 2) ).

9 H-PICARD GROUPS 9 Assume that Ω A, and fx t 0 Ω A wth convoluton nverse u 0. Now consder the bjecton F : C A 1, 1) = HomH, B) C A 1, 2) = Hom H H, A), F v) = v t 0, F 1 t) = t u 0. It s then easy to show that F v) Ω A f and only f 13) vh) = vh 1) )ω t0 h 2) v)) and v1 H ) = 1 B. If 13) holds, then v1 H ) = 1 B f and only f v s convoluton nvertble. Moreover, F v) t 0 f and only f vh) = ω t0 h b)b 1 for some nvertble b B. We wll now dscuss when F 1 Ω A ) s a subgroup of HomH, B). Proposton 4.2. Let H be cocommutatve, and let A be an H-cleft rght H-comodule algebra. Assume that B = A coh s commutatve. Choose t Hom H H, A) wth convoluton nverse u, such that t1) = 1 and, a fortor, u1 H ) = 1 A. Then we have the followng propertes. 1) ω t s ndependent of the choce of t; 2) ab = ω t a [1] b)a [0], for all a A and b B. If Ω A, then we have an algebra map t Hom H H, A), and then the map ω t defnes a left H-module algebra structure on B, and we can consder the Sweedler cohomology groups H n H, B), see [20]. We then denote h b = ω t h b). Proposton 4.3. Assume that Ω A. Then Ω A = Z 1 H, B) and Ω A = H 1 H, B). Proof. setch) If H s cocommutatve and B s commutatve, then 13) s equvalent to vh) = h 1) v))vh 2) ), whch s precsely the condton that v s a Sweedler 1-cocycle. Proposton 4.4. Now assume that B = ; t s not necessary that H s cocommutatve. If Ω A, then Ω A = AlgH, ). Proof. In ths stuaton, ω t h b) = εh)b, for every choce of t. Then 13) s equvalent to vh) = vh)v), and the result follows. Suppose that A s H-cleft. Pc a convoluton nvertble t Hom H H, A) such that t1) = 1. Then consder σ : H H B, σh ) = th 1) )t 1) )uh 2) 2) ). Let B# σ H be equal to B H as a vector space, wth rght H-coacton ρ = B, and wth multplcaton b#h)c#) = bh c)σh 1) 1) )h 2) 2). Proposton 4.5. The map φ : B# σ H A, φb#h) = bth) s an somorphsm of rght H-comodule algebras. The nverse of φ s gven by the formula φ 1 a) = a [0] ua [1] )#a [2]. Let σ Z 2 H, B). The followng statements are equvalent: 1) σ B 2 H, B);

10 10 S. CAENEPEEL AND A. MARCUS 2) there exsts an algebra map t Hom H H, A); 3) A = B# ε ε H. The Mltaru-Stefan lftng Theorem. Let A be a fathfully flat H- Galos extenson of B = A coh. A M H wll denote the category of left-rght) relatve Hopf modules. Let P, Q A M H. A left A-lnear map f : P Q s called ratonal f there exsts a unque) element f [0] f [1] A HomP, Q) H such that f [0] p) f [1] = fp [0] ) [0] S 1 p [1] )fp [0] ) [1], or, equvalently, 14) ρfp)) = f [0] p [0] ) p [1] f [1], for all p P. The subset of A HomP, Q) consstng of ratonal maps s denoted by A HOMP, Q). Ths s a rght H-comodule, and A ENDP ) op s a rght H-comodule algebra. Now tae M B M. Then A B M A M H, and E = A ENDA B M) op s a rght H-comodule algebra. From the category equvalence between B M and A M H, t follows that F := E coh = A End H A B M) op = B EndM) op. B can be vewed as a rght H-comodule algebra, wth trval coacton ρb) = b 1, for all b B, so we can consder the category of relatve Hopf modules BM H. If M s a left B-module, then A B M and M H are objects of BM H. D M wll be the full subcategory of B M H, wth two objects A B M and M H. We then have the followng result. Theorem 4.6. Let A be a fathfully flat H-Galos extenson of B = A coh and M B M. Then the categores C E and D M are ant-somorphc. Proof. setch) We defne a contravarant functor α : C E D M at the objects level n the followng obvous way: α1) = M H and α2) = A B M. Before we state the defnton at the morphsms level, we observe that we have two natural somorphsms defned as follows: β 1 : BHomA B M, M) B Hom H A B M, M H); β 2 : BHomM H, M) B End H M H) β 1 φ)a B m) = φa [0] B m) a [1] ; β1 1 ϕ) = M ε) ϕ; β 2 Θ)m h) = Θm h 1) ) h 2) ; β2 1 θ) = M ε) θ. Consder η M : M A B M) coh, the unt of the adjuncton F 2, G 2 ) see Secton 2) evaluated at M. Snce F 2 s an equvalence of categores, η M s an somorphsm. We have an somorphsm α 11 : C E 1, 1) = HomH, E coh ) B HomM H, M), gven by the formulas α 11 v)m h) = η 1 M vh)1 B m)); α 1 11 Θ)h)a B m) = a B Θm h).

