Cyclic homology of crossed products

CYCLIC HOMOLOGY OF CROSSED P R ODUCTS GRACI ELA CARBONI, JOR GE A. GUCCIONE, AND JUAN J. GUCCIONE Abstract. W e o btain a mixed c omplex, simpler that the canonical one, given the Hochsc hild, cyclic, negativ e and p erio dic homology of a crossed pro duct E = A # f H , where H is an arbitrary Hopf algebra and f is a con v olution inv e rtible co cycle w i th v alues i n A . Actually , we work i n the more general con text of relative cyclic homology . Specifically , w e consider a subalgebra K of A which is stable und er the action of H , and we fi nd a mixed complex computing the Hochsc hild, cyclic, negativ e and p eriodic homology of E r el ative to K . As an appli cat ion w e obt ain t w o spectral sequence s con v erging to t he cyclic homolog y of E relative to K . The first one in the gene ral setting an d the second one (which generalizes those previously f ound b y several authors) when f tak es i ts v alues in K . Introduction Let G be a group ac ting on a differential o r algebra ic manifo ld M . Then G acts natur ally on the r ing A of regular functions o f M , and the algebr a G A o f inv ariants of this action cons is ts of the functions that a r e constants on each of the orbits of M . This sug gest to consider G A as a r e pla cemen t for M /G in non- commutativ e g e ometry . Under suitable conditions the in v a r ian t alg ebra G A and the s ma sh pro duct A # k [ G ], asso ciated with the a ction of G on A , are Morita equiv a len t. Since K -theory , Ho chsc hild homo logy and cyclic ho mology are Mor ita inv ariant, there is no lo s s of informa tio n if G A is replaced by A # k [ G ]. In the general case the exp erience had shown that s ma sh pro ducts are b etter choices than inv ariants r ings for alg e bras playing the role of noncommutativ e quotients. This fact was a motiv atio n for the interest in to develop to ols to compute the cyclic homology of s ma sh pr oducts alg ebras. This pr oblem was c onsidered in [F-T], [N ] and [G-J]. In the first pap er it was o btained a sp ectral sequence conv erg ing to the cyclic homolog y of the sma sh pr oduct a lgebra A # k [ G ]. In [G-J], this re sult was derived fr o m the theory of paracy clic mo dules a nd cy lindrical mo dules developed by the authors . The main too l for this computation was a version for cylindrical mo dules of E ilen b erg-Zilb er theorem. In [A-K] this theory was use d to obtain a F eigin-Tsygan t yp e spectr al sequence f or smash pro ducts A # H , of a Hopf algebra H with an H -mo dule algebra A . A t this po int it is natural to try to e x tend this result to the genera l cros sed pro ducts A # f H intro duced in [B-C-M] and [D-T]. Cro ssed Pro ducts, and mor e general alg ebras such a s Hopf Galois ex tensions, hav e b een homolog ically studied in several papers (se e for instance [L], [S], [G-G] and [J-S]) but almost all of them deal with its Ho chsc hild (co)homology . In [J-S] the relative to A cyclic ho mology of a Galois H extension C / A was s tudied, and the obtained r esults a pply to the Hopf cross ed pro ducts A # f H , giving the abs olute cyclic homolo gy when A is a separable algebr a. As far as we know, the unique w o rk dealing with the a bs olute 2000 Mathematics Subje ct Classific ation. primary 16E40;secondary 16W30. Supported by UBACYT 0294. Supported by UBACYT 0294 and PIP 5617 (CONICET). Supported by UBACYT 0294 and PIP 5617 (CONICET). 1 2 GRA CIELA CARBONI, JORGE A. GUCCIONE, AND JUAN J. GUCCIONE cyclic homolo gy of a crossed pro duct A # f H , with A non separable a nd f non triv ia l is [K-R]. In this pap er the authors get a F eigin-Tsyga n type s pectral se quence for a cro ssed pro ducts A # f H , under the h yp othesis that H is co commutativ e a nd f takes v alues in k . The goal of this article is to present a mixed complex  X , d, D  , simpler than the canonica l o ne , g iving the Ho c hschild, cy clic, negative a nd per iodic homolo gy of a crossed pro duct E = A # f H . Under the assumptions of [K-R] o ur complex is isomorphic to the one obtain there. Our main result is Theo rem 3.2, in which is proved that  X , d, D  is homotopically equiv ale nt to the cano nical normalized mixed complex  E ⊗ E ⊗ ∗ , b, B  . Actually , we work in the more genera l cont ext of r e la tiv e cyclic homology . Sp ecif- ically , w e consider a subalgebra K of A which is stable under the action of H , a nd we find a mixed complex co mputing the Ho chsc hild, cyclic , neg ativ e and p erio dic homology of E r e la tiv e to K (which we simply call the Ho chsc hild, cyclic, nega - tive and p erio dic homology of the K -algebra E ). As an application we obtain t wo sp ectral sequences conv erging to t he cyclic ho mo logy of the K -a lgebra E . The fir st