(Co)homology of crossed products in weak contexts
((CO)HOMOLOGY OF CROSSED PRODUCTS IN WEAK CONTEXTS
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 the weak crossed products E := A × F χ V introduced in [2]. Thiscomplex is provided with a canonical filtration, whose spectral sequence generalizes the spectralsequence obtained in [6]. Under suitable conditions, the above mentioned mixed complex isprovided with another filtration, whose spectral sequence generalize the Feigin-Tsygan spectralsequence. These results apply in particular to the crossed products of algebras by weakbialgebras. Contents
Introduction
Due to the relation of the actions of groups on algebras with non-commutative geometry, theproblem of 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 [9, 12, 18]. For instance, in the firstpaper the authors obtained a spectral sequence converging to the cyclic homology of A k [ G ],and in [12], this result was derived from the theory of paracyclic modules and cylindrical mod-ules developed by the authors. The main tool for this computation was a version for cylindri-cal modules of the Eilenberg-Zilber theorem. More recently, and also due to its connections 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 with non-commutative geometry, the cyclic homology of algebras obtained through more generalconstructions (Hopf crossed products, Hopf Galois extensions, Braided Hopf crossed products,etcetera) has been extensively studied. See for instance [1, 6, 7, 13, 15, 19, 20]. The method de-veloped in [6, 7, 13] has the advantage over others that it works for arbitrary cocycles. In thispaper we use it to compute the Hochschild (co)homology and the type cyclic homologies of theweak crossed products introduced in [2]. Specifically, for such a crossed product E := A × F χ V ,we construct a mixed complex, simpler than the canonical one, that computes the Hochschild,cyclic, negative and periodic homologies of E . The Hochschild and cyclic complexes of thismixed complex are provided with canonical filtrations whose spectral sequences generalize theHochschild-Serre spectral sequence and the Feigin and Tsygan spectral sequence. We also studythe Hochschild cohomology of these algebras. In the subsequent paper [14] we use the mainresults of this paper in order to compute the Hochschild (co)homology and the type cyclic ho-mologies of crossed products of algebras by weak Hopf algebras with invertible cocycle. We hopethat our method works also for partial crossed products (see [3])The paper is organized as follows:A crossed product system with preunit is a tuple ( A, V, χ, F , ν ), where A a k -algebra, V a k -vector space and χ : V ⊗ k A → A ⊗ k V and F : V ⊗ k A → A ⊗ k V are maps satisfying suitableconditions. Each such a tuple has associated an algebra E := A × F χ V . If F is a cocycle thatsatisfied the twisted module condition (see Definition 1.5), then E is an associative algebra withunit, which is named the unitary crossed product of A by V associated with χ and F . In Section 1we give a review of the these crossed products (that includes the crossed products introducedin [5]), we also recall the concept of mixed complex and the perturbation lemma. In Section 2we construct a resolution of E as an E -bimodule. In Section 3 and 4 we use this resolution toobtain complexes, simpler than the canonical ones, that compute the Hochschild homology andthe Hochschild cohomology of E with coefficients in an E -bimodule M . Then, in Section 5 westudy the cup and the cap products of E , and, finally, in Section 6 we obtain a mixed complex,simpler that the canonical one, that computes the type cyclic homologies of E . 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 . By an algebrawe understand and associative algebra over k . Given an arbitrary 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 consideredas 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 . Given a algebra C , we let µ : C ⊗ k C → C denote its multiplication map. When C is unitary we let η : k → C , denote its unit.In some parts of this article we use the nowadays well known graphic calculus for monoidaland braided categories. As usual, morphisms will be composed from top to bottom and tensorproducts will be represented by horizontal concatenation from left to right. The identity map ofa k -vector space will be represented by a vertical line. Given an algebra A , the diagrams, , and CO)HOMOLOGY OF CROSSED PRODUCTS IN WEAK CONTEXTS 3 stand for the multiplication map, the unit (when C is unital), the action of C on a left C -moduleand the action of C on a right C -module, respectively. We will also use the diagrams, , F and ν to denote the flip, a twisting map (See Definition 1.1), and the maps F and ν of a crossed productsystem with preunit (See Definition 1.3). In this subsection we recall a very general notion of crossed product developed in [2, 10], and wereview its basic properties. Also we compare this construction with the one given in [5].
Definition 1.1.
A triple (
A, V, χ ), consisting of an associative algebra A , a k -vector space V and a map χ : V ⊗ k A −→ A ⊗ k V , is a twisted space if χ ◦ ( V ⊗ k µ A ) = ( µ A ⊗ k V ) ◦ ( A ⊗ k χ ) ◦ ( χ ⊗ k A ) . (1.1)In such a case we say that χ is a twisting map .Throughout this paper we assume that A is a unitary algebra and ( A, V, χ ) is a twisted space.A direct computation shows that A ⊗ k V is a non unitary A -bimodule via a (cid:48) · ( a ⊗ k v ) = a (cid:48) a ⊗ k v and ( a ⊗ k v ) · a (cid:48) = a · χ ( v ⊗ k a (cid:48) ) . Let ∇ χ be the endomorphism of A ⊗ k V defined by ∇ χ ( a ⊗ k v ) := a · χ ( v ⊗ k A ). It is easyto see that ∇ χ is a left and right A -linear idempotent, and that χ ( V ⊗ k A ) ⊆ A × V , where A × V := ∇ χ ( A ⊗ k V ). Let p χ and ı χ be the corestriction of ∇ χ to A × V and the canonicalinclusion of A × V in A ⊗ k V , respectively. By the above discussion p χ ◦ ı χ = id A × V . Moreover A × V is an unitary A -subbimodule of A ⊗ k V and both p χ and ı χ are A -bimodule morphisms. Remark . Note that ∇ χ ( x ) = x · A for all x ∈ A ⊗ k V . So, A × V is the set of all the x ’s in A ⊗ k V such that x · A = x . The group X χ := (cid:8) x ∈ A ⊗ k V : x · A = 0 (cid:9) . is an A -subbimoduleof A ⊗ k V and A ⊗ k V = A × V ⊕ X χ . Moreover the projection of A ⊗ k V onto A × V along X χ coincides with p χ .Let C be a k -vector space and let ∇ C : C → C be an idempotent map. An associative product µ C : C ⊗ k C −→ C is said to be normalized with respect to ∇ C if ∇ C ( cc (cid:48) ) = cc (cid:48) = ∇ C ( c ) ∇ C ( c (cid:48) ),for all c, c (cid:48) ∈ C .Let C be an algebra. A map ν : k → C is a preunit of µ C if ν (1) is a central idempotent of C . Definition 1.3.
We say that a tuple (
A, V, χ, F , ν ) is a crossed product system with preunit if(1) ( A, V, χ ) is a twisted space,(2) F : V ⊗ k V −→ A ⊗ k V is a map with Im( F ) ⊆ A × V ,(3) ν : k −→ A ⊗ k V is a map satisfying( µ A ⊗ k V ) ◦ ( A ⊗ k F ) ◦ ( χ ⊗ k V ) ◦ ( V ⊗ k ν ) = ∇ χ ◦ ( η A ⊗ k V ) , (1.2)( µ A ⊗ k V ) ◦ ( A ⊗ k F ) ◦ ( ν ⊗ k V ) = ∇ χ ◦ ( η A ⊗ k V ) , (1.3)( µ A ⊗ k V ) ◦ ( A ⊗ k χ ) ◦ ( ν ⊗ k A ) = ( µ A ⊗ k V ) ◦ ( A ⊗ k ν ) . (1.4) Notation 1.4.
Given a crossed product system with preunit (
A, V, χ, F , ν ), we let A ⊗ F χ V denote A ⊗ k V endowed with the (non necessarily associative) multiplication map µ A ⊗ F χ V defined by µ A ⊗ F χ V := ( µ A ⊗ k V ) ◦ ( µ A ⊗ k F ) ◦ ( A ⊗ k χ ⊗ k V ) , JORGE A. GUCCIONE, JUAN J. GUCCIONE, AND C. VALQUI and we let A × F χ V denote A × V endowed with the (non necessarily associative) multiplicationmap µ A × F χ V , defined as the restriction and corestriction of µ A ⊗ F χ V to A × V (this is correct sinceclearly Im (cid:0) µ A ⊗ F χ V (cid:1) ⊆ A × V ). In the sequel for simplicity we will write E instead of A × F χ V and E instead of A ⊗ F χ V . Definition 1.5.
Let (
A, V, χ, F , ν ) be a crossed product system with preunit. We say that F isa cocycle that satisfies the twisted module condition if F = F and F F = FF More precisely, the first equality says that F satisfies the twisted module condition, and thesecond one says that F is a cocycle.Let ( A, V, χ, F , ν ) be a crossed product system with preunit and let ∇ ν : E −→ E , (cid:48) ν : A → E , ν : A → E and γ : V → E be the arrows defined by ∇ ν ( a ⊗ k v ) := ( a ⊗ k v ) ν (1 k ) , (cid:48) ν ( a ) := a · ν (1 k ) , ν ( a ) := ∇ χ ( (cid:48) ν ( a )) and γ ( v ) := ∇ χ (1 A ⊗ k v ) . Theorem 1.6.
Let ( A, V, χ, F , ν ) be a crossed product system with preunit. If F is a cocyclethat satisfies the twisted module condition, then the following facts hold: (1) µ E is a left and right A -linear associative product, that is normalized with respect to ∇ χ . (2) ν is a preunit of µ E , ∇ ν = ∇ χ and ν ( k ) ⊆ E . (3) µ E is left and right A -linear, associative and has unit E := ν (1 k ) . (4) The maps ı χ and p χ are multiplicative. (5) (cid:48) ν is left and right A -linear, multiplicative, and (cid:48) ν ( A ) ⊆ E . (6) ν is left and right A -linear, multiplicative and unitary. (7) ν ( a ) x = a · x and x ν ( a ) = x · a , for all a ∈ A and x ∈ E . (8) χ ( v ⊗ k a ) = (1 A ⊗ k v ) (cid:48) ν ( a ) and F ( v ⊗ k w ) = (1 A ⊗ k v )(1 A ⊗ k w ) . (9) χ ( v ⊗ k a ) = γ ( v ) ν ( a ) and F ( v ⊗ k w ) = γ ( v ) γ ( w ) .Proof. Except for items 4), 7) and the assertions about the right A -linearity in items 3), 5)and 6), whose proofs we leave to the reader, this follows immediately from [10, Remark 3.10, oneimplication of Theorem 3.11 and Corollary 3.12]. (cid:3) Remark . By item 5) of the previous theorem, ν ( a ) = (cid:48) ν ( a ) for all a ∈ A .When the hypotheses of Theorem 1.6 are fulfilled we say that E is the unitary crossed productof A with V associated with χ and F . Example 1.8.
Let (
A, V, χ ) and F be as in items 1) and 2) of Definition 1.3 and let 1 V ∈ V . If χ (1 V ⊗ k a ) = a · χ (1 V ⊗ k A ) and F is a twisted module cocycle satisfying F (1 V ⊗ k v ) = F ( v ⊗ k V ) = χ ( v ⊗ k A ) , then the tuple ( A, V, χ, F , ν ), where ν is the map defined by ν (1 k ) := χ (1 V ⊗ k A ), is a crossedproduct system with preunit. CO)HOMOLOGY OF CROSSED PRODUCTS IN WEAK CONTEXTS 5
Example 1.9.
The crossed product systems introduced by Brzezi´nski in [5] are the crossedproduct systems with unit (
A, V, χ, F , V ), such that χ ( v ⊗ k A ) = 1 A ⊗ k v and F is a normalcocycle that satisfies the twisted module condition. The crossed product constructed from thedatum ( A, V, χ, F , V ) is called the Brzezi´nski crossed product of A with V associated with χ , F and V . If ( A, V, χ, F , V ) is a Brzezi´nski’s crossed product system, then χ (1 V ⊗ k a ) = a ⊗ k V and ∇ = id A ⊗ V , which implies E = E . Suppose now that V is an algebra with unit 1 V . A Brzezi´nski’s crossedproduct in which F is the trivial cocycle given by F ( v ⊗ k w ) := 1 A ⊗ k vw , is called a twisted tensorproduct . In this case F is automatically a normal cocycle and the twisted module condition saysthat χ ◦ ( µ V ⊗ k A ) = ( A ⊗ k µ V ) ◦ ( χ ⊗ k V ) ◦ ( V ⊗ k χ ) . When we deal with twisted tensor products we write A ⊗ χ V instead A ⊗ F χ V .In the rest of the section we assume that ( A, V, χ, F , ν ) is a crossed product system withpreunit, in which F is a cocycle that satisfies the twisted module condition. Remark . By definition ∇ χ ( a ⊗ k v ) = a · γ ( v ). Consequently; if a · γ ( v ) = (cid:80) l a ( l ) ⊗ k v ( l ) , then ν ( a ) γ ( v ) = a · γ ( v ) = (cid:88) l a ( l ) · γ ( v ( l ) ) = (cid:88) l ν ( a ( l ) ) γ ( v ( l ) ); (1.5)if χ ( v ⊗ k a ) = (cid:80) l a ( l ) ⊗ k v ( l ) , then γ ( v ) ν ( a ) = χ ( v ⊗ k a ) = (cid:88) l ν ( a ( l ) ) γ ( v ( l ) ); (1.6)and if F ( v ⊗ k w ) = (cid:80) l a ( l ) ⊗ k u ( l ) , then γ ( v ) γ ( w ) = F ( v ⊗ k w ) = (cid:88) l ν ( a ( l ) ) γ ( u ( l ) ) . (1.7) Proposition 1.11.
For each subalgebra R of A , ( R ⊗ k V ) ∩ E ⊆ R · γ ( V ) = ν ( R ) γ ( V ) . Moreover, if γ ( V ) ⊆ R ⊗ k V , then the equality holds.Proof. Since ∇ χ is a left A -linear projection with image E , we have( R ⊗ k V ) ∩ E = ∇ χ (cid:0) ( R ⊗ k V ) ∩ E (cid:1) ⊆ ∇ χ ( R ⊗ k V ) = R · γ ( V ) = ν ( R ) γ ( V ) . The last assertion holds since γ ( V ) ⊆ R ⊗ k V implies R · γ ( V ) ⊆ R · ( R ⊗ k V ) = R ⊗ k V . (cid:3) Definition 1.12.
We say that a subalgebra R of A is stable under χ if χ ( V ⊗ k R ) ⊆ R ⊗ k V . Lemma 1.13. If R is a stable under χ subalgebra of A , then γ ( V ) ν ( R ) ⊆ ( R ⊗ k V ) ∩ E = ν ( R ) γ ( V ) . Proof.
Since R is stable under χ , we know that γ ( V ) ⊆ R ⊗ k V . So, γ ( V ) ν ( R ) = χ ( V ⊗ k R ) ⊆ ( R ⊗ k V ) ∩ E = ν ( R ) γ ( V ) , where the first equality holds by Theorem 1.6(9); and the last one, by Proposition 1.11. (cid:3) Lemma 1.14.
Let R be a k -subalgebra of A . If F ( V ⊗ k V ) ⊆ R ⊗ k V , then γ ( V ) γ ( V ) ⊆ ( R ⊗ k V ) ∩ E ⊆ ν ( R ) γ ( V ) . Proof.
By Theorem 1.6(9), the fact that F ( V ⊗ k V ) ⊆ ( R ⊗ k V ) ∩ E and Proposition 1.11. (cid:3) JORGE A. GUCCIONE, JUAN J. GUCCIONE, AND C. VALQUI
Notation 1.15.
We let χ jl : V ⊗ jk ⊗ k A ⊗ lk −→ A ⊗ lk ⊗ k V ⊗ jk denote the map defined by: χ := χ,χ j +1 , := (cid:0) χ j ⊗ k V (cid:1) ◦ (cid:0) V ⊗ jk ⊗ k χ (cid:1) for j ≤ ,χ j,l +1 := (cid:0) A ⊗ lk ⊗ k χ j (cid:1) ◦ (cid:0) χ jl ⊗ k A (cid:1) for j, l ≤ . Furthermore, we set χ j := id V ⊗ jk and χ l := id A ⊗ lk for all j, l ≥ Proposition 1.16.
Let K be a subalgebra of A and let A := A/K . If K is stable under χ , theneach χ jl induces maps ¯ χ jl : V ⊗ jk ⊗ k A ⊗ lK −→ A ⊗ lK ⊗ k V ⊗ jk and χ jl : V ⊗ jk ⊗ k A ⊗ lK −→ A ⊗ lK ⊗ k V ⊗ jk . Proof.
Straightforward. (cid:3)
In this subsection we recall briefly the notion of mixed complex. For more details about thisconcept we refer to [4, 16].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 ). CO)HOMOLOGY OF CROSSED PRODUCTS IN WEAK CONTEXTS 7
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 ∗ ). Next, we recall the perturbation lemma. We present the version given in [8].A homotopy equivalence data ( Y, ∂ ) (
X, d ) pi X ∗ X ∗ +1 h (1.8)consists of the following:(1) Chain complexes ( Y, ∂ ), (
X, d ) and quasi-isomorphisms i , p between them,(2) A homotopy h from i ◦ p to id.A perturbation of (1.8) is a map δ : X ∗ → X ∗− such that ( d + δ ) = 0. We call it small ifid − δ ◦ h is invertible. In this case we write A := (id − δ ◦ h ) − ◦ δ and we consider de diagram( Y, ∂ ) ( X, d ) p i X ∗ X ∗ +1 , h (1.9)where ∂ := ∂ + p ◦ A ◦ i, i := i + h ◦ A ◦ i, p := p + p ◦ A ◦ h, h := h + h ◦ A ◦ h. A deformation retract is a homotopy equivalence data such that p ◦ i = id. A deformation retractis called special if h ◦ i = 0, p ◦ h = 0 and h ◦ h = 0.In all the cases considered in this paper the morphism δ ◦ h is locally nilpotent (in other for all x ∈ X ∗ there exists n ∈ N such that ( δ ◦ h ) n ( x ) = 0). Consequently, (id − δ ◦ h ) − = (cid:80) ∞ n =0 ( δ ◦ h ) n . Theorem 1.17 ([8]) . If δ is a small perturbation of (1.8) , then the diagram (1.9) is an homotopyequivalence data. Furthermore, if (1.8) is a special deformation retract, then so it is (1.9) . Let A be an algebra, V a k -vector space, K a subalgebra of A and ( A, V, χ, F , ν ) a crossedproduct system with preunit. Assume that F is a cocycle that satisfies the twisted modulecondition and that K is stable under χ . Recall that E = A × F χ V . By the discussions aboveRemark 1.2 we know that E is an A -bimodule, and so it is also a K -bimodule. Moreover, byTheorem 1.6(7) the left and right actions of A on E coincide with those obtained through themorphism ν : A → E . Let Υ denote the family of all the epimorphisms of E -bimodules whichsplit as ( E, K )-bimodule maps. In this section we construct a Υ-relative projective resolution
JORGE A. GUCCIONE, JUAN J. GUCCIONE, AND C. VALQUI ( X ∗ , d ∗ ), of E as an E -bimodule, simpler than the normalized bar resolution of E . Also we willconstruct comparison maps between both resolutions. We let ⊗ denote ⊗ K and we set A := AK , E := E ν ( K ) and (cid:101) E := E ν ( A ) . Notations 2.1.