11 We then defne α 11 = β 2 α 11. The somorphsm H-PICARD GROUPS 11 α 12 : C E 1, 1) = Hom H H, E) B Hom H M H, A B M) s gven by the formulas α 12 t)m h) = th)1 B m) ; α 1 12 ψ)h))a B m) = aψm h). We have an somorphsm gven by the formulas α 21 : C E 2, 1) B HomA B M, M), α 21 u)a B m) = η 1 M ua [1])a [0] B m)); 1 α 21 φ)h)) a B m) = al h) B φr h) B m). We then defne α 21 = β 1 α 21. Fnally, the somorphsm s gven by the formulas α 22 : C E 2, 1) B End H A B M) op α 22 w)a B m) = wa [1] )a [0] B m); α22 1 κ))h)a B m) = al h)κr h) B m). A long computaton shows that α 22 s a well-defned somorphsm, and that α s a functor. Recall from [19] that M B M s called H-stable f A B M and M H are somorphc as left B-modules and rght H-comodules, or, equvalently, the two objects of D M are somorphc. From Theorem 4.6 we mmedately deduce the followng result. Corollary 4.7. M B M s H-stable f and only f there exsts a convoluton nvertble t Hom H H, E). Assume that M B M s H-stable. Then there s an somorphsm ϕ : A B M M H n B M H. Let ψ = ϕ 1, φ = M ε) ϕ, t = α12 1 ψ), u = α21 1 ϕ). Then the followng assertons are equvalent. 1) t1) = 1; 2) u1) = 1; 3) ψm 1) = 1 B m, for all m M; 4) φ1 B m) = m, for all m M. Indeed, the equvalences 1) 2) and 3) 4) are obvous, and 1) 3) follows mmedately from the defnton of α 12 and α21 1. We have seen cf. comments followng Proposton 4.1) that t and u gven by t h) = th) u1) and u h) = t1) uh) are convoluton nverses, satsfyng the addtonal condton t 1) = u 1) = 1. Thus ψ = α 1 t ) satsfes 3), and φ = α 2 u ) satsfes 4). ψ and φ can be computed explctly, usng the formulas gven n the proof of Theorem 4.6: ψ m h) = ψφ1 B m) h) ; 1 B φ a B m) = ψφa B m) 1). ψ and ϕ are composton nverses. The proof of the followng result s now a straghtforward exercse.

12 12 S. CAENEPEEL AND A. MARCUS Proposton 4.8. Tae φ B HomA B M, M), and let u = α 1 21 φ) C2, 1) and t = u S 1 C1, 2) = Hom H H, E). Then the followng statements are equvalent: 1) φ : A B M M, φa B m) = a m s an assocatve left A-acton on M; 2) u s an ant-algebra map; 3) t s an algebra map. Proposton 4.9. For = 1, 2, tae φ B HomA B M, M), and consder u = α 21 1 φ ) Hom S H, E) and t = u S 1 Hom H H, E). Let M = M as a left B-module, wth left A-acton defned by φ. Then M 1 = M2 f and only f t 1 t 2. Proof. We have that t 1 t 2 f and only f there exsts an nvertble map f B EndM) = E coh such that t 1 h) A B f) = A B f) t 2 h), or, equvalently, u 1 h) A B f) = A B f) u 2 h), for all h H. Ths mples that 1 B φ 1 a B fm)) = u 1 a [1] )a [0] B fm)) = u 1 a [1] ) A B f))a [0] B m) = A B f) u 2 a [1] ))a [0] B m) = 1 B fφ 2 a B fm))), and φ 1 a B fm)) = fφ 2 a B fm))), for all a A and m M, whch means that f : M 2 M 1 s an somorphsm of left A-modules. Conversely, let f : M 2 M 1 s an somorphsm of left A-modules. Then f : M M s left B-lnear, so f B EndM). Then we have, for all h H, a A and m M, that u 1 h)a B fm)) = al h) B φ 1 r h) B fm)) = al h) B fφ 2 r h) B m)) = A B f)u 2 h)a B m)), hence u 1 h) A B f) = A B f) u 2 h), as needed. As an mmedate consequence, we obtan the Mltaru-Ştefan lftng Theorem. Corollary Let A be a fathfully flat H-Galos extenson of B = A coh and M B M. There s a bjectve correspondence between the somorphsm classes of left A-module structures on M extendng the B-module structure on M and the elements of Ω E. Example Let A be an H-Galos object, that s, A coh =, and M =. Then E = A EndA) op = A as an H-comodule algebra, and E coh = A coh =. The map α 21 : C A 1, 2) and ts nverse are gven by the formulas α 21 u)a) = ua [1] )a [0] A coh = ; α 1 21 φ)h) = l h)φr h)).

13 H-PICARD GROUPS 13 φ A defnes an A-acton A f and only f φ s an algebra map. It follows from Corollary 4.10 that Ω A = AlgA, ). If Ω A, then t follows from Proposton 4.4 that AlgA, ) = AlgH, ). The correspondence goes as follows. Fx φ 0 AlgA, ). φ AlgA, ) correspondng to v AlgH, ) s gven by the formula φa) = vsa [1] ))l a [2] )φ 0 r a [2] ))a [0]. 5. Pcard groups The Pcard group of an H-comodule algebra. Consder a Hopf algebra H wth bjectve antpode and an H-comodule algebra A. Let Pc H A) be the category wth strct H-Morta contexts of the form A, A, P, Q, α, β) as objects. A morphsm between A, A, P 1, Q 1, α 1, β 1 ) and A, A, P 2, Q 2, α 2, β 2 ) conssts of a couple f, g), wth f : P 1 P 2, g : Q 1 Q 2 H-colnear A- bmodule somorphsms such that α 1 = α 2 f B g) and β 1 = β 2 g A f). Note that Pc H A) has the structure of monodal category, where the tensor product s gven by the formula A, A, P 1, Q 1, α 1, β 1 ) A, A, P 2, Q 2, α 2, β 2 ) = A, A, P 1 A P 2, Q 2 A Q 1, α 1 P 1 A α 2 A Q 1 ), β 2 Q 2 A β 1 A P 1 ) ). The unt object s A, A, A, A, A, A). Every object A, A, P 1, Q 1, α 1, β 1 ) of Pc H A) has an nverse, namely A, A, Q 1, P 1, β 1, α 1 ). Up to somorphsm, a strct H-Morta context s completely determned by one of ts underlyng bmodules; therefore, we use the shorter notaton P 1 = A, A, P 1, Q 1, α 1, β 1 ). Pc H A) = K 0 Pc H A), the set of somorphsm classes n Pc H A), s a group under the operaton nduced by the tensor product, and s called the H-Pcard group of A. If H =, and B s a -algebra, then Pc B) = PcB) s the classcal Pcard group of B. The -Pcard group of B. Let M, N A em. In [6], t s shown that M B N A em. We wll need an explct formula for the A e -acton on M B N, gven n Proposton 5.1 below. In the proof [6, Theorem 2.4], t s shown that we have an somorphsm α N : A B N A e A e N, α N a B n) = a 1) A e n. We clam that the nverse α 1 N of α N s gven by the formula α 1 N d e) A e n) = dl Se [1] )) B r Se [1] )) e [0] ) n. It follows from Lemma 6.4 that α 1 N s well-defned. Usng the property that γ A 1 H ) = 1 A B 1 A, we fnd that α 1 N α N)a B n) = α 1 N a 1) A e n) = a B n.

