(Co)homology of Crossed Products by Weak Hopf Algebras
((CO)HOMOLOGY OF CROSSED PRODUCTS BY WEAK HOPF ALGEBRAS
JORGE A. GUCCIONE, JUAN J. GUCCIONE, AND CHRISTIAN VALQUI
Abstract.
We obtain a mixed complex simpler than the canonical one the computes thetype cyclic homologies of a crossed product with invertible cocycle A × fρ H , of a weak modulealgebra A by a weak Hopf algebra H . This complex is provided with a filtration. The spectralsequence of this filtration generalizes the spectral sequence obtained in [13]. When f takes itsvalues in a separable subalgebra of A that satisfies suitable conditions, the above mentionedmixed complex is provided with another filtration, whose spectral sequence generalize theFeigin-Tsygan spectral sequence. Contents
Introduction
Given a differential or algebraic manifold M , each group G acting of M acts in a natural wayon the ring A of regular functions of M , and the algebra G A of invariants of this action consistsof the functions that are constants on each of the orbits of M . This suggest to consider G A asa replacement for M/G in noncommutative geometry. Under suitable conditions the invariantalgebra G A and the smash product A k [ G ], associated with the action of G on A , are Moritaequivalent. Since K -theory, Hochschild homology and cyclic homology are Morita invariant, thereis no loss of information if G A is replaced by A k [ G ]. In the general case the experience hasshown that smash products are better choices than invariants rings for algebras playing the role ofnoncommutative quotients. In fact, except when the invariant algebra and the smash product areMorita equivalent, the first one never is considered in noncommutative geometry. The problemof developing tools to compute the cyclic homology of smash products algebras A k [ G ], where A is an algebra and G is a group, was considered in [16, 19, 29]. For instance, in the first paper theauthors obtained a spectral sequence converging to the cyclic homology of A k [ G ], and in [19] Mathematics Subject Classification. primary 16E40; secondary 16T05.
Key words and phrases.
Crossed products, Hochschild (co)homology, Cyclic homology, Weak Hopf algebras. a r X i v : . [ m a t h . K T ] J u l JORGE A. GUCCIONE, JUAN J. GUCCIONE, AND C. VALQUI this result was derived from the theory of paracyclic modules and cylindrical modules developedby the authors. The main tool for this computation is a version for cylindrical modules of theEilenberg-Zilber theorem. More recently, and also due to its connections with noncommutativegeometry, the cyclic homology of algebras obtained through more general constructions involvingHopf algebras (Hopf crossed products, Hopf Galois extensions, Braided Hopf crossed products,etcetera) has been extensively studied. See for instance [1, 14, 24, 31, 33]. Weak Hopf algebras(also called quantum groupoids) are an important generalization of Hopf algebras in which thecounit is not required to be an algebra homomorphism and the unit is not required to be acoalgebra homomorphism, these properties being replaced by weaker axioms. Examples of weakHopf algebras are groupoid algebras and their duals, face algebras [23], quantum groupoidsconstructed from subfactors [28], generalized Kac algebras of Yamanouchi [32], etcetera. It isnatural to try to extend the results of [16] to noncommutative quotients A H of algebras A by actions of weak Hopf algebras H . More generally, in this paper we use the results obtainedin [22] to study the Hochschild (co)homology and the cyclic type homologies of weak crossedproducts with invertible cocycle. Specifically, for a unitary crossed product E := A × fρ H , of analgebra A by a weak Hopf algebra H , we construct a cochain complex and a mixed complex,simpler than the canonical ones (and also simpler that the ones constructed in [22]), that computethe Hochschild cohomology and the Hochschild, cyclic, negative and periodic homologies of E ,respectively. These complexes are provided with canonical filtrations whose spectral sequencesgeneralize the Hochschild-Serre spectral sequence and the Feigin and Tsygan spectral sequence.The paper is organized as follows:In Section 1 we review the notions of weak Hopf algebra and of crossed products of algebras byweak Hopf algebras, and we recall the concept of mixed complex and the perturbation lemma. Insections 2 and 3 we obtain complexes that compute the Hochschild homology and the Hochschildcohomology of a weak crossed products with invertible cocycle E := A × fρ H with coefficientsin an E -bimodule M . Then, in Section 5 we study the cup and the cap products of E , and inSection 6 we obtain a mixed complex that computes the type cyclic homologies of E . Remark.
In this paper we consider the notion of weak crossed products introduced in [6], butthis is not the unique concept of weak crossed product of algebras by weak Hopf algebras in theliterature. There is a notion of crossed product of an algebra A with a Hopf algebroid introducedby B¨ohm and Brzezi´nski in [7]. It is well known that weak Hopf algebras H provide examples ofHopf algebroids. The crossed products considered by us in this paper are canonically isomorphicto the B¨ohm-Brzezi´nski crossed products A f H with invertible cocycle whose action satisfies h · ( l · A ) = hl · A for all h ∈ H and l ∈ H L . So, our results also apply to these algebras. In this article we work in the category of vector spaces over a field k . Hence we assume implicitlythat all the maps are k -linear maps. The tensor product over k is denoted by ⊗ k . Given anarbitrary algebra K , a K -bimodule V and a n ≥
0, we let V ⊗ nK denote the n -fold tensor product V ⊗ K · · · ⊗ K V , which is considered as a K -bimodule via λ · ( v ⊗ K · · · ⊗ K v n ) · λ (cid:48) := λ · v ⊗ K · · · ⊗ K v n · λ (cid:48) . Given k -vector spaces U , V , W and a map g : V → W we write U ⊗ k g for id U ⊗ k g and g ⊗ k U for g ⊗ k id U . We assume that the reader is familiar with the notions of weak Hopf algebra introducedin [8, 9] and of weak crossed products introduced in [6] and studied in a series of papers (see forinstance [2–4, 17, 21, 30]). We are specifically interested in the case in which A is a weak modulealgebra and the cocycle of the crossed product is convolution invertible (see [21, Sections 4–6]). CO)HOMOLOGY OF CROSSED PRODUCTS BY WEAK HOPF ALGEBRAS 3
Weak bialgebras and weak Hopf algebras are generalizations of bialgebras and Hopf algebras,introduced in [8, 9], in which the axioms about the unit, the counit and the antipode are replacedby weaker properties. Next we give a brief review of the basic properties of these structures.Let k be a field. A weak bialgebra is a k -vector space H , endowed with an algebra structureand a coalgebra structure, such that ∆( hl ) = ∆( h )∆( l ) for all h, l ∈ H , and the equalities∆ (1) = 1 (1) ⊗ k (2) (1 (cid:48) ) ⊗ k (2 (cid:48) ) = 1 (1) ⊗ k (1 (cid:48) ) (2) ⊗ k (2 (cid:48) ) (1.1)and (cid:15) ( hlm ) = (cid:15) ( hl (1) ) (cid:15) ( l (2) m ) = (cid:15) ( hl (2) ) (cid:15) ( l (1) m ) for all h, l, m ∈ H , (1.2)are fulfilled, where we are using the Sweedler notation for the coproduct, with the summationsymbol omitted. A weak bialgebra morphism is a function g : H → L that is an algebra and acoalgebra map. For each weak bialgebra H , the maps Π L , Π R , Π L , Π R ∈ End k ( H ), defined byΠ L ( h ) := (cid:15) (1 (1) h )1 (2) , Π R ( h ) := 1 (1) (cid:15) ( h (2) ) , Π L ( h ) := 1 (1) (cid:15) (1 (2) h ) , Π R ( h ) := (cid:15) ( h (1) )1 (2) , respectively, are idempotent (for a proof see [8, 12]). We set H L := Im(Π L ) and H R := Im(Π R ).In [12] it was also proven that Im(Π R ) = H L and Im(Π L ) = H R . Remark . Arguing as in [8, equality (2.8b)], we obtain that Π L ( h (1) ) ⊗ k h (2) = 1 (1) ⊗ k (2) h ,for all h ∈ H . Proposition 1.2.
For each weak bialgebra H , the subspaces H L and H R are separable unitarysubalgebras of H and hl = lh for all l ∈ H L and h ∈ H R .Proof. This is [8, Propositions 2.4 and 2.11]. (cid:3)
Proposition 1.3.
Let H be a weak bialgebra. We have ∆( H L ) ⊆ H ⊗ k H L and ∆( H R ) ⊆ H R ⊗ k H. Proof.
This is [8, (2.6a) and (2.6b)]. (cid:3)
Proposition 1.4.
Let H be a weak bialgebra. (1) For each h ∈ H and l ∈ H R , l (1) h (1) ⊗ k l (2) h (2) = h (1) ⊗ k lh (2) and h (1) l (1) ⊗ k h (2) l (2) = h (1) ⊗ k h (2) l. (2) For each h ∈ H and l ∈ H L , h (1) l (1) ⊗ k h (2) l (2) = h (1) l ⊗ k h (2) and l (1) h (1) ⊗ k l (2) h (2) = lh (1) ⊗ k h (2) . Proof.
This follows from [8, (2.7a) and (2.7b)]. (cid:3)
Let H be a weak bialgebra. An antipode of H is a map S : H → H (or S H if necessary toavoid confusion), such that(1) h (1) S ( h (2) ) = Π L ( h ),(2) S ( h (1) ) h (2) = Π R ( h ),(3) S ( h (1) ) h (2) S ( h (3) ) = S ( h ),for all h ∈ H . As it was shown in [8], an antipode S , if there exists, is unique. It was also shownin [8] that S is antimultiplicative, anticomultiplicative and leaves the unit and counit invariant.A weak Hopf algebra is a weak bialgebra that has an antipode. A morphism of weak Hopf algebras g : H → L is simply a bialgebra morphism from H to L . In [5, Proposition 1.4] it was proventhat if g : H → L is a weak Hopf algebra morphism, then g ◦ S H = S L ◦ g . JORGE A. GUCCIONE, JUAN J. GUCCIONE, AND C. VALQUI
Proposition 1.5.
Let H be a weak Hopf algebra. We have Π L = S ◦ Π L and Π R = S ◦ Π R .Proof. This is [8, equalities (2.24a) and (2.24b)]. (cid:3)
Proposition 1.6.
Let H be a weak Hopf algebra. For all h, l ∈ H and m ∈ H L , we have Π R ( hl ) = Π R (Π R ( h ) l ) , Π R ( hl ) = Π R (Π R ( h ) l ) and Π R ( hm ) = Π R ( h ) m. Proof.
Left to the reader. (cid:3)
Proposition 1.7.
Let H be a weak bialgebra. For h, m ∈ H and l ∈ H L , we have Π L ( h (1) l )Π L ( h (2) m ) = Π L ( hlm ) . Proof.
In fact, we haveΠ L ( h (1) l )Π L ( h (2) m ) = Π L (cid:0) Π L ( h (1) l (1) )Π L ( h (2) l (1) m ) (cid:1) = Π L (cid:0) Π L ( h (1) l (1) ) h (2) l (1) m ) (cid:1) = Π L ( hlm ) , where the first equality holds by Proposition 1.4(2); the second one, by [8, equality (2.5a)] andthe last one, because Π L ∗ id = id. (cid:3) Consider H R as a right H -module via l · h := Π R ( lh ) (by [8, equality (2.5b)] this is an action).The homology of H with coefficients in a left H -module N is H ∗ ( H, N ) := Tor H ∗ ( H R , N ), and the cohomology of H with coefficients in a right H -module N is H ∗ ( H, N ) := Ext ∗ H ( H R , N ). Notations 1.8.
We will use the following notations:(1) We set H := H/H L . Moreover, given h ∈ H we let h denote its class in H .(2) Given h , . . . , h s ∈ H we set h s := h ⊗ H L · · · ⊗ H L h s . Proposition 1.9.
The chain complex H R H H ⊗ H L H H ⊗ HL ⊗ H L H H ⊗ HL ⊗ H L H · · · , Π R d (cid:48) d (cid:48) d (cid:48) d (cid:48) (1.3) where H ⊗ sHL ⊗ H L H is a right H -module via the canonical action and d (cid:48) s (cid:0) h s ⊗ H L h s +1 (cid:1) := Π R ( h ) h ⊗ H L h s ⊗ H L h s +1 + s − (cid:88) i =1 ( − i h ,i − ⊗ H L h i h i +1 ⊗ H L h i +2 ,s ⊗ H L h s +1 + ( − s h ,s − ⊗ H L h s h s +1 , is contractible as a right H L -module.Proof. An easy argument using the third equality in Proposition 1.6 shows that the maps d (cid:48) s arewell defined. Using now the first two equalities in Proposition 1.6 it is easy to see that (1.3) is acomplex. We claim that the family of morphisms (cid:126) : H R −→ H, (cid:126) s +1 : H ⊗ sHL ⊗ H L H −→ H ⊗ s +1 HL ⊗ H L H ( s ≥ (cid:126) ( h ) := h and (cid:126) s +1 ( h s ⊗ H L h s +1 ) := ( − s +1 h ,s +1 ⊗ H L s ≥
0, is a contractinghomotopy of (1.3) as a complex of right H L -modules. In fact, this follows immediately from theequalitiesΠ R ◦ (cid:126) ( h ) = h, (cid:126) ◦ Π R ( h ) = Π R ( h ) , CO)HOMOLOGY OF CROSSED PRODUCTS BY WEAK HOPF ALGEBRAS 5 d (cid:48) s +1 ◦ (cid:126) s +1 ( h s ⊗ H L h s +1 ) = ( − s +1 Π R ( h ) h ⊗ H L h ,s +1 ⊗ H L h s ⊗ H L h s +1 + s − (cid:88) i =1 ( − s + i +1 h ,i − ⊗ H L h i h i +1 ⊗ H L h i +2 ,s +1 ⊗ H L (cid:126) s ◦ d (cid:48) s ( h s ⊗ H L h s +1 ) = ( − s Π R ( h ) h ⊗ H L h ,s +1 ⊗ H L s (cid:88) i =1 ( − s + i h ,i − ⊗ H L h i h i +1 ⊗ H L h i +2 ,s +1 ⊗ H L . Finally, since H L is separable, the right H L -modules H ⊗ sHL are projective. Hence, H ⊗ sHL ⊗ H L H is a right projective H -module for each s ≥ (cid:3) Proposition 1.10.