We will use the following notations:(1) For each x ∈ E we let x and (cid:101) x denote its class in E and (cid:101) E , respectively. Similarly, for a ∈ A we let a denote its class in A .(2) Given x , . . . , x s ∈ E and 1 ≤ i ≤ j ≤ s , we set x ij := x i ⊗ · · · ⊗ x j and (cid:101) x ij := (cid:101) x i ⊗ A · · · ⊗ A (cid:101) x j . (3) Given v , . . . , v s ∈ V and 1 ≤ i ≤ j ≤ s , we set v ij := v i ⊗ k · · · ⊗ k v j .(4) Given a , . . . , a r ∈ A and 1 ≤ i ≤ j ≤ r , we set a ij := a i ⊗ · · · ⊗ a j , and we let a ij denotethe class of a ij in A ⊗ j − i +1 .(5) We let γ : V → E and (cid:101) γ : V → (cid:101) E denote the maps induced by γ .(6) Given v , . . . , v s ∈ V and 1 ≤ i ≤ j ≤ s , we write γ (v ij ), γ A (v ij ) and (cid:101) γ A (v ij ) to mean γ ( v i ) ⊗ · · · ⊗ γ ( v j ) , γ ( v i ) ⊗ A · · · ⊗ A γ ( v j ) and (cid:101) γ ( v i ) ⊗ A · · · ⊗ A (cid:101) γ ( v j ) , respectively.(7) We let ν : A → E denote the map induced by ν .(8) Given a , . . . , a r ∈ A and 1 ≤ i < j ≤ r , we set ν ( a ij ) := ν ( a i ) ⊗ · · · ⊗ ν ( a j ).(9) We let (cid:101) µ E : E ⊗ A E → E, µ E : E ⊗ E → E and µ A : A ⊗ A → A denote the maps induced by the multiplication maps µ E and µ A .For all s ≥
0, we set Y (cid:48) s := E ⊗ A E ⊗ sA ⊗ A E and Y s := E ⊗ A (cid:101) E ⊗ sA ⊗ A E, while, for all r, s ≥
0, we set X (cid:48) rs := E ⊗ A E ⊗ sA ⊗ A ⊗ r ⊗ E and X rs := E ⊗ A (cid:101) E ⊗ sA ⊗ A ⊗ r ⊗ E. Remark . For each s ≥
0, we consider E ⊗ k V ⊗ sk as a right A -module via( x ⊗ k v s ) · a := (cid:88) x · a ( l ) ⊗ k v ( l )1 s , where (cid:80) l a ( l ) ⊗ k v ( l )1 s := χ (v s ⊗ k a ). Proposition 2.3.
The following assertions hold: (1)
For each s ≥ the map θ s : (cid:0) E ⊗ k V ⊗ sk (cid:1) ⊗ A E −→ Y (cid:48) s , given by θ s (cid:0) (1 E ⊗ k v s ) ⊗ A E (cid:1) := 1 E ⊗ A γ A (v s ) ⊗ A E , is an isomorphism of E -bimodules. (2) For each r, s ≥ the map θ rs : (cid:0) E ⊗ k V ⊗ sk (cid:1) ⊗ A ⊗ r ⊗ E −→ X (cid:48) rs , given by θ rs (cid:0) (1 E ⊗ k v s ) ⊗ a r ⊗ E (cid:1) := 1 E ⊗ A γ A (v s ) ⊗ a r ⊗ E , is an isomorphism of E -bimodules. CO)HOMOLOGY OF CROSSED PRODUCTS IN WEAK CONTEXTS 9
Proof.
We prove item (2) and leave the proof of item (1), which is similar, to the reader. ByRemark 1.2 we have a canonical isomorphism X (cid:48) rs = E ⊗ A ( A ⊗ k V ) ⊗ sA ⊗ A ⊗ r ⊗ E (cid:39) (cid:0) E ⊗ k V ⊗ sk (cid:1) ⊗ A ⊗ r ⊗ E. (2.10)Let a , . . . , a r ∈ A and v , . . . , v s ∈ V . Since, for all v ∈ V , γ ( v ) = (1 ⊗ k v ) · A ≡ ⊗ k v (mod X χ ) , the isomorphism (2.10) maps 1 E ⊗ A γ A (v s ) ⊗ a r ⊗ E to (1 E ⊗ k v s ) ⊗ a r ⊗ E , and so it isthe inverse of θ rs . (cid:3) Remark . The following assertions hold:(1) Y (cid:48) = Y (cid:39) E ⊗ A E and X (cid:48) r = X r (cid:39) E ⊗ A ⊗ r ⊗ E as E -bimodules.(2) X (cid:48) s (cid:39) E ⊗ A E ⊗ sA ⊗ E and X s (cid:39) E ⊗ A (cid:101) E ⊗ sA ⊗ E as E -bimodules.Consider the diagram of E -bimodules and E -bimodule maps... Y X X · · · Y X X · · · Y X X · · · , − ∂ − ∂ − ∂ υ d d υ d d υ d d where ( Y ∗ , ∂ ∗ ) is the normalized bar resolution of the A -algebra E , introduced in [11]; for each s ≥
0, the complex ( X ∗ s , d ∗ s ) is ( − s -times the normalized bar resolution of the K -algebra A ,tensored on the left over A with E ⊗ A (cid:101) E ⊗ sA , and on the right over A with E ; and for each s ≥ υ s is the canonical surjection. Proposition 2.5.
Each one of the rows of the above diagram is contractible as a ( E, K ) -bimodulecomplex. A contracting homotopy σ s : Y s → X s and σ r +1 ,s : X rs → X r +1 ,s for r ≥ ,of the s -th row, is given by σ s ( x ⊗ A (cid:101) x s ⊗ A x s +1 ) := (cid:88) x ⊗ A (cid:101) x s · a ( l ) ⊗ γ ( v ( l ) ) and σ r +1 ,s ( x ⊗ A (cid:101) x s ⊗ a r ⊗ x s +1 ) := ( − r + s +1 (cid:88) x ⊗ A (cid:101) x s ⊗ a r ⊗ a ( l ) ⊗ γ ( v ( l ) ) , where (cid:80) l a ( l ) ⊗ k v ( l ) = x s +1 .Proof. To begin not that the morphism σ s is well defined, since˜ σ s ( x ⊗ A (cid:101) x s ⊗ k a · x s +1 ) = (cid:88) x ⊗ A (cid:101) x s · aa ( l ) ⊗ γ ( v ( l ) ) = ˜ σ s ( x ⊗ A (cid:101) x s · a ⊗ k x s +1 ) . In order to finish the proof of the proposition we must check that υ s ◦ σ s = id Y s , (2.11) σ s ◦ υ s + d s ◦ σ s = id X s (2.12) and σ rs ◦ d rs + d r +1 ,s ◦ σ r +1 ,s = id X rs for r >
0. (2.13)A direct computation shows that υ (cid:0) σ ( x ⊗ A (cid:101) x s ⊗ A x s +1 ) (cid:1) = (cid:88) x ⊗ A (cid:101) x s ⊗ A a ( l ) · γ ( v ( l ) ) ,σ (cid:0) υ ( x ⊗ A (cid:101) x s ⊗ x s +1 ) (cid:1) = (cid:88) l x ⊗ A (cid:101) x s · a ( l ) ⊗ γ ( v ( l ) ) ,σ (cid:0) d ( x ⊗ A (cid:101) x s ⊗ a r ⊗ x s +1 ) (cid:1) = (cid:88) ( − r x ⊗ A (cid:101) x s ⊗ A b (cid:48) (cid:0) A ⊗ a r ⊗ A (cid:1) ⊗ A a ( l ) ⊗ γ ( v ( l ) )and d (cid:0) σ ( x ⊗ A (cid:101) x s ⊗ a r ⊗ x s +1 ) (cid:1) = (cid:88) ( − r +1 x ⊗ A (cid:101) x s ⊗ A b (cid:48) (cid:0) A ⊗ a r ⊗ a ( l ) ⊗ A (cid:1) ⊗ A γ ( v ( l ) ) , where b (cid:48)∗ is the boundary map of the normalized Hochschild resolution of the K -algebra A . So,equalities (2.11), (2.12) and (2.13) follow immediately from equality (1.5). (cid:3) Remark . Using equality (1.5) it is easy to see that if x s +1 ∈ ( K ⊗ k V ) ∩ E , then, for all a ∈ A , σ s ( x ⊗ A (cid:101) x s ⊗ A a · x s +1 ) = x ⊗ A (cid:101) x s · a ⊗ x s +1 (2.14)and σ r +1 ,s ( x ⊗ A (cid:101) x s ⊗ a r ⊗ a · x s +1 ) = ( − r + s +1 x ⊗ A (cid:101) x s ⊗ a r ⊗ a ⊗ x s +1 . (2.15) Remark . The complex of E -bimodules E Y Y Y Y Y Y · · · − (cid:101) µ E − ∂ − ∂ − ∂ − ∂ − ∂ − ∂ is contractible as a complex of ( E, A )-bimodules. A chain contracting homotopy σ − : E → Y and σ − s +1 : Y s → Y s +1 for s ≥ , is given by σ − s +1 ( x ⊗ A (cid:101) x s ⊗ A x s +1 ) := ( − s x ⊗ A (cid:101) x ,s +1 ⊗ A E . Notation 2.8.
In the sequel we will use the following notations:(1) Let r, s ≥ ≤ u ≤ r . For each k -subalgebra R of A , we let L urs ( R ) denote the( A, K )-subbimodule of X rs generated by the simple tensors of the form1 E ⊗ A (cid:101) γ A (v s ) ⊗ a r ⊗ E , where v , . . . , v s ∈ V , a , . . . , a r ∈ A and at least u of the a j ’s are in R .(2) Let r, s ≥ ≤ u ≤ r . For each k -subalgebra R of A , we let L urs ( R ) denote the( A, K )-subbimodule of X (cid:48) rs generated by the simple tensors of the form1 E ⊗ A γ A (v s ) ⊗ a r ⊗ E , where v , . . . , v s ∈ V , a , . . . , a r ∈ A and at least u of the a j ’s are in R .(3) Let r, s ≥ ≤ u ≤ r . For each k -subalgebra R of A we set U urs ( R ) := L urs ( R ) · γ ( V ) , W urs ( R ) := L urs ( R ) · A and U urs ( R ) := L urs ( R ) · γ ( V ) Remark . Note that for 0 ≤ u ≤ r and all R ⊆ A , L urs ( A ) = L rs ( R ) , U urs ( A ) = U rs ( R ) and W urs ( A ) = W rs ( R ) . To abbreviate expressions we will write L rs , U rs and W rs instead of L rs ( A ), U rs ( A ) and W rs ( A ),respectively. CO)HOMOLOGY OF CROSSED PRODUCTS IN WEAK CONTEXTS 11
For r ≥ ≤ l ≤ s , we define E -bimodule maps d lrs : X rs → X r + l − ,s − l recursively on l and r , by: d l ( z ) := σ ◦ ∂ ◦ υ ( z ) if l = 1 and r = 0, − σ ◦ d ◦ d ( z ) if l = 1 and r > − (cid:80) l − j =1 σ ◦ d l − j ◦ d j ( z ) if 1 < l and r = 0, − (cid:80) l − j =0 σ ◦ d l − j ◦ d j ( z ) if 1 < l and r > z ∈ E · L rs . Theorem 2.10.
The chain complex
E X X X X X · · · , µ E d d d d d where X n := (cid:77) r + s = n X rs 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 , is a Υ -relative projective resolution of E .Proof. This is an immediate consequence of [13, Corollary A2]. (cid:3)
In order to carry out our computations we also need to give an explicit contracting homotopyof this resolution. For this we define (
E, K )-bimodule maps σ ll,s − l : Y s −→ X l,s − l and σ lr + l +1 ,s − l : X rs −→ X r + l +1 ,s − l recursively on l , by: σ lr + l +1 ,s − l := − l − (cid:88) i =0 σ ◦ d l − i ◦ σ i (0 < l ≤ s and r ≥ − . Proposition 2.11.
A contracting homotopy σ : E → X and σ n +1 : X n → X n +1 for n ≥ , of the resolution introduced in Theorem 2.10, is given by σ ( x ) := − σ ◦ σ − ( x ) and σ n +1 ( x ) := − n +1 (cid:88) l =0 σ ll,n − l +1 ◦ σ − n +1 ◦ υ n ( x ) + n (cid:88) l =0 σ ll +1 ,n − l ( x ) if x ∈ X n , n − r (cid:88) l =0 σ lr + l +1 ,n − r − l ( x ) if x ∈ X r,n − r with r > .Proof. This is a direct consequence of [13, Corollary A2]. (cid:3)
Notation 2.12.
Given a right A -submodule E of E and a left A -submodule E of E , we let E ¯ ⊗ A (cid:101) E ⊗ sA ¯ ⊗ A E denote the image of the canonical map E ⊗ A (cid:101) E ⊗ sA ⊗ A E −→ E ⊗ A (cid:101) E ⊗ sA ⊗ A E . Remark . By its very definition, σ ( ν ( A ) ¯ ⊗ A (cid:101) E ⊗ sA ¯ ⊗ A E ) ⊆ U s and σ ( L rs · E ) ⊆ U r +1 ,s for all r, s ≥ d l ( E · L rs ) ⊆ E · U r + l − ,s − l for all r ≥ ≤ l ≤ s . (2.16) Remark . By Remark 2.13 and the definition of σ l , we have σ l ( Y s ) ⊆ E · U l,s − l and σ l ( X rs ) ⊆ E · U r + l +1 ,s − l for all r ≥ ≤ l ≤ s . Remark . By Remark 2.14 and the definition of σ , we have σ ( X r,n − r ) ⊆ (cid:40)(cid:76) nl = − E · U l +1 ,n − l if r = 0, (cid:76) n − rl =0 E · U r + l +1 ,n − r − l if r > K is stable under χ and that 1 E ∈ K ⊗ k V . By the first conditionwe have γ ( V ) ⊆ K ⊗ k V . So, by equality (2.15), σ ( E · U rs ) = 0 for all r, s ≥
0. (2.17)On the other hand, by the second condition and equalities (2.14) and (2.15), σ (cid:0) ν ( A ) ¯ ⊗ A (cid:101) E ⊗ sA ¯ ⊗ A ν ( A ) (cid:1) ⊆ L s for all r, s ≥
0. (2.18)and σ ( W rs ) ⊆ L r +1 ,s for all s ≥
0. (2.19)
Remark . By Remark 2.14 and equality (2.17), we have σ ◦ σ = 0. Remark . By Remark 2.14, the definitions of σ , υ and σ − , and the inclusion in (2.18), σ ( X n ) ⊆ E · L ,n +1 ⊕ n (cid:77) l =0 E · U l +1 ,n − l . Proposition 2.18.
The homotopy σ found in Proposition 2.11 satisfies σ n +1 ( x ) = − σ ,n +1 ◦ σ − n +1 ◦ υ n ( x ) + n (cid:88) l =0 σ ll +1 ,n − l ( x ) for all x ∈ X n .Proof. By the definitions of σ , υ and σ − , it suffices to prove that σ l (cid:0) E ¯ ⊗ A (cid:101) E ⊗ n +1 A ¯ ⊗ A ν ( A ) (cid:1) = 0 for all l ≥ l ≥ σ l ( x ) = − l − (cid:88) i =0 σ ◦ d l − i ◦ σ i ( x ) = − σ ◦ d l ◦ σ ( x ) for each x ∈ E ¯ ⊗ A (cid:101) E ⊗ n +1 A ¯ ⊗ A ν ( A ).But, by the inclusions in (2.16) and (2.18) σ ◦ d l ◦ σ ( x ) ∈ σ ◦ d l ( E · L n ) ⊆ σ (cid:0) E · U l +1 ,n − l ) , which, by equality (2.17) is zero. This contradiction shows that the proposition is true. (cid:3) Proposition 2.19.
The contracting homotopy σ satisfies σ ◦ σ = 0 .Proof. By Proposition 2.18 it will be sufficient to see that σ ◦ σ − ◦ υ ◦ σ ◦ σ − ◦ υ = 0 and σ l ◦ σ l (cid:48) = 0 for all l, l (cid:48) ≥
0. The first equality is true because υ ◦ σ = id and σ − ◦ σ − = 0.Since σ ◦ σ = 0, in order to prove the second one it suffices to show that there exists a map ς l such that σ l = σ ◦ ς l ◦ σ for all l ≥
1. But this follows by an easy inductive argument. (cid:3)
Proposition 2.20.
For all r ≥ and ≤ l ≤ s , we have σ l (cid:0) E · W rs ) = 0 . CO)HOMOLOGY OF CROSSED PRODUCTS IN WEAK CONTEXTS 13
Proof.
We proceed by induction on l . Suppose that this assertion is true for all l < l . By thedefinition of σ l and the inductive hypothesis, σ l ( x ) = − l − (cid:88) i =0 σ ◦ d l − i ◦ σ i ( x ) = σ ◦ d l ◦ σ ( x ) for each x ∈ E · W rs .But, by the inclusions in (2.16) and (2.19) σ ◦ d l ◦ σ ( x ) ∈ σ ◦ d l ( E · L r +1 ,s ) ⊆ σ ( E · U r + l +1 ,s − l ) , which by equality (2.17) is zero. (cid:3) Remark . Since σ − ◦ υ ( E · W n ) = 0, by the definition of σ and Proposition 2.20, σ ( x ) = σ ( x ) for all 0 ≤ r ≤ n and all x ∈ E · W r,n − r . Consequently, by the inclusion in (2.19), σ ( W r,n − r ) ⊆ L r +1 ,n − r for all 0 ≤ r ≤ n . (2.20) Definition 2.22.
For each r ≥ s >
0, we define E -bimodule maps µ (cid:48) i : X (cid:48) rs −→ X (cid:48) r,s − for 0 ≤ i ≤ s ,by µ (cid:48) i ( x s ⊗ a s +1 ,s + r ⊗ E ) := x x ⊗ A x s ⊗ a s +1 ,s + r ⊗ E if i = 0, x ,i − ⊗ A x i x i +1 ⊗ A x i +2 ,s ⊗ a s +1 ,s + r ⊗ E if 0 < i < s , (cid:80) l x ,s − ⊗ A x s − ν ( a s ) ⊗ a ( l ) s +1 ,s + r ⊗ γ ( v ( l ) s ) if i = s ,where a , . . . , a s + r ∈ A , v , . . . , v s ∈ V , x j := ν ( a j ) γ ( v j ) for 0 ≤ j ≤ s , and (cid:88) l a ( l ) s +1 ,s + r ⊗ k v ( l ) s := χ (cid:0) v s ⊗ k a s +1 ,s + r (cid:1) . Theorem 2.23.