14 14 S. CAENEPEEL AND A. MARCUS We also compute that α N α 1 N )d e) Ae n) ) = α N dl Se [1] )) B r Se [1] )) e [0] ) n = dl Se [1] )) 1) A e r Se [1] )) e [0] ) n = ) dl Se [1] ))r Se [1] )) e [0] A e n 5) =d e) A e n. Usng α N, the left A e -acton on A e A e N can be transported to a left A e -acton on A B N: d e)a B n) = α 1 N d e)αn a B n) ) = α 1 N da e) A e n ) = dal Se [1] )) B r Se [1] )) e [0] ) n. If M, N A em, then A e A e M, A e A e N A M H A, hence A e A e M) A A e A e N) = A B M) A A B N) = A B M B N n the category A M H A. On A B M) A A B N), the A-bmodule structure or left A e -module structure) s gven by the formula d e) a B m) A a B n) ) = d 1) a B m) A a e) a B n) = da B m) A a ) l Se [1] )) B r Se [1] )) e [0] n ). We transport ths left A e -module structure to A B M B N: d e) a B m B n) = =,j 4) =,j 1 l Se [1] ))) da B m) B r Se [1] )) e [0] ) n dal j S l Se [1] )) [1] ) ) B r j S ) ) ) l Se [1] )) [1] r Se [1] )) [0] m ) B r Se [1] )) e [0] n dal j Se [1] )) B rj Se [1] )) l Se [2] )) ) m B r Se [2] ) e [0] ) n. Now tae a a Ae. Usng the above formula, we compute that a a ) 1 B m B n) =,j, a l j Sa [1] )) B rj Sa [1] )) l Sa [2] ))) m B r Sa [2] )) a [0] ) n

15 H-PICARD GROUPS 15 =,j, a [0] l j Sa [1] )) B rj Sa [1] )) l Sa [2] )) ) m B r Sa [2] )) a ) n 6) = 1 B a [0] l a [1] )) m B r a [1] ) a ) n The map M B N A B M B N) coh, m B n 1 B m B n s an somorphsm. Hence the left A e -acton on A B M B N restrcts to an acton on A B M B N) coh, and defnes an acton on M B N. We can summarze ths as follows. Proposton 5.1. Let M, N A em. Then we have the followng acton on M B N: 15) a a ) m B n) = a [0] l a [1] )) m B r a [1] ) a ) n. Now let Pc H B) be the category wth strct H -Morta contexts of the form B, B, M, N, γ, δ) as objects. A morphsm between the H -Morta contexts B, B, M 1, N 1, γ 1, δ 1 ) and B, B, M 2, N 2, γ 2, δ 2 ) conssts of a couple f, g) wth f : M 1 M 2 and g : N 1 N 2 left A e -module somorphsms such that γ 1 = γ 2 f B g) and δ 1 = δ 2 g B f). It follows from Proposton 5.1 that Pc H B) s a monodal category, wth tensor product nduced by the tensor product over B, and unt object B, B, B, B, B, B). Every object n Pc H B) has an nverse, and we call K 0 Pc H B) = Pc H B) the H -Pcard group of B. From Theorem 2.5 and the constructon precedng Proposton 5.1, t follows that Pc H A) and Pc H B) are equvalent monodal categores, so we conclude that Pc H A) = Pc H B). 6. The H-stable part of the Pcard group Throughout ths Secton, we assume that H s cocommutatve. Now let A be a rght H-Galos extenson of B. Our next am s to ntroduce the H- nvarant subgroup PcB) H of PcB); roughly spoen, an object of PcB) represents an element of PcB) H f ts connectng modules M and N are H-stable. Frst we need to fx some techncal detals. We consder the category B M H B. Its objects are B-bmodules and rght H- comodules M, such that the rght H-coacton ρ s left and rght B-lnear, that s, ρbmb ) = bm [0] b m [1], for all b, b B and m M. The morphsms are the H-colnear B-bmodule maps. For M, N B M H B, we consder the generalzed cotensor product M H B N = { m B n M B N m [0] B n m [1] = m B n [0] n [1] }.

16 16 S. CAENEPEEL AND A. MARCUS Then M H B N s an object of BM H B, wth rght H-coacton ρ m n ) = m [0] B n m [1] = m B n [0] n [1]. We have a functor H : BM B B M H B. For M BM B, the structure on M H s gven by the formulas ρm h) = m h), bm h)b = bmb h. In partcular, B H B M H B. The functor H s monodal n the sense of our next Lemma. Lemma 6.1. For M, M B M B, we have a natural somorphsm n B M H B. M H) H B M H) = M B M ) H Proof. It s easy to see that the map κ : M B M ) H M H) H B M H), s well-defned and rght H-colnear. nverse gven by the formula κ 1 j m B m h m h 1) ) B m h 2) ) m j h j ) B m j h j)) = j We clam that κ s bjectve, wth m j m j h j εh j). It s clear that κ 1 κ = M B M H. If x := j m j h j ) B m j h j) M H) H B M H), then m j h j1) ) B m j h j) h j2) = m j h j ) B m j h j1) ) h j2). j j Applyng ε to the thrd tensor factor, we fnd κ κ 1 )x) = m j h j1) ) B m j εh j)h j2) ) = x, j hence the clam s verfed. Lemma 6.2. For all P B M H B, we have that n B M H B. P H B B H) = B H) H B P = P Proof. We have a well-defned morphsm α : P P H B B H), αp) = p [0] B 1 p [1] ) n B M H B. The nverse of α s gven by the formula α 1 p B b h )) = p b εh ).