Let N be a left H -module. The homology of H with coefficients in N is thehomology of the chain complex N H ⊗ H L N H ⊗ HL ⊗ H L N H ⊗ HL ⊗ H L N H ⊗ HL ⊗ H L N · · · , d d d d d where d (cid:0) h ⊗ H L n (cid:1) := Π R ( h ) · n − h · n and, for s > , d s (cid:0) h s ⊗ H L n (cid:1) := Π R ( h ) h ⊗ H L h s ⊗ H L n + s − (cid:88) i =1 ( − i h ,i − ⊗ H L h i h i +1 ⊗ H L h i +2 ,s ⊗ H L n + ( − s h ,s − ⊗ H L h s · n. Proof.
Use that (cid:0) H ⊗ sHL ⊗ H L H (cid:1) ⊗ H N (cid:39) H ⊗ sHL ⊗ H L N and Proposition 1.9. (cid:3) Proposition 1.11.
Let N be a right H -module. The cohomology of H with coefficients in N isthe cohomology of the cochain complex N Hom H L ( H, N ) Hom H L ( H ⊗ HL , N ) Hom H L ( H ⊗ HL , N ) · · · , d d d d where d ( n )( h ) := n · Π R ( h ) − n · h and, for s > , d s ( β ) (cid:0) h s (cid:1) := β (cid:0) Π R ( h ) h ⊗ H L h s (cid:1) + s − (cid:88) i =1 ( − i β (cid:0) h ,i − ⊗ H L h i h i +1 ⊗ H L h i +2 ,s (cid:1) + ( − s β (cid:0) h ,s − (cid:1) · h s . Proof.
Use that Hom H (cid:0) H ⊗ sHL ⊗ H L H, N (cid:1) (cid:39)
Hom H L (cid:0) H ⊗ sHL , N (cid:1) and Proposition 1.9. (cid:3) Let H be a weak-Hopf algebra, A be an algebra and ρ : H ⊗ k A → A a linear map. For h ∈ H and a ∈ A , we set h · a := ρ ( h ⊗ k a ). We say that ρ is a weak measure of H on A if h · ( aa (cid:48) ) = ( h (1) · a )( h (2) · a (cid:48) ) for all h ∈ H and a, a (cid:48) ∈ A . (1.4)From now on ρ is a weak measure of H on A . Let χ ρ : H ⊗ k A −→ A ⊗ k H be the map definedby χ ρ ( h ⊗ k a ) := h (1) · a ⊗ k h (2) . By (1.4) the triple ( A, H, χ ρ ) is a twisted space (see [21, Defi-nition 1.6]). By [21, Subsection 1.2] we know that A ⊗ k H is a non unitary A -bimodule via a (cid:48) · ( a ⊗ k h ) = a (cid:48) a ⊗ k h and ( a ⊗ k h ) · a (cid:48) = a ( h (1) · a (cid:48) ) ⊗ k h (2) , JORGE A. GUCCIONE, JUAN J. GUCCIONE, AND C. VALQUI and that the map ∇ ρ : A ⊗ k H −→ A ⊗ k H , defined by ∇ ρ ( a ⊗ k h ) := a · χ ρ ( h ⊗ k A ), is a left andright A -linear idempotent. In the sequel we will write a × h := ∇ ρ ( a ⊗ k h ). It is easy to checkthe A -subbimodule A × H := ∇ ρ ( A ⊗ k H ) of A ⊗ k H is unitary. Let γ : H → A × H, ν : k → A ⊗ k H and ν : A → A × H be the maps defined by γ ( h ) := 1 A × h , ν ( λ ) := λ A × ν ( a ) := a ×
1, respectively. Givena morphism f : H ⊗ k H → A , we define F f : H ⊗ k H −→ A ⊗ k H by F f ( h ⊗ k l ) := f ( h (1) ⊗ k l (1) ) ⊗ k h (2) l (2) . Assume that f ( h ⊗ k l ) = f ( h (1) ⊗ k l (1) )( h (2) l (2) · A ) for all h, l ∈ H .By [21, Propositions 2.4] we know that ( A, H, χ ρ , F f ) is a crossed product system (see [21, Defi-nition 1.7]). We say that f satisfy the twisted module condition if f ( h (1) ⊗ k l (1) )( h (2) l (2) · a ) = ( h (1) · ( l (1) · a )) f ( h (2) ⊗ k l (2) ) for all h, l ∈ H and a ∈ A and that f is a cocycle if f ( h (1) ⊗ k l (1) ) f ( h (2) l (2) ⊗ k m ) = ( h (1) · f ( l (1) ⊗ k m (1) )) f ( h (2) ⊗ k l (2) m (2) ) for all h, l, m ∈ H .Let E be A × H endowed with the multiplication map µ E introduced in [21, Notation 1.9]. Adirect calculation using the twisted module condition shows that( a × h )( b × l ) = a ( h (1) · b ) f ( h (2) ⊗ k l (1) ) × h (3) l (2) . Theorem 1.12.
Assume that (1) f ( h ⊗ k l ) = f ( h (1) ⊗ k l (1) )( h (2) l (2) · A ) for all h, l ∈ H , (2) h · A = ( h (1) · (1 (1) · A )) f ( h (2) ⊗ k (2) ) for all h ∈ H , (3) h · A = (1 (1) · A ) f (1 (2) ⊗ k h ) for all h ∈ H , (4) a × (1) · a ⊗ k (2) for all a ∈ A . (5) f is a cocycle that satisfies the twisted module condition,Then, (6) ( A, H, χ ρ , F f , ν ) is a crossed product system with preunit (see [21, Definition 1.11] ), (7) F f is a cocycle that satisfies the twisted module condition (see [21, Definitions 1.8] ), (8) µ E is left and right A -linear, associative and has unit A × , (9) The morphism ν : A → E is left and right A -linear, multiplicative and unitary. (10) ν ( a ) x = a · x and x ν ( a ) = x · a , for all a ∈ A and x ∈ E . (11) χ ρ ( h ⊗ k a ) = γ ( h ) ν ( a ) and F f ( h ⊗ k l ) = γ ( h ) γ ( l ) , for all h, l ∈ H and a ∈ A .Proof. by [21, Propositions 2.6 an 2.10, and Theorems 1.12(7) and 2.11]. (cid:3) In the rest of this subsection we assume that the hypotheses of Theorem 1.12 are fulfilled, andwe say that E is the unitary crossed product of A with H associated with ρ and f . Note that byitem (10) of that theorem we have ν ( a ) γ ( h ) = a · ∇ ρ (1 A ⊗ k h ) = ∇ ρ ( a ⊗ k h ) = a × h . Proposition 1.13.
The following equality holds: f ( hl ⊗ k m ) = f ( h ⊗ k lm ) for all h, m ∈ H and l ∈ H L ∪ H R .Proof. This is [21, Propositions 2.7 and 2.8]. (cid:3)
Proposition 1.14.
For all h, l ∈ H and a ∈ A , γ ( h ) ν ( a ) = ν ( h (1) · a ) γ ( h (2) ) and γ ( h ) γ ( l ) = ν (cid:0) f ( h (1) ⊗ k l (1) ) (cid:1) γ ( h (2) l (2) ) . CO)HOMOLOGY OF CROSSED PRODUCTS BY WEAK HOPF ALGEBRAS 7
Proof.
This follows from [22, Equalities (1.6) and (1.7)] and the definition of χ ρ . (cid:3) Proposition 1.15.
We have ν ( a ) γ ( hS (1 (1) )) ν ( b ) γ (1 (2) l ) = ν ( a ) γ ( h (1) ) ν ( b ) γ ( S (1 (2) ) l ) = ν ( a ) γ ( h ) ν ( b ) γ ( l ) , for all a, b ∈ A and l, h ∈ H .Proof. This is [21, Proposition 2.21]. (cid:3)
Proposition 1.16.
For all h ∈ H and l ∈ H L ∪ H R , we have γ ( h ) γ ( l ) = γ ( hl ) and γ ( l ) γ ( h ) = γ ( lh ) . Proof.
This is [21, Proposition 2.22]. (cid:3) E Let B a right H -comodule. The tensor product B ⊗ k B is a (not necessarily counitary) right H -comodule, via δ B ⊗ k B ( b ⊗ k c ) := b (0) ⊗ k c (0) ⊗ k b (1) c (1) . Definition 1.17.
An unitary algebra B , which is also a right H -comodule, is a right H -comodulealgebra if µ B is H -colinear (or, equivalently, δ B is multiplicative). Proposition 1.18 (Comodule algebra structure on E ) . Assume that the hypotheses of Theo-rem 1.12 are satisfied. Then E is a weak H -comodule algebra via the map δ E : E → E ⊗ k H defined by δ E (cid:16)(cid:88) a i ⊗ k h i (cid:17) := (cid:88) ∇ ρ (cid:0) a i ⊗ k h (1) i (cid:1) ⊗ k h (2) i . Proof.
See [21, Proposition 2.12]. (cid:3)
Proposition 1.19.
For all a ∈ A and h ∈ H it is true that δ E ( ν ( a ) γ ( h )) = ν ( a ) γ ( h (1) ) ⊗ k h (2) . Proof.
By the definitions of γ , ν and δ E , and the fact that ν ( A ) γ ( H ) ⊆ E , δ E ( ν ( a ) γ ( h )) = ∇ ρ (cid:0) a × h (1) (cid:1) ⊗ k h (2) = ∇ ρ (cid:0) ν ( a ) γ ( h (1) ) (cid:1) ⊗ k h (2) = ν ( a ) γ ( h (1) ) ⊗ k h (2) , as desired. (cid:3) Remark . Since ν (1 A ) = 1 E , the previous proposition says in particular that γ is H -collinear. Let H be a weak bialgebra, A and algebra and ρ : H ⊗ A → A a map. In this subsection westudy the weak crossed products of A with H in which A is a left weak H -module algebra. Definition 1.21.
We say that A is a left weak H -module algebra via ρ , if(1) 1 · a = a for all a ∈ A ,(2) h · ( aa (cid:48) ) = ( h (1) · a )( h (2) · a (cid:48) ) for all h ∈ H and a, a (cid:48) ∈ A ,(3) h · ( l · A ) = ( hl ) · A for all h, l ∈ H and a ∈ A ,In this case we say that ρ is a weak left action of H on A . If also h · ( l · a ) = ( hl ) · a for all h, l ∈ H and all a ∈ A, (1.5)then we say that A is a left H -module algebra . Proposition 1.22.
For each weak H -module algebra A the following assertions hold: JORGE A. GUCCIONE, JUAN J. GUCCIONE, AND C. VALQUI (1) Π L ( h ) · a = ( h · A ) a for all h ∈ H and a ∈ A . (2) Π L ( h ) · a = a ( h · A ) for all h ∈ H and a ∈ A . (3) Π L ( h ) · A = h · A for all h ∈ H . (4) Π L ( h ) · A = h · A for all h ∈ H . (5) h · ( l · A ) = ( h (1) · A ) (cid:15) ( h (2) l ) for all h, l ∈ H . (6) h · ( l · A ) = ( h (2) · A ) (cid:15) ( h (1) l ) for all h, l ∈ H .Proof. In [12] it was proven that these items are equivalent and the proof of item (3) it waskindly communicated to us by Jos´e Nicanor Alonso ´Alvarez y Ram´on Gonz´alez Rodr´ıguez (forthe proof see [21, Proposition 4.2]). (cid:3)
Example 1.23.
The algebra H L is a left H -module algebra via h · l := Π L ( hl ). In fact item (1)of Definition 1.21 is trivial, item (2) follows from Proposition 1.7, and equality (1.5) followsfrom [8, equality (2.5a)]. This example is known as the trivial representation. Example 1.24.
Let A be a left H -module algebra. The map f ( h ⊗ k l ) := hl · A , named the trivial cocycle of A , satisfies the hypotheses of Theorem 1.12. Definition 1.25.
A subalgebra K of A is stable under ρ if h · λ ∈ K for all h ∈ H and λ ∈ K . Remark . A subalgebra K of A is stable under ρ if and only if χ ρ ( H ⊗ k K ) ⊆ K ⊗ k H . Example 1.27.
Let K := { h · A : h ∈ H } . By Definition 1.21 and Proposition 1.22, we knowthat K is stable under ρ subalgebra of A and K = { h · A : h ∈ H L } = { h · A : h ∈ H R } .Moreover, the map π L : H L → K , defined by π L ( h ) := h · A , is a surjective morphism ofalgebras, and the map π R : H R → K , defined by the same formula, is a surjective anti-morphismof algebras. By Proposition 1.2 this implies that K is separable. Remark . Each stable under ρ subalgebra of A includes { h · A : h ∈ H } .From here to the end of this subsection A is a left weak H -module algebra and E is the unitarycrossed product of A by H associated with ρ and a map f : H ⊗ H → A . Thus, we assume thatthe hypotheses of Theorem 1.12 are fulfilled. In particular ν , ν an γ are as in that theorem. Proposition 1.29.
The following assertions hold: (1) γ ( l ) = ν ( l · A ) , for all l ∈ H L . (2) ν ( a ) γ ( l ) = γ ( l ) ν ( a ) for all l ∈ H R and a ∈ A .Proof. These are [21, Proposition 4.6 and 4.7]. (cid:3)
Let u : H ⊗ k H → A be the map defined by u ( h ⊗ k l ) := hl · A . We say that the cocyle f isinvertible if there exists a (unique) map f − : H ⊗ k H → A such that(1) u ∗ f − = f − ∗ u = f − ,(2) f − ∗ f = f ∗ f − = u . Remark . Condition (1) in Theorem 1.12 says that f ∗ u = f . By [21, Remark 2.17]) andthe comment above [21, Proposition 4.3]), this implies that u ∗ f = f . Example 1.31.
Assume that A is a left H -module algebra and that f is the trivial cocycle. Bythe previous remark and Definition 1.21(2), the cocycle f is invertible and f − = f . CO)HOMOLOGY OF CROSSED PRODUCTS BY WEAK HOPF ALGEBRAS 9
Definition 1.32.
A map g : H ⊗ k H → A is normal if g (1 ⊗ k h ) = g ( h ⊗ k
1) = h · A for all h . Remark . By [21, Proposition 2.17 and Remark 2.19] we know that f is normal if and onlyif the equalities in items (2) and (3) above of Theorem 1.12 hold.When the cocycle f is invertible, we define γ − : H → E by γ − ( h ) := ν (cid:0) f − (cid:0) S ( h (2) ) ⊗ k h (3) (cid:1)(cid:1) γ (cid:0) S ( h (1) ) (cid:1) . (1.6) Example 1.34.