one in the gener al setting and the second o ne (which gener alizes those o f [A-K] and [K-R]) when f takes v alues in K . Our metho d of pro of is different of the o ne used in [G-J], [A-K] and [K-R], being based in the results obtained in [G-G] and the P erturba tion Lemma. The pap er is orga nized in the following wa y: in Section 1 we summarize the material on mixed complexes, p erturbation lemma and Ho ch schild homolo g y of Hopf cros sed pro ducts necessary for o ur purp ose. More over we set up nota tio n and terminology . F or pro ofs w e refer to [C] and [G-G]. In Section 2 we obtain a mixed complex  b X , b d, b D  , more simpler tha t the canonical one, g iving that Ho ch schild, cyclic, per iodic and negative ho mology of the K -alg ebra E = A # f H , whic h works without the usual as sumption that f is co n volution inv ertible. Fina lly in Section 3 , we show that when f is conv olutio n inv e rtible, then  b X , b d, b D  is isomorphic to a simpler mixed complex  X , d, D  . Finally , as an application we de r iv e the above men tioned spectra l sequences. 1. Preliminaries In this sectio n we fix the ge ne r al terminology and notation used in the following, and give a brief review of the background necessar y for the understa nding of this pap er. Let k b e a commutativ e ring, A a k -algebr a and H a Ho pf k -algebra . W e will use the Sweedler notation ∆( h ) = h (1) ⊗ h (2) , with the summation understo o d and sup e rindices instead o f subindices. Rec all fro m [B-C-M] and [D-T] that a we ak action of H on A is a bilinear map ( h, a ) 7→ a h , from H × A to A , such that for h ∈ H , a, b ∈ A (1) ( ab ) h = a h (1) b h (2) , (2) 1 h = ǫ ( h )1, (3) a 1 = a . Given a weak action o f H on A and a k - linea r ma p f H ⊗ H → A , we let A # f H denote the k -algebra (in genera l non-as sociative a nd without 1) with underlying k -module A ⊗ H and multiplication map ( a ⊗ h )( b ⊗ l ) = ab h (1) f ( h (2) , l (1) ) ⊗ h (3) l (2) , for all a, b ∈ A , h, l ∈ H . The element a ⊗ h of A # f H will usually b e written a # h to remind us H is weakly acting on A . The algebra A # f H is called a cr osse d CYCLIC HOMOLOGY OF CROSSED PRODUCTS 3 pr o duct if it is asso ciative with 1#1 as iden tit y e le men t. It is easy to c heck that this happ ens if and only if f and the weak actio n satisfy the f ollowing conditio ns : (i) (Normality of f ) for all h ∈ H , we have f ( h, 1) = f (1 , h ) = ǫ ( h )1 A , (ii) (Co cycle condition) for all h , l, m ∈ H , we ha ve f  l (1) , m (1)  h (1) f  h (2) , l (2) m (2)  = f  h (1) , l (1)  f  h (2) l (2) , m  , (iii) (Twisted mo dule condition) f or all h, l ∈ H , a ∈ A we ha ve  a l (1)  h (1) f  h (2) , l (2)  = f  h (1) , l (1)  a h (2) l (2) . Next w e establis h so me no tations that w e will use trough the paper . Notations 1 .1. Let K b e a subalgebra of A and let B = A or B = E . (1) W e set D = B /K and H = H /k . (2) W e use the unadorned tensor symbol ⊗ to denote the tensor pro duct ⊗ K . (3) W e wr ite H ⊗ l k = H ⊗ k · · · ⊗ k H ( l times), D ⊗ l = D ⊗ · · · ⊗ D ( l times) a nd B l ( B ) = B ⊗ D ⊗ l ⊗ B . (4) Given b 0 ⊗ · · · ⊗ b r ∈ B ⊗ r +1 and 0 ≤ i < j ≤ r , we write b ij = b i ⊗ · · · ⊗ b j . (5) Given h ij ∈ H ⊗ j − i +1 k , we set h (1) ij ⊗ k h (2) ij = h (1) i ⊗ k · · · ⊗ k h (1) j ⊗ k h (2) i ⊗ k · · · ⊗ k h (2) j . (6) Given a ∈ A and h ij ∈ H ⊗ j − i +1 k , w e write a h ij =  · · · ( a h j ) h j − 1 · · ·  h i . (7) Given a ij ∈ A ⊗ j − i +1 and h ∈ H , we wr ite a h ij = a h (1) i ⊗ · · · ⊗ a ( h ( j − i +1) ) j . (8) The symbol γ ( h ) sta nds fo r 1# h . (9) Given h ij ∈ H ⊗ j − i +1 k , we set γ ( h ij ) = γ ( h i ) ⊗ · · · ⊗ γ ( h j ) and γ ( h ij ) = γ ( h i ) ⊗ A · · · ⊗ A γ ( h j ) . (10) W e will denote by H the imag e of the canonical inclusion of H int o A # H . (11) Given h 1 , . . . , h i ∈ H , we will denote by h h 1 , . . . , h i i the Hopf subalgebra of H gener ated by h 1 , . . . , h i . 1.1. A simp le resol ution. Let Υ b e the family of a ll epimorphisms of E - bimo dules which split as left E -mo dule ma ps. In this subsection we revie w the construction of the Υ -pro jective resolution ( X ∗ , d ∗ ), o f E as an E -bimo dule, given in Section 1 of [G-G]. W e are going to m o dify the sign of so me maps in order to obtain expr es- sions for the bo undary maps d ∗ and the comparis on maps b et ween ( X ∗ , d ∗ ) and the normalized bar resolution of E , simpler than those of the a bov e mentioned pap er. Let K b e a subalgebra of A , closed under the weak action of H on A . Since we wan t to co nsider the Cyclic ho mology of the K -alg ebra E , in the sequel Υ will b e the family of a ll epimorphisms o f E -bimo dules which split as a ( E , K )-bimo dule map. F or all r , s ≥ 0, let Y s = E ⊗ A ( E / A ) ⊗ s A ⊗ A E and X r s = E ⊗ A ( E / A ) ⊗ s A ⊗ A ⊗ r ⊗ E . 