17 H-PICARD GROUPS 17 It s clear that α 1 α = P. If p B b h ) P H B B H), then p [0] B b h ) p [1] = p B b h 1) ) h 2). Then we fnd α α 1 ) p B b h )) = p [0] b εh ) B p [1] = p b εh 1) ) B 1 h 2) ) = p B b h ). Observe that A e = A H A op B em H B e, wth left and rght Be -acton gven by the formula b b ) a a )c c ) = ba c c a b. Hence we have a second functor A e B e : BM B B M H B. Tae M B M B. A e B e M s a left B e -module, and, a fortor, a B- bmodule. The rght H-coacton on A e B e M s gven by the formula ρ a a ) B e m) = a [0] a ) B e m a [1] = a a [0] ) B e m a [1]. Our next am s to show that the functor A e B e s also monodal. Before we can show ths, we need a few techncal Lemmas. Let M B M B. Then A op B M B M H B, wth the rght H-coacton nduced by the coacton on A op. Lemma 6.3. Suppose that M B M B s flat as a left B-module. Then the map f : A e B M A H A op B M), a a ) Bm a a Bm) s an somorphsm. In a smlar way, f M s flat as a rght B-module, then M B A e = M B A) H A op. Proof. Consder the commutatve dagram 0 A e B M f A e B M 0 A H A op B M) A A op B M = A e B M H A e B M H The top row s exact because M s left B-flat, and because of the defnton of the generalzed cotensor product. The exactness of the bottom row also follows from the defnton of the generalzed cotensor product. It follows from the Fve Lemma that f s an somorphsm.

18 18 S. CAENEPEEL AND A. MARCUS Lemma 6.4. For all a A, the element x := l Sa [1] )) B r Sa [1] )) a [0] A B A e. Proof. By Lemma 6.3, t suffces to show that x A B A) H A op. Indeed, l Sa [1] )) B r Sa [1] )) [0] a [0] r Sa [1] )) [1] 3) = l Sa [2] )) B r Sa [2] )) a [0] Sa [1] ). Lemma 6.5. If a a Ae, then the element x =, =, a [0] l a [1] ) B r a [1] ) a a l Sa [1] )) B r Sa [1] )) a [0] Ae H B A e. Proof. It follows from Proposton 3.5 that A e s flat as a left B e -module. Snce B e s flat as a left B-module, we have that A e s flat as a left B- module. We have shown n Lemma 6.4 that x A e B A e. Now a [0] l a [2] ) B r a [2] ) a a [1], 4) =, a [0] l a [1] ) [0] B r a [1] ) a Sl a [1] ) [1] ), so x A H A op B A e ) = A e B A e, by Lemma 6.3. It then follows mmedately that x A e H B Ae. Lemma 6.6. We have an somorphsm of vector spaces f : A e H B Ae, gven by the formula A e B f a a b) =, a [0] bl a [1] ) B r a [1] ) a. Proof. It follows from Lemma 6.5 that f s well-defned. The nverse of f s defned as follows. For y = a a B a a Ae H B Ae, we let f 1 y) = a a a a. Let us show that f 1 s well-defned. Frst we show that f 1 y) A e B. Snce y A e H B Ae, we have that a a a [0] a [0] a [1] a [1] = a a a a [0] Sa [2] )a [1] = a a a a 1.

19 H-PICARD GROUPS 19 For any vector space V, we have that V A) coh = V B B s flat over ), so the above computaton shows that f 1 y) A e B. Let us next show that f 1 y) A e B: snce y A e H B Ae, we have that hence a [0] a B a a a [1] = a [0] a a a a [1] = a a B a a [0] Sa [1] ), a a [0] a a Sa [1] ). Let us fnally verfy that f and f 1 are nverses. f 1 f) a a b) = f 1 a [0] bl a [1] ) B r a [1] ) a ), = a [0] a bl a [1] )r a [1] ) 5) = a a b;, f f 1 ) a a B a a ) = f a a a a ) =, =, 6) = a [0] a a l a [1] ) B r a [1] ) a a a a [0] l a [1] ) B r a [1] ) a a a B a a. Tae M, M B M B and consder the composton g = d can 1 d) ρ A d): We compute that A e B e M B M ) = A B M B B B M B A A B M B B H B M B A A B M B A B A B M B A = A e B e M B A e B e M. g a a B e m B m )) =, a [0] l a [1] )) B e m B r a [1] ) a ) B e m. It follows from Lemma 6.6 that g restrcts to a map g : A e B e M B M ) A e B e M) H B A e B e M ).

20 20 S. CAENEPEEL AND A. MARCUS It s obvous that g B M H B, and that g s bjectve wth nverse g 1 16) a a ) B e m B a a ) B e m ) = a a ) B e ma a B e m ) = a a ) B e m B e a a m ). As a concluson, we obtan the followng Lemma. Lemma 6.7. For M, M B M B, we have an somorphsm g : A e B e M B M ) A e B e M) H B A e B e M ). Remar 6.8. It follows from Lemmas 6.1 and 6.7 that, for M, M, M BM B, we have somorphsms M H) H B M H) ) H B M H) = M H) H B M H) H B M H) ), A e B e M) H B A e B e M ) ) H B A e B e M ) = A e B e M) H B A e B e M ) H B A e B e M ) ) n B M H B that are natural n M, M, M. We now consder the noton of H-stablty, as ntroduced before Corollary 4.7, but wth B replaced by B e and A by A e. The B, B)-bmodule M s H-stable f there exsts an somorphsm n the category B M H B. ϕ M : A e B e M M H Proposton 6.9. If M, M B M B are H-stable, then M B M s also H-stable. Proof. We defne ϕ M B M by the commutatvty of the followng dagram: 17) A e B e M B M ) g A e B e M) H B Ae B e M ) ϕ M B M M B M H ϕ M H B ϕ M κ M H) H B M H) Suppose that M, M B M B are H-stable, and let ψ M = ϕ 1 M, ψ M = ϕ 1 M, t M = α12 1 ψ M), t M = α12 1 ψ M ). For later use, we compute t M B M = α12 1 ψ M B M ) n terms of t M and t M. To ths end, we frst ntroduce the followng Sweedler-type notaton for the map t M : t M h)1 A e B e m) = mh) + mh) ) B e mh) 0.