Assume that A is a left H -module algebra and that f is the trivial cocycle. ByExample 1.31 and the fact that S ∗ id = Π R , we have γ − ( h ) = ν (cid:0) S ( h (2) ) h (3) · A (cid:1) γ (cid:0) S ( h (1) ) (cid:1) = ν (cid:0) Π R ( h (2) ) · A (cid:1) γ (cid:0) S ( h (1) ) (cid:1) . Using this, Proposition 1.22(3) and the fact that Π L ◦ Π R = Π L ◦ Π R ◦ S = Π L ◦ S , we obtain γ − ( h ) = ν (cid:0) Π L ( S ( h (2) )) · A (cid:1) γ (cid:0) S ( h (1) ) (cid:1) = γ (cid:0) Π L ( S ( h (2) )) S ( h (1) ) (cid:1) = γ (cid:0) S ( h ) (cid:1) , where the second equality holds by Proposition 1.16 and Proposition 1.29(1); and the last one,by the fact that Π L ∗ id = id and S is an anticomultiplicative map. Proposition 1.35.
Assume that the cocycle f is invertible. Then ( γ ∗ γ − )( h ) = γ (Π L ( h )) and ( γ − ∗ γ )( h ) = γ (Π R ( h )) for all h ∈ H .Proof. This is [21, Proposition 5.19]. (cid:3)
Proposition 1.36.
Let δ E be as in Proposition 1.18. Then δ E (cid:0) γ − ( h ) (cid:1) = γ − ( h (2) ) ⊗ k S ( h (1) ) for all h ∈ H .Proof. By the definition of γ − and Proposition 1.19, we have δ E (cid:0) γ − ( h ) (cid:1) = δ E (cid:16) ν (cid:0) f − (cid:0) S ( h (2) ) ⊗ k h (3) (cid:1)(cid:1) γ (cid:0) S ( h (1) ) (cid:1)(cid:17) = ν (cid:0) f − (cid:0) S ( h (2) ) ⊗ k h (3) (cid:1)(cid:1) γ (cid:0) S ( h (1) ) (1) (cid:1) ⊗ k S ( h (1) ) (2) = ν (cid:0) f − (cid:0) S ( h (3) ) ⊗ k h (4) (cid:1)(cid:1) γ (cid:0) S ( h (2) ) (cid:1) ⊗ k S ( h (1) )= γ − (cid:0) h (2) (cid:1) ⊗ k S ( h (1) ) , as desired. (cid:3) In this subsection we recall briefly the notion of mixed complex. For more details about thisconcept we refer to [11] and [25].A mixed complex X := ( X, b, B ) is a graded k -module ( X n ) n ≥ , endowed with morphisms b : X n −→ X n − and B : X n −→ X n +1 , such that b ◦ b = 0 , B ◦ B = 0 and B ◦ b + b ◦ B = 0 . A morphism of mixed complexes g : ( X, b, B ) −→ ( Y, d, D ) is a family of maps g : X n → Y n , suchthat d ◦ g = g ◦ b and D ◦ g = g ◦ B . Let u be a degree 2 variable. A mixed complex X := ( X, b, B ) determines a double complexBP( X ) = ... ... ... ... · · · X u − X u X u X u · · · X u − X u X u · · · X u − X u · · · X u − b b b bB B B Bb b bB B Bb bB BbB ,where b ( x u i ) := b ( x ) u i and B ( x u i ) := B ( x ) u i − . By deleting the positively numbered columnswe obtain a subcomplex BN( X ) of BP( X ). Let BN (cid:48) ( X ) be the kernel of the canonical surjectionfrom BN( X ) to ( X, b ). The quotient double complex BP( X ) / BN (cid:48) ( X ) is denoted by BC( X ).The homology groups HC ∗ ( X ), HN ∗ ( X ) and HP ∗ ( X ), of the total complexes of BC( X ), BN( X )and BP( X ) respectively, are called the cyclic , negative and periodic homology groups of X . Thehomology HH ∗ ( X ), of ( X, b ), is called the
Hochschild homology of X . Finally, it is clear that amorphism f : X → Y of mixed complexes induces a morphism from the double complex BP( X )to the double complex BP( Y ).Let C be an algebra. If K is a subalgebra of C we will say that C is a K -algebra. Given a K -bimodule M , we let M ⊗ K denote the quotient M/ [ M, K ], where [
M, K ] is the k -submodule of M generated by all the commutators mλ − λm , with m ∈ M and λ ∈ K . Moreover, for m ∈ M ,we let [ m ] denote the class of m in M ⊗ K .By definition, the normalized mixed complex of the K -algebra C is ( C ⊗ K C ⊗ ∗ K ⊗ K , b ∗ , B ∗ ),where C := C/K , b ∗ is the canonical Hochschild boundary map and the Connes operator B ∗ isgiven by B (cid:0) [ c ⊗ K · · · ⊗ K c r ] (cid:1) := r (cid:88) i =0 ( − ir [1 ⊗ K c i ⊗ K · · · ⊗ K c r ⊗ K c ⊗ K c ⊗ K · · · ⊗ K c i − ] . The cyclic , negative , periodic and Hochschild homology groups HC K ∗ ( C ), HN K ∗ ( C ), HP K ∗ ( C ) andHH K ∗ ( C ) of C are the respective homology groups of ( C ⊗ K C ⊗ ∗ K ⊗ K , b ∗ , B ∗ ). Let H be a weak bialgebra, A an algebra, ρ : H → A a weak measure and f : H ⊗ k H → A alinear map. Let χ ρ , γ , ν , ν and F f be as at the beginning of Subsection 1.3. Assume that thehypotheses of Theorem 1.12 are fulfilled. Let E be the crossed product associated with ρ and f ,and let M be an E -bimodule. Let K be a stable under ρ subalgebra of A . For instance we cantake K as the minimum stable under ρ subalgebra of A (see Example 1.27). By Theorem 1.12,the tuple ( A, χ ρ , F f , ν ) is a crossed product system with preunit, F f is a cocycle that satisfies thetwisted module condition and E is an associative algebra with unit 1 E := 1 A ×
1. Moreover, byRemark 1.28 we know that 1 E ∈ K ⊗ k H . So, the hypothesis of [22, Section 3] are satisfied. Let CO)HOMOLOGY OF CROSSED PRODUCTS BY WEAK HOPF ALGEBRAS 11 M be an E -bimodule. In [22, Section 3] was obtained a chain complex ( (cid:98) X ∗ ( M ) , (cid:98) d ∗ ) simpler thanthe canonical one, that gives the Hochschild homology of E with coefficients in M . From now onwe assume that H is a weak Hopf algebra, A is a left weak H -module algebra and the cocycle f is convolution invertible. In this Section we prove that ( (cid:98) X ∗ ( M ) , (cid:98) d ∗ ) is isomorphic to a simplercomplex ( X ∗ ( M ) , d ∗ ). If K is separable (for instance when K := H · A ), then ( X ∗ ( M ) , d ∗ ) givesthe absolute Hochschild homology of E with coefficients in M . Recall that M is an A -bimodulevia the map ν : A → E . For the sake of simplicity in the sequel we will write ⊗ instead of ⊗ K .Let A := A/K , E := E/ ν ( K ) and (cid:101) E := E/ ν ( A ). We recall from [22, Section 3] that (cid:98) X n ( M ) = (cid:77) r,s ≥ r + s = n (cid:98) X rs ( M ) , where (cid:98) X rs ( M ) := M ⊗ A (cid:101) E ⊗ sA ⊗ A ⊗ r ⊗ ,and that the exist maps (cid:98) d lrs : (cid:98) X rs ( M ) → (cid:98) X r − l,s + l − ( M ) such that (cid:98) d n := n (cid:88) l =1 (cid:98) d l n + n (cid:88) r =1 n − r (cid:88) l =0 (cid:98) d lr,n − r for all n ≥ Notations 2.1.
We will use the following notations:(1) Given h , . . . , h s ∈ H we set h s := h · · · h s and h s := h ⊗ k · · · ⊗ k h s .(2) Given h , . . . , h s ∈ H we set h (1)1 s ⊗ k h (2)1 s := (cid:0) h (1)1 ⊗ H L · · · ⊗ H L h (1) s (cid:1) ⊗ k (cid:0) h (2)1 ⊗ H L · · · ⊗ H L h (2) s (cid:1) . (3) Given h , . . . , h s ∈ H we set γ × (h s ) := γ ( h ) · · · γ ( h s ) and γ − × (h s ) := γ − ( h s ) · · · γ − ( h ) , where γ − is as in (1.6). Lemma 2.2.
Let a, a (cid:48) ∈ A and h ∈ H . The following equalities hold: (1) δ E (cid:0) ν ( a (cid:48) ) γ ( h ) ν ( a ) γ ( l ) (cid:1) = ν ( a (cid:48) ) γ ( h (1) ) ν ( a ) γ ( l ) ⊗ k h (2) , for all l ∈ H L . (2) δ E (cid:0) ν ( a ) γ ( l ) ν ( a (cid:48) ) γ ( h ) (cid:1) = ν ( a ) γ ( l ) ν ( a (cid:48) ) γ ( h (1) ) ⊗ k h (2) , for all l ∈ H L . (3) δ E (cid:0) ν ( A ) (cid:1) ⊆ E ⊗ k H L . (4) ν ( a ) γ − ( h ) = γ − ( h (1) ) ν ( h (2) · a ) .Proof.
1) By Propositions 1.4(2), 1.14, 1.16 and 1.19, δ E (cid:0) ν ( a (cid:48) ) γ ( h ) ν ( a ) γ ( l ) (cid:1) = δ E (cid:0) ν ( a (cid:48) ) ν ( h (1) · a ) γ ( h (2) l ) (cid:1) = ν ( a (cid:48) ) ν ( h (1) · a ) γ ( h (2) l (1) ) ⊗ k h (3) l (2) = ν ( a (cid:48) ) ν ( h (1) · a ) γ ( h (2) l ) ⊗ k h (3) = ν ( a (cid:48) ) γ ( h (1) ) ν ( a ) γ ( l ) ⊗ k h (2) .
2) Mimic the proof of item (3).3) Since, for all a ∈ A , we have δ E ( ν ( a )) = ν ( a ) γ (1 (1) ) ⊗ k (2) ∈ E ⊗ k H L .4) This is [21, Proposition 5.23]. (cid:3) Lemma 2.3.
For all h ∈ H and l ∈ H L , we have γ − ( hl ) = γ (cid:0) S ( l ) (cid:1) γ − ( h ) and γ − ( lh ) = γ − ( h ) γ (cid:0) S ( l ) (cid:1) . Proof.
We prove the first equality and leave the second one, which is similar to the reader. Wehave γ − ( hl ) = ν (cid:0) f − ( S ( h (2) l (2) ) ⊗ k h (3) l (3) (cid:1) γ (cid:0) S ( h (1) l (1) ) (cid:1) = ν (cid:0) f − ( S ( h (2) ) ⊗ k h (3) (cid:1) γ (cid:0) S ( h (1) l ) (cid:1) = ν (cid:0) f − ( S ( h (2) ) ⊗ k h (3) (cid:1) γ (cid:0) S ( l ) (cid:1) γ (cid:0) S ( h (1) (cid:1) = γ (cid:0) S ( l ) (cid:1) ν (cid:0) f − ( S ( h (2) ) ⊗ k h (3) (cid:1) γ (cid:0) S ( h (1) (cid:1) = γ (cid:0) S ( l ) (cid:1) γ − ( h ) , where the first and last equality hold by the definition of γ − ; the second one, by Proposi-tion 1.4(2); the third one, by Proposition 1.16 and the fact that S is antimultiplicative; and thefourth one, by Proposition 1.29(2). (cid:3) Lemma 2.4.
Let h , . . . , h s ∈ H and a ∈ A . The following equality holds: γ − × (h (1)1 s ) ⊗ A (cid:101) γ A (h (2)1 s ) · ν ( a ) = ν ( a ) γ − × (h (1)1 s ) ⊗ A (cid:101) γ A (h (2)1 s ) . Proof.
We proceed by induction on s . By Proposition 1.14 and Lemma 2.2(4), for s = 1 we have γ − ( h (1)1 ) ⊗ A (cid:101) γ ( h (2)1 ) ν ( a ) = γ − ( h (1)1 ) ν ( h (2)1 · a ) ⊗ A (cid:101) γ ( h (3)1 ) = ν ( a ) γ − ( h (1)1 ) ⊗ A (cid:101) γ ( h (2)1 ) . Assume s > s −
1. Let T := γ − × (h (1)1 s ) ⊗ A (cid:101) γ A (h (2)1 s ) · ν ( a ). ByProposition 1.14, T = γ − × (h (1)1 s ) ⊗ A (cid:101) γ A (h (2)1 ,s − ) ⊗ A ν ( h (2) s · a ) (cid:101) γ ( h (3) s ) = γ − × (h (1)1 s ) ⊗ A (cid:101) γ A (h (2)1 ,s − ) · ν ( h (2) s · a ) ⊗ A (cid:101) γ ( h (3) s ) . Consequently, by the inductive hypothesis and Lemma 2.2(4), T = γ − ( h (1) s ) ν ( h (2) s · a ) γ − × (h (1)1 ,s − ) ⊗ A (cid:101) γ A (h (2)1 ,s − ) ⊗ A (cid:101) γ ( h (3) s ) = ν ( a ) γ − × (h (1)1 s ) ⊗ A (cid:101) γ A (h (2)1 s ) , as desired. (cid:3) Let r, s ≥
0. By Proposition 1.16 and Proposition 1.29(2), we know that H ⊗ sHL is a right H L -module and M ⊗ A ⊗ r ⊗ is a left H L -module via h s · l := h s − ⊗ H L h s l and l · [ m ⊗ a r ] := [ m · γ ( S ( l )) ⊗ a r ] , (2.7)respectively, where [ m ⊗ a r ] denotes the class of m ⊗ a r in M ⊗ A ⊗ r ⊗ , etcetera. Write X rs ( M ) := H ⊗ sHL ⊗ H L (cid:0) M ⊗ A ⊗ r ⊗ (cid:1) . Since H ⊗ HL = H L and A ⊗ = K , we have X r ( M ) (cid:39) M ⊗ A ⊗ r ⊗ and X s ( M ) (cid:39) H ⊗ sHL ⊗ H L (cid:0) M ⊗ (cid:1) . (2.8)Let Θ (cid:48) rs : M ⊗ k E ⊗ sk ⊗ k A ⊗ r −→ X rs ( M ) and Λ (cid:48) rs : H ⊗ sk ⊗ k M ⊗ k A ⊗ r −→ (cid:98) X rs ( M )be the maps defined byΘ (cid:48) ( x ) := ( − rs h (2)1 s ⊗ H L (cid:2) m · ν ( a ) γ ( h (1)1 ) · · · ν ( a s ) γ ( h (1) s ) ⊗ a s +1 ,s + r (cid:3) and Λ (cid:48) ( y ) := ( − rs (cid:2) m · γ − × (h (1)1 s ) ⊗ A (cid:101) γ A (h (2)1 s ) ⊗ a r (cid:3) , where x := m ⊗ k ν ( a ) γ ( h ) ⊗ k · · · ⊗ k ν ( a s ) γ ( h s ) ⊗ k a s +1 ,s + r and y := h s ⊗ k m ⊗ k a r . CO)HOMOLOGY OF CROSSED PRODUCTS BY WEAK HOPF ALGEBRAS 13
Proposition 2.5.