The map d : X rs → X r,s − is induced by (cid:80) si =0 ( − i µ (cid:48) i .Proof. For 0 ≤ i ≤ s , let x i = ν ( a i ) γ ( v i ) with a i ∈ A and v i ∈ V . Assume that r = 0. Since1 E ∈ K ⊗ k V and γ ( v s ) ⊆ K ⊗ k V , from equality (2.14) it follows that d ( x ⊗ A (cid:101) x s ⊗ E ) = σ ◦ ∂ ◦ υ ( x ⊗ A (cid:101) x s ⊗ E )= x x ⊗ A (cid:101) x s ⊗ E + s − (cid:88) i =1 ( − i x ⊗ A (cid:101) x ,i − ⊗ A x i x i +1 (cid:102) ⊗ A (cid:101) x i +2 ,s ⊗ E + ( − s x ⊗ A (cid:101) x ,s − ⊗ A x s ν ( a s ) (cid:102) ⊗ γ ( v s ) . Assume now that r > r − s . Let T ∈ E ⊗ A (cid:101) E ⊗ sA ⊗ A ⊗ r − ⊗ E be defined by T := x ⊗ A (cid:101) x ,s − ⊗ A x s ν ( a s +1 ) (cid:102) ⊗ a s +2 ,s + r ⊗ E + r − (cid:88) i =1 ( − i x ⊗ A (cid:101) x s ⊗ a s +1 ,s + i − ⊗ a s + i a s + i +1 ⊗ a i +2 ,r ⊗ E , where a s +1 , . . . , a s + r ∈ A . By the very definition of d , σ ◦ d ( T ) = (cid:40) σ ◦ σ ◦ ∂ ◦ υ ( T ) if r = 1, − σ ◦ σ ◦ d ◦ d ( T ) if r > which, by Remark 2.16 and the definition of d , implies d (cid:0) x ⊗ A (cid:101) x s ⊗ a s +1 ,s + r ⊗ E (cid:1) = ( − s + r +1 σ ◦ d (cid:0) x ⊗ A (cid:101) x s ⊗ a s +1 ,s + r − ⊗ ν ( a s + r ) (cid:1) . Now the theorem for d rs can be obtained easily by induction on r , using that 1 E ∈ K ⊗ k V and γ ( V ) ⊆ K ⊗ k V , the first equality in Theorem 1.6(9) and equality (2.15). (cid:3) In the following theorem we assume that V is a non unitary associative algebra, a non counitarycoassociative coalgebra such that( vw ) (1) ⊗ k ( vw ) (2) = v (1) w (1) ⊗ k v (2) w (2) for all v, w ∈ V ,where we are using the Sweedler notation. Assume also that there exist maps ρ : V ⊗ k A → A and f : V ⊗ k V → A such that γ ( v ) ν ( a ) = v (1) · a ⊗ k v (2) and γ ( v ) γ ( w ) = f ( v (1) ⊗ k w (1) ) ⊗ k v (2) w (2) , (2.21)where v · a := ρ ( v ⊗ k a ). We will use the following notation: Given v ∈ V and a , . . . , a r ∈ A , weset v · a r := v (1) · a ⊗ · · · ⊗ v ( r ) · a r ∈ A ⊗ r . The previous conditions are satisfied for the crossed products of algebras by weak bialgebras. SoTheorems 3.6 and 4.6 (which are immediate consequences of the following result) apply in thiscontext. We will use these results in [14].
Theorem 2.24.
The map d : X rs → X r +1 ,s − is given by d ( z rs ) = ( − s +1 E ⊗ A (cid:101) γ A (v ,s − ) ⊗ T ( v (1) s − , v (1) s , a r ) ⊗ γ ( v (2) s − v (2) s ) , where z rs := 1 E ⊗ A (cid:101) γ A (v s ) ⊗ a r ⊗ E , with v , . . . , v s ∈ V and a , . . . , a r ∈ A , and T ( v s − , v s , a r ) := r (cid:88) i =0 ( − i v (1) s − · ( v (1) s · a i ) ⊗ f ( v (2) s − ⊗ k v (2) s ) ⊗ v (3) s − v (3) s · a i +1 ,r . Proof.
We proceed by induction on r . Assume r = 0. By definition d ( z s ) = − σ ◦ d ◦ d ( z s ).Since 1 E ∈ K ⊗ k V and γ ( V ) ⊆ K ⊗ k V , by Theorem 2.23 and equality (2.15), we have d ( z s ) = σ (cid:0) E ⊗ A (cid:101) γ A (v ,s − ) ⊗ γ ( v s − ) γ ( v s ) (cid:1) . Consequently, by the second equalities (2.15) and (2.21), d ( z s ) = ( − s +1 E ⊗ A (cid:101) γ A (v ,s − ) ⊗ f ( v (1) s − ⊗ k v (1) s ) ⊗ γ (cid:0) v (2) s − v (2) s (cid:1) , as desired. Suppose now that the result is true for r . By definition d ( z r +1 ,s ) = − σ ◦ ( d ◦ d + d ◦ d )( z r +1 ,s ) . (2.22)A similar computation as above (using the inductive hypothesis) shows that σ ◦ d ◦ d ( z r +1 ,s ) = ( − r σ (cid:0) E ⊗ A (cid:101) γ A (v ,s − ) ⊗ T ( v (1) s − , v (2) s , a r ) ⊗ γ ( v (2) s − v (2) s ) ν ( a r +1 ) (cid:1) and σ ◦ d ◦ d ( z r +1 ,s ) = − σ (cid:0) E ⊗ A (cid:101) γ A (v ,s − ) ⊗ v (1) s − · ( v (1) s · a ,r +1 ) ⊗ γ ( v (1) s − ) γ ( v (1) s ) (cid:1) . The formula for d ( z r +1 ,s ) follows now from equalities (2.15), (2.21) and (2.22). (cid:3) Proposition 2.25.
Let R be a k -subalgebra of A . If R is stable under χ , then µ (cid:48) i ( L urs ( R )) ⊆ E · L ur,s − ( R ) if i = 0 , L ur,s − ( R ) if < i < s , U ur,s − ( R ) if i = s . CO)HOMOLOGY OF CROSSED PRODUCTS IN WEAK CONTEXTS 15
Proof.
For i = 0 this is trivial, while for 0 < i < s it follows using equality (1.6) repeatedly.Finally for i = s , it follows from the very definition of µ (cid:48) s using that R is stable under χ . (cid:3) Theorem 2.26.
Let R be a k -subalgebra of A . If R is stable under χ and F ( V ⊗ k V ) ⊆ R ⊗ k V ,then the following assertions hold: (1) d ( L urs ( R )) ⊆ L u − r − ,s ( R ) · R + L ur − ,s ( R ) · A , for each r ≥ , s ≥ and < u ≤ r . (2) d ( L urs ( R )) ⊆ E · L ur,s − ( R ) + U ur,s − ( R ) , for each r ≥ , s ≥ and ≤ u ≤ r . (3) d l ( L urs ( R )) ⊆ U u + l − r + l − ,s − l ( R ) , for each r ≥ , s ≥ , ≤ u ≤ r and ≤ l ≤ s .Proof. Item (1) is trivial and item (2) is an immediate consequence of Theorem 2.23 and Propo-sition 2.25. It remains to prove item (3). Assume that s ≥ l > r ≥ d jr (cid:48) s (cid:48) with j ≤ min( l − , s (cid:48) ), or with j = l ≤ s (cid:48) and r (cid:48) < r , or with j = l ≤ s (cid:48) < s and r (cid:48) = r . We claim that l − (cid:88) j =1 σ ◦ d l − j ◦ d j (cid:0) L urs ( R ) (cid:1) ⊆ U u + l − r + l − ,s − l ( R ) . By item (2) and the inductive hypothesis l − (cid:88) j =1 σ ◦ d l − j ◦ d j (cid:0) L urs ( R ) (cid:1) ⊆ σ ◦ d l − (cid:0) E · L ur,s − ( R ) (cid:1) + l − (cid:88) j =1 σ ◦ d l − j (cid:0) U u + j − r + j − ,s − j ( R ) (cid:1) . So, by the inclusion in (2.16) and equality (2.17), l − (cid:88) j =1 σ ◦ d l − j ◦ d j (cid:0) L urs ( R ) (cid:1) ⊆ l − (cid:88) j =1 σ ◦ d l − j (cid:0) U u + j − r + j − ,s − j ( R ) (cid:1) . (2.23)Since, by item (2), d (cid:0) U u + l − r + l − ,s − l +1 ( R ) (cid:1) ⊆ E · U u + l − r + l − ,s − l ( R ) + L u + l − r + l − ,s − l ( R ) · γ ( V ) γ ( V ) , and, by the inductive hypothesis, d l − j (cid:0) U u + j − r + j − ,s − j ( R ) (cid:1) ⊆ L u + l − r + l − ,s − l ( R ) · γ ( V ) γ ( V ) for l − j > , from the inclusion (2.23) we obtain that l − (cid:88) j =1 σ ◦ d l − j ◦ d j (cid:0) L urs ( R ) (cid:1) ⊆ σ (cid:0) E · U u + l − r + l − ,s − l ( R ) (cid:1) + σ (cid:0) L u + l − r + l − ,s − l ( R ) · γ ( V ) γ ( V ) (cid:1) . Thus, the claim follows using equality (2.17) and the fact that, by the second equality in Theo-rem 1.6(9), Proposition 1.11 and equality (2.15), σ ( L u + l − r + l − ,s − l ( R ) · γ ( V ) γ ( V )) ⊆ U u + l − r + l − ,s − l ( R ) . Consequently, by the very definition of d l we have d l ( L urs ( R )) ⊆ (cid:40) U u + l − r + l − ,s − l ( R ) if r = 0 σ ◦ d l ◦ d ( L urs ( R )) + U u + l − r + l − ,s − l ( R ) if r > σ ◦ d l ◦ d ( L urs ( R )) ⊆ U u + l − r + l − ,s − l ( R ) for each r > σ ◦ d l ◦ d ( L urs ( R )) ⊆ σ (cid:16) U u + l − r + l − ,s − l ( R ) · R + U u + l − r + l − ,s − l ( R ) · A (cid:17) ⊆ σ (cid:16) L u + l − r + l − ,s − l ( R ) · γ ( V ) ν ( R ) + L u + l − r + l − ,s − l ( R ) · E (cid:17) , and so, by the definition of σ , Lemma 1.13, the fact that γ ( V ) ⊆ K ⊗ k V and equality (2.15), σ ◦ d l ◦ d ( L urs ( R )) ⊆ U u + l − r + l − ,s − l ( R ) , as we want. (cid:3) Corollary 2.27. If F takes its values in K ⊗ k V , then ( X ∗ , d ∗ ) is the total complex of the doublecomplex ( X ∗∗ , d ∗∗ , d ∗∗ ) . Let ( E ⊗ E ⊗ ∗ ⊗ E, b (cid:48)∗ ) be the normalized bar resolution of the K -algebra E . As it is well known,the complex E E ⊗ E E ⊗ E ⊗ E E ⊗ E ⊗ ⊗ E · · · , µ E b (cid:48) b (cid:48) b (cid:48) is contractible as a complex of ( E, K )-bimodules, with contracting homotopy ξ : E −→ E ⊗ E and ξ n +1 : E ⊗ E ⊗ n ⊗ E −→ E ⊗ E ⊗ n +1 ⊗ E for n ≥ ξ n +1 ( x ) := ( − n +1 x ⊗ E . Let φ ∗ : ( X ∗ , d ∗ ) −→ ( E ⊗ E ⊗ ∗ ⊗ E, b (cid:48)∗ ) and ψ ∗ : ( E ⊗ E ⊗ ∗ ⊗ E, b (cid:48)∗ ) −→ ( X ∗ , d ∗ )be the morphisms of E -bimodule complexes, recursively defined by φ := id , ψ := id ,φ n ( x ⊗ E ) := ξ n ◦ φ n − ◦ d n ( x ⊗ E ) for n > ψ n ( y ⊗ E ) := σ n ◦ ψ n − ◦ b (cid:48) ( y ⊗ E ) for n > Notation 2.28.
Given a right K -submodule E of E , we let E ⊗ E ⊗ n ¯ ⊗ E denote the image ofthe canonical map E ⊗ E ⊗ n ⊗ E −→ E ⊗ E ⊗ n ⊗ E . Proposition 2.29. ψ ∗ ◦ φ ∗ = id ∗ and φ ∗ ◦ ψ ∗ is homotopically equivalent to the identity map.More precisely, the one degree map ω ∗ +1 : E ⊗ E ⊗ ∗ ⊗ E −→ E ⊗ E ⊗ ∗ +1 ⊗ E, recursively defined by ω := 0 and ω n +1 ( y ⊗ E ) := ξ n +1 ◦ ( φ n ◦ ψ n − id − ω n ◦ b (cid:48) n )( y ⊗ E ) for n ≥ ,is a homotopy from φ ∗ ◦ ψ ∗ to id ∗ .Proof. We prove both assertions by induction. Clearly ψ ◦ φ = id. Assume that ψ n ◦ φ n = id.Since φ n +1 ( E · L n +1 − s,s ) = ξ n +1 ◦ φ n ◦ d n +1 ( E · L n +1 − s,s ) ⊆ E ⊗ E ⊗ n +1 ¯ ⊗ ν ( K ) , on E · L n +1 − s,s , we have ψ n +1 ◦ φ n +1 = σ n +1 ◦ ψ n ◦ b (cid:48) n +1 ◦ ξ n +1 ◦ φ n ◦ d n +1 = σ n +1 ◦ ψ n ◦ φ n ◦ d n +1 − σ n +1 ◦ ψ n ◦ ξ n ◦ b (cid:48) n ◦ φ n ◦ d n +1 = σ n +1 ◦ d n +1CO)HOMOLOGY OF CROSSED PRODUCTS IN WEAK CONTEXTS 17 = id n +1 − d n +2 ◦ σ n +2 , So, to conclude that ψ n +1 ◦ φ n +1 = id it suffices to check that σ n +2 ( E · L n +1 − s,s ) = 0. But bythe definition of σ n +2 and the arguments in the proof of Proposition 2.19, this will follow if weprove that σ − ◦ υ ( E · L ,n +1 ) = 0 and σ ( E · L n +1 − s,s ) = 0 . The first equality is clear and the second one is a particular case of equality (2.17).Now we prove the second assertion. Clearly φ ◦ ψ − id = 0 = b (cid:48) ◦ w . Let U n := φ n ◦ ψ n − id and T n := U n − ω n ◦ b (cid:48) n . Assuming that b (cid:48) n ◦ ω n + ω n − ◦ b (cid:48) n − = U n − , we get that, on E ⊗ E ⊗ n ¯ ⊗ ν ( K ), b (cid:48) n +1 ◦ ω n +1 + ω n ◦ b (cid:48) n = b (cid:48) n +1 ◦ ξ n +1 ◦ T n + ω n ◦ b (cid:48) n = T n − ξ n ◦ b (cid:48) n ◦ T n + ω n ◦ b (cid:48) n = U n − ξ n ◦ b (cid:48) n ◦ U n + ξ n ◦ b (cid:48) n ◦ ω n ◦ b (cid:48) n = U n − ξ n ◦ U n − ◦ b (cid:48) n + ξ n ◦ b (cid:48) n ◦ ω n ◦ b (cid:48) n = U n − ξ n ◦ U n − ◦ b (cid:48) n + ξ n ◦ U n − ◦ b (cid:48) n − ξ n ◦ w n − ◦ b (cid:48) n − ◦ b (cid:48) n = U n . Hence, b (cid:48) n +1 ◦ ω n +1 + ω n ◦ b (cid:48) n = U n on E ⊗ E ⊗ n ⊗ E . (cid:3) Proposition 2.30.
We have ψ ( x ⊗ x n ⊗ E ) = ( − n σ ◦ ψ ( x ⊗ x ,n − ⊗ x n ) for all n ≥ .Proof. By definition ψ ( E ⊗ E ⊗ n − ¯ ⊗ ν ( K )) ⊆ Im( σ ). So, by Proposition 2.19, ψ ( x ⊗ x n ⊗ E ) = σ ◦ ψ ◦ b (cid:48) ( x ⊗ x n ⊗ E ) = ( − n σ ◦ ψ ( x ⊗ x ,n − ⊗ x n ) , as desired. (cid:3) Remark . Let x , . . . , x n ∈ E and let x := x ⊗ x n ⊗ E . Since ω (cid:0) E ⊗ E ⊗ n − ¯ ⊗ ν ( K ) (cid:1) ⊆ E ⊗ E ⊗ n ¯ ⊗ ν ( K )and ξ vanishes on E ⊗ E ⊗ n ¯ ⊗ ν ( K ), ω ( x ) = ξ (cid:0) φ ( ψ ( x )) − x − ω ( b (cid:48) ( x )) (cid:1) = ξ (cid:0) φ ( ψ ( x )) − ( − n ω ( x ⊗ x ,n − ⊗ x n ) (cid:1) . Notation 2.32.
For each n, i ≥ F i ( X n ) := (cid:77) ≤ s ≤ i X n − s,s and we let F i (cid:0) E ⊗ n (cid:1) denote the K -subbimodule of E ⊗ n generated by the tensors x n such thatat least n − i of the x j ’s belong to ν ( A ). Furthermore, given a right K -submodule E of E anda left K -submodule E of E , we let E ¯ ⊗ F i ( E ⊗ n ) ¯ ⊗ E denote the image of the canonical map E ⊗ F i (cid:0) E ⊗ n (cid:1) ⊗ E −→ E ⊗ E ⊗ n ⊗ E . Moreover we write F i (cid:0) E ⊗ E ⊗ n ⊗ E (cid:1) := E ¯ ⊗ F i ( E ⊗ n ) ¯ ⊗ E . The normalized bar resolution (cid:0) E ⊗ E ⊗ ∗ ⊗ E, b (cid:48)∗ (cid:1) and the resolution ( X ∗ , d ∗ ) are filtered by F (cid:0) E ⊗ E ⊗ ∗ ⊗ E (cid:1) ⊆ F (cid:0) E ⊗ E ⊗ ∗ ⊗ E (cid:1) ⊆ F (cid:0) E ⊗ E ⊗ ∗ ⊗ E (cid:1) ⊆ . . . and F ( X ∗ ) ⊆ F ( X ∗ ) ⊆ F ( X ∗ ) ⊆ F ( X ∗ ) ⊆ F ( X ∗ ) ⊆ F ( X ∗ ) ⊆ . . . , respectively. Notation 2.33.
Let 0 ≤ i ≤ n and 0 ≤ u ≤ n − i . For each k -subalgebra R of A , we let F iR,u (cid:0) E ⊗ n (cid:1) denote the K -subbimodule of E ⊗ n generated by all the simple tensors x n such that { j : x j / ∈ ν ( A ) ∪ γ ( V ) } = 0 , { j : x j / ∈ ν ( A ) } ≤ i and { j : x j ∈ ν ( R ) } ≥ u. Furthermore, given a right K -submodule E of E and a left K -submodule E of E , we let E ¯ ⊗ F iR,u ( E ⊗ n ) ¯ ⊗ E denote the image of the canonical map E ⊗ F iR,u ( E ⊗ n ) ⊗ E −→ E ⊗ E ⊗ n ⊗ E. Remark . Note that F iR, (cid:0) E ⊗ n (cid:1) and E ¯ ⊗ F iR, (cid:0) E ⊗ n ) ¯ ⊗ E do not depend on R . Definition 2.35.
For s ≥ i ≥
0, we define (cid:122) si : V ⊗ sk ⊗ k A ⊗ iK −→ V ⊗ s − k ⊗ k A ⊗ iK ⊗ K E , by (cid:122) si (v s ⊗ k a i ) := ( − i (cid:88) v ,s − ⊗ k a ( l )1 i ⊗ K γ (cid:0) v ( l ) s (cid:1) , where (cid:80) l a ( l )1 i ⊗ k v ( l ) s := ¯ χ ( v s ⊗ k a i ). Remark . Note that (cid:122) s (v s ) = v ,s − ⊗ k γ ( v s ). Definition 2.37.
For s, r ≥
0, we let Sh sr : V ⊗ sk ⊗ k A ⊗ r −→ E ⊗ r + s denote the map recursivelydefined by:Sh r := ⊗ r ν and Sh sr := r (cid:88) i =0 (cid:16) Sh s − ,i ⊗ E ⊗ r − i +1 (cid:17) ◦ (cid:16) (cid:122) si ⊗ ⊗ r − i ν (cid:17) for s ≥ Remark . Let R be a k -subalgebra of A and let v , . . . , v s ∈ V and a , . . . , a r ∈ A . It is evidentthat if R is stable under χ , then Sh sr (v s ⊗ k a r ) ∈ F sR,u (cid:0) E ⊗ r + s (cid:1) , where u = { i : a i ∈ R } . Proposition 2.39.