21 H-PICARD GROUPS 21 Summaton s mplctly understood. Usng the defnton of α 12 and the commutatvty of 17), we compute t M B M h)1 A e B e m B m )) = ψ M B M m B m h) = g 1 ψ M H B ψ M ) ) m B m h) = g 1 ψ M H B ψ M ) ) m h 1) ) B m h 2) ) ) = g 1 th 1) )1 A e B e m) B th 2) )1 A e B e m ) ) = g 1 mh1) ) + mh 1) ) ) B e mh 1) ) 0) B m h 2) ) + m h 2) ) ) B e m h 2) ) 0)) = mh 1) ) + m h 2) ) ) B e mh 1) ) 0 mh 1) ) m h 2) ) + B m h 1) ) 0 ) = mh 1) ) + m h 2) ) ) B e mh 1) ) 0 B mh 1) ) m h 2) ) + m h 1) ) 0 ). We wll need a slght mprovement of ths formula. For b B, we have t M h)1 A e B e mb) = t M h)1 b) B e m) hence 18) = 1 b)t M h)1 A e B e m) = mh) + mh) b) B e mh) 0, t M B M h)1 A e B e mb B m )) = mh 1) ) + m h 2) ) ) B emh 1) ) 0 mh 1) ) bm h 2) ) + B m h 1) ) 0 ). We have that B s a left A e -module, wth acton φ B a a ) B e b) = a ba. The correspondng map ϕ B = β 1 φ B ) : A e B e B B H s gven by ϕ B a a ) B e b) = a [0] ba a [1] = a ba [0] Sa [1]). It follows from Corollary 4.7 and Proposton 4.8 that ϕ B s an somorphsm n B M H B. Now tae M = B, B, M, N, α, β) PcB). We call M H-stable f there exst somorphsms ϕ M : A e B e M M H and ϕ N : A e B e N N H such that the followng dagrams commute: 19) A e B e M B N) Ae B eα A e B e B ϕ M B N M B N H α H ϕ B B H 20) A e B e N B M) Ae B eβ A e B e B ϕ N B M N B M H β H ϕ B B H

22 22 S. CAENEPEEL AND A. MARCUS Theorem Let H be a cocommutatve Hopf algebra, and let A be a fathfully flat Hopf-Galos extenson of A coh = B. Then PcB) H = {[M] PcB) M s H-stable} s a subgroup of PcB), called the H-stable part of PcB). Proof. Assume that M 1 and M 2 are H-stable. It follows from Proposton 6.9 that M 1 B M 2 and N 2 B N 1 are H-stable. A commutatve dagram argument tang Remar 6.8 nto account shows that the dagrams 19-20), wth M replaced by M 1 B M 2 and N by N 2 B N 1, commute. Ths mples that M 1 B M 2 s H-stable. Fnally, f M s H-stable, then t s clear from the defnton that M 1 = B, B, N, M, β, α) s also H-stable. 7. A Hopf algebra verson of the Beatte-del Río exact sequence As n the prevous Secton, let H be a cocommutatve Hopf algebra, and A a fathfully flat H-Galos extenson of B. Tae M PcB) H. Then we have an somorphsm ϕ : A e B e M M H n B M H B. We have that E = A eenda e B e M) op s an H-comodule algebra. Lemma 7.1. E coh = ZB). Proof. We frst observe that E coh = A eend H A e B e M) = B eendm) = B End B M). The second somorphsm s due to the fact that A e B e : B em A em H s a category equvalence, by Theorem 2.1 and Proposton 3.5. Snce M s a strct Morta context, we have that B M s an autoequvalence of M B. B M and ts adjont send B-bmodules to B-bmodules, so B M also defnes an autoequvalence of B M B. Consequently B End B M) = BEnd B B) = ZB). For later use, we gve an explct descrpton of the somorphsm λ : ZB) E coh = A eend H A e B e M), x λ x : 21) λ x a a ) B e m) = a a ) B e xm. We have seen n Theorem 4.6 that there are somorphsms α 12 : Hom H H, E) B Hom H B M H, A e B e M), α 21 : C2, 1) B Hom H B A e B e M, M H). Usng Proposton 3.5, we compute u = α21 1 ϕ) and t = α 1 12 ϕ 1 ): 22) th) a a ) B e m) = a a )ϕ 1 m h); 23) uh) a a ) B e m) = a l h 1) ) r j h 2) )a,j, ) B eφ r h 1) ) l j h 2) )) B e m ).