For each r, s ≥ the maps Θ (cid:48) rs and Λ (cid:48) rs induce morphisms Θ rs : (cid:98) X rs ( M ) −→ X rs ( M ) and Λ rs : X rs ( M ) −→ (cid:98) X rs ( M ) . Proof.
First we show that the map Θ rs is well defined. Let x be as in the definition of Θ (cid:48) rs . Wemust prove that(1) If some h i equals 1, then Θ (cid:48) ( x ) = 0,(2) Θ (cid:48) is A -balanced in the first s tensors of M ⊗ k E ⊗ sk ⊗ k A ⊗ r ,(3) Θ (cid:48) is K -balanced in the s + 1-tensor of M ⊗ k E ⊗ sk ⊗ k A ⊗ r ,(4) Θ (cid:48) ( ν ( λ ) · x ) = Θ (cid:48) ( x · ν ( λ )) for all λ ∈ K .Condition (1) follows from Lemma 2.2(3). Next we prove that Condition (2) is satisfied at thefirst tensor. Let a ∈ A . By Lemma 2.2(2), we haveΘ (cid:48) ( m · ν ( a ) ⊗ k ν ( a ) γ ( h ) ⊗ k · · · ⊗ k ν ( a s ) γ ( h s ) ⊗ k a s +1 ,s + r )= ( − rs h (2)1 s ⊗ H L (cid:2) m · ν ( a ) ν ( a ) γ ( h (1)1 ) · · · ν ( a s ) γ ( h (1) s ) ⊗ a s +1 ,s + r (cid:3) = Θ (cid:48) ( m ⊗ k ν ( a ) ν ( a ) γ ( h ) ⊗ k · · · ⊗ k ν ( a s ) γ ( h s ) ⊗ k a s +1 ,s + r ) , A similar argument using items (1) and (2) of Lemma 2.2 proves Condition (2) at the i -th tensorwith 2 ≤ i ≤ s . We now prove Condition (3). Let λ ∈ K . By Lemma 2.2(1), we haveΘ (cid:48) ( m ⊗ k ν ( a ) γ ( h ) ⊗ k · · · ⊗ k ν ( a s ) γ ( h s ) ν ( λ ) ⊗ k a s +1 ,s + r )= ( − rs h (2)1 s ⊗ H L (cid:2) m · ν ( a ) γ ( h (1)1 ) · · · ν ( a s ) γ ( h (1) s ) ν ( λ ) ⊗ a s +1 ,s + r (cid:3) = ( − rs h (2)1 s ⊗ H L (cid:2) m · ν ( a ) γ ( h (1)1 ) · · · ν ( a s ) γ ( h (1) s ) ⊗ λa s +1 ⊗ a s +2 ,s + r (cid:3) = Θ (cid:48) ( m ⊗ k ν ( a ) γ ( h ) ⊗ k · · · ⊗ k ν ( a s ) γ ( h s ) ⊗ k λa s +1 ⊗ k a s +2 ,s + r ) . Finally, when r ≥ r = 0 it follows from Lemma 2.2(1).We next show that Λ rs is well defined. Let y be as in the definition of Λ (cid:48) rs . We must provethat(1) If some h i ∈ H L , then Λ (cid:48) ( y ) = 0,(2) Λ (cid:48) is H L -balanced in the first ( s − H ⊗ sk ⊗ k M ⊗ k A ⊗ r ,(3) If r >
0, then Λ (cid:48) is K -balanced in the ( s + 1)-th tensor of H ⊗ sk ⊗ k M ⊗ k A ⊗ r ,(4) Λ (cid:48) (h s ⊗ k ν ( λ ) · m ⊗ k a r ) = Λ (cid:48) rs (h s ⊗ k m ⊗ k a r · λ ) for all λ ∈ K ,(5) If s >
0, then Λ (cid:48) (h s ⊗ k m · γ ( S ( l )) ⊗ k a r ) = Λ (cid:48) (h s · l ⊗ k m ⊗ k a r ) for all l ∈ H L ,Item (1) follows from the fact that, by Propositions 1.3 and 1.29(1), γ − ( l (1) ) ⊗ k γ ( l (2) ) ∈ E ⊗ k ν ( A ) for all l ∈ H L .In order to prove item (2) we must check thatΛ (cid:48) (h i · l ⊗ k h i +1 ,s ⊗ k m ⊗ k a r ) = Λ (cid:48) (h i ⊗ k l · h i +1 ,s ⊗ k m ⊗ k a r ) , for all i ≤ s and l ∈ H L . But this follows from Proposition 1.4(2) and Lemma 2.3. Item (3)holds since, by Lemma 2.4,Λ (cid:48) (h s ⊗ k m · ν ( λ ) ⊗ k a r ) = (cid:2) m · ν ( λ ) γ − × (h (1)1 s ) ⊗ A (cid:101) γ A (h (2)1 s ) ⊗ a r (cid:3) = (cid:2) m · γ − × (h (1)1 s ) ⊗ A (cid:101) γ A (h (2)1 s ) · ν ( λ ) ⊗ a r (cid:3) = (cid:2) m · γ − × (h (1)1 s ) ⊗ A (cid:101) γ A (h (2)1 s ) ⊗ λ · a r (cid:3) = Λ (cid:48) (h s ⊗ k m ⊗ k λ · a r ) . When r ≥
1, Item (4) is trivial, while, when r = 0, it holds sinceΛ (cid:48) (h s ⊗ k ν ( λ ) · m ) = (cid:2) ν ( λ ) · m · γ − × (h (1)1 s ) ⊗ A (cid:101) γ A (h (2)1 s ) (cid:3) = (cid:2) m · γ − × (h (1)1 s ) ⊗ A (cid:101) γ A (h (2)1 s ) · ν ( λ ) (cid:3) = (cid:2) m · ν ( λ ) γ − × (h (1)1 s ) ⊗ A (cid:101) γ A (h (2)1 s ) (cid:3) = Λ (cid:48) rs (h s ⊗ k m · ν ( λ )) , by Lemma 2.4. Finally, for Item (5) we haveΛ (cid:48) rs (h s ⊗ k m · γ ( S ( l )) ⊗ k a r ) = (cid:2) m · γ ( S ( l )) γ − × (h (1)1 s ) ⊗ A (cid:101) γ A (h (2)1 s ) ⊗ a r (cid:3) = (cid:2) m · γ − ( h (1) s l ) γ − × (h (1)1 ,s − ) ⊗ A (cid:101) γ A (h (2)1 s ) ⊗ a r (cid:3) = (cid:2) m · γ − ( h (1) s l (1) ) γ − × (h (1)1 ,s − ) ⊗ A (cid:101) γ A (h (2)1 ,s − ) ⊗ A (cid:101) γ ( h (2) s l (2) ) ⊗ a r (cid:3) = Λ (cid:48) rs (h s · l ⊗ k m ⊗ k a r ) , where the second equality holds by Lemma 2.3; and the third one, by Proposition 1.4(2). (cid:3) Remark . Under the first identifications in [22, (3.28)] and (2.8), the morphisms Θ ∗ and Λ ∗ become the identity maps.Our next purpose is to show that the maps Θ rs and Λ rs are inverse one of each other. Lemma 2.7.
Let s ≥ . for all h , . . . , h s ∈ H , we have γ × (h (1)1 s ) γ − × (h (2)1 s ) ⊗ A (cid:101) γ A (h (3)1 s ) = 1 E ⊗ A (cid:101) γ A (h s ) . Proof.
We proceed by induction on s . For s = 1 we have γ ( h (1)1 ) γ − ( h (2)1 ) ⊗ A (cid:101) γ ( h (3)1 ) = γ (Π L ( h (1)1 )) ⊗ A (cid:101) γ ( h (2)1 ) = γ ( S (1 (1) )) ⊗ A (cid:101) γ (1 (2) h ) = 1 E ⊗ A (cid:101) γ ( h ) , where the first equality holds by Proposition 1.35; the second one, by Remark 1.1 and Propo-sition 1.5; and the last one, by Propositions 1.15 and 1.29(1). Assume now that s > s −
1. Set T := γ × (h (1)1 s ) γ − × (h (2)1 s ) ⊗ A (cid:101) γ A (h (3)1 s ). By Remark 1.1 andPropositions 1.5, 1.29(1) and 1.35, γ ( h (1) s ) γ − ( h (2) s ) ⊗ k h (3) s = γ (Π L ( h (1) s )) ⊗ k h (2) s = γ ( S (1 (1) )) ⊗ k (2) h s = ν ( S (1 (1)) · A ) ⊗ k (2) h s . Hence T = γ × (h (1)1 ,s − ) ν ( S (1 (1) ) · A ) γ − × (h (2)1 ,s − ) ⊗ A (cid:101) γ A (h (3)1 ,s − ) ⊗ A (cid:101) γ (1 (2) h s ) . Using now the equality in Definition 1.21(3) and Proposition 1.14 again and again, we obtain T = ν ( h (1)1 ,s − S (1 (1) ) · A ) γ × (h (2)1 ,s − ) γ − × (h (3)1 ,s − ) ⊗ A (cid:101) γ A (h (4)1 ,s − ) ⊗ A (cid:101) γ (1 (2) h s ) . Consequently, by the inductive hypothesis, T = ν ( h (1)1 ,s − S (1 (1) ) · A ) ⊗ A (cid:101) γ A (h (2)1 ,s − ) ⊗ A (cid:101) γ (1 (2) h s )= 1 E ⊗ A (cid:101) γ A (h ,s − ) ⊗ A ν ( S (1 (1) ) · A ) (cid:101) γ (1 (2) h s ) , where the last equality follows using the equality in Definition 1.21(3) and Proposition 1.14 againand again. Hence, by Propositions 1.15 and 1.29(1) we have T = 1 E ⊗ A (cid:101) γ A (h s ), as desired. (cid:3) Lemma 2.8.
Let s ≥ . For all z ∈ H R and h , . . . , h s ∈ H , we have γ − × (h (1)1 s ) γ ( z ) γ × (h (2)1 s ) ⊗ k h (3)1 s = γ (1 (1) ) ⊗ k Π R ( z ) · h s · (2) . CO)HOMOLOGY OF CROSSED PRODUCTS BY WEAK HOPF ALGEBRAS 15
Proof.
Set T := γ − × (h (1)1 s ) γ ( z ) γ × (h (2)1 s ) ⊗ k h (3)1 s . We proceed by induction on s . Consider the case s = 1. By Propositions 1.16, 1.4(1) and 1.35, we have T = γ − ( h (1)1 ) γ ( zh (2)1 ) ⊗ k h (3)1 = γ − ( z (1) h (1)1 ) γ ( z (2) h (2)1 ) ⊗ k h (3)1 = γ (cid:0) Π R ( zh (1)1 ) (cid:1) ⊗ k h (2)1 . Consequently, T = γ (1 (1) ) ⊗ k (cid:15) ( zh (1)1 (2) ) h (2)1 = γ (1 (1) ) ⊗ k (cid:15) ( z (1 (cid:48) ) ) (cid:15) (1 (2 (cid:48) ) h (1)1 (2) )1 (3 (cid:48) ) h (2)1 (3) = γ (1 (1) ) ⊗ k (cid:15) ( z (1 (cid:48) ) )1 (2 (cid:48) ) h (2) = γ (1 (1) ) ⊗ k Π R ( z ) h (2) , where the first equality holds by the definition of Π R , the second one, by equality (1.2) andPropositions 1.3 and 1.4(2); and the last one, by the definition of Π R . Assume now that s > s −
1. By the inductive hypothesis, we have T = γ − ( h (1) s ) γ (1 (1) ) γ ( h (2) s ) ⊗ k (cid:0) Π R ( z ) · h ,s − · (2) ⊗ H L h (3) s (cid:1) . Thus, by the case s = 1 and Propositions 1.3 and 1.16, we have T = γ − ( h (1) s ) γ (1 (1) h (2) s ) ⊗ k (cid:0) Π R ( z ) · h ,s − ⊗ H L (2) h (3) s (cid:1) = γ − ( h (1) s ) γ ( h (2) s ) ⊗ k (cid:0) Π R ( z ) · h ,s − ⊗ H L h (3) s (cid:1) = γ (1 (1) ) ⊗ k Π R ( z ) · h s · (2) , as desired. (cid:3) Proposition 2.9.