4 GRA CIELA CARBONI, JORGE A. GUCCIONE, AND JUAN J. GUCCIONE Consider the diagram of E -bimo dules and E -bimo dule maps . . . − ∂ 2   Y 2 − ∂ 2   X 02 µ 2 o o X 12 d 0 12 o o . . . d 0 22 o o Y 1 − ∂ 1   X 01 µ 1 o o X 11 d 0 11 o o . . . d 0 21 o o Y 0 X 00 µ 0 o o X 10 d 0 10 o o . . . , d 0 20 o o where ( Y ∗ , ∂ ∗ ) is the normalized bar reso lution of the A -alg e br a E , introduced in [G-S]; for each s ≥ 0 , the complex ( X ∗ s , d 0 ∗ s ) is ( − 1) s times the no rmalized bar r esolution of the alg ebra inclusion K ⊆ A , tensored on the left ov er A with E ⊗ A ( E / A ) ⊗ s A , and on the right over A with E ; and for eac h s ≥ 0, the map µ s is the canonical pro jection. Note that X r s ≃ E ⊗ k H ⊗ s k ⊗ A ⊗ r ⊗ E , where the right action of K on E ⊗ k H ⊗ s k is the one o btained by transla tion o f structure through the canonica l bijection from E ⊗ k H ⊗ s k to E ⊗ A ( E / A ) ⊗ s A . Mor e o ver, each o ne of the rows of this diagra m is contractible as a ( E , K )-bimo dule complex. A con tr acting homo top y σ 0 0 s : Y s → X 0 s and σ 0 r +1 ,s : X r s → X r +1 ,s , of the s -th row, is given b y σ 0 0 s  γ ( h 0 ,s +1 )  = γ ( h 0 s ) ⊗ γ ( h s +1 ) and σ 0 r +1 ,s  γ ( h 0 s ) ⊗ a 1 r ⊗ a r +1 γ ( h )  = ( − 1) r + s +1 γ ( h 0 s ) ⊗ a 1 ,r +1 ⊗ γ ( h ) , Let e µ : Y 0 → E b e the m ultiplication ma p. The complex of E -bimo dules E Y 0 − e µ o o Y 1 − ∂ 1 o o Y 2 − ∂ 2 o o Y 3 − ∂ 3 o o Y 4 − ∂ 4 o o Y 5 − ∂ 5 o o . . . − ∂ 6 o o is also contractible as a co mplex of ( E , K )-bimo dules. A chain contracting homo- topy σ − 1 0 : E → Y 0 and σ − 1 s +1 : Y s → Y s +1 ( s ≥ 0) , is given by σ − 1 s +1 ( x ) = ( − 1) s x ⊗ A 1 E . F or r ≥ 0 and 1 ≤ l ≤ s , we define E -bimo dule maps d l r s : X r s → X r + l − 1 ,s − l recursively on l and r , by: d l ( x ) =          σ 0   ∂   µ ( x ) if l = 1 and r = 0, − σ 0   d 1   d 0 ( x ) if l = 1 and r > 0, − P l − 1 j =1 σ 0   d l − j   d j ( x ) if 1 < l and r = 0, − P l − 1 j =0 σ 0   d l − j   d j ( x ) if 1 < l and r > 0, for x ∈ E ⊗ A ( E / A ) ⊗ s A ⊗ A ⊗ r ⊗ K . Theorem 1. 2 ([G-G]) . Ther e is a Υ -pr oje ctive reso lu tion of E (1.1) E X 0 − µ o o X 1 d 1 o o X 2 d 2 o o X 3 d 3 o o X 4 d 4 o o . . . , d 5 o o CYCLIC HOMOLOGY OF CROSSED PRODUCTS 5 wher e µ : X 00 → E is the multiplic ation map, X n = M r + s = n X r s and d n = n X l =1 d l 0 n + n X r =1 n − r X l =0 d l r,n − r . In order to carr y o ut our computations we also need to give an explicit co n trac t- ing homotopy of the resolutio n (1.1). F or this w e define maps σ l l,s − l : Y s → X l,s − l and σ l r + l +1 ,s − l : X r s → X r + l +1 ,s − l recursively on l , b y: σ l r + l +1 ,s − l = − l − 1 X i =0 σ 0   d l − i   σ i (0 < l ≤ s and r ≥ − 1) . Prop osition 1.3 ([G-G]) . The family σ 0 : E → X 0 , σ n +1 : X n → X n +1 ( n ≥ 0 ) , define d by σ 0 = σ 0 00   σ − 1 0 and σ n +1 = − n +1 X l =0 σ l l,n − l +1   σ − 1 n +1   µ n + n X r =0 n − r X l =0 σ l r + l +1 ,n − r − l ( n ≥ 0 ) , is a c ontr acting homotopy of (1.1) . Let e f h h 1 , . . . , h i i b e the minimal K -subbimo dule o f A including f h h 1 , . . . , h i i and closed under the w ea k actio n of h h 1 , . . . , h i i on A . Theorem 1. 4 ([G-G]) . L et x = γ ( h 0 s ) ⊗ a 1 r ⊗ 1 . The fol lowing assertions hold : d 1 ( x ) = s − 1 X i =0 ( − 1) i γ ( h 0 ,i − 1 ) ⊗ A γ ( h i ) γ ( h i +1 ) ⊗ A γ ( h i +1 ,s ) ⊗ a 1 r ⊗ 1 + ( − 1) s γ ( h 0 ,s − 1 ) ⊗ a h (1) s 1 r ⊗ γ ( h (2) s ) and d 2 ( x ) = ( − 1) s − 1 γ ( h 0 ,s − 2 ) ⊗ f ( h (1) s − 1 , h (1) s ) ∗ a 1 r ⊗ γ ( h (2) s − 1 h (2) s ) , wher e f ( h, l ) ∗ a 1 r = P r i =0 ( − 1) i ( a l (1) 1 i ) h (1) ⊗ f ( h (2) , l (2) ) ⊗ a h (3) l (3) i +1 ,r . Mor e over, for e ach l ≥ 2 , t he map d l r s takes x into the E -subbimo dule of X r + l − 1 ,s − l gener ate d by al l the simple tensors 1 ⊗ x 1 ⊗ A · · · ⊗ A x s − l ⊗ a 1 ⊗ · · · ⊗ a r + l − 1 ⊗ 1 with one a j in f h h 1 , . . . , h s i and l − 2 of t he others a j ’s in e f h h 1 , . . . , h s i . 1.1.1. Comp arison with t he normalize d b ar r esolut ion. Let ( B ∗ ( E ) , b ′ ∗ ) b e the nor- malized bar resolution of the a lgebra inclusion A ⊆ E . As it is well known, the complex E B 0 ( E ) µ o o B 1 ( E ) b ′ 1 o o B 2 ( E ) b ′ 2 o o B 3 ( E ) b ′ 3 o o . . . b ′ 4 o o is con tractible as a complex of ( E , K )-bimodules, with contracting homotopy ξ 0 : E → B 0 ( E ) , ξ n +1 : B n ( E ) → B n +1 ( E ) ( n ≥ 0), given by ξ n ( x ) = ( − 1) n x ⊗ 1. Let φ ∗ : ( X ∗ , d ∗ ) → ( B ∗ ( E ) , b ′ ∗ ) and ψ ∗ : ( B ∗ ( E ) , b ′ ∗ ) → ( X ∗ , d ∗ ) be the mo rphisms of E -bimo dule complexes, recursively defined by φ 0 = id , ψ 0 = id , φ n +1 ( x ⊗ 1) = ξ n +1   φ n   d n +1 ( x ⊗ 1) 6 GRA CIELA CARBONI, JORGE A. GUCCIONE, AND JUAN J. GUCCIONE and ψ n +1 ( y ⊗ 1) = σ n +1   ψ n   b ′ n +1 ( y ⊗ 1) . Prop osition 1.5 ([G-G]) . ψ   φ = id and φ   ψ is homotopic al ly e quivalent to the identity map. A homotopy ω ∗ +1 : φ ∗   ψ ∗ → id ∗ is r e cursively define d by ω 1 = 0 and ω n +1 ( x ) = ξ n +1   ( φ n   ψ n − id − ω n   b ′ n )( x ) , for x ∈ E ⊗ E ⊗ n ⊗ K . R emark 1.6 . Since ω  E ⊗ E ⊗ n − 1 ⊗ K  ⊆ E ⊗ E ⊗ n ⊗ K a nd ξ v anishes on E ⊗ E ⊗ n ⊗ K , ω ( x 0 n ⊗ 1) = ξ  φ   ψ ( x 0 n ⊗ 1) − ( − 1 ) n ω ( x 0 n )  . 