23 H-PICARD GROUPS 23 Snce E coh = ZB) s commutatve, we can apply Proposton 4.2, and we fnd that ZB) s a left H-module algebra. We wll show n Proposton 7.3 that the left H-acton on ZB) s ndependent of the choce of M PcB) H, and s gven by the Myashta-Ulbrch acton 8). Lemma 7.2. For x ZB), m M and h H, we have that 24) λ x ϕ 1 m h)) = ϕ 1 h 2) x)m h 1) ). Proof. Wrte 25) ϕ 1 m h) = s s ) B e m A e B e M. Snce ϕ 1 s rght H-colnear, we have that 26) ϕ 1 m h 1) ) h 2) = Then we compute ϕ 1 xm h) = xϕ 1 m h) 25) = xs s ) B e m 9) = s [0] s ) B e m s [1]. s [0] Ss [1] ) x) s ) B e m = s [0] s ) B e Ss [1]) x)m 21,26) = λ Sh2) ) xϕ 1 m h 1) ). and t follows that ϕ 1 h 2) x)m h 1) ) = λ Sh2) ) h 3) x)ϕ 1 m h 1) )) = λ x ϕ 1 m h)). Proposton 7.3. Assume that M PcB) s H-stable. The correspondng left H-acton on E coh s gven by the formula h λ x = λ h x, for all x ZB). Ths means that the transported acton on ZB) s the Myashta-Ulbrch acton gven by 9). Proof. Tae x ZB) and the correspondng λ x E coh. The acton of h H on λ x s gven by see Proposton 4.2) and we have Now wrte h λ x = uh 1) ) λ x th 2) ), := h λ x ) a a ) B e m)) 22) = uh 1) ) λ x ) a a )ϕ 1 h 3) x)m h 2) )) 24) = uh 1) ) a a )ϕ 1 h 3) x)m h 2) ) ). 27) ϕ 1 h 2) x)m h 1) ) = q s q s q) B e m q.

24 24 S. CAENEPEEL AND A. MARCUS Snce ϕ 1 s rght H-colnear, we have that ϕ 1 h 3) x)m h 2) ) h 1) = q hence = us q[1] ) a s q[0] s qa ) B e m q) 23) =,j,,q q, s q[0] s q) B e m q s q[1], q s q[0] l s q[1] ) r j s q[2] )s qa ) B e φ r s q[1] ) l j s q[2] )) B e m q. Usng 6,12), we fnd s q[0] s q)l s q[1] ) r j s q[2] )) B e r s q[1] ) l j s q[2] )),q = q Snce φ s left B e -lnear, we fnd = a a ) B e,j,,q =,q 1 A e B e s q s q) B e B e A e. φ s q[0] l s q[1] )r s q[1] ) l j s q[2] )r j s q[2] )s q) B e m q a a ) B e φ s q s ) q) B e m q 27) = a a ) B e M ε) ϕ ϕ 1 )h 2) x)m h 1) ),q = a a ) B e h x)m = λ h x a a ) B e m).,q Ths shows that h λ x = λ h x, for all x ZB). It follows from the dscusson n Secton 5 that the functor Pc H B) PcB) restrctng the A e -module structure on the connectng bmodules to the B-bmodule structure s strongly monodal. Ths mples that we have a group homomorphsm g 2 : Pc H B) PcB). Proposton 7.4. The groups Ker g 2 ) and H 1 H, ZB)) are somorphc. Proof. Tae [M] = [B, B, M, N, α, β)] Ker g 2 ). Then M and N are somorphc to B as B-bmodules. M s descrbed completely once we now the left A e -module structure on M = B, by Theorem 2.5 2). Isomorphsm classes of left A e -module structures on B are n bjectve correspondence to the elements of Ω E, cf. Corollary It follows from Proposton 4.3 that Ω E = H 1 H, ZB)), hence we have a bjecton between H 1 H, ZB)) and Ker g 2 ), and an njecton g 1 : H 1 H, ZB)) Pc H B). We wll now descrbe ths njecton explctly, and show that t preserves multplcaton.

25 H-PICARD GROUPS 25 Let φ 0 be the left A e -acton on B correspondng to the trval element n Pc H B): φ 0 a a ) B e b) = a ba. Let u 0 = α 1 21 be the correspondng element n C E2, 1). Usng the formulas n the proof of Theorem 4.6 we obtan that u 0 h) a a ) B e b) = l h 1) ) r j h 2) )a,j,a ) B e r h 1) )bl j h 2) ). Let α Z 1 H, ZB)), and tae Gα) = t = α t 0 Ω E see Proposton 4.3). Then th) = t 0 h 1) ) αh 2) ), and uh) = tsh)) = u 0 h 1) ) αsh 2) )). We compute φ α = α 21 u), usng the formulas gven n the proof of Theorem 4.6: 1 B e φ α a a ) B e b) = ua [1] ) a [0] a ) B e b) = u 0 a [1] ) a [0] a ) B e αsa [2]))b ) =,j, a[0] l a [1] ) r j a [2] )a ) B e r a [1] )αsa [3] ))bl j a [2] ) =,j, 1 A e B e a [0] l a [1] )r a [1] )αsa [3] ))bl j a [2] )r j a [2] )a = 1 A e B e a [0] αsa [1] ))ba Ths means that g 1 α) s represented by B, wth left A e -acton gven by 28) a a ) α b = φ α a a ) B e b) = a [0] αsa [1] ))ba.

26 26 S. CAENEPEEL AND A. MARCUS Let β Z 1 H, ZB)) be another cocycle. Then g 1 α) B g 1 β) = B B B = B as a B, B)-bmodule, wth left A e -acton a a ) b = a a ) 1 B b) 15) = a [0] l a [1] )) α 1 B r a [1] ) a ) β b, 28) =, 3) =, a [0] αsa [1] ))l a [2] ) B r a [2] ) [0] βsr a [2] ) [1] ))ba a [0] αsa [1] ))l a [2] ) B r a [2] )βsa [3] ))ba =, 5) = a [0] αsa [1] ))l a [2] )r a [2] )βsa [3] ))ba a [0] αsa [1] ))βsa [2] ))ba = = a [0] α β)sa [1] ))ba a a ) α β b. Ths shows that g 1 α) B g 1 β) = g 1 α β), that s, g 1 s a group monomorphsm. Let M PcB) be H-stable. Then there exsts an somorphsm ψ : M H A e B e M n B M H B such that ψm 1) = 1 A e Be m, for all m M see the arguments gven after Corollary 4.7). Then t := α1 1 ψ) HomH H, E) s convoluton nvertble and satsfes the condton t1) = 1. In Proposton 4.5, we constructed a cocycle σ Z 2 H, ZB)). Now let g 3 [M]) = [σ] H 2 H, ZB)). Ths defnes a map g 3 : PcB) H H 2 H, ZB)). It follows from Proposton 4.5 that g 3 [M]) = 1 f and only f there exsts an algebra map t Hom H H, E). By Proposton 4.8, ths s equvalent to the exstence of an assocatve left A e -acton φ : A e B e M M, whch s equvalent to [M] Im g 2 ). We conclude that Im g 2 ) = Ker g 3 ). Our observatons can be summarzed as follows. Theorem 7.5. Let H be a cocommutatve Hopf algebra over a feld, and A a fathfully flat Hopf-Galos extenson of B = A coh. Then we have an exact sequence 1 H 1 H, ZB)) g 1 Pc H B) = Pc H A) g 2 PcB) H g 3 H 2 H, ZB)). Observe that Pc H B) = Pc H A) and PcB) H are non-abelan groups. The category of groups s not an abelan category, so t maes no sense to tal about exact sequences of groups. In the statement n Theorem 7.5, exactness means that g 1 s an njectve map, and that Im g ) = {x g +1 x) = 1},