The morphisms Θ rs and Λ rs are inverse one of each other.Proof. We leave the case s = 0 to the reader (use Remark 2.6). Assume s ≥ m ∈ M , h , . . . , h s ∈ H and a , . . . , a r ∈ A . Set x := [ m ⊗ A (cid:101) γ A (h s ) ⊗ a r ] ∈ (cid:98) X rs ( M ) and y := h s ⊗ H L [ m ⊗ a r ] ∈ X rs ( M ) . By Lemma 2.7 Λ (cid:0) Θ( x ) (cid:1) = ( − rs Λ (cid:0) h (2)1 s ⊗ H L [ m · γ × (h (1)1 s ) ⊗ a r ] (cid:1) = (cid:2) m · γ × (h (1)1 s ) γ − × (h (2)1 s ) ⊗ A (cid:101) γ A (h (3)1 s ) ⊗ a r (cid:3) = [ m ⊗ A (cid:101) γ A (h s ) ⊗ a r ] , as desired. For the other composition, by Lemma 2.8 and Proposition 1.15, we haveΘ (cid:0) Λ( y ) (cid:1) = ( − rs Θ (cid:0)(cid:2) m · γ − × (h (1)1 s ) ⊗ A (cid:101) γ A (h (2)1 s ) ⊗ a r (cid:3)(cid:1) = h (3)1 s ⊗ H L (cid:2) m · γ − × (h (1)1 s ) γ × (h (2)1 s ) ⊗ a r (cid:3) = h s · (2) ⊗ H L [ m · γ (1 (1) ) ⊗ a r ]= h s ⊗ H L [ m · γ (1 (1) ) γ ( S (1 (2) )) ⊗ a r ]= h s ⊗ H L [ m ⊗ a r ] , which finishes the proof. (cid:3) For each 0 ≤ l ≤ s and r ≥ r + l ≥
1, let d lrs : X rs ( M ) −→ X r + l − ,s − l ( M ) be themap d lrs := Θ r + l − ,s − l ◦ (cid:98) d lrs ◦ Λ rs . Theorem 2.10.
The Hochschild homology of the K -algebra E with coefficients in M is thehomology of the chain complex ( X ∗ ( M ) , d ∗ ) , where X n ( M ) := (cid:77) r + s = n X rs ( M ) and d n := n (cid:88) l =1 d l n + n (cid:88) r =1 n − r (cid:88) l =0 d lr,n − r . Proof.
By Proposition 2.9 and the definition of ( X ∗ ( M ) , d ∗ ), the mapsΘ ∗ : ( (cid:98) X ∗ ( M ) , (cid:98) d ∗ ) −→ ( X ∗ ( M ) , d ∗ ) and Λ ∗ : ( X ∗ ( M ) , d ∗ ) −→ ( (cid:98) X ∗ ( M ) , (cid:98) d ∗ ) , given by Θ n := (cid:76) r + s = n Θ rs and Λ n := (cid:76) r + s = n Λ rs , are inverse one of each other. (cid:3) Remark . By [22, Remark 3.2] if f takes its values in K , then ( X ∗ ( M ) , d ∗ ) is the totalcomplex of the double complex ( X ∗∗ ( M ) , d ∗∗ , d ∗∗ ). Remark . If K = A , then ( X ∗ ( M ) , d ∗ ) = ( X ∗ ( M ) , d ∗ ). Remark . For each i, n ≥
0, let F i ( X n ( M )) := (cid:76) s ≤ i X n − s,s ( M ). Clearly the chain complex( X ∗ ( M ) , d ∗ ) is filtrated by F ( X ∗ ( M )) ⊆ F ( X ∗ ( M )) ⊆ F ( X ∗ ( M )) ⊆ F ( X ∗ ( M )) ⊆ . . . . (2.9)Moreover the isomorphism Θ ∗ : ( (cid:98) X ∗ ( M ) , (cid:98) d ∗ ) → ( X ∗ ( M ) , d ∗ ) preserve filtrations. So, the spectralsequence of [22, (3.30)] coincide with the spectral sequence determined by the filtration (2.9). Lemma 2.14.
Let m ∈ M , a, a . . . , a r ∈ A , h , . . . , h s ∈ H and z ∈ H R . (1) For x := (cid:2) m · γ − × (h (1)1 s ) ⊗ A (cid:101) γ A (h (2)1 ,s − ) ⊗ A γ ( h (2) s ) ν ( a ) (cid:103) ⊗ a r (cid:3) , we have Θ( x ) = ( − rs h s ⊗ H L [ m · ν ( a ) ⊗ a r ] . (2) For x := (cid:2) m · γ − × (h (1)1 s ) ⊗ A (cid:101) γ A (h (2)1 ,i − ) ⊗ A γ ( h (2) i ) γ ( h (2) i +1 ) (cid:93) ⊗ A (cid:101) γ A (h (2) i +2 ,s ) ⊗ a r (cid:3) , we have Θ( x ) = ( − rs (cid:0) h ,i − ⊗ H L h i h i +1 ⊗ H L h i +2 ,s (cid:1) ⊗ H L [ m ⊗ a r ] (of course, we are assuming that s ≥ and ≤ i < s ). (3) For x := (cid:2) m · γ − × (h (1)1 s ) γ ( z ) ⊗ A (cid:101) γ A (h (2)1 s ) ⊗ a r (cid:3) , we have Θ( x ) = ( − rs Π R ( z ) · h s ⊗ H L [ m ⊗ a r ] . Proof.
1) Under the first identifications in [22, (3.28)] and (2.8), for s = 0 the equality in item (1)becomes Θ([ m · ν ( a ) ⊗ a r ]) = [ m · ν ( a ) ⊗ a r ], which follows immediately from Remark 2.6.Assume now that s ≥
1. By Lemma 2.8 and Propositions 1.14 and1.15, we haveΘ( x ) = Θ (cid:16)(cid:104) m · γ − × (h (1)1 s ) ⊗ A (cid:101) γ A (h (2)1 ,s − ) ⊗ A ν ( h (2) s · a ) γ ( h (3) s ) (cid:94) ⊗ a r (cid:105)(cid:17) = ( − rs h (3)1 s ⊗ H L (cid:2) m · γ − × (h (1)1 s ) γ × (h (2)1 s ) ν ( a ) ⊗ a r (cid:3) = ( − rs h s · (2) ⊗ H L (cid:2) m · γ (1 (1) ) ν ( a ) ⊗ a r (cid:3) = ( − rs h s ⊗ H L (cid:2) m · γ (1 (1) ) ν ( a ) γ ( S (1 (2) )) ⊗ a r (cid:3) = ( − rs h s ⊗ H L (cid:2) m · ν ( a ) ⊗ a r (cid:3) , as desired.2) By Lemma 2.8 and Propositions 1.14 and 1.15, we haveΘ( x ) = Θ (cid:16)(cid:104) m · γ − × (h (1)1 s ) ⊗ A (cid:101) γ A (h (2)1 ,i − ) ⊗ A ν (cid:0) f (cid:0) h (2) i ⊗ k h (2) i +1 (cid:1)(cid:1) γ (cid:0) h (3) i h (3) i +1 (cid:1) (cid:94) ⊗ A (cid:101) γ A (h (2) i +2 ,s ) ⊗ a r (cid:105)(cid:17) CO)HOMOLOGY OF CROSSED PRODUCTS BY WEAK HOPF ALGEBRAS 17 = ( − rs (cid:0) h (3)1 ,i − ⊗ H L h (3) i h (3) i +1 ⊗ H L h (3) i +2 ,s (cid:1) ⊗ H L (cid:2) m · γ − × (h (1)1 s ) γ × (h (2)1 s ) ⊗ a r (cid:3) = ( − rs (cid:0) h ,i − ⊗ H L h i h i +1 ⊗ H L h i +2 ,s · (2) (cid:1) ⊗ H L [ m · γ (1 (1) ) ⊗ a r ]= ( − rs (cid:0) h ,i − ⊗ H L h i h i +1 ⊗ H L h i +2 ,s (cid:1) ⊗ H L (cid:2) m · γ (1 (1) ) γ ( S (1 (2) )) ⊗ a r (cid:3) = ( − rs (cid:0) h ,i − ⊗ H L h i h i +1 ⊗ H L h i +2 ,s (cid:1) ⊗ H L [ m ⊗ a r ] , as desired.3) Under the first identifications in [22, (3.28)] and (2.8), for s = 0 the equality in item (3)becomes Θ([ m · γ ( z ) ⊗ a r ]) = [ m · γ ( S (Π R ( z ))) ⊗ a r ], which follows immediately from Proposi-tion 1.5 and Remark 2.6. Assume now that s ≥
1. We haveΘ( x ) = ( − rs Π R ( z ) · h s · (2) ⊗ H L (cid:2) m · γ (1 (1) ) ⊗ a r (cid:3) = ( − rs Π R ( z ) · h s ⊗ H L (cid:2) m · γ (1 (1) ) γ ( S (1 (2) )) ⊗ a r (cid:3) = ( − rs Π R ( z ) · h s ⊗ H L (cid:2) m ⊗ a r (cid:3) , where the first equality holds by the definition of Θ and Lemma 2.8; the second one, by Propo-sition 1.3 and the definition of the actions in (2.7); and the last one, by Proposition 1.15 (cid:3) Notation 2.15.
Given a k -subalgebra R of A and 0 ≤ u ≤ r , we let X urs ( R, M ) denote the k -sub-module of X rs ( M ) generated by all the elements h s ⊗ H L [ m ⊗ a r ] with m ∈ M , a , . . . , a r ∈ A , h , . . . , h s ∈ H , and at least u of the a j ’s in R . Theorem 2.16.
Let y := h s ⊗ H L [ m ⊗ a r ] ∈ X rs ( M ) , where m ∈ M , a , . . . , a r ∈ A and h , . . . , h s ∈ H . The following assertions hold: (1) For r ≥ and s ≥ , we have d ( y ) = h s ⊗ H L [ m · ν ( a ) ⊗ a r ]+ r − (cid:88) i =1 ( − i h s ⊗ H L [ m ⊗ a ,i − ⊗ a i a i +1 ⊗ a i +2 ,r ]+ ( − r h s ⊗ H L [ ν ( a r ) · m ⊗ a ,r − ] . (2) For r ≥ and s = 1 , we have d ( y ) = ( − r (cid:2) m · γ (cid:0) Π R ( h ) (cid:1) ⊗ a r (cid:3) − ( − r (cid:2) γ ( h (3)1 ) · m · γ − ( h (1)1 ) ⊗ h (2)1 · a r (cid:3) , while for r ≥ and s > , we have d ( y ) = ( − r Π R ( h ) · h s ⊗ H L [ m ⊗ a r ]+ s − (cid:88) i =1 ( − r + i (cid:0) h ,i − ⊗ H L h i h i +1 ⊗ H L h i +2 ,s (cid:1) ⊗ H L [ m ⊗ a r ]+ ( − r + s h ,s − ⊗ H L [ γ ( h (3) s ) · m · γ − ( h (1) s ) ⊗ h (2) s · a r ] . (3) For r ≥ and s ≥ , we have d ( y ) = − h (2)1 ,s − ⊗ H L (cid:2) γ ( h (3) s − h (3) s ) · m · γ − ( h (1) s ) γ − ( h (1) s − ) ⊗ T ( h (2) s − , h (2) s , a r ) (cid:3) , where T ( h s − , h s , a r ) := r (cid:88) i =0 ( − i h (1) s − · ( h (1) s · a i ) ⊗ f ( h (2) s − ⊗ k h (2) s ) ⊗ h (3) s − h (3) s · a i +1 ,r . (4) Let R be a k -subalgebra of A . If R is stable under ρ and f takes its values in R , then d l (cid:0) X rs ( M ) (cid:1) ⊆ X l − r + l − ,s − l ( R, M ) for each r ≥ and < l ≤ s .Proof.
1) By the definition of Λ and [22, Theorem 3.5(1)], we have d ( y ) = ( − rs Θ ◦ (cid:98) d (cid:0)(cid:2) m · γ − × (h (1)1 s ) ⊗ A (cid:101) γ A (h (2)1 s ) ⊗ a r (cid:3)(cid:1) = ( − rs + s Θ (cid:0)(cid:2) m · γ − × (h (1)1 s ) ⊗ A (cid:101) γ A (h (2)1 s ) · ν ( a ) ⊗ a r (cid:3)(cid:1) + r − (cid:88) i =1 ( − rs + s + i Θ (cid:0)(cid:2) m · γ − × (h (1)1 s ) ⊗ A (cid:101) γ A (h (2)1 s ) ⊗ a ,i − ⊗ a i a i +1 ⊗ a i +2 ,r (cid:3)(cid:1) + ( − rs + s + r Θ (cid:0)(cid:2) ν ( a r ) · m · γ − × (h (1)1 s ) ⊗ A (cid:101) γ A (h (2)1 s ) ⊗ a ,r − (cid:3)(cid:1) . The formula for d follows from this using Lemma 2.14(1).2) By the definition of Λ, [22, Theorem 3.5(2)] and Proposition 1.35, we have d ( y ) = ( − rs Θ ◦ (cid:98) d (cid:0)(cid:2) m · γ − × (h (1)1 s ) ⊗ A (cid:101) γ A (h (2)1 s ) ⊗ a r (cid:3)(cid:1) = ( − rs Θ (cid:0)(cid:2) m · γ − × (h (1)2 s ) γ (Π R ( h )) ⊗ A (cid:101) γ A (h (2)2 s ) ⊗ a r (cid:3)(cid:1) + r − (cid:88) i =1 ( − rs + i Θ (cid:0)(cid:2) m · γ − × (h (1)1 s ) ⊗ A (cid:101) γ A (h (2)1 ,i − ) ⊗ A γ ( h (2) i ) γ ( h (2) i +1 ) (cid:103) ⊗ A (cid:101) γ A (h (2) i +2 ,s ) ⊗ a r (cid:3)(cid:1) + ( − rs + r Θ (cid:0)(cid:2) γ ( h (3) s ) · m · γ − × (h (1)1 s ) ⊗ A (cid:101) γ A (h (2)1 ,s − ) ⊗ h (2) s · a r (cid:3)(cid:1) . The formula for d follows from this using Lemma 2.14 and the fact that Π R ◦ Π R = Π R .3) By the definition of Λ and [22, Theorem 3.6], we have d ( y ) = ( − rs Θ ◦ (cid:98) d (cid:0)(cid:2) m · γ − × (h (1)1 s ) ⊗ A (cid:101) γ A (h (2)1 s ) ⊗ a r (cid:3)(cid:1) = ( − rs + s +1 Θ (cid:0)(cid:2) γ ( h (3) s − h (3) s ) · m · γ − × (h (1)1 s ) ⊗ A (cid:101) γ A (h (2)1 ,s − ) ⊗ T ( h (2) s − , h (2) s , a r ) (cid:3)(cid:1) . The formula for d follows from this using Lemma 2.14(1).4) Let (cid:98) X l − r + l − ,s − l ( R, M ) be as in [22, Remark 3.4]. By Remark 1.26 and [22, Theorem 3.5(3)]this item follows from the fact that X l − r + l − ,s − l ( R, M ) = Θ (cid:0) (cid:98) X l − r + l − ,s − l ( R, M ) (cid:1) . (cid:3) Proposition 2.17.