Let R be a k -subalgebra of A . If R is stable under χ and F takes its valuesin R ⊗ k V , then φ (cid:0) E ⊗ A (cid:101) γ A (v i ) ⊗ a ,n − i ⊗ E (cid:1) ≡ E ⊗ Sh(v i ⊗ k a ,n − i ) ⊗ E mod ν ( A ) ¯ ⊗ F i − R, ( E ⊗ n ) ¯ ⊗ ν ( K ) , for all v , . . . , v i ∈ V and a , . . . , a n − i ∈ A .Proof. We proceed by induction on n . For n = 0 the proposition is trivial. Assume that n > n −
1. Write x := 1 E ⊗ A (cid:101) γ A (v i ) ⊗ a ,n − i ⊗ E . By item (3) of Theorem 2.26, Remark 2.38, the inductive hypothesis and the definition of ξ , ξ ◦ φ ◦ d l ( x ) ∈ ν ( A ) ¯ ⊗ F i − l +1 R, ( E ⊗ n ) ¯ ⊗ ν ( K ) for all l > φ ( x ) = ξ ◦ φ ◦ d ( x ) + ξ ◦ φ ◦ d ( x ) mod ν ( A ) ¯ ⊗ F i − R, ( E ⊗ n ) ¯ ⊗ ν ( K ) . Note now that, since by the inductive hypothesis, φ ( E · L i (cid:48) ,n − i (cid:48) − ) ⊆ Im( ξ ) ⊆ ker( ξ ), from thedefinition of d it follows that ξ ◦ φ ◦ d ( x ) = ( − n ξ ◦ φ (cid:0) E ⊗ A (cid:101) γ A (v i ) ⊗ a ,n − i − ⊗ ν ( a n − i ) (cid:1) , CO)HOMOLOGY OF CROSSED PRODUCTS IN WEAK CONTEXTS 19 while, by Theorem 2.23 and Proposition 2.25, we have ξ ◦ φ ◦ d ( x ) = (cid:88) ( − i ξ ◦ φ (cid:0) E ⊗ A (cid:101) γ A (v ,i − ) ⊗ a ( l )1 ,n − i ⊗ γ ( v ( l ) i ) (cid:1) , where (cid:80) l a ( l )1 ,n − i ⊗ k v ( l ) i := χ ( v i ⊗ k a ,n − i ). Now the proof can be finished using the inductivehypothesis. (cid:3) Remark . If the hypothesis of Proposition 2.39 are satisfied, then there exist maps φ (cid:48) in : V ⊗ ik ⊗ k A ⊗ n − iK −→ ν ( A ) ¯ ⊗ F i − R, (cid:0) E ⊗ n (cid:1) ¯ ⊗ ν ( K ) for 0 ≤ i ≤ n, such that φ n (cid:0) E ⊗ A (cid:101) γ A (v i ) ⊗ a ,n − i ⊗ E (cid:1) = 1 E ⊗ Sh(v i ⊗ k a ,n − i ) ⊗ E + φ (cid:48) in (v i ⊗ k a ,n − i ) , for all v , . . . , v i ∈ V and a , . . . , a n − i ∈ A (of course φ (cid:48) n = 0 for all n ). Notation 2.41.
Given v , . . . , v i ∈ V and a , . . . , a n − i ∈ A we will write φ (cid:48) in (v i ⊗ k a ,n − i ) inthe form φ (cid:48)(cid:48) in (v i ⊗ k a ,n − i ) ⊗ E . Theorem 2.42.
The maps φ , ψ and ω preserve filtrations.Proof. Since for φ this follows from Proposition 2.39, we only must check it for ψ and ω . Inorder to abbreviate expressions we set F iQ ( X n ) := i (cid:77) s =0 E · U n − s,s for 0 ≤ i ≤ n .Let 0 ≤ i ≤ n . By Remark 2.15 σ (cid:0) F i ( X n ) (cid:1) ⊆ F iQ ( X n +1 ) if i < n , (2.24)while, by Remark 2.17, σ ( X n ) ⊆ E · L ,n +1 + F nQ ( X n +1 ) . (2.25)Furthermore, by the inclusion (2.20), σ ( E · W n − i,i ) ⊆ E · L n +1 − i,i for 0 ≤ i ≤ n . (2.26)We claim that ψ (cid:0) E ¯ ⊗ F i (cid:0) E ⊗ n (cid:1) ¯ ⊗ ν ( K ) (cid:1) ⊆ E · L n − i,i + F i − Q ( X n ) . (2.27)For n = 0 this is trivial. Suppose (2.27) is valid for n . Let x ⊗ x ,n +1 ⊗ E ∈ E ¯ ⊗ F i (cid:0) E ⊗ n +1 (cid:1) ¯ ⊗ ν ( K ) , where 0 ≤ i ≤ n + 1. By Proposition 2.30, to prove (2.27) for n + 1, we only must check that σ ( ψ ( x ⊗ x n ⊗ x n +1 )) ⊆ E · L n +1 − i,i + F i − Q ( X n +1 ) . If x n +1 ∈ ν ( A ), then i ≤ n and using (2.24), (2.26) and the inductive hypothesis, we get σ (cid:0) ψ ( x ⊗ x n ⊗ x n +1 ) (cid:1) = σ (cid:0) ψ ( x ⊗ x n ⊗ E ) x n +1 (cid:1) ⊆ σ (cid:0) E · W n − i,i + F i − ( X n ) (cid:1) ⊆ E · L n +1 − i,i + F i − Q ( X n +1 ) , while if x n +1 / ∈ ν ( A ), then x ⊗ x n ⊗ x n +1 ∈ F i − (cid:0) E ⊗ E ⊗ n ⊗ E (cid:1) , which together with (2.24),(2.25) and the inductive hypothesis, implies that σ (cid:0) ψ ( x ⊗ x n ⊗ x n +1 ) (cid:1) ⊆ σ (cid:0) F i − ( X n ) (cid:1) ⊆ E · L n +1 − i,i + F i − Q ( X n +1 ) , which finishes the proof of the claim. From (2.27) it follows immediately that ψ preserves fil-trations. Next, we prove that ω also does it. More concretely we claim that w (cid:0) E ¯ ⊗ F i (cid:0) E ⊗ n (cid:1) ¯ ⊗ ν ( K ) (cid:1) ⊆ E ¯ ⊗ F i (cid:0) E ⊗ n +1 (cid:1) ¯ ⊗ ν ( K ) . This is trivial for ω , since ω = 0. Assume that ω n does it. Let x := x ⊗ x n ⊗ E ∈ E ¯ ⊗ F i (cid:0) E ⊗ n (cid:1) ¯ ⊗ ν ( K ) . By Remark 2.31, we know that ω ( x ) = ξ ◦ φ ◦ ψ ( x ) + ( − n ξ ◦ ω ( x ⊗ x ,n − ⊗ x n ) . Since, by inclusion (2.27), Proposition 2.39 and the definition of ξ , ξ ◦ φ ◦ ψ ( x ) ∈ ξ ◦ φ (cid:0) E · L n − i,i + F i − Q ( X n ) (cid:1) ⊆ ξ (cid:16) E ¯ ⊗ F iA, ( E ⊗ n ) ¯ ⊗ ν ( K ) + ν ( A ) ¯ ⊗ F i − A, ( E ⊗ n ) ¯ ⊗ ν ( K ) · γ ( V ) (cid:17) ⊆ E ¯ ⊗ F iA, ( E ⊗ n +1 ) ¯ ⊗ ν ( K ) , in order to finish the proof of the claim it remains to check that ξ (cid:0) ω ( x ⊗ x ,n − ⊗ x n ) (cid:1) ∈ E ¯ ⊗ F i (cid:0) E ⊗ n +1 (cid:1) ¯ ⊗ ν ( K ) . If x n ∈ ν ( A ), then, by the inductive hypothesis, ξ (cid:0) ω ( x ⊗ x ,n − ⊗ x n ) (cid:1) = ξ (cid:0) ω ( x ⊗ x ,n − ⊗ E ) x n (cid:1) ⊆ ξ (cid:0) E ¯ ⊗ F i (cid:0) E ⊗ n (cid:1) ¯ ⊗ ν ( A ) (cid:1) ⊆ E ¯ ⊗ F i (cid:0) E ⊗ n +1 (cid:1) ¯ ⊗ ν ( K ) , while if x n / ∈ ν ( A ), then x ⊗ x ,n − ⊗ x n ∈ F i − (cid:0) E ⊗ E ⊗ n − ⊗ E (cid:1) , and so ξ (cid:0) ω ( x ⊗ x ,n − ⊗ x n ) (cid:1) ∈ ξ (cid:0) F i − (cid:0) E ⊗ E ⊗ n − ⊗ E (cid:1)(cid:1) ⊆ E ¯ ⊗ F i (cid:0) E ⊗ n +1 (cid:1) ¯ ⊗ ν ( K ) , as we want. (cid:3) In this section we use freely the notations introduced in Section 2. We assume that all the proper-ties required in that section are fulfilled (for the convenience of the reader we recall that all theseproperties were established at the beginning of Section 2, with the only exception of the fact that K is stable under χ and 1 E ∈ K ⊗ k V , which were established below Remark 2.15). Let M be an E -bimodule. Since ( X ∗ , d ∗ ) is a Υ-relative projective resolution of E , the Hochschild homologyH K ∗ ( E, M ), of the K -algebra E with coefficients in M , is the homology of M ⊗ E e ( X ∗ , d ∗ ). It iswell known that when K is separable, H K ∗ ( E, M ) is the absolute Hochschild homology of E withcoefficients in M . We consider M as an A -bimodule through the map ν : A → E . For r, s ≥ (cid:98) X rs ( M ) := M ⊗ A (cid:101) E ⊗ sA ⊗ A ⊗ r ⊗ . Since (cid:101) E ⊗ A = ν ( A ) and A ⊗ = K , we have (cid:98) X r ( M ) (cid:39) M ⊗ A ⊗ r ⊗ and (cid:98) X s ( M ) (cid:39) M ⊗ A (cid:101) E ⊗ sA ⊗ . (3.28) CO)HOMOLOGY OF CROSSED PRODUCTS IN WEAK CONTEXTS 21
Throughout this section we let [ m ⊗ A (cid:101) x s ⊗ a r ] denote the class of m ⊗ A (cid:101) x s ⊗ a r in (cid:98) X rs ( M ).It is easy to check that the map (cid:98) X rs ( M ) M ⊗ E e X rs [ m ⊗ A (cid:101) x s ⊗ a r ] m ⊗ E e (1 E ⊗ A (cid:101) x s ⊗ a r ⊗ E )is an isomorphism. Let (cid:98) d lrs : (cid:98) X rs ( M ) −→ (cid:98) X r + l − ,s − l ( M ) be the map induced by id M ⊗ E e d lrs .Via the above identifications, M ⊗ E e ( X ∗ , d ∗ ) becomes the complex ( (cid:98) X ∗ ( M ) , (cid:98) d ∗ ), where (cid:98) X n ( M ) := (cid:77) r + s = n (cid:98) X rs ( M ) and (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 . Consequently, we have the following result:
Theorem 3.1.
The Hochschild homology H K ∗ ( E, M ) , of the K -algebra E with coefficients in M ,is the homology of ( (cid:98) X ∗ ( M ) , (cid:98) d ∗ ) .Remark . It follows immediately from Corollary 2.27 that, if F takes its values in K ⊗ k V ,then ( (cid:98) X ∗ ( M ) , (cid:98) d ∗ ) is the total complex of the double complex ( (cid:98) X ∗∗ ( M ) , (cid:98) d ∗∗ , (cid:98) d ∗∗ ). Remark . If K = A , then ( (cid:98) X ∗ ( M ) , (cid:98) d ∗ ) = ( (cid:98) X ∗ ( M ) , (cid:98) d ∗ ).For r, s ≥
0, let X rs ( M ) := M ⊗ A E ⊗ sA ⊗ A ⊗ r ⊗ . Likewise for (cid:98) X rs ( M ), we have canonicalidentifications X rs ( M ) (cid:39) M ⊗ E e X (cid:48) rs . For 0 ≤ i ≤ s , let µ i : X rs ( M ) −→ X r,s − ( M ) be the mapinduced by µ (cid:48) i . It is easy to see that µ i ([ m ⊗ A x s ⊗ a s +1 ,s + r ]) = [ m · x ⊗ A x s ⊗ a s +1 ,s + r ] if i = 0,[ m ⊗ A x ,i − ⊗ A x i x i +1 ⊗ A x i +2 ,s ⊗ a s +1 ,s + r ] if 0 < i < s , (cid:80) l [ γ ( v ( l ) s ) · m ⊗ A x ,s − ⊗ A x s − ν ( a s ) ⊗ a ( l ) s +1 ,s + r ] if i = s ,where m ∈ M , a , . . . , a s + r ∈ A , v , . . . , v s ∈ V , x j := ν ( a j ) γ ( v j ) for 0 ≤ j ≤ s and (cid:88) l a ( l ) s +1 ,s + r ⊗ k v ( l ) s := χ (cid:0) v s ⊗ k a s +1 ,s + r (cid:1) . Notation 3.4.
Given a k -subalgebra R of A and 0 ≤ u ≤ r , we let (cid:98) X urs ( R, M ) denote the k -submodule of (cid:98) X rs ( M ) generated by all the circular simple tensors (cid:2) m ⊗ A (cid:101) γ A (v s ) ⊗ a r (cid:3) , with m ∈ M , v , . . . , v s ∈ V , a , . . . , a r ∈ A , and at least u of the a j ’s in R . Theorem 3.5.
The following assertions hold: (1)
The morphism (cid:98) d : (cid:98) X rs ( M ) → (cid:98) X r − ,s ( M ) is ( − s -times the boundary map of the nor-malized chain Hochschild complex of the K -algebra A with coefficients in M ⊗ A (cid:101) E ⊗ sA ,considered as an A -bimodule via the left and right canonical actions. (2) The morphism (cid:98) d : (cid:98) X rs ( M ) → (cid:98) X r,s − ( M ) is induced by (cid:80) si =0 ( − i (cid:98) µ (cid:48) i . (3) Let R be a k -subalgebra of A . If R is stable under χ and F takes its values in R ⊗ k V ,then (cid:98) d l (cid:0) (cid:98) X rs ( M ) (cid:1) ⊆ (cid:98) X l − r + l − ,s − l ( R, M ) , for each r ≥ and < l ≤ s .Proof. Item (1) follows from the definition of d , item (2), from Theorem 2.23, and item (3),from Theorem 2.26(3). (cid:3) Theorem 3.6.
Under the hypothesis of Theorem 2.24 the map (cid:98) d : (cid:98) X rs ( M ) → (cid:98) X r +1 ,s − ( M ) isgiven by (cid:98) d (cid:0) [ m ⊗ A (cid:101) γ A (v s ) ⊗ a r ] (cid:1) = ( − s +1 (cid:2) γ ( v (2) s − v (2) s ) · m ⊗ A (cid:101) γ A (v ,s − ) ⊗ T ( v (1) s − , v (1) s , a r ) (cid:3) , where m ∈ M , v , . . . , v s ∈ V , a , . . . , a r ∈ A and T ( v s − , v s , a r ) is as in Theorem 2.24.Proof. This follows immediately from Theorem 2.24. (cid:3)
Proposition 3.7. If F takes its values in K ⊗ k V , then A ⊗ A ⊗ r ⊗ M is an E -bimodule via ν ( a ) γ ( v ) · T · ν ( a (cid:48) ) γ ( v (cid:48) ) := (cid:88) aa ( l )0 ⊗ a ( l )1 r ⊗ γ ( v ( l ) ) · m · ν ( a (cid:48) ) γ ( v (cid:48) ) , where r ≥ , T := a ⊗ a r ⊗ m and (cid:80) l a ( l )0 ⊗ a ( l )1 r ⊗ k v ( l ) = ( A ⊗ χ ) ◦ ( χ ⊗ A ⊗ rK )( v ⊗ k a ⊗ a r ) .Proof. It is clear that A ⊗ A ⊗ r ⊗ M is a right E -module. We next prove it is also a left E -module.For this it suffices to show that the left action is unitary and that γ ( v ) ν ( a ) · T = γ ( v ) · (cid:0) ν ( a ) · T (cid:1) and γ ( v ) γ ( v ) · T = γ ( v ) · (cid:0) γ ( v ) · T (cid:1) . (3.29)The first equality in (3.29) follows easily using equality (1.6), while for the second one, it sufficesto check that F γ = F γ = F γ = F γ ν ,where the domain is V ⊗ k V ⊗ k A ⊗ A ⊗ r ⊗ M and the codomain is A ⊗ A ⊗ r ⊗ M . But thisfollows easily by an inductive argument using the twisted module condition and the fact that F takes its values in K ⊗ k V . The proof that the left action is unitary follows the same pattern,but using equality (1.4) instead of the twisted module condition. (cid:3) Remark . Note that H r (cid:0) (cid:98) X ∗ s ( M ) , (cid:98) d ∗ s (cid:1) = H Kr (cid:0) A, M ⊗ A (cid:101) E ⊗ sA (cid:1) and that if F takes its values in K ⊗ k V , then H s (cid:0) (cid:98) X r ∗ ( M ) , (cid:98) d r ∗ (cid:1) = H As (cid:0) E, A ⊗ A ⊗ r ⊗ M (cid:1) . Let (cid:0) M ⊗ E ⊗ ∗ ⊗ , b ∗ (cid:1) be the normalized Hochschild chain complex of the K -algebra E with coe-fficients in M . Recall that there is a canonical identification (cid:0) M ⊗ E ⊗ ∗ ⊗ , b ∗ (cid:1) (cid:39) M ⊗ E e (cid:0) E ⊗ E ⊗ ∗ ⊗ E, b (cid:48)∗ (cid:1) . From now on we let [ m ⊗ x n ] denote the class of m ⊗ x n in M ⊗ E ⊗ n ⊗ . Let (cid:98) φ ∗ : ( (cid:98) X ∗ ( M ) , (cid:98) d ∗ ) −→ (cid:0) M ⊗ E ⊗ ∗ ⊗ , b ∗ (cid:1) and (cid:98) ψ ∗ : (cid:0) M ⊗ E ⊗ ∗ ⊗ , b ∗ (cid:1) −→ ( (cid:98) X ∗ ( M ) , (cid:98) d ∗ )be the morphisms of complexes induced by φ ∗ and ψ ∗ , respectively. By Proposition 2.29 we knowthat (cid:98) ψ ∗ ◦ (cid:98) φ ∗ = id ∗ and (cid:98) φ ∗ ◦ (cid:98) ψ ∗ is homotopically equivalent to the identity map. More precisely ahomotopy (cid:98) ω ∗ +1 , from (cid:98) φ ∗ ◦ (cid:98) ψ ∗ to id ∗ , is the family of maps (cid:16)(cid:98) ω n +1 : M ⊗ E ⊗ n ⊗ −→ M ⊗ E ⊗ n +1 ⊗ (cid:17) n ≥ , induced by (cid:0) ω n +1 : E ⊗ E ⊗ n ⊗ E −→ E ⊗ E ⊗ n +1 ⊗ E (cid:1) n ≥ . CO)HOMOLOGY OF CROSSED PRODUCTS IN WEAK CONTEXTS 23
The complex ( (cid:98) X ∗ ( M ) , (cid:98) d ∗ ) is filtered by F ( (cid:98) X ∗ ( M )) ⊆ F ( (cid:98) X ∗ ( M )) ⊆ F ( (cid:98) X ∗ ( M )) ⊆ F ( (cid:98) X ∗ ( M )) ⊆ F ( (cid:98) X ∗ ( M )) ⊆ . . . , where F i ( (cid:98) X n ( M )) := (cid:76) is =0 (cid:98) X n − s,s ( M ). Hence, by Remark 3.8 there is a spectral sequence E lrs = ⇒ H Kr + s ( E, M ) , with E rs = H Kr (cid:0) A, M ⊗ A (cid:101) E ⊗ sA (cid:1) . (3.30)Let F i (cid:0) M ⊗ E ⊗ n ⊗ (cid:1) be the image of the canonical map M ⊗ F i (cid:0) E ⊗ n (cid:1) ⊗ −→ M ⊗ E ⊗ n ⊗ . Thenormalized Hochschild complex (cid:0) M ⊗ E ⊗ ∗ ⊗ , b ∗ (cid:1) is filtered by F (cid:0) M ⊗ E ⊗ ∗ ⊗ (cid:1) ⊆ F (cid:0) M ⊗ E ⊗ ∗ ⊗ (cid:1) ⊆ F (cid:0) M ⊗ E ⊗ ∗ ⊗ (cid:1) ⊆ . . . . The spectral sequence associated with this filtration is called the homological Hochschild-Serrespectral sequence of E with coefficients in M . Theorem 3.9.