27 H-PICARD GROUPS 27 for = 1, 2. The maps g 1 and g 2 are group homomorphsms. An example gven n [4] shows that g 3 s not a group homomorphsm n general, even n the case of group graded algebras. We wll dscuss n Secton 8 the property satsfed by g g 3 s a 1-cocycle We recall from [10] that PcB) acts on ZB) as follows. For [M] PcB), we have a map ξ M : ZB) ZB) characterzed by the property 29) ξ M x) = y mx = ym, for all m M. It s easy to show that ξ M xy) = ξ M x)ξ M y). We wll show that ths acton defnes an acton of PcB) H on H n H, ZB)), so that we can consder the group of cocycles Z 1 PcB) H, H 2 H, ZB))). We wll then show that g 3 s such a 1-cocycle. Our frst am s to show that the acton PcB) H on ZB) commutes wth the acton of H on ZB). Frst, we need some Lemmas. Lemma 8.1. Tae [M] PcB) H. For all x ZB), m M and a a Ae, we have that 30) a a x ) B e m = ξ M x)a a ) B e m n A e B e M. Proof. Ths follows mmedately from the fact that mx h = ξ M x)m h n M H, for all m M, x ZB) and h H, and the fact that we a B, B)-bmodule somorphsm ψ M : M H A e B e M. Lemma 8.2. The map l : A e B e B A H B A op, s an somorphsm. a a ) B e b a b B a Proof. Observe frst that A e B eb and A H B Aop are objects of the category A emh. It follows from Theorem 2.1 and Proposton 3.5 that t suffces to show that A H B A op ) coh = B = A e B e B) coh. Tae a B a A H B A op ) coh A H B A op. Then a [0] B a a [1] = a b B a ) 1. From the fact that A B M s fathfully flat, we deduce that a B a A coh B A = B B A, hence a B a = 1 B a a = 1 B a. Snce a B a A H B Aop, we also have that 1 B a [0] Sa [1] ) = 1 B a 1.

28 28 S. CAENEPEEL AND A. MARCUS Apply ρ A to the second tensor factor ρ A s left B-lnear), and then multply the second and thrd tensor factor. Ths gves 1 B a [0] a [1] = 1 B a B 1, and t follows that a B. Ths shows that the map s an somorphsm. f : B A H B A op ) coh, fb) = 1 B b Lemma 8.2 tells us that the map A e A H B Aop nduced by the canoncal surjecton A e A B A op s surjectve. Proposton 8.3. Let M = B, B, M, N, α, β) represent an H-stable element of PcB). Then for all h H and x ZB). ξ M h x) = h ξ M x)), Proof. For a a Ae, x ZB) and m M, we compute that ξ M x)a a ) B e m 30) = a a x) B e m 9) = = 29) = = 9) = = a a [1] x)a [0] ) B e m a a [0] ) B e ma [1] x) a a [0] ) B e ξ Ma [1] x)m a ξ M a [1] x) a [0] ) B e m a [1] ξ M a [1] x)a [0] a [0] ) B e m a [1] ξ M Sa [2] ) x)a [0] a ) B e m. Now tae an arbtrary n N. Applyng Lemma 6.5, we fnd ξm x)a [0] l a [1] ) ) r ) B em) B a [1] ) a ) B en, Now we apply = a[1] ξ M Sa [2] ) x))a [0] l a [3] ) ) ) B em, r ) B a [3] ) a ) B en. g 1 : A e B e M) H B A e B e N) A e B e M B N) to both sdes see 16)). Usng 5), we obtan ξ M x)a a ) B e m B n) = a [1] ξ M Sa [2] ) x)a [0] a ) B e m B n).

29 Now M B N = B. It follows that ξ M x)a a ) B e b = H-PICARD GROUPS 29 a [1] ξ M Sa [2] ) x)a [0] a ) B e b, for all a a Ae, x ZB) and b B. Usng Lemma 8.2, we fnd that ξ M x)a B a = a [1] ξ M Sa [2] ) x)a [0] B a = Sa [1] ) ξ Ma [2] x)a B a [0] for all a B a A H B Aop and x ZB). Now tae h H. It follows from 3-4) that γ A h) = l h) B r h) A H B Aop. Therefore ξ M x)l h) B r h) = 3) = Sr h) [1] ) ξ M r h) [1] x))l h) B r h) [0] Sh 2) ) ξ M h 3) x) l h 1) ) B r h 1) ). We apply A ε) γ A to both sdes; ths gves ξ M x)εh) = Sh 1) ) ξ M h 2) x), and, fnally, h ξ M x) = h 1) ξ M x)εh 2) ) = h 1) Sh 2) )) ξ M h 3) x) = ξ M h x), whch gves the desred formula. Proposton 8.4. The acton of PcB) on ZB) nduces an acton of PcB) H on Z n H, ZB)), B n H, ZB)) and H n H, ZB)). More precsely, f f : H n ZB) s a cocycle resp. a coboundary), then ξ M f s also a cocycle resp. a coboundary). Proof. Ths follows mmedately from Proposton 8.3 and the defnton of Sweedler cohomology, see [20] or [5, Sec. 9.1]. Snce PcB) H acts on H 2 H, ZB)), we can consder the cohomology group H 1 PcB) H, H 2 H, ZB))). Theorem 8.5. g 3 Z 1 PcB) H, H 2 H, ZB))). Proof. Let [M], [M ] PcB) H, and consder the correspondng total ntegrals t M : H E := A eenda e B e M), t M : H E. We recall from Secton 4 that [σ M ] = g 3 [M] s defned by the formula t M ) t M h) = σ M h 1) 1) )t M h 2) 2) ).