For each h ∈ H , the map F h ∗ : (cid:0) M ⊗ A ⊗ ∗ ⊗ , b ∗ (cid:1) −→ (cid:0) M ⊗ A ⊗ ∗ ⊗ , b ∗ (cid:1) , defined by F hr ([ m ⊗ a r ]) := (cid:2) γ ( h (3) ) · m · γ − ( h (1) ) ⊗ h (2) · a r (cid:3) , is a morphism of complexes.Proof. It follows using the equality in Definition 1.21(2), Proposition 1.14 and Lemma 2.2(4). (cid:3)
Proposition 2.18.
For each h, l ∈ H , the endomorphism of H K ∗ ( A, M ) induced by F h ∗ ◦ F l ∗ and F hl ∗ coincide. Moreover F ∗ is the identity map. Consequently H K ∗ ( A, M ) is a left H -module.Proof. We claim that the family of maps (cid:0) h r : M ⊗ A ⊗ r ⊗ −→ M ⊗ A ⊗ r +1 ⊗ (cid:1) r ≥ , CO)HOMOLOGY OF CROSSED PRODUCTS BY WEAK HOPF ALGEBRAS 19 defined by h r ([ m ⊗ a r ]) := − (cid:2) γ ( h (3) l (3) ) · m · γ − ( l (1) ) γ − ( h (1) ) ⊗ T ( h, l, a r ) (cid:3) , where T ( h, l, a r )is as in Theorem 2.16(3), is a homotopy from F hl ∗ to F h ∗ ◦ F l ∗ . In order to check this we mustprove that (cid:0) F hlr − F hr ◦ F lr (cid:1) ([ m ⊗ a r ]) = (cid:40)(cid:0) b ◦ h (cid:1) ([ m ]) if r = 0, (cid:0) b ◦ h r + h r − ◦ b (cid:1) ([ m ⊗ a r ]) if r >
0. (2.10)Let y := ( h ⊗ H L l ) ⊗ H L (cid:2) m ⊗ a r (cid:3) ∈ X r ( M ). Since ( X ∗ ( M ) , d ∗ ) is a chain complex, d ( d ( y )) = (cid:40) d ( d ( y )) if r = 0, d ( d ( y )) + d ( d ( y )) if r > d ( d ( y )) = (cid:2) γ (cid:0) h (3) l (3) · m · γ − (cid:0) h (1) l (1) (cid:1) ⊗ h (2) l (2) · a r (cid:3) − (cid:2) γ ( h (3) ) γ ( l (3) ) · m · γ − ( l (1) ) γ − ( h (1) ) ⊗ h (2) · (cid:0) l (2) · a r (cid:1)(cid:3) . (2.11)Now, a direct computation shows that d (cid:0) d ( y ) (cid:1) = ( − r d (cid:0) Π R ( h ) l ⊗ H L [ m ⊗ a r ] (cid:1) − ( − r d (cid:0) hl ⊗ H L [ m ⊗ a r ] (cid:1) + ( − r d (cid:0) h ⊗ H L [ γ ( l (3) ) · m · γ − ( l (1) ) ⊗ l (2) · a r ] (cid:1) = (cid:2) m · γ (cid:0) Π R (cid:0) Π R ( h ) l (cid:1)(cid:1) ⊗ a r (cid:3) − (cid:2) γ (cid:0) Π R ( h ) (3) l (3) (cid:1) · m · γ − (cid:0) Π R ( h ) (1) l (1) (cid:1) ⊗ Π R ( h ) (2) l (2) · a r (cid:3) − (cid:2) m · γ (cid:0) Π R ( hl ) (cid:1) ⊗ a r (cid:3) + (cid:2) γ (cid:0) h (3) l (3) (cid:1) · m · γ − (cid:0) h (1) l (1) (cid:1) ⊗ h (2) l (2) · a r (cid:3) + (cid:2) γ ( l (3) ) · m · γ − ( l (1) ) γ (cid:0) Π R ( h ) (cid:1) ⊗ l (2) · a r (cid:3) − (cid:2) γ ( h (3) ) γ ( l (3) ) · m · γ − ( l (1) ) γ − ( h (1) ) ⊗ h (2) · ( l (2) · a r ) (cid:3) . Using the second equality in Proposition 1.6 and the fact that, by Lemma 2.3 and Proposi-tions 1.4(2) and 1.5, (cid:2) γ (cid:0) l (3) (cid:1) · m · γ − (cid:0) l (1) (cid:1) γ (Π R ( h )) ⊗ l (2) · a r (cid:3) = (cid:2) γ (cid:0) l (3) (cid:1) · m · γ − (cid:0) Π R ( h ) l (1) (cid:1) ⊗ l (2) · a r (cid:3) = (cid:2) γ (cid:0) Π R ( h ) (3) l (3) (cid:1) · m · γ − (cid:0) Π R ( h ) (1) l (1) (cid:1) ⊗ (cid:0) Π R ( h ) (2) l (2) (cid:1) · a r (cid:3) , we obtain that equality (2.11) holds. Finally we prove that F ∗ is the identity map. By theequality in Definition 1.21(2), Proposition 1.22(1) and Remark 1.28, we have1 (1) ⊗ k (2) · a r · (1 (3) · A ) = 1 (1) ⊗ k (2) · a ,r − ⊗ (1 (3) · a r )(1 (4) · A ) = 1 (1) ⊗ k (2) · a ,r − · (1 (3) · A ) ⊗ a r , for all r >
1. An inductive argument using this fact, shows that, for all r ≥ γ − (1 (1) ) ⊗ (2) · a r · (1 (3) · A ) = γ − (1 (1) ) ⊗ (1 (2) · A ) · a r = γ (1) ⊗ a r = 1 E ⊗ a r , where the third equality holds by Propositions 1.29(1) and 1.35. Combining this with Proposi-tion 1.29(1) we obtain that F r ([ m ⊗ a r ]) = (cid:2) ν (1 (3) · A ) · m · γ − (1 (1) ) ⊗ (2) · a r (cid:3) = [ m ⊗ a r ] , as desired. (cid:3) Example 2.19. If A = K , then M ⊗ is a left H -module via h · [ m ] := [ γ ( h (2) ) · m · γ − ( h (1) )]and H K ∗ ( E, M ) = H ∗ (cid:0) H, M ⊗ (cid:1) . Proposition 2.20.
The spectral sequence of [22, (3.30)] satisfies E rs = H Kr ( A, M ) ⊗ H L H ⊗ sHL and E rs = H s (cid:0) H, H Kr ( A, M ) (cid:1) . Proof.
By Remark 2.13, Proposition 2.18 and Theorem 2.16. (cid:3)
Let H , A , ρ , χ ρ , f , F f , E , K , M , ν , ν , γ and γ − be as in the previous section. Assume thatthe hypotheses of that section are fulfilled. In particular H is a weak Hopf algebra, A is a weakmodule algebra and f is convolution invertible. In [22, Section 4] was obtained a cochain complex( (cid:98) X ∗ ( M ) , (cid:98) d ∗ ) simpler than the canonical one, that gives the Hochschild cohomology of E withcoefficients in M . In this Section we prove that ( (cid:98) X ∗ ( M ) , (cid:98) d ∗ ) is isomorphic to a simpler complex( X ∗ ( M ) , d ∗ ). When K is separable, the complex ( X ∗ ( M ) , d ∗ ) gives the absolute Hochschildcohomology of E with coefficients in M . We recall from [22, Section 4] that (cid:98) X n ( M ) = (cid:77) r,s ≥ r + s = n (cid:98) X rs ( M ) , where (cid:98) X rs ( M ) := Hom ( A,K ) (cid:0) (cid:101) E ⊗ sA ⊗ A ⊗ r , M (cid:1) ,and that there exist maps (cid:98) d rsl : (cid:98) X r + l − ,s − l ( M ) −→ (cid:98) X rs ( M ) such that (cid:98) d n := n (cid:88) l =1 (cid:98) d nl + n (cid:88) r =1 n − r (cid:88) l =0 (cid:98) d r,n − rl for all n ≥ M is a ( K, K ⊗ k H L )-bimodule via λ · m · ( λ (cid:48) ⊗ k l ) := ν ( λ ) γ ( S ( l )) · m · ν ( λ (cid:48) ) . For each r, s ≥
0, we set X rs ( M ) := Hom ( K,K ⊗ k H L ) (cid:0) H ⊗ sHL ⊗ k A ⊗ r , M (cid:1) , where we consider H ⊗ sHL ⊗ k A ⊗ r as a ( K, K ⊗ k H L )-bimodule via λ · (cid:0) h s ⊗ k a r (cid:1) · ( λ (cid:48) ⊗ k l ) := h s · l ⊗ k λ · a r · λ (cid:48) . Let M K := { m ∈ M : λ · m = m · λ for all λ ∈ K } . Since H ⊗ HL = H L and A ⊗ = K , we have X r ( M ) (cid:39) Hom K e (cid:0) A ⊗ r , M (cid:1) and X s ( M ) (cid:39) Hom H L (cid:0) H ⊗ sHL , M K (cid:1) , (3.12)where M K is considered as a right H L -module via m · l := S ( l ) · m . Remark . For each r, s ≥
0, we have X rs ( M ) (cid:39) Hom H L (cid:0) H ⊗ sHL , Hom K e (cid:0) A ⊗ r , M (cid:1)(cid:1) , where Hom K e (cid:0) A ⊗ r , M (cid:1) is considered as right H L -module via ( β · l )( a r ) := γ ( S ( l )) · β ( a r ). Proposition 3.2.
For each r, s ≥ there exist maps Θ rs : X rs ( M ) −→ (cid:98) X rs ( M ) and Λ rs : (cid:98) X rs ( M ) −→ X rs ( M ) , such that for each x ∈ (cid:98) X rs ( M ) and y ∈ X rs ( M ) , of the forms x := ν ( a ) γ ( h ) (cid:103) ⊗ A · · · ⊗ A ν ( a s ) γ ( h s ) (cid:103) ⊗ a s +1 ,s + r and y := h s ⊗ k a r , the equalities Θ rs ( β )( x ) := ( − rs ν ( a ) γ ( h (1)1 ) · · · ν ( a s ) γ ( h (1) s ) · β (cid:0) h (2)1 s ⊗ k a s +1 ,s + r (cid:1) CO)HOMOLOGY OF CROSSED PRODUCTS BY WEAK HOPF ALGEBRAS 21 and Λ rs ( α )( y ) := ( − rs γ − × (h (1)1 s ) · α (cid:0)(cid:101) γ A (h (2)1 s ) ⊗ a r (cid:1) , hold.Proof. Mimic the proof of Proposition 2.5. (cid:3)
Remark . Under the first identifications in [22, (4.32)] and (3.12) the maps Θ r and Λ r become the identity maps. Similarly, under the last identifications in [22, (4.32)] and (3.12), themaps Θ s and Λ s becomeΘ s ( β ) (cid:0) ν ( a ) γ ( h ) (cid:103) ⊗ A · · · ⊗ A ν ( a s ) γ ( h s ) (cid:103) (cid:1) = ν ( a ) γ ( h (1)1 ) · · · ν ( a s ) γ ( h (1) s ) · β (cid:0) h (2)1 s (cid:1) and Λ s ( α )( h s ) = γ − × (h (1)1 s ) · α (cid:0)(cid:101) γ A (h (2)1 s ) (cid:1) . Proposition 3.4.
The morphisms Θ rs and Λ rs are inverse one of each other.Proof. Mimic the proof of Proposition 2.9. (cid:3)
For each 0 ≤ l ≤ s and r ≥ r + l ≥
1, let d rsl : X r + l − ,s − l ( M ) → X rs ( M ) be themap d rsl := Λ r + l − ,s − l ◦ (cid:98) d rsl ◦ Θ rs . Theorem 3.5.
The Hochschild cohomology of the K -algebra E is the cohomology of ( X ∗ ( M ) , d ∗ ) ,where X n ( M ) := (cid:77) r + s = n X rs ( M ) and d n := n (cid:88) l =1 d nl + n (cid:88) r =1 n − r (cid:88) l =0 d r,n − rn . Proof.
By Proposition 3.4 and the definition of ( X ∗ ( M ) , d ∗ ), the mapΘ ∗ : ( X ∗ ( M ) , d ∗ ) −→ ( (cid:98) X ∗ ( M ) , (cid:98) d ∗ ) , given by Θ n := (cid:76) r + s = n Θ rs , is an isomorphism of complexes. (cid:3) Remark . By [22, Remark 4.2], if f takes its values in K , then ( X ∗ ( M ) , d ∗ ) is the totalcomplex of the double complex ( X ∗∗ ( M ) , d ∗∗ , d ∗∗ ). Remark . If K = A , then ( X ∗ ( M ) , d ∗ ) = ( X ∗ ( M ) , d ∗ ). Remark . For each s, n ≥
0, let F i ( X n ( M )) := (cid:76) s ≥ i X n − s,s ( M ). Clearly the cochain complex( X ∗ ( M ) , d ∗ ) is filtrated by F ( X ∗ ( M )) ⊇ F ( X ∗ ( M )) ⊇ F ( X ∗ ( M )) ⊇ F ( X ∗ ( M )) ⊇ . . . . (3.13)Since the isomorphism Θ ∗ : ( X ∗ ( M ) , d ∗ ) → ( (cid:98) X ∗ ( M ) , (cid:98) d ∗ ) preserve filtrations, the spectral se-quence ov of [22, (4.33)] coincide with the one determined by the filtration (3.13). Lemma 3.9.
Let β ∈ X rs ( M ) , a, a . . . , a r ∈ A , h , . . . , h s ∈ H and z ∈ H R . (1) We have γ − × (h (1)1 s ) · Θ( β ) (cid:0)(cid:101) γ A (h (2)1 s ) · ν ( a ) ⊗ a r (cid:1) = ( − rs ν ( a ) · β (cid:0) h s ⊗ k a r (cid:1) . (2) For s ≥ and ≤ i < s , we have γ − × (h (1)1 s ) · Θ( β ) (cid:16)(cid:101) γ A (h (2)1 ,i − ) ⊗ A γ ( h (2) i ) γ ( h (2) i +1 ) (cid:93) ⊗ A (cid:101) γ A (h (2) i +2 ,s ) ⊗ a r (cid:17) = ( − rs β (cid:0) h ,i − ⊗ H L h i h i +1 ⊗ H L h i +2 ,s ⊗ k a r (cid:1) . (3) We have γ − × (h (1)1 s ) γ ( z ) · Θ( β ) (cid:0)(cid:101) γ A (h (2)1 s ) ⊗ a r (cid:1) = ( − rs β (cid:0) Π R ( z ) · h s ⊗ k a r (cid:1) , Proof.