1.1.2. The filtr ations of ( B ∗ ( E ) , b ′ ∗ ) and ( X ∗ , d ∗ ) . Let F i ( X n ) = M 0 ≤ s ≤ i E ⊗ A ( E / A ) ⊗ s A ⊗ A ⊗ n − s ⊗ E and let F i ( B n ( E )) b e the E -subbimo dule of B n ( E ) genera ted by the tensors 1 ⊗ x 1 ⊗ · · · ⊗ x n ⊗ 1 such that a t least n − i of the x j ’s belo ng to A . The normaliz ed bar res olution ( B ∗ ( E ) , b ′ ∗ ) and the resolution ( X ∗ , d ∗ ) are filtered by F 0 ( B ∗ ( E )) ⊆ F 1 ( B ∗ ( E )) ⊆ F 2 ( B ∗ ( E )) ⊆ . . . and F 0 ( X ∗ ) ⊆ F 1 ( X ∗ ) ⊆ F 2 ( X ∗ ) ⊆ . . . , resp ectiv e ly . In [G-G, Prop osition 1 .2.2] it w a s proven that the maps φ ∗ , ψ ∗ and ω ∗ +1 preserve filtr ations. In Appendix A we a re going to improve this result. 1.2. Mi xed comp lexes. In this subsection we recall briefly the notion of mixed complex. F or more details ab out this concept w e refer to [Ka] and [B]. A mixed co mplex ( X , b, B ) is a graded k -mo dule ( X n ) n ≥ 0 , endowed with mo r- phisms b : X n → X n − 1 and B : X n → X n +1 , suc h that b   b = 0 , B   B = 0 and B   b + b   B = 0 . A morphism of mixed complexes f : ( X , b, B ) → ( Y , d, D ) is a family of maps f : X n → Y n , such that d   f = f   b and D   f = f   B . Le t u be a deg ree 2 v ar i- able. A mixed complex X = ( X, b, B ) determines a double complex BP ( X ) = . . . b   . . . b   . . . b   . . . b   . . . X 3 u − 1 B o o b   X 2 u 0 B o o b   X 1 u B o o b   X 0 u 2 B o o . . . X 2 u − 1 B o o b   X 1 u 0 B o o b   X 0 u B o o . . . X 1 u − 1 B o o b   X 0 u 0 B o o . . . X 0 u − 1 , B o o CYCLIC HOMOLOGY OF CROSSED PRODUCTS 7 where b ( x u i ) = b ( x ) u i and B ( x u i ) = B ( x ) u i − 1 . By deleting the p ositively num- ber ed columns we obtain a sub complex BN ( X ) of BP ( X ). Let BN ′ ( X ) b e the kernel of the canonical surjection from BN ( X ) to ( X , b ). The quotien t double com- plex BP ( X ) / BN ′ ( X ) is denoted by BC ( X ). T he homolo gies HC ∗ ( X ), HN ∗ ( X ) a nd HP ∗ ( X ), of the total co mplexes of BC ( X ), BN ( X ) a nd BP ( X ) resp ectively , are called the cyclic, negative and p erio dic homologies of X . The homology HH ∗ ( X ), of ( X , b ), is called the Ho c hschild homolog y of X . Fina lly , it is clear that a mor- phism f : X → Y of mixed c o mplexes induces a morphis m from the double complex BP ( X ) to the double complex BP ( Y ). As usual, given a K -bimo dule M , we let M ⊗ denote the q uo tien t M / [ M , K ], where [ M , K ] is the k -mo dule g e nerated b y the commutators mλ − λm , with λ ∈ K and m ∈ M . Moreov er [ m ] will b e deno te the cla ss of an element m ∈ M in M ⊗ . Let C b e a k algebra and K ⊆ C a suba lg ebra. The norma liz ed mixed co mplex of the K -algebr a C is the mixe d complex ( C ⊗ C ⊗ ∗ ⊗ , b, B ), where b is the cano nical Ho c hschild boundary map and the Connes op erator B is given by B ([ c 0 ⊗ · · · ⊗ c r ]) = r X i =0 ( − 1) ir [1 ⊗ c i ⊗ · · · ⊗ c r ⊗ c 0 ⊗ · · · ⊗ c i − 1 ] . The cyclic, negative, per iodic and Ho chschild ho mo logy groups HC K ∗ ( C ), HN K ∗ ( C ), HP K ∗ ( C ) and HH K ∗ ( C ), of the K -a lgebra C , are the res p ective ho mology groups of ( C ⊗ C ⊗ ∗ ⊗ , b, B ). 1.3. The p erturbation l emma. Next, we recall the p erturbatio n lemma. W e give the mor e gener al v ersio n introduced in [C]. A homotop y equiv alence data (1.2) ( Y , ∂ ) i / / ( X, d ) p o o , h : X ∗ → X ∗ +1 , consists of the follo w ing : (1) Chain complexes ( Y , ∂ ), ( X, d ) and quasi-isomor phisms i , p betw een them, (2) A homoto p y h from i   p to id . A p erturbation δ of (1.2) is a map δ : X ∗ → X ∗− 1 such that ( d + δ ) 2 = 0. W e call it small if id − δ   h is inv ertible. In this case w e write A = ( id − δ   h ) − 1   δ and w e consider (1.3) ( Y , ∂ 1 ) i 1 / / ( X, d + δ ) p 1 o o , h 1 : X ∗ → X ∗ +1 , with ∂ 1 = ∂ + p   A   i, i 1 = i + h   A   i, p 1 = p + p   A   h, h 1 = h + h   A   h. A deformation retract is a homoto py equiv alence data such that p   i = id . A defor- mation retract is called spe c ial if h   i = 0, p   h = 0 and h   h = 0. In all the cases considered in this pap er the map δ   h is lo cally nilpotent, and so ( id − δ   h ) − 1 = P ∞ n =0 ( δ   h ) n . Theorem 1.7 ([C ]) . If δ is a smal l p ert urb ation of the homotopy e quivalenc e data (1.2) , then the p erturb e d data (1.3) is a homotopy