The maps (cid:98) φ ∗ , (cid:98) ψ ∗ and (cid:98) ω ∗ preserve filtrations.Proof. This follows immediately from Theorem 2.42. (cid:3)
Corollary 3.10.
The homological Hochschild-Serre spectral sequence of E with coefficients in M is isomorphic to the spectral sequence (3.30) . Notation 3.11.
For 0 ≤ u ≤ n , i ≥ k -subalgebra R of A , we let M ¯ ⊗ F iR,u ( E ⊗ n ) ¯ ⊗ denote the image of the canonical map M ⊗ F iR,u (cid:0) E ⊗ n (cid:1) ⊗ −→ M ⊗ E ⊗ n ⊗ . Proposition 3.12.
Let R be a k -subalgebra of A . If R is stable under χ and F takes its valuesin R ⊗ k V , then (cid:98) φ (cid:0) [ m ⊗ A (cid:101) γ A (v i ) ⊗ a ,n − i ] (cid:1) = (cid:2) m ⊗ Sh(v i ⊗ k a ,n − i ) (cid:3) + (cid:2) m ⊗ A φ (cid:48)(cid:48) in (v i ⊗ k a ,n − i ) (cid:3) , where m ∈ M , a , . . . , a n − i ∈ A , v , . . . , v i ∈ V and φ (cid:48)(cid:48) in (v i ⊗ k a ,n − i ) is as in Notation 2.41.Proof. This is an immediate consequence of Remark 2.40. (cid:3)
To prove Propositions 3.13 and 3.14 we will need some technical results that we establish inan appendix. For the sake of brevity we set F iR (cid:0) (cid:98) X n ( M ) (cid:1) := (cid:76) is =0 (cid:98) X s,n − s ( R, M ). Proposition 3.13.
Let R be a k -subalgebra of A . Assume that R is stable under χ and that F takes its values in R ⊗ k V . Let m ∈ M , v, v , . . . , v i ∈ V and a, a i +1 , . . . , a n ∈ A . The map (cid:98) ψ n has the following properties: (1) (cid:98) ψ ([ m ⊗ γ (v i ) ⊗ ν ( a i +1 ,n )]) = [ m ⊗ A (cid:101) γ A (v i ) ⊗ a i +1 ,n ] . (2) Let x , . . . , x n ∈ ν ( A ) ∪ γ ( V ) . If there exist indices j < j such that x j ∈ ν ( A ) and x j ∈ γ ( V ) , then (cid:98) ψ ([ m ⊗ x n ]) = 0 . (3) If x = (cid:2) m ⊗ γ (v ,i − ) ⊗ a · γ ( v ) ⊗ ν ( a i +1 ,n ) (cid:3) , then (cid:98) ψ ( x ) ≡ (cid:2) m ⊗ A (cid:101) γ A (v ,i − ) ⊗ A a · (cid:101) γ ( v ) ⊗ a i +1 ,n (cid:3) + (cid:88)(cid:2) γ ( v ( l ) ) · m ⊗ A (cid:101) γ A (v ,i − ) ⊗ a ⊗ a ( l ) i +1 ,n (cid:3) , mod F i − R (cid:0) (cid:98) X n ( M ) (cid:1) , where (cid:80) l a ( l ) i +1 ,n ⊗ k v ( l ) := χ ( v ⊗ k a i +1 ,n ) . (4) If x = (cid:2) m ⊗ γ (v ,j − ) ⊗ a · γ ( v ) ⊗ γ (v j +1 ,i ) ⊗ ν ( a i +1 ,n ) (cid:3) with j < i , then (cid:98) ψ ( x ) ≡ (cid:2) m ⊗ A (cid:101) γ A (v ,j − ) ⊗ A a · (cid:101) γ ( v ) ⊗ A (cid:101) γ A (v j +1 ,i ) ⊗ a i +1 ,n (cid:3) mod F i − R (cid:0) (cid:98) X n ( M ) (cid:1) . (5) If x = (cid:2) m ⊗ γ (v ,i − ) ⊗ ν ( a i,j − ) ⊗ a · γ ( v ) ⊗ ν ( a j +1 ,n ) (cid:3) with j > i , then (cid:98) ψ ( x ) ≡ (cid:88) γ ( v ( l ) ) · m ⊗ A (cid:101) γ A (v ,i − ) ⊗ a i,j − ⊗ a ⊗ a ( l ) j +1 ,n ] mod F i − R (cid:0) (cid:98) X n ( M ) (cid:1) , where (cid:80) l a ( l ) j +1 ,n ⊗ k v ( l ) := χ ( v ⊗ k a j +1 ,n ) . (6) Let x , . . . , x n ∈ E satisfying { l : x l / ∈ ν ( A ) ∪ γ ( V ) } = 1 . If there exist j < j suchthat x j ∈ ν ( A ) and x j ∈ γ ( V ) , then (cid:98) ψ ([ m ⊗ x n ]) ∈ F i − R (cid:0) (cid:98) X n ( M ) (cid:1) , where i := { l : x l / ∈ ν ( A ) } .Proof. This follows immediately from Proposition A.3. (cid:3)
Proposition 3.14.
Let x , . . . , x n ∈ E be such that { l : x l / ∈ ν ( A ) ∪ γ ( V ) } = 1 , let m ∈ M and let i := { l : x l / ∈ ν ( A ) } . There exists y ∈ ν ( A ) ⊗ F iA, (cid:0) E ⊗ n +1 (cid:1) such that (cid:98) ω ([ m ⊗ x n ]) = [ m ⊗ A y ] .Proof. This is an immediate consequence of Proposition A.5. (cid:3)
In this section we are in the same conditions as in the previous one. Since ( X ∗ , d ∗ ) is a Υ-relativeprojective resolution of E , the Hochschild cohomology H ∗ K ( E, M ), of the K -algebra E with co-efficients in an E -bimodule M , is the cohomology of the cochain complex Hom E e (cid:0) ( X ∗ , d ∗ ) , M (cid:1) .It is well known that when K is separable, H ∗ K ( E, M ) is the absolute Hochschild cohomology of E with coefficients in M . For each s ≥
0, we will denote by Hom A ( (cid:101) E ⊗ sA , M ) the abelian group ofleft A -linear maps from (cid:101) E ⊗ sA to M . As it is well known, Hom A ( (cid:101) E ⊗ sA , M ) is an A -bimodule via( a · α )( (cid:101) x s ) := α ( (cid:101) x s · a ) and ( α · a )( (cid:101) x s ) := α ( (cid:101) x s ) · a. For each r, s ≥
0, write (cid:98) X rs ( M ) := Hom ( A,K ) (cid:0) (cid:101) E ⊗ sA ⊗ A ⊗ r , M (cid:1) (cid:39) Hom K e (cid:0) A ⊗ r , Hom A (cid:0) (cid:101) E ⊗ sA , M (cid:1)(cid:1) . (4.31)Note that, since (cid:101) E ⊗ A = ν ( A ) and A ⊗ = K , we have (cid:98) X r ( M ) (cid:39) Hom K e (cid:0) A ⊗ r , M (cid:1) and (cid:98) X s ( M ) (cid:39) Hom ( A,K ) (cid:0) (cid:101) E ⊗ sA , M (cid:1) . (4.32)It is easy to check that the k -linear map ζ rs : Hom E e ( X rs , M ) −→ (cid:98) X rs ( M ) , given by ζ ( α )( (cid:101) x s ⊗ a r ) := α (1 E ⊗ A (cid:101) x s ⊗ a r ⊗ E ), is an isomorphism. For each l ≤ s , welet (cid:98) d rsl : (cid:98) X r + l − ,s − l ( M ) −→ (cid:98) X rs ( M ) denote the map induced by Hom E e ( d lrs , M ). Via the aboveidentifications, Hom E e (( X ∗ , d ∗ ) , M ) becomes the complex ( (cid:98) X ∗ ( M ) , (cid:98) d ∗ ), where (cid:98) X n ( M ) := (cid:77) r + s = n (cid:98) X rs ( M ) and (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 . We have thus proven the following result:
Theorem 4.1.
The Hochschild cohomology H ∗ K ( E, M ) , of the K -algebra E with coefficients in M , is the cohomology of ( (cid:98) X ∗ ( M ) , (cid:98) d ∗ ) .Remark . It follows immediately from Corollary 2.27 that, if F takes its values in K ⊗ k V ,then ( (cid:98) X ∗ ( M ) , (cid:98) d ∗ ) is the total complex of the double complex ( (cid:98) X ∗∗ ( M ) , (cid:98) d ∗∗ , (cid:98) d ∗∗ ). CO)HOMOLOGY OF CROSSED PRODUCTS IN WEAK CONTEXTS 25
Remark . If K = A , then ( (cid:98) X ∗ ( M ) , (cid:98) d ∗ ) = ( (cid:98) X ∗ ( M ) , (cid:98) d ∗ ).For each s ≥
0, we let Hom A ( E ⊗ sA , M ) denote the abelian group of left A -linear maps from E ⊗ sA to M . Likewise Hom A ( (cid:101) E ⊗ sA , M ), the group Hom A ( E ⊗ sA , M ) is an A -bimodule via( a · α )( x s ) := α ( x s · a ) and ( α · a )( x s ) := α ( x s ) · a, For r, s ≥
0, let X rs ( M ) := Hom ( A,K ) (cid:0) E ⊗ sA ⊗ A ⊗ r , M (cid:1) (cid:39) Hom K e (cid:0) A ⊗ r , Hom A (cid:0) E ⊗ sA , M (cid:1)(cid:1) . Similarly like for (cid:98) X rs ( M ), we have canonical identifications X rs ( M ) (cid:39) Hom E e ( X (cid:48) rs , M ). Let µ i : X r,s − ( M ) −→ X rs ( M ) (0 ≤ i ≤ s ),be the map induced by µ (cid:48) i . It is easy to see that µ i ( α )( γ A (v s ) ⊗ a r ) = γ ( v ) · α (cid:0) γ A (v s ) ⊗ a r (cid:1) if i = 0, α (cid:0) γ A (v ,i − ) ⊗ A γ ( v i ) γ ( v i +1 ) ⊗ A γ A (v i +2 ,s ) ⊗ a r (cid:1) if 0 < i < s , (cid:80) α (cid:0) γ A (v ,s − ) ⊗ a ( l )1 r (cid:1) · γ (cid:0) v ( l ) s (cid:1) if i = s ,where a , . . . , a r ∈ A , v , . . . , v s ∈ V and (cid:80) l a ( l )1 r ⊗ k v ( l ) s := χ ( v s ⊗ k a r ). Notation 4.4.
Given a k -subalgebra R of A and 0 ≤ u ≤ r , we set (cid:98) X rsu ( R, M ) := ζ rs ( T urs ( M )),where T urs ( M ) denote the k -submodule of Hom E e (cid:0) X rs , M (cid:1) consisting of all the E -bimodule mapsthat factorize throughout the E -subbimodule of X r + u,s − u − generated by all the simple tensors1 E ⊗ A (cid:101) x ,s − u − ⊗ a ,r + u ⊗ E , with at least u of the a j ’s in R . Theorem 4.5.
The following assertions hold: (1)
Via the identifications (4.31) , the morphism (cid:98) d : (cid:98) X r − ,s ( M ) → (cid:98) X rs ( M ) is ( − s -timesthe coboundary map of the normalized cochain Hochschild complex of the K -algebra A with coefficients in Hom A ( (cid:101) E ⊗ sA , M ) , considered as an A -bimodule as at the beginning ofthis section. (2) The morphism (cid:98) d : (cid:98) X r,s − ( M ) → (cid:98) X rs ( M ) is induced by (cid:80) si =0 ( − i µ i . (3) Let R be a k -subalgebra of A . If R is stable under χ and F takes its values in R ⊗ k V ,then (cid:98) d l (cid:0) (cid:98) X r + l − ,s − l ( M ) (cid:1) ⊆ (cid:98) X rsl − ( R, M ) , for each r ≥ and < l ≤ s .Proof. Item (1) follows from the definition of d ; item (2), from Theorem 2.23; and item (3),from Theorem 2.26(3). (cid:3) Theorem 4.6.
Under the hypothesis of Theorem 2.24 the map (cid:98) d : (cid:98) X r +1 ,s − ( M ) → (cid:98) X rs ( M ) isgiven by (cid:98) d ( α )( (cid:101) γ A (v s ) ⊗ a r ) = ( − s +1 α (cid:0)(cid:101) γ A (v ,s − ) ⊗ T ( v (1) s − v (1) s , a r ) (cid:1) · γ ( v (2) s − v (2) s ) , where v , . . . , v s ∈ H , a , . . . , a r ∈ A and T ( v s − , v s , a r ) is as in Theorem 2.24.Proof. This follows immediately from Theorem 2.24. (cid:3)
Remark . Note that H r (cid:0) (cid:98) X ∗ s ( M ) , (cid:98) d ∗ s (cid:1) = H rK (cid:0) A, Hom A (cid:0) (cid:101) E ⊗ sA , M (cid:1)(cid:1) . Let (cid:0)
Hom K e (cid:0) E ⊗ ∗ , M (cid:1) , b ∗ (cid:1) be the normalized Hochschild cochain complex of the K -algebra E with coefficients in M . Recall that there is a canonical identification (cid:0) Hom K e (cid:0) E ⊗ ∗ , M (cid:1) , b ∗ (cid:1) (cid:39) Hom E e (cid:0)(cid:0) E ⊗ E ⊗ ∗ ⊗ E, b (cid:48)∗ (cid:1) , M (cid:1) . Let (cid:98) φ ∗ : (cid:0) Hom K e (cid:0) E ⊗ ∗ , M (cid:1) , b ∗ (cid:1) −→ (cid:0) (cid:98) X ∗ ( M ) , (cid:98) d ∗ (cid:1) and (cid:98) ψ ∗ : (cid:0) (cid:98) X ∗ ( M ) , (cid:98) d ∗ (cid:1) −→ (cid:0) Hom K e (cid:0) E ⊗ ∗ , M (cid:1) , b ∗ (cid:1) be the morphisms of complexes induced by φ ∗ and ψ ∗ respectively. Proposition 2.29 implies that (cid:98) φ ∗ ◦ (cid:98) ψ ∗ = id ∗ and (cid:98) ψ ∗ ◦ (cid:98) φ ∗ is homotopically equivalent to the identity map. An homotopy (cid:98) ω ∗ +1 from (cid:98) ψ ∗ ◦ (cid:98) φ ∗ to id ∗ , is the family of maps (cid:16)(cid:98) ω n +1 : Hom K e (cid:0) E ⊗ n +1 , M (cid:1) −→ Hom K e (cid:0) E ⊗ n , M (cid:1)(cid:17) n ≥ , induced by (cid:0) ω n +1 : E ⊗ E ⊗ n ⊗ E −→ E ⊗ E ⊗ n +1 ⊗ E (cid:1) n ≥ . The complex ( (cid:98) X ∗ ( M ) , (cid:98) d ∗ ) is filtered by F ( (cid:98) X ∗ ( M )) ⊇ F ( (cid:98) X ∗ ( M )) ⊇ F ( (cid:98) X ∗ ( M )) ⊇ F ( (cid:98) X ∗ ( M )) ⊇ F ( (cid:98) X ∗ ( M )) ⊇ . . . , where F i ( (cid:98) X n ( M )) := (cid:76) ns = i (cid:98) X n − s,s ( M ). Hence, by Remark 4.7 there is an spectral sequence E rsl = ⇒ H r + sK ( E, M ) , with E rs = H rK ( A, Hom A ( (cid:101) E ⊗ sA , M )) . (4.33)We let F i (cid:0) Hom K e (cid:0) E ⊗ ∗ , M (cid:1)(cid:1) denote the k -submodule of Hom K e (cid:0) E ⊗ ∗ , M (cid:1) consisting of all homo-morphisms α ∈ Hom K e (cid:0) E ⊗ ∗ , M (cid:1) , such that α (cid:0) F i (cid:0) E ⊗ ∗ (cid:1)(cid:1) = 0. The normalized Hochschild co-chain complex (cid:0) Hom K e (cid:0) E ⊗ ∗ , M (cid:1) , b ∗ (cid:1) is filtered by F (cid:0) Hom K e ( E ⊗ ∗ , M ) (cid:1) ⊇ F (cid:0) Hom K e ( E ⊗ ∗ , M ) (cid:1) ⊇ F (cid:0) Hom K e ( E ⊗ ∗ , M ) (cid:1) ⊇ . . . . (4.34)The spectral sequence associated to this filtration is called the cohomological Hochschild-Serrespectral sequence of E with coefficients in M . Theorem 4.8.
The maps (cid:98) φ ∗ , (cid:98) ψ ∗ and (cid:98) ω ∗ preserve filtrations.Proof. This follows immediately from Theorem 2.42. (cid:3)
Corollary 4.9.
The cohomological Hochschild-Serre spectral sequence is isomorphic to the spec-tral sequence (4.33) . Recall that the cup product in HH ∗ K ( E ), of β ∈ Hom K e ( E ⊗ m , E ) and β (cid:48) ∈ Hom K e ( E ⊗ n , E ),is given by ( β (cid:94) β (cid:48) )( x ,m + n ) := β ( x m ) β (cid:48) ( x m +1 ,m + n ). Corollary 4.10.
When M = E the spectral sequence (4.33) is multiplicative.Proof. By Corollary 4.9 and the fact that the filtration (4.34) satisfies F i (cid:94) F j ⊆ F i + j . (cid:3) Proposition 4.11.
Let R be a k -subalgebra of A and let ρ l be the left action of A on M . If R is stable under χ and F takes its values in R ⊗ k V , then for all β ∈ Hom K e ( E ⊗ n , M ) , (cid:98) φ ( β ) (cid:0)(cid:101) γ A (v i ) ⊗ a ,n − i (cid:1) = ( β ◦ Sh)(v i ⊗ k a ,n − i )+ (cid:0) ρ l ◦ ( A ( A ) ⊗ β ) ◦ φ (cid:48)(cid:48) in (cid:1) (v i ⊗ k a ,n − i ) , where a , . . . , a n − i ∈ A , v , . . . , v i ∈ V and φ (cid:48)(cid:48) in (v i ⊗ k a ,n − i ) is as in Notation 2.41. CO)HOMOLOGY OF CROSSED PRODUCTS IN WEAK CONTEXTS 27
Proof.