Mimic the proof of Lemma 2.14. (cid:3)
Notation 3.10.
For each k -subalgebra R of A and 0 ≤ u ≤ r , we set X rsu ( R, M ) := Λ (cid:0) (cid:98) X rsu ( R, M ) (cid:1) ,where (cid:98) X rsu ( R, M ) is as in [22, Notation 4.4]
Theorem 3.11.
Let y := h s ⊗ k a r , where h , . . . , h s ∈ H and a , . . . , a r ∈ A . The followingassertions hold: (1) For r ≥ and s ≥ , we have d ( β )( y ) = ν ( a ) β (cid:0) h s ⊗ k a r (cid:1) + r − (cid:88) i =1 ( − i β (cid:0) h s ⊗ k ( a ,i − ⊗ a i a i +1 ⊗ a i +2 ,r ) (cid:1) + β (cid:0) h s ⊗ k a ,r − (cid:1) · ν ( a r ) . (2) For r ≥ and s = 1 , we have d ( β )( y ) = ( − r (cid:16) γ (Π R ( h )) · β (cid:0) H L ⊗ k a r (cid:1) − γ − ( h (1)1 ) · β (cid:0) H L ⊗ k h (2)1 · a r (cid:1) · γ ( h (3)1 ) (cid:17) , while for r ≥ and s > , we have d ( β )( y ) = ( − r β (cid:0) (Π R ( h ) h ⊗ H L h s ) ⊗ k a r (cid:1) + s − (cid:88) i =1 ( − r + i β (cid:0) ( h ,i − ⊗ H L h i h i +1 ⊗ H L h i +2 ,s ) ⊗ k a r (cid:1) + ( − r + s γ − ( h (1) s ) · β (cid:0) h ,s − ⊗ k h (2) s · a r (cid:1) · γ ( h (3) s ) . (3) For r ≥ and s ≥ , we have d ( β )( y ) = − γ − ( h (1) s ) γ − ( h (1) s − ) · β (cid:0) h (2)1 ,s − ⊗ k T ( h (2) s − , h (2) s , a r ) (cid:1) · γ ( h (3) s − h (3) s ) , where T ( h s − , h s , a r ) is as in Theorem 2.16(3). (4) Let R be a k -subalgebra of A . If R is stable under ρ and f takes its values in R , then d l (cid:0) X r + l − ,s − l ( M ) (cid:1) ⊆ X rsl − ( R, M ) , for each r ≥ and < l ≤ s .Proof. Mimic the proof of Theorem 2.16. (cid:3)
Proposition 3.12.
For each h ∈ H the map F ∗ h : (cid:16) Hom K e (cid:0) A ⊗ r , M (cid:1) , b ∗ (cid:17) −→ (cid:16) Hom K e (cid:0) A ⊗ r , M (cid:1) , b ∗ (cid:17) , defined by F rh ( β ) (cid:0) a r (cid:1) := γ − ( h (1) ) · β (cid:0) h (2) · a r (cid:1) · γ ( h (3) ) , is a morphism of complexes.Proof. It follows by a direct computation using Lemma 2.2(2), the equality in Definition 1.21(2)and Proposition 1.14. (cid:3)
Proposition 3.13.
For each h, l ∈ H , the endomorphism of H ∗ K ( A, M ) induced by F ∗ l ◦ F ∗ h and F ∗ hl coincide. Moreover F ∗ is the identity map. Consequently H ∗ K ( A, M ) is a right H -module.Proof. Mimic the proof of Proposition 2.18. (cid:3)
CO)HOMOLOGY OF CROSSED PRODUCTS BY WEAK HOPF ALGEBRAS 23
Example 3.14. If A = K , then M K is a right H -module via [ m ] · h := [ γ ( h (2) ) · m · γ − ( h (1) )]and H ∗ K ( E, M ) = H ∗ (cid:0) H, M K ). Proposition 3.15.
The spectral sequence of [22, (4.33)] satisfies E rs = Hom H L (cid:0) H ⊗ sHL , H rK ( A, M ) (cid:1) and E rs = H s (cid:0) H, H rK ( A, M ) (cid:1) . Proof.
By Remark 3.8, Proposition 3.13 and Theorem 3.11. (cid:3)
Let H , A , ρ , f , E , K , M , ν , γ and γ − be as in Section 2. Thus H is a weak Hopf algebra, A is aweak module algebra and f is convolution invertible. In this section we obtain formulas involvingthe complexes ( X ∗ ( E ) , d ∗ ) and ( X ∗ ( M ) , d ∗ ) that induce the cup product of HH ∗ K ( E ) and thecap product of H K ∗ ( E, M ). We will use freely the operators • and (cid:5) introduced in [22, Section 5]. Notations 4.1.
Given h , . . . , h s ∈ H and a , . . . , a r ∈ A we seth s · a r := h · ( h · ( . . . ( h s · a r )) . . . ) . Definition 4.2.
Let β ∈ X rs ( E ) and β (cid:48) ∈ X r (cid:48) s (cid:48) ( E ). We define β (cid:5) β (cid:48) ∈ X r (cid:48)(cid:48) ,s (cid:48)(cid:48) ( E ) by( β (cid:5) β (cid:48) )( y ) := ( − r (cid:48) s γ − × (h (1) s +1 ,s (cid:48)(cid:48) ) β (cid:0) h (3)1 s ⊗ k h (2) s +1 ,s (cid:48)(cid:48) · a r (cid:1) γ × (h (2) s +1 ,s (cid:48)(cid:48) ) β (cid:48) (cid:0) h (3) s +1 ,s (cid:48)(cid:48) ⊗ k a r +1 ,r (cid:48)(cid:48) (cid:1) , where r (cid:48)(cid:48) := r + r (cid:48) , s (cid:48)(cid:48) := s + s (cid:48) , h , . . . , h s (cid:48)(cid:48) ∈ H , a , . . . , a r (cid:48)(cid:48) ∈ A and y := h s (cid:48)(cid:48) ⊗ k a r (cid:48)(cid:48) . Proposition 4.3.
For each β ∈ X rs ( E ) , β (cid:48) ∈ X r (cid:48) s (cid:48) ( E ) , we have Θ (cid:0) β (cid:5) β (cid:48) (cid:1) = Θ( β ) • Θ( β (cid:48) ) .Proof. Let x := (cid:101) γ A (cid:0) h ,s + s (cid:48) ) ⊗ a ,r + r (cid:48) , where a , . . . , a r + r (cid:48) ∈ A and h , . . . , h s + s (cid:48) ∈ H . In orderto abbreviate expressions we set α := Θ( β ) and α (cid:48) := Θ( β (cid:48) ). By Lemma 2.7 and the definitionsof Θ, • and (cid:5) , (cid:0) Θ( β ) • Θ( β (cid:48) ) (cid:1) ( x ) = ( − rs (cid:48) Θ( β ) (cid:0)(cid:101) γ A (cid:0) h s ) ⊗ h (1) s +1 ,s + s (cid:48) · a r (cid:1) Θ( β (cid:48) ) (cid:0)(cid:101) γ A (cid:0) h (2) s +1 ,s + s (cid:48) ) ⊗ a r +1 ,r + r (cid:48) (cid:1) = ( − rs (cid:48) + rs + r (cid:48) s (cid:48) γ × (h (1)1 s ) β (cid:0) h (2)1 s ⊗ k h (1) s +1 ,s + s (cid:48) · a r (cid:1) γ × (h (2) s +1 ,s + s (cid:48) ) β (cid:0) h (3) s +1 ,s + s (cid:48) ⊗ k a r +1 ,r + r (cid:48) (cid:1) = ( − ( r + r (cid:48) )( s + s (cid:48) ) γ × (h (1)1 ,s + s ) (cid:0) β (cid:5) β (cid:48) (cid:1)(cid:0) h (2)1 ,s + s (cid:48) ⊗ k a ,r + r (cid:48) (cid:1) = Θ (cid:0) β (cid:5) β (cid:48) (cid:1) ( x ) , as desired. (cid:3) Corollary 4.4. If f takes its values in K , then the cup product of HH ∗ K ( E ) is induced by theoperation (cid:5) in ( X ∗ ( E ) , d ∗ ) .Proof. This follows from [22, Corollary 5.3] and Proposition 4.3. (cid:3)
Definition 4.5.
Let β ∈ X r (cid:48) s (cid:48) ( E ) and let y := h s ⊗ H L [ m ⊗ a r ] ∈ X rs ( M ), where m ∈ M , a , . . . , a r ∈ A and h , . . . , h s ∈ H . Assume that r ≥ r (cid:48) and s ≥ s (cid:48) . We define y ∗ β by y ∗ β := ( − rs (cid:48) + r (cid:48) s (cid:48) h (4) s (cid:48) +1 ,s ⊗ H L (cid:104) m · γ − × (h (1) s (cid:48) +1 ,s ) β (cid:0) h (1)1 s (cid:48) ⊗ k h (2) s (cid:48) +1 ,s · a r (cid:48) (cid:1) γ × (h (3) s (cid:48) +1 ,s ) ⊗ a r (cid:48) +1 ,r (cid:105) . Proposition 4.6.
For each y ∈ X rs ( M ) and β ∈ X r (cid:48) s (cid:48) ( E ) , we have Λ (cid:0) y ∗ β (cid:1) = Λ( y ) (cid:5) Θ( β ) . Proof.
Let β and y be as in Definition 4.5. We have,Λ( y ) (cid:5) Θ( β ) = ( − rs (cid:2) m · γ − × (h (1)1 s ) ⊗ A (cid:101) γ A (h (2)1 s ) ⊗ a r (cid:3) (cid:5) Θ( β )= ( − rs + r (cid:48) ( s − s (cid:48) ) (cid:2) m · γ − × (h (1)1 s )Θ( β ) (cid:0)(cid:101) γ A (h (2)1 s (cid:48) ) ⊗ h (2) s (cid:48) +1 ,s · a r (cid:48) (cid:1) ⊗ A (cid:101) γ A (h (3) s (cid:48) +1 ,s ) ⊗ a r (cid:48) +1 ,r (cid:3) = ( − rs + r (cid:48) s (cid:2) m · γ − × (h (1)1 s ) γ × (h (2)1 s (cid:48) ) β (cid:0) h (3)1 s (cid:48) ⊗ k h (2) s (cid:48) +1 ,s · a r (cid:48) (cid:1) ⊗ A (cid:101) γ A (h (3) s (cid:48) +1 ,s ) ⊗ a r (cid:48) +1 ,r (cid:3) = ( − rs + r (cid:48) s (cid:2) m · γ − × (h (1) s (cid:48) +1 ,s ) γ (1 (1) ) β (cid:0) h (1)1 s (cid:48) · (2) ⊗ k h (2) s (cid:48) +1 ,s · a r (cid:48) (cid:1) ⊗ A (cid:101) γ A (h (3) s (cid:48) +1 ,s ) ⊗ a r (cid:48) +1 ,r (cid:3) = ( − rs + r (cid:48) s (cid:2) m · γ − × (h (1) s (cid:48) +1 ,s ) β (cid:0) h (1)1 s (cid:48) ⊗ k h (2) s (cid:48) +1 ,s · a r (cid:48) (cid:1) ⊗ A (cid:101) γ A (h (3) s (cid:48) +1 ,s ) ⊗ a r (cid:48) +1 ,r (cid:3) = ( − rs (cid:48) + r (cid:48) s (cid:48) Λ (cid:16) h (4) s (cid:48) +1 ,s ⊗ H L (cid:104) m · γ − × (h (1) s (cid:48) +1 ,s ) β (cid:0) h (1)1 s (cid:48) ⊗ k h (2) s (cid:48) +1 ,s · a r (cid:48) (cid:1) γ × (h (3) s (cid:48) +1 ,s ) ⊗ a r (cid:48) +1 ,r (cid:105)(cid:17) = Λ( y ∗ β ) , where the first equality holds by the definition of Λ; the second one, by the definition of (cid:5) ;the third one, by the definition of β ; the fourth one, by Lemma 2.8; the fifth one, since β is aright H L -module morphism and γ (1 (1) ) γ ( S (1 (2) )) = 1 E ; the fifth one, by the definition of λ andLemma 2.7; and the last one, by the definition of ∗ . (cid:3) Corollary 4.7. If f takes its values in K , then in terms of the complexes ( X ∗ ( M ) , d ∗ ) and ( X ∗ ( E ) , d ∗ ) , the cap product is induced by the operation ∗ .Proof. This follows from [22, Corollary 5.7] and Proposition 4.6. (cid:3)
Let H , A , ρ , f , E , K , M , ν , γ and γ − be as in Section 2. Thus H is a weak Hopf algebra, A isa weak module algebra and f is convolution invertible. In this section we prove that the mixedcomplex ( (cid:98) X ∗ , (cid:98) d ∗ , (cid:98) D ∗ ) of [22, Section 6] is isomorphic to a simpler mixed complex ( X ∗ , d ∗ , D ∗ ).Let Θ ∗ : ( (cid:98) X ∗ ( E ) , (cid:98) d ∗ ) −→ ( X ∗ ( E ) , d ∗ ) and Λ ∗ : ( X ∗ ( E ) , d ∗ ) −→ ( (cid:98) X ∗ ( E ) , (cid:98) d ∗ ) , be the maps introduced in the proof of Theorem 2.10. For each n ≥
0, let D n : X n ( E ) → X n +1 ( E )be the map defined by D n := Θ n +1 ◦ (cid:98) D n ◦ Λ n . Theorem 5.1.