e quivalenc e. Mor e over, if (1.2 ) is a sp e cial deformation r etr act, then (1.3 ) is also. 8 GRA CIELA CARBONI, JORGE A. GUCCIONE, AND JUAN J. GUCCIONE 2. A mixed complex giving the cyclic homology o f a crossed product Recall that Υ is the family of all epimorphisms of E -bimo dules which split as a ( E , K )-bimodule map. Since ( X ∗ , d ∗ ) is a Υ-pr o jective reso lutio n of E , the Ho c hschild homolog y of the K -algebra E , is the homolo gy of E ⊗ E e ( X ∗ , d ∗ ). W rite b X r s = E ⊗ A ( E / A ) ⊗ s A ⊗ A ⊗ r ⊗ . It is e asy to chec k that b X r s ≃ E ⊗ E e X r s . Let b d l r s : b X r s → b X r + l − 1 ,s − l be the map induced by id E ⊗ E e d l r s . Clea rly b d 0 r s is ( − 1) s times the b oundary map o f the norma liz ed chain Ho c hschild complex of the K - algebra A , with co efficients in E ⊗ A ( E / A ) ⊗ s A . Moreover, fro m Theo rem 1.4, it follows e a sily that b d 1 ( x ) =  a 0 γ ( h 0 ) γ ( h 1 ) ⊗ A γ ( h 2 s ) ⊗ a 1 r  + s − 1 X i =1 ( − 1) i  a 0 γ ( h 0 ) ⊗ A γ ( h 1 ,i − 1 ) ⊗ A γ ( h i ) γ ( h i +1 ) ⊗ A γ ( h i +2 ,s ) ⊗ a 1 r  + ( − 1) s  γ ( h (2) s ) a 0 γ ( h 0 ) ⊗ A γ ( h 1 ,s − 1 ) ⊗ a h (1) s 1 r  and b d 2 ( x ) = ( − 1) s − 1  γ ( h (2) s − 1 h (2) s ) a 0 γ ( h 0 ) ⊗ A γ ( h 0 ,s − 2 ) ⊗ f ( h (1) s − 1 , h (1) s ) ∗ a 1 r  , where x =  a 0 γ ( h 0 ) ⊗ A γ ( h 1 s ) ⊗ a 1 r  and f ( h, l ) ∗ a 1 r is as in Theo r em 1.4. With the ab o ve identifications the complex E ⊗ E e ( X ∗ , d ∗ ) beco mes ( b X ∗ , b d ∗ ), where b X n = M r + s = n b X r s and b d n := n X l =1 b d l 0 n + n X r =1 n − r X l =0 b d l r,n − r . Let b φ ∗ : ( b X ∗ , b d ∗ ) → ( E ⊗ E ⊗ ∗ ⊗ , b ∗ ) and b ψ ∗ : ( E ⊗ E ⊗ ∗ ⊗ , b ∗ ) → ( b X ∗ , b d ∗ ) be the morphis ms of co mplexes induced by φ and ψ resp ectively . By Prop osition 1.5, we hav e b ψ   b φ = id and b φ   b ψ is homotopica lly equiv alent to the iden tity map, b eing an homotopy b ω ∗ +1 : b φ ∗   b ψ ∗ → id ∗ , the family of maps  b ω n +1 : E ⊗ E ⊗ n ⊗ → E ⊗ E ⊗ n +1 ⊗  n ≥ 0 , induced b y  ω n +1 : B n ( E ) → B n +1 ( E )  n ≥ 0 . 2.0.1. The filtr ations of ( E ⊗ E ⊗ ∗ ⊗ , b ∗ ) and ( b X ∗ , b d ∗ ) . Let F i ( b X n ) = M 0 ≤ s ≤ i b X n − s,s . and let F i ( E ⊗ E ⊗ n ⊗ ) be the k -submo dule o f E ⊗ E ⊗ n ⊗ generated b y the classes of the simple tenso rs x 0 ⊗ · · · ⊗ x n such that a t least n − i of the elements x 1 , . . . , x n belo ng to A . The normalize d Ho chsc hild complex ( E ⊗ E ⊗ ∗ ⊗ , b ∗ ) and t he complex ( b X ∗ , b d ∗ ) are filtered by F 0 ( E ⊗ E ⊗ ∗ ⊗ ) ⊆ F 1 ( E ⊗ E ⊗ ∗ ⊗ ) ⊆ F 2 ( E ⊗ E ⊗ ∗ ⊗ ) ⊆ . . . and F 0 ( b X ∗ ) ⊆ F 1 ( b X ∗ ) ⊆ F 2 ( b X ∗ ) ⊆ . . . , resp ectiv e ly . F ro m [G-G, P ropo s ition 1.2.2 ] it follows immediately that the maps b φ ∗ , b ψ ∗ and b ω ∗ +1 preserve filtrations. In App endix A we are going to improve this result. CYCLIC HOMOLOGY OF CROSSED PRODUCTS 9 Let b V n ⊆ b V ′ n be the k -submo dules of E ⊗ E ⊗ n ⊗ ge ne r ated by the simple tensors x 0 n such that #( { j ≥ 1 : x j / ∈ A ∪ H} ) = 0 and #( { j ≥ 1 : x j / ∈ A ∪ H} ) ≤ 1, resp ectiv e ly . Let h 1 , . . . , h i ∈ H . Recall tha t e f h h 1 , . . . , h i i is the minimal K -subbimo dule of A including f h h 1 , . . . , h i i and closed under the weak action of H . W e will denote by b C n ( h 1 , . . . , h i ) the k -submo dule of E ⊗ E ⊗ n ⊗ ge ne r ated by the classe s of all the simple tensors x 0 ⊗ · · · ⊗ x n with some x 1 , . . . , x n in e f h h 1 , . . . , h i i . Prop osition 2.1. The map b φ satisfies b φ  a 0 γ ( h 0 ) ⊗ A γ ( h 1 i ) ⊗ a 1 ,n − i  ≡  a 0 γ ( h 0 ) ⊗ γ ( h 1 i ) ∗ a 1 ,n − i  +  a 0 γ ( h 0 ) ⊗ A x  , wher e  a 0 γ ( h 0 ) ⊗ A x  ∈ F i − 1 ( E ⊗ E ⊗ n ⊗ ) ∩ b V n ∩ b C n ( h 1 , . . . , h i ) . Pr o of. See App endix A.  Prop osition 2.2. If x = [1 ⊗ x 1 n ] ∈  F i ( E ⊗ E ⊗ n ⊗ ) ∩ b V ′ n  , then b ω ( x ) ∈ ( K ⊗ E ⊗ n +1 ) ∩ F i ( E ⊗ E ⊗ n +1 ⊗ ) ∩ b V n +1 . Pr o of. See App endix A.  Lemma 2.3. L et B ∗ : E ⊗ E ⊗ ∗ ⊗ → E ⊗ E ⊗ ∗ +1 ⊗ b e t he Connes op er ator. The c omp osition B   b ω   B   b φ is the zer o map. Pr o of. Let x =  a 0 γ ( h 0 ) ⊗ A γ ( h 1 i ) ⊗ a 1 ,n − i  ∈ b X n − i,i . By Pr opositio n 2.1, b φ ( x ) ∈ F i ( E ⊗ E ⊗ n ⊗ ) ∩ b V n . Hence B   b φ ( x ) ∈ ( K ⊗ E ⊗ n +1 ) ∩ F i +1 ( E ⊗ E ⊗ n +1 ⊗ ) ∩ b V ′ n +1 , a nd so, by P ropo sition 2 .2, b ω   B   b φ ( x ) ∈ ( K ⊗ E ⊗ n +1 ⊗ ) ∩ F i +1 ( E ⊗ E ⊗ n +1 ⊗ ) ∩ b V n +2 ⊆ ker B , as desired.  F or each n ≥ 0, let b D n : b X n → b X n +1 be the ma p b D = b ψ   B   b φ . Theorem 2. 4.  