This is an immediate consequence of Remark 2.40. (cid:3)
We will use the following proposition in the next section.
Proposition 4.12.
Let ≤ s ≤ n and α ∈ (cid:98) X n − s,s ( M ) . Let a , . . . , a i ∈ A , v i +1 , . . . , v n ∈ V and x , . . . , x n ∈ ν ( A ) ∪ γ ( V ) . The map (cid:98) ψ ( α ) have the following properties: (1) If i = s , then (cid:98) ψ ( α ) (cid:0) γ (v s ) ⊗ ν ( a s +1 ,n ) (cid:1) = α (cid:0)(cid:101) γ A (v s ) ⊗ a s +1 ,n (cid:1) . (2) If i (cid:54) = s , then (cid:98) ψ ( α ) (cid:0) γ (v i ) ⊗ A ( a i +1 ,n ) (cid:1) = 0 . (3) If there exist j < j such that x j ∈ ν ( A ) and x j ∈ γ ( V ) , then (cid:98) ψ ( α )( x n ) = 0 .Proof. This follows immediately from items (1) and (2) of Proposition A.3. (cid:3)
In this section we obtain formulas involving the complexes ( (cid:98) X ∗ ( E ) , (cid:98) d ∗ ) and ( (cid:98) X ∗ ( M ) , (cid:98) d ∗ ) thatinduce the cup product of HH ∗ K ( E ) and the cap product of H K ∗ ( E, M ). First of all recall thatfor m ≤ n , the cap product (cid:97) : H Kn ( E, M ) × HH mK ( E ) −→ H Kn − m ( E, M ), is induced by the map (cid:0) M ⊗ E ⊗ n ⊗ (cid:1) × Hom K e ( E ⊗ m , E ) −→ M ⊗ E ⊗ n − m ⊗ , defined by [ m ⊗ x n ] (cid:97) β := [ m · β ( x m ) ⊗ x m +1 ,n ]. When m > n we set [ m ⊗ x n ] (cid:97) β := 0. Definition 5.1.
For α ∈ (cid:98) X rs ( E ) and α (cid:48) ∈ (cid:98) X r (cid:48) s (cid:48) ( E ) we define α • α (cid:48) ∈ (cid:98) X r (cid:48)(cid:48) ,s (cid:48)(cid:48) ( E ) by( α • α (cid:48) ) (cid:0)(cid:101) γ A (v s (cid:48)(cid:48) ) ⊗ a r (cid:48)(cid:48) (cid:1) := (cid:88) ( − rs (cid:48) α (cid:0)(cid:101) γ A (v s ) ⊗ a ( l )1 r (cid:1) α (cid:48) (cid:0)(cid:101) γ A (v ( l ) s +1 ,s (cid:48)(cid:48) ) ⊗ 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) , v , . . . , v s (cid:48)(cid:48) ∈ V , a , . . . , a r (cid:48)(cid:48) ∈ A and (cid:88) l a ( l )1 r ⊗ k v ( l ) s +1 ,s (cid:48)(cid:48) := χ (cid:0) v s +1 ,s (cid:48)(cid:48) ⊗ k a r (cid:1) . Theorem 5.2.
Let α ∈ (cid:98) X rs ( E ) and α (cid:48) ∈ (cid:98) X r (cid:48) s (cid:48) ( E ) . If F takes its values in K ⊗ k V , then, for all v , . . . , v i ∈ V and a i +1 , . . . , a n ∈ A , we have (cid:98) φ (cid:0) (cid:98) ψ ( α ) (cid:94) (cid:98) ψ ( α (cid:48) ) (cid:1)(cid:0)(cid:101) γ A (v i ) ⊗ a i +1 ,n (cid:1) = (cid:40) ( α • α (cid:48) ) (cid:0)(cid:101) γ A (v s (cid:48)(cid:48) ) ⊗ a s (cid:48)(cid:48) +1 ,n (cid:1) if i = s (cid:48)(cid:48) , otherwise,where s (cid:48)(cid:48) := s + s (cid:48) and n := r + r (cid:48) + s + s (cid:48) .Proof. For the sake of brevity we set T := Sh(v i ⊗ k a i +1 ,n ). By Proposition 4.11, (cid:98) φ (cid:0) (cid:98) ψ ( α ) (cid:94) (cid:98) ψ ( α (cid:48) ) (cid:1)(cid:0)(cid:101) γ A (v i ) ⊗ a i +1 ,n (cid:1) = (cid:0) (cid:98) ψ ( α ) (cid:94) (cid:98) ψ ( α (cid:48) ) (cid:1) ( T ) . Since, by the definition of (cid:94) and Proposition 4.12,- if i (cid:54) = s (cid:48)(cid:48) , then (cid:0) (cid:98) ψ ( α ) (cid:94) (cid:98) ψ ( α (cid:48) ) (cid:1) ( T ) = 0,- if i = s (cid:48)(cid:48) , then (cid:0) (cid:98) ψ ( α ) (cid:94) (cid:98) ψ ( α (cid:48) ) (cid:1) ( T ) = (cid:88) ( − s (cid:48) r α (cid:0)(cid:101) γ A (v s ) ⊗ a ( l ) i +1 ,i + r (cid:1) α (cid:48) (cid:0)(cid:101) γ A (v ( l ) s +1 ,i ) ⊗ a i + r +1 ,n (cid:1) , where (cid:80) l a ( l ) i +1 ,i + r ⊗ k v ( l ) s +1 ,i := χ (cid:0) v s +1 ,i ⊗ k a i +1 ,i + r (cid:1) ,the result follows. (cid:3) Corollary 5.3. If F takes its values in K ⊗ k V , then the cup product of HH ∗ K ( E ) is induced bythe operation • in ( (cid:98) X ∗ ( E ) , (cid:98) d ∗ ) . Proof.
This follows immediately from Theorem 5.2. (cid:3)
Remark . Let M be and E -bimodule. There is an evident generalization of the definitionof the cup products that makes up H ∗ K ( E, M ) in a HH ∗ K ( E )-bimodule. It is clear that obviousgeneralizations of Theorem 5.2 and Corollary 5.3 hold. Definition 5.5.
Let α ∈ (cid:98) X r (cid:48) s (cid:48) ( E ) and let x := [ m ⊗ A γ A (v s ) ⊗ a r ] ∈ (cid:98) X rs ( M ), where m ∈ M , v , . . . , v s ∈ V and a , . . . , a r ∈ A . If r (cid:48) ≤ r and s (cid:48) ≤ s , then we define x (cid:5) α ∈ (cid:98) X r − r (cid:48) ,s − s (cid:48) ( M ) by x (cid:5) α := (cid:88) ( − r (cid:48) ( s − s (cid:48) ) (cid:2) m · α (cid:0)(cid:101) γ A (v s (cid:48) ) ⊗ a ( l )1 r (cid:48) (cid:1) ⊗ A (cid:101) γ A (v ( l ) s (cid:48) +1 ,s ) ⊗ a r (cid:48) +1 ,r (cid:3) , where (cid:80) l a ( l )1 r (cid:48) ⊗ k v ( l ) s (cid:48) +1 ,s := χ (cid:0) v s (cid:48) +1 ,s ⊗ k a r (cid:48) (cid:1) . If r (cid:48) > r or s (cid:48) > s , the we set x (cid:5) α := 0. Theorem 5.6.
Let m ∈ M , v , . . . , v s ∈ V , a , . . . , a r ∈ A and α ∈ (cid:98) X r (cid:48) s (cid:48) ( E ) . If F takes itsvalues in K ⊗ k V , then (cid:98) ψ (cid:0) (cid:98) φ ([ m ⊗ A (cid:101) γ A (v s ) ⊗ a r ]) (cid:97) (cid:98) ψ ( α ) (cid:1) = [ m ⊗ A (cid:101) γ A (v s ) ⊗ a r ] (cid:5) α. Proof.
By Proposition 3.12, (cid:98) ψ (cid:0) (cid:98) φ ([ m ⊗ A γ A (v s ) ⊗ a r ]) (cid:97) (cid:98) ψ ( α ) (cid:1) = (cid:98) ψ (cid:0) [ m ⊗ T ] (cid:97) (cid:98) ψ ( α ) (cid:1) , where T := Sh(v s ⊗ k a r ). Now, by the definition of (cid:97) and Proposition 4.12, we know that- If s (cid:48) > s or r (cid:48) > r , then [ m ⊗ T ] (cid:97) (cid:98) ψ ( α ) = 0.- If s (cid:48) ≤ s and r (cid:48) ≤ r , then[ m ⊗ T ] (cid:97) (cid:98) ψ ( α ) = (cid:88) ( − r (cid:48) s − r (cid:48) s (cid:48) m · α (cid:0)(cid:101) γ A (v s (cid:48) ) ⊗ a ( l )1 r (cid:48) (cid:1) ⊗ Sh (cid:0) v ( l ) s (cid:48) +1 ,s ⊗ k a r (cid:48) +1 ,r (cid:1) , where (cid:80) l a ( l )1 r (cid:48) ⊗ k v ( l ) s (cid:48) +1 ,s := χ (v s (cid:48) +1 ,s ⊗ k a r (cid:48) ).The result follows now immediately from items (1) and (2) of Proposition 3.13. (cid:3) Corollary 5.7. If F takes its values in K ⊗ k V , then in terms of the complexes ( (cid:98) X ∗ ( M ) , (cid:98) d ∗ ) and ( (cid:98) X ∗ ( E ) , (cid:98) d ∗ ) , the cap product is induced by the operation (cid:5) .Proof. This follows from Theorem 5.6. (cid:3)
In this section we construct a mixed complex computing the cyclic homology of E , whose under-lying Hochschild complex is ( (cid:98) X ∗ ( E ) , (cid:98) d ∗ ). The notation (cid:98) X n +1 ( M ) is not only meaningful in thecase where M is an E -bimodule, but also when M is an A -bimodule. Throughout this sectionwe will use it with M = ν ( A ). Notation 6.1.
For i ≤ n , we let F i, A, (cid:0) E ⊗ n (cid:1) denote the K -subbimodule of E ⊗ n generated byall the simple tensors x n such that { j : x j / ∈ ν ( A ) ∪ γ ( V ) } ≤ { j : x j / ∈ ν ( A ) } ≤ i .Furthermore we let ν ( K ) ¯ ⊗ F i, A, (cid:0) E ⊗ n (cid:1) ¯ ⊗ denote the image of the canonical map ν ( K ) ⊗ F i, A, (cid:0) E ⊗ n (cid:1) ⊗ −→ E ⊗ E ⊗ n ⊗ . Lemma 6.2.
Let (cid:98) φ and (cid:98) ω be as in Subsection 3.1. The composition B ◦ (cid:98) ω ◦ B ◦ (cid:98) φ , where the map B ∗ : E ⊗ E ⊗ ∗ ⊗ −→ E ⊗ E ⊗ ∗ +1 ⊗ is the Connes operator, is the zero map. CO)HOMOLOGY OF CROSSED PRODUCTS IN WEAK CONTEXTS 29
Proof.
By Proposition 3.12, Remark 2.38 and the very definition of B , B ◦ (cid:98) φ (cid:0) (cid:98) X n − i,i ( E ) (cid:1) ⊆ ν ( K ) ¯ ⊗ F i +1 , A, (cid:0) E ⊗ n +1 (cid:1) ¯ ⊗ . So, by Proposition 3.14 we have (cid:98) ω ◦ B ◦ (cid:98) φ (cid:0) (cid:98) X n − i,i ( E ) (cid:1) ⊆ ν ( K ) ¯ ⊗ F i +1 A, (cid:0) E ⊗ n +2 (cid:1) ¯ ⊗ ⊆ ker B, as desired. (cid:3) For each n ≥
0, let (cid:98) D n : (cid:98) X n ( E ) → (cid:98) X n +1 ( E ) be the map (cid:98) D := (cid:98) ψ ◦ B ◦ (cid:98) φ . Theorem 6.3. (cid:0) (cid:98) X ∗ ( E ) , (cid:98) d ∗ , (cid:98) D ∗ (cid:1) is a mixed complex that yields the Hochschild, cyclic, negativeand periodic homologies of the K -algebra E . Moreover we have chain complexes maps Tot (cid:0)
BP( (cid:98) X ∗ ( E ) , (cid:98) d ∗ , (cid:98) D ∗ ) (cid:1) Tot (cid:0)
BP( E ⊗ E ⊗ ∗ ⊗ , b ∗ , B ∗ ) (cid:1) (cid:98) Ψ (cid:98) Φ given by (cid:98) Φ n ( x u i ) := (cid:98) φ ( x ) u i + (cid:98) ω ◦ B ◦ (cid:98) φ ( x ) u i − and (cid:98) Ψ n ( x u i ) := (cid:80) j ≥ (cid:98) ψ ◦ ( B ◦ (cid:98) ω ) j ( x ) u i − j . Thesemaps satisfy (cid:98) Ψ ◦ (cid:98) Φ = id and (cid:98) Φ ◦ (cid:98) Ψ is homotopically equivalent to the identity map. A homotopy (cid:98) Ω ∗ +1 : (cid:98) Φ ∗ ◦ (cid:98) Ψ ∗ → id ∗ is given by (cid:98) Ω n +1 ( x u i ) := (cid:80) j ≥ (cid:98) ω ◦ ( B ◦ (cid:98) ω ) j ( x ) u i − j .Proof. For each i ≥
0, let (cid:98) φu i : (cid:98) X n − i ( E ) u i −→ (cid:0) E ⊗ E ⊗ n − i ⊗ (cid:1) u i , (cid:98) ψu i : (cid:0) E ⊗ E ⊗ n − i ⊗ (cid:1) u i −→ (cid:98) X n − i ( E ) u i and (cid:98) ωu i : (cid:0) E ⊗ E ⊗ n − i ⊗ (cid:1) u i −→ (cid:0) E ⊗ E ⊗ n +1 − i ⊗ (cid:1) u i , be the maps defined by (cid:98) φu i ( x u i ) := (cid:98) φ ( x ) u i , etcetera. By the results in Subsection 2.1 we have aspecial deformation retractTot (cid:0) BC( (cid:98) X ∗ ( E ) , (cid:98) d ∗ , (cid:1) Tot (cid:0)
BC( E ⊗ E ⊗ ∗ ⊗ , b ∗ , (cid:1) (cid:76) i ≥ (cid:98) ψu i (cid:76) i ≥ (cid:98) φu i (cid:76) i ≥ (cid:98) ωu i .Applying the perturbation lemma to this datum endowed with the perturbation induced by B ,and taking into account Lemma 6.2, we obtain the special deformation retractTot (cid:0) BP( (cid:98) X ∗ ( E ) , (cid:98) d ∗ , (cid:98) D ∗ ) (cid:1) Tot (cid:0)
BP( E ⊗ E ⊗ ∗ ⊗ , b ∗ , B ∗ ) (cid:1) (cid:98) Ψ (cid:98) Φ (cid:98) Ω ∗ +1 . It is easy to see that (cid:98) Φ, (cid:98) Ψ and (cid:98)
Ω commute with the canonical surjectionsTot (cid:0)
BC( (cid:98) X ∗ ( E ) , (cid:98) d ∗ , (cid:98) D ∗ ) (cid:1) −→ Tot (cid:0)
BC( (cid:98) X ∗ ( E ) , (cid:98) d ∗ , (cid:98) D ∗ ) (cid:1) [2]and Tot (cid:0) BC( E ⊗ E ⊗ ∗ ⊗ , b ∗ , B ∗ ) (cid:1) −→ Tot (cid:0)
BC( E ⊗ E ⊗ ∗ ⊗ , b ∗ , B ∗ ) (cid:1) [2] . A standard argument, from these facts, finishes the proof. (cid:3)
Remark . If K is a separable algebra, then mixed complex (cid:0) (cid:98) X ∗ ( E ) , (cid:98) d ∗ , (cid:98) D ∗ (cid:1) gives the Hochs-child, cyclic, negative and periodic absolute homologies of E . Definition 6.5.
For each r, s ≥
0, let (cid:98) D rs : (cid:98) X rs → (cid:98) X r,s +1 and (cid:98) D rs : (cid:98) X rs → (cid:98) X r +1 ,s be the mapsdefined by - If x = [ ν ( a ) γ ( v ) ⊗ A (cid:101) γ A (v s ) ⊗ a r ], with a , . . . , a r ∈ A , v , . . . , v s ∈ V , then (cid:98) D ( x ) := s (cid:88) j =0 (cid:88) l ( − js + s (cid:2) E ⊗ A (cid:101) γ A (v ( l ) j +1 ,s ) ⊗ A ν ( a ) γ ( v ) (cid:103) ⊗ A (cid:101) γ A (v j ) ⊗ a ( l )1 r (cid:3) and (cid:98) D ( x ) := r (cid:88) j =0 (cid:88) l ( − jr + r + s (cid:2) γ ( v ( l )0 ) ⊗ A (cid:101) γ A (v ( l )1 s ) ⊗ a j +1 ,r ⊗ a ⊗ a ( l )1 j (cid:3) , where (cid:88) l a ( l )1 r ⊗ k v ( l ) j +1 ,s := χ (v j +1 ,s ⊗ k a r ) and (cid:88) l a ( l )1 j ⊗ k v ( l )0 ⊗ k v ( l )1 s := χ (v s ⊗ k a j ) . - If x = [ ν ( a ) ⊗ A (cid:101) γ A (v s ) ⊗ a r ] with a , . . . , a r ∈ A , v , . . . , v s ∈ V , then (cid:98) D ( x ) := 0 and (cid:98) D ( x ) := r (cid:88) j =0 (cid:88) l ( − jr + r + s (cid:2) E ⊗ A (cid:101) γ A (v ( l )1 s ) ⊗ a j +1 ,r ⊗ a ⊗ a ( l )1 j (cid:3) , where (cid:80) l a ( l )1 j ⊗ k v ( l )1 s := χ (v s ⊗ k a j ). Proposition 6.6.