30 30 S. CAENEPEEL AND A. MARCUS Ths means that t M ) t M h))1 A e B e m) equals = t M )mh) + mh) ) B e mh) 0 ) = mh) + mh) 0 ) + mh) 0 ) mh) ) B e mh) 0 ) 0 σ M h 1) 1) )t M h 2) 2) )1 A e B e m) = σ M h 1) 1) )mh 2) 2) ) + mh 2) 2) ) ) B e mh 2) 2) ) 0. Then we compute tm B M ) t M B M h)) 1 A e B m B m )) 18) = t M B M ) mh 1) ) + m h 2) ) ) B emh 1) ) 0 mh 1) ) m h 2) ) + B m h 2) ) 0 ) ) 18) = mh 1) ) + mh 1) ) 0 1) ) + m h 2) ) 0 2) ) m h 2) ) ) mh1) ) 0 1) ) 0 mh 1) ) 0 1) ) mh 1) ) B e m h 2) ) + m h 2) ) 0 2) ) + B m h 2) ) 0 2) ) 0), hence tm B ) g M ) t M B M h)) 1 A e B m B m )) = mh 1) ) + mh 1) ) 0 1) ) + mh 1) ) 0 1) ) mh 1) ) ) B e mh 1) ) 0 1) ) 0) B m h 2) ) + m h 2) ) 0 2) ) + m h 2) ) 0 2) ) m h 2) ) ) B e m h 2) ) 0 2) ) 0) = σh 1) 1) )mh 2) 2) ) + mh 2) 2) ) ) B e mh 2) 2) ) 0) B σ h 3) 3) )m h 4) 4) ) + m h 4) 4) ) ) B e m h 4) 4) ) 0) 30) = σh 1) 1) )ξ M σ h 2) 2) ))mh 3) 3) ) + mh 3) 3) ) ) B e mh 3) 3) ) 0) B m h 4) 4) ) + m h 4) 4) ) ) B e m h 4) 4) ) 0) = σh 1) 1) )ξ M σ h 2) 2) ))g t M B M h)1 A e B m B m )) ). Ths shows that t M B M ) t M B M h) = σh 1) 1) )ξ M σ h 2) 2) ))t M B M h). Consequently, σ M B M = σ M ξ M σ M ), whch proves the Theorem.

31 H-PICARD GROUPS Galos objects over noncocommutatve Hopf algebras Let H be a possbly non-cocommutatve) Hopf algebra wth bjectve antpode, and A an H-Galos extenson of B = A coh. We can stll defne the Pcard groups Pc H A), PcB) and Pc H B), and we stll have that Pc H A) = Pc H B), cf. Secton 5. We can therefore as whether the exact sequence from Theorem 7.5 can be generalzed to non-cocommutatve Hopf algebras. The obstructons are the followng. 1) We need the property that A H A op s an H-Galos extenson see Theorem 3.3 and Proposton 3.5) n order to apply Corollary 4.10 wth H replaced by A H A op ); 2) We used the fact that H s cocommutatve when we defned the H-stable part of PcB) see Secton 6); 3) We want to have a group structure on Ω AH A op. These problems can be fxed n the case where the algebra of convarants B concdes wth the groundfeld, that s, when A s a Galos object. Examples of Galos objects are for example classcal Galos feld extensons then H = G), wth G a fnte group); other examples of Galos objects over noncocommutatve algebras have been studed n [1, 2]. In ths case, Ω AH A op = AlgH, ) s a group, by Proposton 4.4, and problem 3) s fxed. To handle problem 1), we nvoe the theory of Hopf- Bgalos objects, as developed by Schauenburg [17]. If A s a rght H-Galos object, then there exsts another Hopf algebra L = LA, H), unque up to somorphsm, such that A s an L, H)-Bgalos object, that s, A s left L- Galos object, a rght H-Galos object, and an L, H)-bcomodule. For the constructon of L, we refer to [17, Sec. 3]. If H s cocommutatve, then L = H. We can then ntroduce the Harrson groupod [17, Sec. 4]. Objects are Hopf algebras wth bjectve antpode, morphsms are Hopf-Bgalos objects, and the composton of morphsms s gven by the cotensor product. The nverse of a morphsm A between L and H that s, an L, H)-Bgalos object) s A op, wth left H-coacton λ gven by the formula λa) = S 1 a [1] ) a [0]. In partcular, A H A op ) s an L, L)-Bgalos object, and, n partcular, a rght H-Galos object. Applyng Proposton 4.4 and Corollary 4.10, we obtan Ω AH A op = ΩAH A op = AlgAH A op, ) = AlgL, ). The somorphsm AlgA H A op, ) = AlgL, ) can also be obtaned as follows. Snce A op s the nverse of A n the Harrson groupod, we have that A H A op = L as bcomodule algebras. Snce PcB) = 1 s a feld), the map Pc H B) PcB) s trval. Its ernel s Ω AH Aop, so we obtan the followng result. Proposton 9.1. Let H be a Hopf algebra wth bjectve antpode, A a rght H-Galos object, and L = LA, H). Then Pc H A) = Pc H ) = AlgL, ). If H s cocommutatve, then L = H, so Pc H A) = AlgH, ). Ths somorphsm can be descrbed explctely. The somorphsm AlgH, ) AlgA H A op, ) s a partcular case of 28). For an algebra morphsm