b X , b d, b D  is a mixe d c omplex giving the H o chschild, cy clic, ne gative and p erio dic homolo gy o f t he K -algebr a E . Mor e over we have chain c omplexes maps T ot  BP ( b X , b d, b D )  b Φ / / T ot  BP ( E ⊗ E ⊗ ∗ ⊗ , b, B )  b Ψ o o , given by b Φ n ( x u i ) = b φ ( x ) u i + b ω   B   b φ ( x ) u i − 1 and b Ψ n ( x u i ) = X j ≥ 0 b ψ   ( B   b ω ) j ( x ) u i − j . These maps satisfy b Ψ   b Φ = id and and b Φ   b Ψ is homotopic al ly e qu ivalent to the iden- tity map. A homotopy b Ω ∗ +1 : b Φ ∗   b Ψ ∗ → id ∗ is given by b Ω n +1 ( x u i ) = X j ≥ 0 b ω   ( B   b ω ) j ( x ) u i − j . Pr o of. F or each i ≥ 0, let b φu i : b X n − 21 u i →  E ⊗ E ⊗ n − 2 i ⊗  u i , b φu i :  E ⊗ E ⊗ n − 2 i ⊗  u i → b X n − 21 u i 10 GRA CIELA CARBONI, JORGE A. GUCCIONE, AND JUAN J. GUCCIONE and b ω u i :  E ⊗ E ⊗ n − 2 i ⊗  u i →  E ⊗ E ⊗ n +1 − 2 i ⊗  u i , be the maps defined by b φu i ( x u i ) = b φ ( x ) u i , etcetera. By the comment s preceding Lemma 2.3, w e hav e a sp ecial deformation retract T ot  BC ( b X , b d , 0)  L i ≥ 0 b φu i / / T ot  BC ( E ⊗ E ⊗ ∗ ⊗ , b, 0)  L i ≥ 0 b ψ u i o o , M i ≥ 0 b ω u i . By applying the perturbation lemma to this datum endow ed with the p erturbation induced b y B , and taking int o account Lemma 2.3, w e obtain the s p ecial deforma- tion retract (2.4) T ot  BC ( b X , b d , b D )  b Φ / / T ot  BC ( E ⊗ E ⊗ ∗ ⊗ , b, B )  b Ψ o o , b Ω . It is easy to see that b Φ, b Ψ and b Ω commute with the canonica l surjections (2.5) T ot  BC ( b X , b d, b D )  → T ot  BC ( b X , b d, b D )  [2] and (2.6) T ot  BC ( E ⊗ E ⊗ ∗ ⊗ , b, B )  → T ot  BC ( E ⊗ E ⊗ ∗ ⊗ , b, B )  [2] . An standard argument, from these facts, finishes the pro of.  Let h 1 , . . . , h i ∈ H . In the seq ue l we let b J n ( h 1 , . . . , h i ) and H b J n +1 ( h 1 , . . . , h i ) denote the k -submo dules o f b X n generated by all the class e s of simple tensors x 0 s ⊗ a 1 ,n − s with 0 ≤ s < n and some a j in f h h 1 , . . . , h i i , and for all the c lasses of simple tens ors x 0 s ⊗ a 1 ,n − s with 0 ≤ s < n and s ome a j in e f h h 1 , . . . , h i i , resp ec- tively . Prop osition 2.5. L et b R i = F i ( E ⊗ E ⊗ n ⊗ ) \ F i − 1 ( E ⊗ E ⊗ n ⊗ ) . The fol lowing e qualities hold: (1) b ψ  a 0 γ ( h 0 ) ⊗ γ ( h 1 i ) ⊗ a i +1 ,n  =  a 0 γ ( h 0 ) ⊗ A γ ( h 1 i ) ⊗ a i +1 ,n  . (2) If x 0 n ∈ b R i ∩ b V n and ther e is 1 ≤ j ≤ i such that x j ∈ A , then b ψ ( x 0 n ) = 0 . (3) If x =  a 0 γ ( h 0 ) ⊗ γ ( h 1 ,i − 1 ) ⊗ a i γ ( h i ) ⊗ a i +1 ,n  , then b ψ ( x ) ≡  a 0 γ ( h 0 ) ⊗ A γ ( h 1 ,i − 1 ) ⊗ A a i γ ( h i ) ⊗ a i +1 ,n  +  γ ( h (2) i ) a 0 γ ( h 0 ) ⊗ A γ ( h 1 ,i − 1 ) ⊗ a i ⊗ a h (1) i i +1 ,n  , mo dule L i − 2 l =0  b X n − l,l ∩ b J n ( h 1 , . . . , h i )  . (4) If  x = a 0 γ ( h 0 ) ⊗ γ ( h 1 ,j − 1 ) ⊗ a j h j ⊗ γ ( h j +1 ,i ) ⊗ a i +1 ,n  with j < i , then b ψ ( x ) ≡  a 0 γ ( h 0 ) ⊗ A γ ( h 1 ,j − 1 ) ⊗ A a j γ ( h j ) ⊗ A γ ( h j +1 ,i ) ⊗ a i +1 ,n  , mo dule L i − 2 l =0  b X n − l,l ∩ b J n ( h 1 , . . . , h i )  . (5) If x =  a 0 γ ( h 0 ) ⊗ γ ( h 1 ,i − 1 ) ⊗ a i,j − 1 ⊗ a j γ ( h j ) ⊗ a j +1 ,n  with j > i , then b ψ ( x ) ≡  γ ( h (2) j ) a 0 γ ( h 0 ) ⊗ A γ ( h 1 ,i − 1 ) ⊗ a ij ⊗ a h (1) j j +1 ,n  , mo dule L i − 2 l =0  b X n − l,l ∩ b J n ( h 1 , . . . , h i − 1 , h j )  . (6) If x 0 n ∈ b R i ∩ b V ′ n and ther e exists 1 ≤ j 1 < j 2 ≤ n such that x j 1 ∈ A and x j 2 ∈ H , then b ψ ( x 0 n ) = 0 . CYCLIC HOMOLOGY OF CROSSED PRODUCTS 11 Pr o of. See App endix A.  Let b η n : b X n → b X n +1 , b t H,n : b X n → b X n and b t A,n : b X n +1 → b X n +1 be the k -linear maps defined b y b η  a 0 γ ( h 0 ) ⊗ A γ ( h 1 i ) ⊗ a 1 ,n − i  =  γ ( h 0 i ) ⊗ a 1 ,n − i ⊗ a 0  , b t H  a 0 γ ( h 0 ) ⊗ A γ ( h 1 i ) ⊗ a 1 ,n − i  =  γ ( h (2) i ) ⊗ A a 0 γ ( h 0 ) ⊗ A γ ( h 1 ,i − 1 ) ⊗ a h (1) i 1 ,n − i  and b t A  a 0 γ ( h 0 ) ⊗ A γ ( h 1 i ) ⊗ a 1 ,n − i +1  =  γ ( h (2) 0 i ) ⊗ a 2 ,n − i +1 ⊗ a 0 a h (1) 0 i 1  , resp ectiv e ly Prop osition 2.6. The Connes op er ator b D satisfies: (1) If x =  a 0 ⊗ A γ ( h 1 i ) ⊗ a 1 ,n − i  , then b D ( x ) = n − i X j =0 ( − 1) j ( n − i )+ n b t j A   η ( x ) , mo dule F i − 1 ( b X n +1 ) ∩ H b J n +1 ( h 1 , . . . , h i ) . (2) If x =  a 0 γ ( h 0 ) ⊗ A γ ( h 1 i ) ⊗ a 1 ,n − i  with a 0 γ ( h 0 ) / ∈ A , then b D ( x ) = i X j =0 ( − 1) j i 1 ⊗ A b t j H ( x ) + n − i X j =0 ( − 1) j ( n − i )+ n b t j A   η ( x ) mo dule F i ( b X n +1 ) ∩ H b J n +1 ( h 1 , . . . , h i ) . Pr o of. It is a direc t consequence of the definition of B , Prop ositions 2.1 and 2.5. W e leave the details to the reader.  