Let R be a k -subalgebra of A . Assume that R is stable under χ and F takesits values in R ⊗ k V . Let a , . . . , a n − i ∈ A and v , . . . , v i ∈ V . The Connes operator (cid:98) D satisfiesthe following properties: (1) If x = [ ν ( a ) γ ( v ) ⊗ A (cid:101) γ A (v i ) ⊗ a ,n − i ] , then (cid:98) D ( x ) = (cid:98) D ( x ) + (cid:98) D ( x ) mod F i − R (cid:0) (cid:98) X n +1 ( E ) (cid:1) + F iR (cid:0) (cid:98) X n +1 ( ν ( A )) (cid:1) . (2) If x = [ ν ( a ) ⊗ A (cid:101) γ A (v i ) ⊗ a ,n − i ] , then (cid:98) D ( x ) = (cid:98) D ( x ) mod F i − R (cid:0) (cid:98) X n +1 ( ν ( A )) (cid:1) . Proof. (1) We must compute (cid:98) D ( x ) = (cid:98) ψ ◦ B ◦ (cid:98) φ ( x ). By Proposition 3.12 (cid:98) D ( x ) = (cid:98) ψ ◦ B (cid:0)(cid:2) ν ( a ) γ ( v ) ⊗ Sh(v i ⊗ k a ,n − i ) (cid:3)(cid:1) + (cid:98) ψ ◦ B (cid:0) [ ν ( a ) γ ( v ) ⊗ A y ] (cid:1) , where [ ν ( a ) γ ( v ) ⊗ A y ] ∈ E ¯ ⊗ F i − R, ( E ⊗ n ) ¯ ⊗ . Now- B (cid:0)(cid:2) ν ( a ) γ ( v ) ⊗ Sh(v i ⊗ k a ,n − i ) (cid:3)(cid:1) is a sum of classes in E ⊗ E ⊗ n +1 ⊗ of simple tensors1 ⊗ y ,n +1 , with n − i of the y j ’s in ν ( A ), i of the y j ’s in γ ( V ) and one y j / ∈ ν ( A ) ∪ γ ( V ).- B (cid:0) [ ν ( a ) γ ( v ) ⊗ A y ] (cid:1) is a sum of classes in E ⊗ E ⊗ n +1 ⊗ of simple tensors 1 ⊗ z ,n +1 ,with at least one z j in ν ( R ), at least n − i + 1 of the z j ’s in ν ( A ) and at most one z j in E \ ν ( A ) ∪ γ ( V ).The result follows now using items (3)–(6) of Proposition 3.13 and the definitions of Sh and B .(2) As in the proof of item (1) we have (cid:98) D ( x ) = (cid:98) ψ ◦ B (cid:0) [ ν ( a ) ⊗ Sh(v i ⊗ k a ,n − i )] (cid:1) + (cid:98) ψ ◦ B (cid:0) [ ν ( a ) ⊗ A y ] (cid:1) , where [ ν ( a ) ⊗ A y ] ∈ F i − R, (cid:0) E ⊗ E ⊗ n ⊗ (cid:1) . Now- B (cid:0) [ ν ( a ) ⊗ Sh(v i ⊗ k a ,n − i )] (cid:1) is a sum of classes in E ⊗ E ⊗ n +1 ⊗ of simple tensors1 ⊗ y ,n +1 , with n − i + 1 of the y j ’s in ν ( A ) and i of the y j ’s in γ ( V ). CO)HOMOLOGY OF CROSSED PRODUCTS IN WEAK CONTEXTS 31 - B (cid:0) [ ν ( a ) ⊗ A y ] (cid:1) is a sum of classes in E ⊗ E ⊗ n +1 ⊗ of simple tensors 1 ⊗ z ,n +1 , witheach z j in ν ( A ) ∪ γ ( V ), at least one z j in ν ( R ), and at least n − i + 2 of the z j ’s in ν ( A ).The result follows now from items (1)–(2) of Proposition 3.13 and the definition of Sh and B . (cid:3) Corollary 6.7. If K = A , then (cid:0) (cid:98) X ∗ ( E ) , (cid:98) d ∗ , (cid:98) D ∗ (cid:1) = (cid:0) (cid:98) X ∗ ( E ) , (cid:98) d ∗ , (cid:98) D ∗ (cid:1) , where (cid:98) D n (cid:0) [ ν ( a ) γ ( v ) ⊗ A (cid:101) γ A (v n )] (cid:1) = n (cid:88) j =0 ( − n + jn (cid:2) E ⊗ A (cid:101) γ A (v j +1 ,n ) ⊗ A ν ( a ) γ ( v ) (cid:103) ⊗ A (cid:101) γ A (v j ) (cid:3) . In this subsection we study two spectral sequences. The first one generalizes those obtainedin [6, Section 3.1] and [20, Theorem 4.7], while the second one generalizes those obtained in [1,17]and [6, Section 3.2]. Let (cid:98) d rs and (cid:98) d rs be as at the beginning of Section 3 and let (cid:98) D rs and (cid:98) D rs beas in Definition 6.5. Recall from Remark 3.8 that H r (cid:0) (cid:98) X ∗ s , (cid:98) d ∗ s (cid:1) = H Kr (cid:0) A, E ⊗ A (cid:101) E ⊗ sA (cid:1) . Let˘ d rs : H Kr (cid:0) A, E ⊗ A (cid:101) E ⊗ sA (cid:1) −→ H Kr (cid:0) A, E ⊗ A (cid:101) E ⊗ s − A (cid:1) and ˘ D rs : H Kr (cid:0) A, E ⊗ A (cid:101) E ⊗ sA (cid:1) −→ H Kr (cid:0) A, E ⊗ A (cid:101) E ⊗ s +1 A (cid:1) be the maps induced by (cid:98) d and (cid:98) D , respectively. Proposition 6.8.
For each r ≥ , ˘H Kr (cid:0) A, E ⊗ A (cid:101) E ⊗ ∗ A (cid:1) := (cid:16) H Kr (cid:0) A, E ⊗ A (cid:101) E ⊗ ∗ A (cid:1) , ˘ d r ∗ , ˘ D r ∗ (cid:17) is a mixed complex and there is a convergent spectral sequence ( E vsr , ∂ vsr ) v ≥ = ⇒ HC Kr + s ( E ) , such that E sr = HC s (cid:16) ˘H Kr (cid:0) A, E ⊗ A (cid:101) E ⊗ ∗ A (cid:1)(cid:17) for all r, s ≥ .Proof. For each s, n ≥
0, let F s (cid:0) Tot(BC( (cid:98) X, (cid:98) d, (cid:98) D ) n ) (cid:1) := (cid:77) j ≥ F s − j ( (cid:98) X n − j ) u j , where F s − j ( (cid:98) X n − j ) is the filtration introduced in Subsection 3.2. Let ( E vsr , ∂ vsr ) v ≥ be thespectral sequence associated with the filtration F (cid:0) Tot(BC( (cid:98) X ∗ , (cid:98) d ∗ , (cid:98) D ∗ )) (cid:1) ⊆ F (cid:0) Tot(BC( (cid:98) X ∗ , (cid:98) d ∗ , (cid:98) D ∗ )) (cid:1) ⊆ · · · of Tot (cid:0) BC( (cid:98) X ∗ , (cid:98) d ∗ , (cid:98) D ∗ ) (cid:1) . A straightforward computation shows that- E sr = (cid:76) j ≥ (cid:98) X r,s − j u j ,- ∂ sr : E sr −→ E s,r − is (cid:76) j ≥ (cid:98) d r,s − j u j ,- E sr = (cid:76) j ≥ H r (cid:0) (cid:98) X ∗ ,s − j , (cid:98) d ∗ ,s − j (cid:1) u j , - ∂ sr : E sr −→ E s − ,r is (cid:76) j ≥ ˘ d r,s − j u j + (cid:76) j ≥ ˘ D r,s − j u j − .From this it follows easily that ˘H Kr (cid:0) A, E ⊗ A (cid:101) E ⊗ ∗ A (cid:1) is a mixed complex and E sr = (cid:77) j ≥ H Kr (cid:0) A, E ⊗ A (cid:101) E ⊗ s − jA (cid:1) u j and E sr = HC s (cid:16) ˘H Kr (cid:0) A, E ⊗ A (cid:101) E ⊗ ∗ A (cid:1)(cid:17) . In order to finish the proof note that the filtration of Tot (cid:0)
BC( (cid:98) X ∗ , (cid:98) d ∗ , (cid:98) D ∗ ) (cid:1) introduced above iscanonically bounded, and so, by Theorem 6.3, the spectral sequence ( E vsr , ∂ vsr ) v ≥ converges tothe cyclic homology of the K -algebra E . (cid:3) Assume that F takes its values in K ⊗ k V . Letˇ d rs : H As (cid:0) E, A ⊗ A ⊗ r ⊗ E (cid:1) −→ H As (cid:0) E, A ⊗ A ⊗ r − ⊗ E (cid:1) and ˇ D rs : H As (cid:0) E, A ⊗ A ⊗ r ⊗ E (cid:1) −→ H As (cid:0) E, A ⊗ A ⊗ r +1 ⊗ E (cid:1) be the maps induced by (cid:98) d and (cid:98) D , respectively. Proposition 6.9.
For each s ≥ , ˇH As (cid:0) E, A ⊗ A ⊗ ∗ ⊗ E (cid:1) := (cid:16) H As (cid:0) E, A ⊗ A ⊗ ∗ ⊗ E (cid:1) , ˇ d ∗ s , ˇ D ∗ s (cid:17) is a mixed complex and there is a convergent spectral sequence ( E vrs , d vrs ) v ≥ = ⇒ HC Kr + s ( E ) , such that E rs = HC r (cid:16) ˇH As (cid:0) E, A ⊗ A ⊗ ∗ ⊗ E (cid:1)(cid:17) for all r, s ≥ .Proof. For each r, n ≥
0, let F r (cid:0) Tot(BC( (cid:98) X, (cid:98) d, (cid:98) D ) n ) (cid:1) := (cid:77) j ≥ F r − j ( (cid:98) X n − j ) u j , where F r − j ( (cid:98) X n − j ) := (cid:76) i ≤ r − j (cid:98) X i,n − i − j . Consider the spectral sequence ( E vrs , d vrs ) v ≥ , asso-ciated with the filtration F (cid:0) Tot(BC( (cid:98) X ∗ , (cid:98) d ∗ , (cid:98) D ∗ )) (cid:1) ⊆ F (cid:0) Tot(BC( (cid:98) X ∗ , (cid:98) d ∗ , (cid:98) D ∗ )) (cid:1) ⊆ · · · of Tot (cid:0) BC( (cid:98) X ∗ , (cid:98) d ∗ , (cid:98) D ∗ ) (cid:1) . A straightforward computation shows that- E rs = (cid:76) j ≥ (cid:98) X r − j,s u j ,- d rs : E rs −→ E r,s − is (cid:76) j ≥ (cid:98) d r − j,s u j ,- E rs = (cid:76) j ≥ H s (cid:0) (cid:98) X r − j, ∗ , (cid:98) d r − j, ∗ (cid:1) u j ,- d rs : E rs −→ E r − ,s is (cid:76) j ≥ ˇ d r − j,s u j + (cid:76) j ≥ ˇ D r − j,s u s − j .From this and Remark 3.8 it follows that ˇH As (cid:0) E, A ⊗ A ⊗ ∗ ⊗ E (cid:1) is a mixed complex, E rs = (cid:77) j ≥ H As − j (cid:0) E, A ⊗ A ⊗ r − j ⊗ E (cid:1) and E rs = HC r (cid:16) ˇH As (cid:0) E, A ⊗ A ⊗ ∗ ⊗ E (cid:1)(cid:17) . In order to finish the proof note that the filtration of Tot (cid:0)
BC( (cid:98) X ∗ , (cid:98) d ∗ , (cid:98) D ∗ ) (cid:1) introduced above iscanonically bounded, and so, by Theorem 6.3, the spectral sequence ( E vsr , d vsr ) v ≥ converges tothe cyclic homology of the K -algebra E . (cid:3) CO)HOMOLOGY OF CROSSED PRODUCTS IN WEAK CONTEXTS 33
Appendix
For each 0 ≤ i < n and each k -subalgebra R of A we set F iU,R ( X n ) := (cid:76) is =0 U n − s,s ( R ). Lemma A.1.
Let R be a k -subalgebra of A . Assume that R is stable under χ and that F takesits values in R ⊗ k V . Let r, s ≥ and let n := r + s . For each z ∈ X rs it is true that: (1) If z ∈ E · W rs , then σ ( z ) = σ ( z ) ∈ E · L r +1 ,s . (2) If z ∈ L urs ( R ) · E , then σ l ( z ) ∈ U l + ur + l +1 ,s − l ( R ) for all l ≥ . (3) If z ∈ E · U rs , then σ l ( z ) = 0 for l ≥ . (4) If z ∈ L rs · E and r > , then σ ( z ) ≡ σ ( z ) modulo F s − U,R ( X n +1 ) . (5) If z ∈ L n · E , then σ ( z ) ≡ σ ( z ) − σ ◦ σ − ◦ υ ( z ) modulo F n − U,R ( X n +1 ) . (6) If z ∈ E · U rs and r > , then σ ( z ) = 0 . (7) If z ∈ E · U n , then σ ( z ) = − σ ◦ σ − ◦ υ ( z ) .Proof. Item 1) is Remark 2.21 and item (2) holds for l = 0 by the very definition of σ . Assumethat item (2) is true for all the maps σ i with i < l . By items (2) and (3) of Theorem 2.26, σ l ( z ) = − l − (cid:88) i =0 σ ◦ d l − i ◦ σ i ( z ) ∈ l − (cid:88) i =0 σ (cid:0) d l − i ( U u + ir + i +1 ,s − i ( R )) (cid:1) ⊆ σ ( E · U u + l − r + l,s − l ( R )) + σ ( L u + l − r + l,s − l ( R ) · γ ( V ) γ ( V )) . So, σ l ( z ) ∈ U u + lr + l +1 ,s − l ( R ), since by the definition of σ , equality (2.17) and Lemma 1.14, σ ( E · U u + lr + l,s − l ( R )) = 0 and σ ( L u + l − r + l,s − l ( R ) · γ ( V ) γ ( V )) ⊆ U u + lr + l +1 ,s − l ( R ) . Item (3) follows easily by induction on l using the recursive definition of σ l (the initial case l = 0 is equality (2.17)). Items (4), (5), (6) and (7) follow from Proposition 2.18 and items (2)and (3). (cid:3) Remark
A.2 . From the definition of σ and item (2) of the previous lemma it follows that σ (cid:0) F iU,R ( X n ) · E (cid:1) ⊆ F iU,R ( X n +1 ) for i < n . Proposition A.3.
Let R be a k -subalgebra of A . Assume that R is stable under χ and that F takes its values in R ⊗ k V . Let v, v , . . . , v i ∈ V and a, a i +1 , . . . , a n ∈ A . The map ψ n has thefollowing properties: (1) ψ (cid:0) E ⊗ γ (v i ) ⊗ ν ( a i +1 ,n ) ⊗ E (cid:1) = 1 E ⊗ A (cid:101) γ A (v i ) ⊗ a i +1 ,n ⊗ E . (2) Let x , . . . , x n ∈ ν ( A ) ∪ γ ( V ) . If there exist indices j < j such that x j ∈ ν ( A ) and x j ∈ γ ( V ) , then ψ (1 E ⊗ x n ⊗ E ) = 0 . (3) If x = 1 E ⊗ γ (v ,i − ) ⊗ a · γ ( v ) ⊗ ν ( a i +1 ,n ) ⊗ E , then ψ ( x ) ≡ E ⊗ A (cid:101) γ A (v ,i − ) ⊗ A a · (cid:101) γ ( v ) ⊗ a i +1 ,n ⊗ E + (cid:88) E ⊗ A (cid:101) γ A (v ,i − ) ⊗ a ⊗ a ( l ) i +1 ,n ⊗ γ (cid:0) v ( l ) (cid:1) mod F i − U,R ( X n ) , where (cid:80) l a ( l ) i +1 ,n ⊗ k v ( l ) := χ ( v ⊗ k a i +1 ,n ) . (4) If x = 1 E ⊗ γ (v ,j − ) ⊗ a · γ ( v ) ⊗ γ (v j +1 ,i ) ⊗ ν ( a i +1 ,n ) ⊗ E with j < i , then ψ ( x ) ≡ E ⊗ A (cid:101) γ A (v ,j − ) ⊗ A a · (cid:101) γ ( v ) ⊗ A (cid:101) γ A (v j +1 ,i ) ⊗ a i +1 ,n ⊗ E mod F i − U,R ( X n ) . (5) If x = 1 E ⊗ γ (v ,i − ) ⊗ ν ( a i,j − ) ⊗ a · γ ( v ) ⊗ ν ( a j +1 ,n ) ⊗ E with j > i , then ψ ( x ) ≡ (cid:88) E ⊗ A (cid:101) γ A (v ,i − ) ⊗ a i,j − ⊗ a ⊗ a ( l ) j +1 ,n ⊗ γ ( v ( l ) ) mod F i − U,R ( X n ) , where (cid:80) l a ( l ) j +1 ,n ⊗ k v ( l ) := χ ( v ⊗ k a j +1 ,n ) . (6) Let x , . . . , x n ∈ E satisfying { l : x l / ∈ ν ( A ) ∪ γ ( V ) } = 1 . If there exist j < j suchthat x j ∈ ν ( A ) and x j ∈ γ ( V ) , then ψ (1 E ⊗ x n ⊗ E ) ∈ F i − U,R ( X n ) , where i := { l : x l / ∈ ν ( A ) } .Proof. (1) We proceed by induction on n . The case n = 0 is trivial. Suppose n > n −
1. Assume first that i < n . By Proposition 2.30, the inductive hypothesisand item (1) of Lemma A.1, ψ (cid:0) E ⊗ γ (v i ) ⊗ ν ( a i +1 ,n ) ⊗ E (cid:1) = ( − n σ ◦ ψ (cid:0) E ⊗ γ (v i ) ⊗ ν ( a i +1 ,n − ) ⊗ ν ( a n ) (cid:1) = ( − n σ (cid:0) E ⊗ A (cid:101) γ A (v i ) ⊗ a i +1 ,n − ⊗ ν ( a n ) (cid:1) = ( − n σ (cid:0) E ⊗ A (cid:101) γ A (v i ) ⊗ a i +1 ,n − ⊗ ν ( a n ) (cid:1) , and the result follows from equality (2.15), since 1 E ∈ K ⊗ k V . Assume now that i = n . ByProposition 2.30, the inductive hypothesis, item (7) of Lemma A.1 and the definitions of υ and σ − , ψ (cid:0) E ⊗ γ (v n ) ⊗ E (cid:1) = ( − n σ ◦ ψ (cid:0) E ⊗ γ (v ,n − ) ⊗ γ ( v n ) (cid:1) = ( − n +1 σ ◦ σ − ◦ υ (cid:0) E ⊗ A (cid:101) γ A (v ,n − ) ⊗ γ ( v n ) (cid:1) = σ (cid:0) E ⊗ A (cid:101) γ A (v n ) ⊗ A E (cid:1) . The result follows now immediately from equality (2.14), since 1 E ∈ K ⊗ k V .(2) We proceed by induction on n . The case n = 0 is trivial. Suppose n > n −