The triple (cid:0) X ∗ , d ∗ , D ∗ (cid:1) is a mixed complex that gives the Hochschild, cyclic, nega-tive and periodic homology of the K -algebra E . More precisely, the mixed complexes (cid:0) X ∗ , d ∗ , D ∗ (cid:1) and (cid:0) E ⊗ E ⊗ ∗ , b ∗ , B ∗ (cid:1) are homotopically equivalent.Proof. Since Θ ∗ and Λ ∗ are inverse one of each other it is clear that (cid:0) X ∗ , d ∗ , D ∗ (cid:1) is a mixedcomplex and Θ ∗ : (cid:0) (cid:98) X ∗ , (cid:98) d ∗ , (cid:98) D ∗ (cid:1) → (cid:0) X ∗ , d ∗ , D ∗ (cid:1) is an isomorphism of mixed complexes. So theresult follows from [22, Theorem 6.3]. (cid:3) Definition 5.2.
For each r, s ≥
0, let D rs : X rs → X r,s +1 and D rs : X rs → X r +1 ,s be the mapsdefined by D rs := Θ r,s +1 ◦ (cid:98) D rs ◦ Λ rs and D rs := Θ r +1 ,s ◦ (cid:98) D rs ◦ Λ rs , respectively. Proposition 5.3.
Let R be a k -subalgebra of A . Assume that R is stable under ρ and f takesits values in R . Let a , . . . , a n − i ∈ A and h , . . . , h i ∈ H . If y = h s ⊗ H L [ a · γ ( h ) ⊗ a r ] , then D ( y ) = D ( y ) + D ( y ) module (cid:76) is =0 X s,n +1 − s ( R, M ) .Proof. This follows by Remark 1.26 and [22, Proposition 6.6(1)]. (cid:3)
Corollary 5.4. If f takes its values in K , then D = D + D . CO)HOMOLOGY OF CROSSED PRODUCTS BY WEAK HOPF ALGEBRAS 25
We next compute the maps D and D . Notation 5.5.
Given h , . . . , h s ∈ S and 1 ≤ j ≤ s , we set S × (h s ) := S ( h s ) S ( h s − ) · · · S ( h ) and F h j +1 ,s := F h j +1 ◦ · · · ◦ F h s , where F h i is as in Proposition 2.17. Proposition 5.6.
For y := h s ⊗ H L [ ν ( a ) γ ( h ) ⊗ a r ] , we have D ( y ) = s (cid:88) j =0 ( − js + r + s (cid:0) h (3) j +1 ,s ⊗ H L h (2)0 S × (h (1)1 s ) ⊗ H L h (2)1 j (cid:1) ⊗ H L F h (2) j +1 ,s (cid:0)(cid:2) ν ( a ) γ ( h (1)0 ) ⊗ a r (cid:3)(cid:1) and D ( y ) = s (cid:88) j =0 ( − jr + r h (3)1 s ⊗ H L (cid:2) γ × (h (2)0 s ) ⊗ a j +1 ,r ⊗ a ⊗ h (1)0 s · a ( l )1 j (cid:3) . Proof.
By definition Λ( y ) = ( − rs (cid:2) ν ( a ) γ ( h ) γ − × (h (1)1 s ) ⊗ A (cid:101) γ A (h (2)1 s ) ⊗ a r (cid:3) , and so (cid:98) D (cid:0) Λ( y ) (cid:1) = s (cid:88) j =0 ( − rs + js + s (cid:2) E ⊗ A (cid:101) γ A (cid:0) h (3) j +1 ,s (cid:1) ⊗ A ν ( a ) γ ( h ) γ − × (h (1)1 s ) (cid:94) ⊗ A (cid:101) γ A (h (2)1 j ) ⊗ h (2) j +1 ,s · a r (cid:3) and (cid:98) D (cid:0) Λ( y ) (cid:1) = r (cid:88) j =0 ( − rs + jr + r + s (cid:2) γ ( h (2)0 ) ⊗ A (cid:101) γ A (h (2)1 s ) ⊗ a j +1 ,r ⊗ a ⊗ h (1)0 s · a ( l )1 j (cid:3) . Since, by the definition of Θ, Proposition 1.36, Lemma 2.8, the definitions of the actions in (2.7)and Proposition 1.15,Θ (cid:16)(cid:2) E ⊗ A (cid:101) γ A (cid:0) h (3) j +1 ,s (cid:1) ⊗ A ν ( a ) γ ( h ) γ − × (h (1)1 s ) (cid:94) ⊗ A (cid:101) γ A (h (2)1 j ) ⊗ h (2) j +1 ,s · a r (cid:3)(cid:17) = ( − rs + r (cid:0) h (5) j +1 ,s ⊗ H L h (2)0 S × (h (1)1 s ) ⊗ H L h (4)1 j (cid:1) ⊗ H L (cid:2) γ × (h (4) j +1 ,s ) ν ( a ) γ ( h (1)0 ) γ − × (h (2)1 s ) γ × (h (3)1 j ) ⊗ h (3) j +1 ,s · a r (cid:3) = ( − rs + r (cid:0) h (5) j +1 ,s ⊗ H L h (2)0 S × (h (1)1 s ) ⊗ H L h (2)1 j · (2) (cid:1) ⊗ H L (cid:2) γ × (h (4) j +1 ,s ) ν ( a ) γ ( h (1)0 ) γ − × (h (2) j +1 ,s ) γ (1 (1) ) ⊗ h (3) j +1 ,s · a r (cid:3) = ( − rs + r (cid:0) h (5) j +1 ,s ⊗ H L h (2)0 S × (h (1)1 s ) ⊗ H L h (2)1 j (cid:1) ⊗ H L (cid:2) γ × (h (4) j +1 ,s ) ν ( a ) γ ( h (1)0 ) γ − × (h (2) j +1 ,s ) ⊗ h (3) j +1 ,s · a r (cid:3) and Θ (cid:0)(cid:2) γ ( h (2)0 ) ⊗ A (cid:101) γ A (h (2)1 s ) ⊗ a j +1 ,r ⊗ a ⊗ h (1)0 s · a ( l )1 j (cid:3)(cid:1) = ( − rs + s h (3)1 s ⊗ H L (cid:2) γ × (h (2)0 s ) ⊗ a j +1 ,r ⊗ a ⊗ h (1)0 s · a ( l )1 j (cid:3) , the formulas in the statement are true. (cid:3) References [1] R. Akbarpour and M. Khalkhali,
Hopf algebra equivariant cyclic homology and cyclic homology of crossedproduct algebras , J. Reine Angew. Math. (2003), 137–152, DOI 10.1515/crll.2003.046. MR1989648[2] J. N. Alonso ´Alvarez, J. M. Fern´andez Vilaboa, R. Gonz´alez Rodr´ıguez, and A. B. Rodr´ıguez Raposo,
WeakC-cleft extensions and weak Galois extensions , Journal of Algebra (2006), no. 1, 276–293. [3] ,
Weak C -cleft extensions, weak entwining structures and weak Hopf algebras , J. Algebra (2005),no. 2, 679–704, DOI 10.1016/j.jalgebra.2004.07.043. MR2114575[4] J. N. Alonso ´Alvarez, J. M. Fern´andez Vilaboa, R. Gonz´alez Rodr´ıguez, and A. B. Rodr´ıguez Raposo, Crossedproducts in weak contexts , Appl. Categ. Structures (2010), no. 3, 231–258, DOI 10.1007/s10485-008-9139-2. MR2640214 (2011d:18009)[5] J. N. Alonso ´Alvarez, J. M. Fern´andez Vilaboa, J. M. L´opez L´opez, R. Gonz´alez Rodr´ıguez, and A. B.Rodr´ıguez Raposo, Weak Hopf algebras with projection and weak smash bialgebra structures , J. Algebra (2003), no. 2, 701-725.[6] JN Alonso ´Alvarez and R Gonz´alez Rodr´ıguez,
Crossed products for weak Hopf algebras with coalgebrasplitting , Journal of Algebra (2004), no. 2, 731–752.[7] Gabriella B¨ohm and Tomasz Brzezi´nski,
Cleft extensions of Hopf algebroids , Applied Categorical Structures (2006), no. 5-6, 431–469.[8] Gabriella B¨ohm, Florian Nill, and Kornel Szlach´anyi, Weak Hopf Algebras, I. Integral Theory and C ∗ -Structure , J. Algebra (1999), no. 2, 385–438.[9] Gabriella B¨ohm, Florian Nill, and Kornel Szlach´anyi, Weak Hopf Algebras, II. Representation theory, di-mensions and the Markov trace , J. Algebra (2000), 156–212.[10] Tomasz Brzezi´nski,
Crossed products by a coalgebra , Comm. Algebra (1997), no. 11, 3551–3575, DOI10.1080/00927879708826070. MR1468823 (98i:16034)[11] Dan Burghelea, Cyclic homology and the algebraic K -theory of spaces. I , theory, Part I, II (1983), Con-temp. Math., vol. 55, Amer. Math. Soc., Providence, RI, 1986, pp. 89–115, DOI 10.1090/conm/055.1/862632.MR862632 (88i:18009a)[12] S. Caenepeel and E De Groot, Modules over weak entwining structures , Contemporary Mathematics (2000), 31–54.[13] Graciela Carboni, Jorge A. Guccione, and Juan J. Guccione,
Cyclic homology of Hopf crossed products , Adv.Math. (2010), no. 3, 840–872, DOI 10.1016/j.aim.2009.09.008. MR2565551 (2010m:16015)[14] Graciela Carboni, Jorge A. Guccione, Juan J. Guccione, and Christian Valqui,
Cyclic homology of Brzezi´nski’scrossed products and of braided Hopf crossed products , Adv. Math. (2012), no. 6, 3502–3568, DOI10.1016/j.aim.2012.09.006. MR2980507[15] Marius Crainic,
On the perturbation lemma, and deformations (2004), available at arXiv:Math.AT/0403266 .[16] B. L. Fe˘ıgin and B. L. Tsygan,
Additive K -theory , K -theory, arithmetic and geometry (Moscow, 1984),Lecture Notes in Math., vol. 1289, Springer, Berlin, 1987, pp. 67–209, DOI 10.1007/BFb0078368. MR923136[17] J. M. Fern´andez Vilaboa, R. Gonz´alez Rodr´ıguez, and A. B. Rodr´ıguez Raposo, Preunits and weak crossedproducts , Journal of Pure and Applied Algebra (2009), 2244–2261.[18] Murray Gerstenhaber and Samuel D. Schack,
Relative Hochschild cohomology, rigid algebras, and the Bock-stein , J. Pure Appl. Algebra (1986), no. 1, 53–74, DOI 10.1016/0022-4049(86)90004-6. MR862872(88a:16045)[19] Ezra Getzler and John D. S. Jones, The cyclic homology of crossed product algebras , J. Reine Angew. Math. (1993), 161–174, DOI 10.1515/crll.1995.466.19. MR1244971[20] Jorge A. Guccione and Juan J. Guccione,
Hochschild (co)homology of Hopf crossed products , K-Theory (2002), 138–169.[21] Jorge A. Guccione, Juan J. Guccione, and Christian Valqui, Cleft extensions of weak Hopf algebras , Preprint.[22] , (Co)homology of crossed products in weak contexts , Preprint.[23] Takahiro Hayashi,
Face algebras I. A generalization of quantum group theory , Journal of the MathematicalSociety of Japan (1998), no. 2, 293–315.[24] P. Jara and D. S¸tefan, Hopf-cyclic homology and relative cyclic homology of Hopf-Galois extensions , Proc.London Math. Soc. (3) (2006), no. 1, 138–174, DOI 10.1017/S0024611506015772. MR2235945[25] Christian Kassel, Cyclic homology, comodules, and mixed complexes , J. Algebra (1987), no. 1, 195–216,DOI 10.1016/0021-8693(87)90086-X. MR883882 (88k:18019)[26] M. Khalkhali and B. Rangipour,
On the cyclic homology of Hopf crossed products , Galois theory, Hopfalgebras, and semiabelian categories, Fields Inst. Commun., vol. 43, Amer. Math. Soc., Providence, RI, 2004,pp. 341–351. MR2075593[27] Ling Liu, Bing-liang Shen, and Shuan-hong Wang,
On weak crossed products of weak Hopf algebras , Algebrasand Representation Theory (2013), no. 3, 633–657.[28] Dmitri Nikshych and Leonid Vainerman, A Galois correspondence for II1 factors and quantum groupoids ,Journal of Functional Analysis (2000), no. 1, 113–142.[29] V. Nistor,
Group cohomology and the cyclic cohomology of crossed products , Invent. Math. (1990), no. 2,411–424, DOI 10.1007/BF01234426. MR1031908 CO)HOMOLOGY OF CROSSED PRODUCTS BY WEAK HOPF ALGEBRAS 27 [30] Ana Bel´en Rodr´ıguez Raposo,
Crossed products for weak Hopf algebras , Comm. Algebra (2009), no. 7,2274–2289, DOI 10.1080/00927870802620274. MR2536918[31] Christian Voigt, Equivariant periodic cyclic homology , J. Inst. Math. Jussieu (2007), no. 4, 689–763, DOI10.1017/S1474748007000102. MR2337312[32] Takehiko Yamanouchi, Duality for generalized Kac algebras and a characterization of finite groupoid algebras ,Journal of Algebra (1994), no. 1, 9–50.[33] Jiao Zhang and Naihong Hu,
Cyclic homology of strong smash product algebras , J. Reine Angew. Math. (2012), 177–207, DOI 10.1515/CRELLE.2011.098. MR2889710
Departamento de Matem´atica, Facultad de Ciencias Exactas y Naturales-UBA, Pabell´on 1-CiudadUniversitaria, Intendente Guiraldes 2160 (C1428EGA) Buenos Aires, Argentina.Instituto de Investigaciones Matem´aticas “Luis A. Santal´o”, Facultad de Ciencias Exactas y Natu-rales-UBA, Pabell´on 1-Ciudad Universitaria, Intendente Guiraldes 2160 (C1428EGA) Buenos Aires,Argentina.
E-mail address : [email protected] Departamento de Matem´atica, Facultad de Ciencias Exactas y Naturales-UBA, Pabell´on 1-CiudadUniversitaria, Intendente Guiraldes 2160 (C1428EGA) Buenos Aires, Argentina.Instituto Argentino de Matem´atica-CONICET, Savedra 15 3er piso, (C1083ACA) Buenos Aires,Argentina.
E-mail address : [email protected] Pontificia Universidad Cat´olica del Per´u - Instituto de Matem´atica y Ciencias Afines, Secci´onMatem´aticas, PUCP, Av. Universitaria 1801, San Miguel, Lima 32, Per´u.
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