3. The cyclic ho mology of a cr ossed product with inver tible cocycle Let E = A # f H . Assume tha t the co cycle f is inv e r tible. Then, the map γ is con- volution in vertible and its in verse is giv e n by γ − 1 ( h ) = f − 1 ( S ( h (2) ) , h (3) )# S ( h (1) ). In [G-G] it was prov en that under this hypo thesis the complex ( b X ∗ , b d ∗ ) of Section 2 is isomo rphic to a simpler c o mplex ( X ∗ , d ∗ ). In this section we obtain a similar result for the mixed complex  b X , b d, b D  . F or each r , s ≥ 0, let X r s =  E ⊗ A ⊗ r ⊗  ⊗ k H ⊗ s k . The map θ r s : b X r s → X r s , defined b y θ r s ( x ) = ( − 1) r s  a 0 γ ( h 0 ) a 1 γ ( h (1) 1 ) · · · a s γ ( h (1) s ) ⊗ a s +1 ,s + r  ⊗ k h (2) 1 s , where x =  a 0 γ ( h 0 ) ⊗ A · · · ⊗ A a s γ ( h s ) ⊗ a s +1 ,s + r  , is an iso morphism. The in verse map of θ r s is the map given by  a 0 γ ( h 0 ) ⊗ a 1 r  ⊗ k h 1 s 7→ ( − 1) r s  a 0 γ ( h 0 ) γ − 1 ( h (1) s ) · · · γ − 1 ( h (1) 1 ) ⊗ A γ ( h (2) 1 s ) ⊗ a 1 r  . Let d l r s : X r s → X r + l − 1 ,s − l be the map d l r s := θ r + l − 1 ,s − l ◦ b d l r s ◦ θ − 1 r s . In the a bsolute case the following result was obtained in [G-G]. The genera lization to the relative context is dir ect. 12 GRA CIELA CARBONI, JORGE A. GUCCIONE, AND JUAN J. GUCCIONE Theorem 3.1. The Ho chschild homolo gy of the K -algebr a E is t he homolo gy of ( X ∗ , d ∗ ) , wher e X n = M r + s = n X r s and d n := n X l =1 d l 0 n + n X r =1 n − r X l =0 d l r,n − r . Mor e over d 0 r s is the b oundary map of the normalize d chain Ho chschild c omplex of the K -algebr a A , with c o efficients in E , tensor e d on the right over k with id H ⊗ s , d 1 r s ( x ) = ( − 1) r + s  γ ( h (3) s ) a 0 γ ( h 0 ) γ − 1 ( h (1) s ) ⊗ a h (2) s 1 r  ⊗ k h 1 ,s − 1 + s − 1 X i =1 ( − 1) r + i  a 0 γ ( h 0 ) ⊗ a 1 r ⊗ h 1 ,i − 1 ⊗ h i h i +1  ⊗ k h i +2 ,s + ( − 1) r  a 0 γ ( h 0 ) ǫ ( h 1 ) ⊗ a 1 r  ⊗ k h 2 s and d 2 r s ( x ) = r X i =0 ( − 1) i − 1 h γ ( h (5) s − 1 h (5) s ) a 0 γ ( h 0 ) γ − 1 ( h (1) s ) γ − 1 ( h (1) s − 1 ) ⊗ ( a h (2) s 1 i ) h (2) s − 1 ⊗ f ( h (3) s − 1 , h (3) s ) ⊗ a h (4) s − 1 h (4) s i +1 ,r i ⊗ k h 1 ,s − 2 , wher e x = [ a 0 γ ( h 0 ) ⊗ a 1 r ⊗ h 1 s ] . F or each n ≥ 0, let D n = θ n   b D n   θ − 1 n . Theorem 3. 2.  X , d, D  is a mixe d c omplex giving the H o chschild, cy clic, ne gative and p erio dic homolo gy of E . Mor e pr e cisely, the m ix e d c omplexes  X , d, D  and  E ⊗ E ⊗ ∗ , b, B  ar e homotopic al ly e quivalent. Pr o of. Clear ly  X , d, D  is a mixed complex a nd θ :  b X , b d, b D  →  X , d, D  is an isomorphism of mixed complexes. So the result follows from Theorem 2.4.  W e now a re go ing to obtain a formula for D . T o do this we need t o in tro duce a map T : H ⊗ i +1 k → A such that γ ( h 0 ) γ − 1 ( h i ) · · · γ − 1 ( h 1 ) = T  h (1) 0 , S ( h 1 ) (1) , . . . , S ( h i ) (1)  γ  h (2) 0 S ( h i ) (2) · · · S ( h 1 ) (2)  . T o abbre v iate notations we set ζ = γ − 1   S − 1 and U  h 0 i ) = T ( h 0 , S ( h 1 ) , . . . , S ( h i )  . Since γ ( h 0 ) γ − 1 ( h i ) · · · γ − 1 ( h 1 ) = γ ( h 0 ) ζ  S ( h i )  · · · ζ  S ( h 1 )  , we can solve U ( h 0 i ) = γ ( h (1) 0 ) ζ  S ( h i ) (1)  · · · ζ  S ( h 1 ) (1)  γ − 1  h (2) 0 S ( h 1 · · · h i ) (2)  = γ ( h (1) 0 ) ζ  S ( h (2) i )  · · · ζ  S ( h (2) 1 )  γ − 1  h (2) 0 S ( h (1) 1 · · · h (1) i )  = γ ( h (1) 0 ) γ − 1 ( h (2) i ) · · · γ − 1 ( h (2) 1 ) γ − 1  h (2) 0 S ( h (1) 1 · · · h (1) i )  . W e now must c heck that T  h 0 , S ( h 1 ) , . . . , S ( h i )  ∈ A . F or this it s uffices to see that this elemen t is coinv ar ian t under the coaction ν = id ⊗ ∆ of A # f H , which follows e a sily from the fact that ν  γ − 1 ( h )  = γ − 1  h (2)  ⊗ S  h (1)  and A # f H is a como dule algebra. Note that a 0 γ ( h 0 ) γ − 1 ( h i ) · · · γ − 1 ( h 1 ) = a 0 U  h (1) 0 , h (2) 1 i  γ  h (2) 0 S ( h (1) 1 · · · h (1) i )  . CYCLIC HOMOLOGY OF CROSSED PRODUCTS 13 F or each 0 ≤ i ≤ n , let F i ( X n ) = L 0 ≤ s ≤ i X n − s,s . The complex ( X ∗ , d ∗ ) is filtered by F 0 ( X ∗ ) ⊆ F 1 ( X ∗ ) ⊆ F 2 ( X ∗ ) ⊆ . . . . Given h 1 , . . . , h i ∈ H , we let H J n ( h 1 , . . . , h i ) denote the k -submo dule of X n generated by all the elements [ a 0 γ ( h 0 ) ⊗ a 1 r ] ⊗ k h 1 s , with r > 0 and so me a j ∈ e f ( h 1 , . . . , h i ) (for the definition of this last expressio n see the discussion ab ov e Prop osition 1.4). Let η n : X n → X n +1 and t H,n : X n +1 → X n +1 be the k -linear maps defined by η ( x ) =  a 0 γ ( h (1) 0 ) ⊗ a 1 ,n − i  ⊗ k h (2) 0 S ( h (1) 1 · · · h (1) i ) ⊗ k h (2) 1 i and t H ( y ) =  γ ( h (3) i +1 ) a 0 γ ( h 0 ) γ − 1 ( h (1) i +1 ) ⊗ a h (2) i +1 1 ,n − i  ⊗ k h (4) i +1 ⊗ k h 1 i , where x =  a 0 γ ( h 0 ) ⊗ a 1 ,n − i  ⊗ k h 1 i and y =  a 0 γ ( h 0 ) ⊗ a 1 ,n − i  ⊗ k h 1 ,i +1 , resp ectiv e ly . Theorem 3. 3. If x =  a 0 γ ( h 0 ) ⊗ a 1 ,n − i  ⊗ k h 1 i , then D ( x ) = i X j =0 ( − 1) j i + n − i t j H   η ( x ) + n − i X j =0 ( − 1) ( j +1)( n − i ) h γ  h (3) 0 S ( h (1) 1 · · · h (1) i )  γ ( h (5) 1 ) · · · γ ( h (5) i ) ⊗ a j +1 ,n − i ⊗ a 0 U ( h (1) 0 , h (3) 1 i ) ⊗  a h (4) 1 i 1 j  h (2) 0 S ( h (2) 1 ··· h (2) i ) i ⊗ k h (6) 1 i , mo dule F i ( X n +1 ) ∩ H J n +1 ( h 1 , . . . , h i ) . Pr o of. It follows straightforwardly from Pr opos ition 2.6, the fact tha t D = θ   b D   θ − 1 , and the form ula s of θ and θ − 1 .  