1. By Proposition 2.30 and the inductive hypothesis, if there exist j < j < n suchthat x j ∈ ν ( A ) and x j ∈ γ ( V ), then ψ (1 E ⊗ x n ⊗ E ) = ( − n σ ◦ ψ (1 E ⊗ x ,n − ⊗ x n ) = ( − n σ (0) = 0 . On the other hand, if x n = γ (v ,i − ) ⊗ ν ( a i,n − ) ⊗ γ ( v n ), then, by Proposition 2.30 and state-ment (1), ψ (1 E ⊗ x n ⊗ E ) = ( − n σ ◦ ψ (1 E ⊗ x ,n − ⊗ x n ) = ( − n σ (cid:0) E ⊗ A (cid:101) γ A (v ,i − ) ⊗ a i,n − ⊗ γ ( v n ) (cid:1) , and the result follows from item (6) of Lemma A.1.(3) We proceed by induction on n . The case n = 0 is trivial. Suppose n > n −
1. Assume first that i < n . Let y := 1 E ⊗ γ (v ,i − ) ⊗ a · γ ( v ) ⊗ ν ( a i +1 ,n − ) ⊗ ν ( a n ) , z := 1 E ⊗ A (cid:101) γ A (v ,i − ) ⊗ A a · (cid:101) γ ( v ) ⊗ a i +1 ,n − ⊗ ν ( a n )and z (cid:48) := (cid:88) E ⊗ A (cid:101) γ A (v ,i − ) ⊗ a ⊗ a ( l ) i +1 ,n − ⊗ γ (cid:0) v ( l ) (cid:1) ν ( a n ) , CO)HOMOLOGY OF CROSSED PRODUCTS IN WEAK CONTEXTS 35 where (cid:80) l a ( l ) i +1 ,n − ⊗ k v ( l ) := χ ( v ⊗ k a i +1 ,n − ). Clearly z (cid:48) ∈ L n − i,i − · E , while z ∈ W n − i − ,i (useequality (1.6) repeatedly). By Proposition 2.30 and the inductive hypothesis, ψ ( x ) = ( − n σ ◦ ψ ( y ) ≡ ( − n σ ( z + z (cid:48) ) mod σ (cid:0) F i − U,R ( X n − ) · ν ( A ) (cid:1) So, by Remark A.2 and items (1) and (4) of Lemma A.1, ψ ( x ) ≡ ( − n σ ( z + z (cid:48) ) mod F i − U,R ( X n ) . Using the fact that γ ( V ) ⊆ K ⊗ k V and equalities (1.6) and (2.15) in order to compute σ ( z (cid:48) ),and using the fact that 1 E ∈ K ⊗ k V and equality (2.15) in order to compute σ ( z ), we obtainthe desired expression for ψ ( x ). Assume now that i = n . Let y := 1 E ⊗ γ (v ,n − ) ⊗ a · γ ( v ) , z := 1 E ⊗ A (cid:101) γ A (v ,n − ) ⊗ a · γ ( v )and z (cid:48) := 1 E ⊗ A (cid:101) γ A (v ,n − ) ⊗ A a · (cid:101) γ ( v ) ⊗ A E . By Proposition 2.30, item (1) and Lemma A.1(5), ψ ( x ) = ( − n σ ◦ ψ ( y ) = ( − n σ ( z ) ≡ ( − n σ ( z ) − ( − n σ ◦ σ − ◦ υ ( z ) mod F n − U,R ( X n ) . Hence, by the definitions of υ and σ − , ψ ( x ) ≡ ( − n σ ( z ) + σ ( z (cid:48) ) mod F n − U,R ( X n ) . The desired formula for ψ ( x ) follows now easily using the fact that γ ( V ) ⊆ K ⊗ k V and equal-ity (2.15) in order to compute σ ( z ), and using the fact that 1 E ∈ K ⊗ k V and equality (2.14)in order to compute σ ( z (cid:48) ).(4) We proceed by induction on n . The case n = 0 is trivial. Suppose n > n −
1. Assume first that i < n . Let y := 1 E ⊗ γ (v ,j − ) ⊗ a · γ ( v ) ⊗ γ (v j − ,i ) ⊗ ν ( a i +1 ,n − ) ⊗ ν ( a n )and z := 1 E ⊗ A (cid:101) γ A (v ,j − ) ⊗ A a · (cid:101) γ ( v ) ⊗ A (cid:101) γ A (v j − ,i ) ⊗ a i +1 ,n − ⊗ ν ( a n ) . Note that z ∈ W n − i − ,i . By Proposition 2.30 and the inductive hypothesis, ψ ( x ) = ( − n σ ◦ ψ ( y ) ≡ ( − n σ ( z ) mod σ (cid:0) F i − U,R ( X n − ) · ν ( A ) (cid:1) . So, by Remark A.2 and item (1) of Lemma A.1, ψ ( x ) ≡ ( − n σ ( z ) mod F i − U,R ( X n ) . The desired formula for ψ ( x ) follows from equality (2.15), because 1 E ∈ K ⊗ k V . Assume nowthat j < i − i = n . Let y := 1 E ⊗ γ (v ,j − ) ⊗ a · γ ( v ) ⊗ γ (v j +1 ,n − ) ⊗ γ ( v n ) , z := 1 E ⊗ A (cid:101) γ A (v ,j − ) ⊗ A a · (cid:101) γ ( v ) ⊗ A (cid:101) γ A (v j +1 ,n − ) ⊗ γ ( v n )and z (cid:48) := 1 E ⊗ A (cid:101) γ A (v ,j − ) ⊗ A a · (cid:101) γ ( v ) ⊗ A (cid:101) γ A (v j +1 ,n ) ⊗ A E . Clearly z ∈ U ,n − . By Proposition 2.30 and the inductive hypothesis, ψ ( x ) = ( − n σ ◦ ψ ( y ) ≡ ( − n σ ( z ) mod σ (cid:0) F n − U,R ( X n − ) · E (cid:1) . Consequently, by Remark A.2 and item (7) of Lemma A.1, ψ ( x ) ≡ ( − n +1 σ ◦ σ − ◦ υ ( z ) mod F n − U,R ( X n ) . So, the definitions of υ and σ − , ψ ( x ) ≡ σ ( z (cid:48) ) mod F n − U,R ( X n ) , and the formula for ψ ( x ) follows from equality (2.14), because 1 E ∈ K ⊗ k V . Assume finallythat j = i − i = n . Let y := 1 E ⊗ γ (v ,n − ) ⊗ a · γ ( v ) ⊗ γ ( v n ) , z := 1 E ⊗ A (cid:101) γ A (v ,n − ) ⊗ A a · (cid:101) γ ( v ) ⊗ γ ( v n ) , z (cid:48) := 1 E ⊗ A (cid:101) γ A (v ,n − ) ⊗ a ⊗ γ ( v ) γ ( v n )and z (cid:48)(cid:48) := 1 E ⊗ A (cid:101) γ A (v ,n − ) ⊗ A a · (cid:101) γ ( v ) ⊗ A (cid:101) γ ( v n ) ⊗ A E . Clearly z (cid:48) ∈ L ,n − · E and z ∈ U ,n − . By Proposition 2.30 and item (3), ψ ( x ) = ( − n σ ◦ ψ ( y ) ≡ ( − n σ ( z + z (cid:48) ) mod σ (cid:0) F n − U,R ( X n − ) · E (cid:1) . Hence, by Remark A.2 and items (4) and (7) of Lemma A.1, ψ ( x ) ≡ ( − n +1 σ ◦ σ − ◦ υ ( z ) + ( − n σ ( z (cid:48) ) mod F n − U,R ( X n ) . By Lemma 1.14 and the definition of σ we have σ ( z (cid:48) ) ∈ U ,n − ( R ). Consequently, by thedefinitions of υ and σ − , ψ ( x ) ≡ σ ( z (cid:48)(cid:48) ) mod F n − U,R ( X n ) . Since 1 E ∈ K ⊗ k V , the formula for ψ ( x ) follows now immediately from equality (2.14).(5) We proceed by induction on n . The case n = 0 is trivial. Suppose n > n −
1. Assume first that j < n . Let y := 1 E ⊗ γ (v ,i − ) ⊗ ν ( a i,j − ) ⊗ a · γ ( v ) ⊗ ν ( a j +1 ,n − ) ⊗ ν ( a n )and z := (cid:88) E ⊗ A (cid:101) γ A (v ,i − ) ⊗ a i,j − ⊗ a ⊗ a ( l ) j +1 ,n − ⊗ γ ( v ( l ) (cid:1) ν ( a n ) , where (cid:80) l a ( l ) j +1 ,n − ⊗ k v ( l ) := χ ( v ⊗ k a j +1 ,n − ). Note that z ∈ L n − i,i − · E . By Proposition 2.30and the inductive hypothesis, ψ ( x ) = ( − n σ ◦ ψ ( y ) ≡ ( − n σ ( z ) mod σ (cid:0) F i − U,R ( X n − ) · ν ( A ) (cid:1) . Thus, by Remark A.2 and item (4) of Lemma A.1, ψ ( x ) ≡ ( − n σ ( z ) mod F i − U,R ( X n ) . The expression for ψ ( x ) follows now from the fact that γ ( V ) ⊆ K ⊗ k V and equalities (1.6)and (2.15). We next consider the case j = n . Let y := 1 E ⊗ γ (v ,i − ) ⊗ ν ( a i,n − ) ⊗ a · γ ( v )and z := 1 E ⊗ A (cid:101) γ A (v ,i − ) ⊗ a i,n − ⊗ a · γ ( v ) . Note that z ∈ L n − i,i − · E . By Proposition 2.30 and item (1), ψ ( x ) = ( − n σ ◦ ψ ( y ) = ( − n σ ( z ) . Thus, by item (4) of Lemma A.1, ψ ( x ) ≡ ( − n σ ( z ) mod F i − U,R ( X n ) . The expression for ψ ( x ) follows now from the fact that γ ( V ) ⊆ K ⊗ k V and equality (2.15). CO)HOMOLOGY OF CROSSED PRODUCTS IN WEAK CONTEXTS 37 (6) By Proposition 2.30, ψ (1 E ⊗ x n ⊗ E ) = ( − n σ ◦ ψ (1 E ⊗ x ,n − ⊗ x n ) . Assume first that x n / ∈ ν ( A ) ∪ γ ( V ). Then, by item (2), ψ ( x ) = ( − n σ (0) = 0 . Assume now that x n ∈ ν ( A ). By inductive hypothesis, ψ ( x ) ∈ σ (cid:0) F i − U,R ( X n − ) · ν ( A ) (cid:1) , and so,by Remark A.2, ψ ( x ) ∈ F i − U,R ( X n ) , Assume finally that x n ∈ γ ( V ). If j in the statement can be taken lesser than n , then, by inductivehypothesis, ψ ( x ) ∈ σ (cid:0) F i − U,R ( X n − ) · γ ( V ) (cid:1) , and so, again by Remark A.2, ψ ( x ) ∈ F i − U,R ( X n ) ⊆ F i − U,R ( X n ) . If j is necessarily equals n , then, by items (3), (4) and (5), ψ ( x ) ∈ σ (cid:0) U n − i,i − + L n − i +1 ,i − · γ ( V ) γ ( V ) (cid:1) + σ (cid:0) F i − U,R ( X n − ) · γ ( V ) (cid:1) . So, by Remark A.2 and items (4) and (6) of Lemma A.1, ψ ( x ) ∈ σ (cid:0) L n − i +1 ,i − · γ ( V ) γ ( V ) (cid:1) + F i − U,R ( X n ) , and the result follows from Lemma 1.14, the fact that γ ( V ) ⊆ K ⊗ k V and equality (2.15). (cid:3) Proposition A.4.
Let R be a k -subalgebra of A . Assume that R is stable under χ and that F takes its values in R ⊗ k V . Let v, v , . . . , v i ∈ V and a, a i +1 , . . . , a n ∈ A . The map φ n ◦ ψ n hasthe following properties: (1) If x := 1 E ⊗ γ (v i ) ⊗ ν ( a i +1 ,n ) ⊗ E , then φ ◦ ψ ( x ) ≡ E ⊗ Sh(v i ⊗ k a i +1 ,n ) ⊗ E mod ν ( A ) ¯ ⊗ F i − R, ( E ⊗ n ) ¯ ⊗ ν ( K ) . (2) Let x , . . . , x n ∈ ν ( A ) ∪ γ ( V ) . If there exist indices j < j such that x j ∈ ν ( A ) and x j ∈ γ ( V ) , then φ ◦ ψ (1 E ⊗ x n ⊗ E ) = 0 . (3) If x = 1 E ⊗ γ (v ,i − ) ⊗ a · γ ( v ) ⊗ ν ( a i +1 ,n ) ⊗ E , then φ ◦ ψ ( x ) ≡ (cid:88) ν ( a ( j ) ) ⊗ Sh (cid:0) v ( j )1 ,i − ⊗ k v ⊗ k a i +1 ,n (cid:1) ⊗ E + (cid:88) E ⊗ Sh (cid:0) v ,i − ⊗ k a ⊗ a ( l ) i +1 ,n (cid:1) ⊗ γ (cid:0) v ( l ) (cid:1) , modulo ν ( A ) ¯ ⊗ F i − R, ( E ⊗ n ) ¯ ⊗ ν ( K ) + ν ( A ) ¯ ⊗ F i − R, ( E ⊗ n ) ¯ ⊗ ν ( K ) · γ ( V ) , where (cid:88) j a ( j ) ⊗ k v ( j )1 ,i − := χ (v ,i − ⊗ k a ) and (cid:88) l a ( l ) i +1 ,n ⊗ k v ( l ) := χ ( v ⊗ k a i +1 ,n ) . (4) If x = 1 E ⊗ γ (v ,j − ) ⊗ a · γ ( v ) ⊗ γ (v j +1 ,i ) ⊗ ν ( a i +1 ,n ) ⊗ E with j < i , then φ ◦ ψ ( x ) ≡ (cid:88) ν ( a ( l ) ) ⊗ Sh (cid:0) v ( l )1 ,i − ⊗ k v ⊗ k a i +1 ,n (cid:1) ⊗ E modulo ν ( A ) ¯ ⊗ F i − R, ( E ⊗ n ) ¯ ⊗ ν ( K ) + ν ( A ) ¯ ⊗ F i − R, ( E ⊗ n ) ¯ ⊗ ν ( K ) · γ ( V ) , where (cid:80) l a ( l ) ⊗ k v ( l )1 ,i − := χ (v ,i − ⊗ k a ) . (5) If x = 1 E ⊗ γ (v ,i − ) ⊗ ν ( a i,j − ) ⊗ a · γ ( v ) ⊗ ν ( a j +1 ,n ) ⊗ E with j > i , then φ ◦ ψ ( x ) ≡ (cid:88) l E ⊗ Sh (cid:0) v ,i − ⊗ k a i,j − ⊗ a ⊗ a ( l ) j +1 ,n (cid:1) ⊗ γ (cid:0) v ( l ) (cid:1) modulo ν ( A ) ¯ ⊗ F i − R, ( E ⊗ n ) ¯ ⊗ ν ( K ) · γ ( V ) , where (cid:80) l a ( l ) j +1 ,n ⊗ k v ( l ) := χ ( v ⊗ k a j +1 ,n ) . (6) Let x , . . . , x n ∈ E satisfying { l : x l / ∈ ν ( A ) ∪ γ ( V ) } = 1 . If there exist j < j suchthat x j ∈ ν ( A ) and x j ∈ γ ( V ) , then φ ◦ ψ (1 E ⊗ x n ⊗ E ) ∈ ν ( A ) ¯ ⊗ F i − R, ( E ⊗ n ) ¯ ⊗ ν ( K ) · γ ( V ) , where i := { l : x l / ∈ ν ( A ) } .Proof. (1) This is a consequence of Proposition 2.39 and item (1) of Proposition A.3.(2) This is a consequence of item (2) of Proposition A.3.(3) By item (3) of Proposition A.3, φ ◦ ψ ( x ) ≡ φ (cid:0) E ⊗ A (cid:101) γ A (v ,i − ) ⊗ A a · (cid:101) γ ( v ) ⊗ a i +1 ,n ⊗ E (cid:1) + (cid:88) φ (cid:0) E ⊗ A (cid:101) γ A (v ,i − ) ⊗ a ⊗ a ( l ) i +1 ,n ⊗ γ ( v ( l ) ) (cid:1) mod φ (cid:0) F i − U,R ( X n ) (cid:1) , Hence, by equality (1.6) and Proposition 2.39, φ ◦ ψ ( x ) ≡ (cid:88) ν ( a ( j ) ) ⊗ Sh (cid:0) v ( j )1 ,i − ⊗ k v ⊗ k a i +1 ,n (cid:1) ⊗ E + (cid:88) E ⊗ Sh (cid:0) v ,i − ⊗ k a ⊗ a ( l ) i +1 ,n (cid:1) ⊗ γ (cid:0) v ( l ) (cid:1) , modulo ν ( A ) ¯ ⊗ F i − R, ( E ⊗ n ) ¯ ⊗ ν ( K ) + ν ( A ) ¯ ⊗ F i − R, ( E ⊗ n ) ¯ ⊗ ν ( K ) · γ ( V ) + φ (cid:0) F i − U,R ( X n ) (cid:1) . Since, by Remark 2.38, Proposition 2.39, and the definition of F i − U,R ( X n ), φ (cid:0) F i − U,R ( X n ) (cid:1) ⊆ ν ( A ) ¯ ⊗ F i − R, ( E ⊗ n ) ¯ ⊗ ν ( K ) · γ ( V ) , the assertion in the statement is true.(4) Mimic the proof of item (3).(5) Mimic the proof of item (3).(6) Mimic the proof of item (3). (cid:3) Proposition A.5. If x , . . . , x n ∈ E satisfy { j : x j / ∈ ν ( A ) ∪ γ ( V ) } ≤ , then ω ( x ) ∈ ν ( A ) ¯ ⊗ F iA, ( E ⊗ n +1 ) ¯ ⊗ ν ( K ) , where x := 1 E ⊗ x n ⊗ E and i := { j : x j / ∈ ν ( A ) } .Proof. We first claim that if { j : x j / ∈ ν ( A ) ∪ γ ( V ) } = 0, then ω ( x ) = 0. We proceed byinduction on n . The case n = 1 is trivial, since ω = 0 by definition. Assume that n > n −
1. Then, ω ( x ) = ξ (cid:0) φ ◦ ψ ( x ) − ( − n ω (1 E ⊗ x ,n − ⊗ x n ) (cid:1) = ξ ◦ φ ◦ ψ ( x ) = 0 , where the first equality holds by Remark 2.31; the second one, by the inductive hypothesis; andthe last one, by the facts that ν ( A ) ¯ ⊗ F iA, ( E ⊗ n ) ¯ ⊗ ν ( K ) ⊆ ker( ξ ) and, by items (1) and (2) ofProposition A.4, φ ◦ ψ ( x ) ∈ ν ( A ) ¯ ⊗ F iA, ( E ⊗ n ) ¯ ⊗ ν ( K ) . CO)HOMOLOGY OF CROSSED PRODUCTS IN WEAK CONTEXTS 39
We now assume that { j : x j / ∈ ν ( A ) ∪ γ ( V ) } = 1 and we prove the proposition by inductionon n . This is trivial for n = 1 since w = 0. Suppose that n > n −
1. Since by Remark 2.31 ω ( x ) = ξ (cid:0) φ ◦ ψ ( x ) − ( − n ω (1 E ⊗ x ,n − ⊗ x n ) (cid:1) , and, by items (3)–(6) of Proposition A.4, Remark 2.38, Proposition 2.39 and the definition of ξ , ξ ◦ φ ◦ ψ ( x ) ∈ ξ (cid:16) ν ( A ) ¯ ⊗ F iA, ( E ⊗ n ) ¯ ⊗ ν ( K ) + ν ( A ) ¯ ⊗ F i − A, ( E ⊗ n ) ¯ ⊗ ν ( K ) · γ ( V ) (cid:17) ⊆ ν ( A ) ¯ ⊗ F iA, ( E ⊗ n +1 ) ¯ ⊗ ν ( K ) , in order to finish the proof it suffices to check that ξ ◦ ω (1 E ⊗ x ,n − ⊗ x n ) ∈ ν ( A ) ¯ ⊗ F iA, ( E ⊗ n +1 ) ¯ ⊗ ν ( K ) . (A.35)But, since by the inductive hypothesis and the claim,- if x n ∈ ν ( A ), then ω (1 E ⊗ x ,n − ⊗ x n ) ∈ F iA, ( E ⊗ E ⊗ n ⊗ E ) · ν ( A ),- if x n ∈ γ ( V ), then ω (1 E ⊗ x ,n − ⊗ x n ) ∈ F i − A, ( E ⊗ E ⊗ n ⊗ E ) · γ ( V ),- if x n / ∈ ν ( A ) ∪ γ ( V ), then ω (1 E ⊗ x ,n − ⊗ x n ) = 0,the condition (A.35) is satisfied. (cid:3) References [1] R. Akbarpour and M. Khalkhali,
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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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