3.1. Firs t sp ectral sequence. Arguing as in [G-G, Prop osition 3.2] we see that, for each h ∈ H , ther e is a mor phism o f co mplexes ϑ h ∗ : ( E ⊗ A ⊗ ∗ ⊗ , b ∗ ) → ( E ⊗ A ⊗ ∗ ⊗ , b ∗ ) , which is giv en b y ϑ h r ([ a 0 γ ( h 0 ) ⊗ a 1 r ]) = [ γ ( h (3) ) a 0 γ ( h 0 ) γ − 1 ( h (1) ) ⊗ a h (2) 1 r ] a nd that, for ea c h h, l ∈ H , the endomorphisms of H K ∗ ( A, E ) induced by ϑ h ∗ ◦ ϑ l ∗ and by ϑ hl ∗ coincide. So, H K ∗ ( A, E ) is a left H -mo dule. Let e d s : H K r ( A, E ) ⊗ k H ⊗ s → H K r ( A, E ) ⊗ k H ⊗ s − 1 and e D s : H K r ( A, E ) ⊗ k H ⊗ s → H K r ( A, E ) ⊗ k H ⊗ s +1 be the ma ps induced by d 1 r s and P s j =0 ( − 1) j s + r t j H   η r + s , respec tively . Prop osition 3.4. Assu m e that H is a flat k -mo dule. F or e ach r ≥ 0 , ^ H K r ( A, E ) =  H K r ( A, E ) ⊗ k H ⊗ ∗ , e d ∗ , e D ∗  is a mixe d c omplex and ther e is a c onver gent sp e ctr al se quenc e E 2 sr = HC s  ^ H K r ( A, E )  ⇒ HC K r + s ( E ) . 14 GRA CIELA CARBONI, JORGE A. GUCCIONE, AND JUAN J. GUCCIONE Pr o of. Consider the sp ectral sequence ( E v sr , d v sr ) v ≥ 0 , asso ciated with th e filtration F 0  T ot ( BC ( X , d, D ))  ⊆ F 1  T ot ( BC ( X , d, D ))  ⊆ F 2  T ot ( BC ( X , d, D ))  ⊆ · · · of the complex T o t ( BC ( X , d, D )), given b y F i  T ot ( BC ( X , d, D )) n  = M j ≥ 0 F i − 2 j ( X n − 2 j ) u j . An straightforw a rd computation shows that • E 0 sr = M j ≥ 0  E ⊗ A ⊗ r ⊗  ⊗ k H ⊗ s − 2 j  u j and d 0 sr is M j ≥ 0 d 0 r,s − 2 j u j , • E 1 sr = M j ≥ 0  H r ( A, E ) ⊗ k H ⊗ s − 2 j  u j and d 1 sr is e d + e D . F ro m this it follows easily that ^ H K r ( A, E ) is a mixed complex and E 2 sr = HC s  ^ H K r ( A, E )  . In order to finish the pro of it suffices to no te that the filtr ation of T ot ( BC ( X , d, D )) int r oduced ab ov e is canonica lly b ounded, and so, by Theo r em 3 .2, the sp ectral sequence ( E v sr ) v ≥ 0 conv erg es to the cyclic homology of the K - a lgebra E .  Corollary 3.5. If H K i ( A, E ) = 0 for al l i > 0 , then HC K n ( E ) = HC n ( ^ H K 0 ( A, E )) . Prop osition 3.6. Assu m e H is a sep ar able algebr a and let t b e t he inte gr al of H satisfying ǫ ( t ) = 1 . Then E 2 sr = ( H 0  H, ^ H K r ( A, E )  if s is even, 0 if s is o dd, and for s even the map d 2 sr : E 2 sr → E 2 s − 2 ,r +1 is given by d 2  X [ a 0 γ ( h ) ⊗ a 1 r ]  = r X j =0 X ( − 1) ( j +1) r h γ ( h (2) ) ⊗ a j +1 ,r ⊗ a 0 ⊗ a h (1) 1 j i + r X j =0 ( − 1) j X h γ ( t (5) h (4) ) a 0 γ − 1 ( t (1) ) ⊗ ( a h (1) 1 j ) t (2) ⊗ f ( t (3) , h (2) ) ⊗ a t (4) h (3) j +1 ,r i , wher e P [ a 0 γ ( h ) ⊗ a 1 r ] is a r -cycle of  E ⊗ A ⊗ ∗ ⊗ , b ∗  and P [ a 0 γ ( h ) ⊗ a 1 r ] denotes its class in H 0  H, ^ H K r ( A, E )  , etc et era . Pr o of. The first assertion is trivial a nd the sec o nd one follows from a dir ect com- putation using the co nstruction of the spectral sequence of a filtr ated complex . F o r this it is con venient to note that t H   η  [ a 0 γ ( h ) ⊗ a 1 r ]  − d 1  [ a 0 γ ( h (1) ) ⊗ a 1 r ] ⊗ k t ⊗ k h (2)  ∈ Im ( e d s ) . W e leave the details to the reader.  3.2. Second sp ectral sequence. In this subsection we a s sume that f takes v a lue s in K . Under this hypo thesis the maps d l v a nish for all l ≥ 2 and we obtain a sp ectral sequence that generalizes those given in [A-K] and [K-R]. F or each r ≥ 0, we define a map H ⊗ k  E ⊗ A ⊗ r ⊗  / / E ⊗ A ⊗ r ⊗ , h ⊗ x  / / h ◮ x CYCLIC HOMOLOGY OF CROSSED PRODUCTS 15 by h ◮ [ aγ ( u ) ⊗ a 1 r ] =  γ ( h (3) ) aγ ( u ) γ − 1 ( h (1) ) ⊗ a h (2) 1 r  . Prop osition 3.7. F or e ach r ≥ 0 the map ◮ is an action of H on E ⊗ A ⊗ r ⊗ . Pr o of. It is trivial that ◮ is unitary . Next we verify the asso ciative prop ert y . By definition l ◮  h ◮ [ aγ ( u ) ⊗ a 1 r ]  = h γ ( l (3) ) γ ( h (3) ) aγ ( u ) γ − 1 ( h (1) ) γ − 1 ( l (1) ) ⊗  a h (2) 1 r  l (2) i . Since  a h 1 r  l = f ( l (1) , h (1) ) a l (2) h (2) 1 r f − 1 ( l (3) , h (3) ) , γ ( l ) γ ( h ) = f ( l (1) , h (1) ) γ ( l (2) h (2) ) and f − 1 is the con volution in verse of f , we ha ve l ◮  h ◮ [ aγ ( u ) ⊗ a 1 r ]  = h γ ( l (4) h (4) ) aγ ( u ) γ − 1 ( h (1) ) γ − 1 ( l (1) ) ⊗ f ( l (2) , h (2) ) a l (3) h (3) 1 r i . Using no w that, b y the twisted module condition applied twice, γ − 1 ( h ) γ − 1 ( l ) = f − 1  S ( h (2) ) , h (3)  γ  S ( h (1) )  f − 1  S ( l (2) ) , l (3)  γ  S ( l (1) )  = f − 1  S ( h (3) ) , h (4)  f − 1  S ( l (3) ) , l (4)  f  S ( h (2) ) , S ( l (2) )  γ  S ( l (1) h (1) )  = f − 1  S ( h (3) ) , h (4)  f − 1  S ( h (2) ) S ( l (2) ) , l (3)  γ  S ( l (1) h (1) )  = f − 1  S ( l (3) h (3) ) l (4) , h (4)  f − 1  S ( l (2) h (2) ) , l (5)  γ  S ( l (1) h (1) )  = f − 1  S ( l (2) h (2) ) , l (3) h (3)  f − 1  l (4) , h (4)  γ  S ( l (1) h (1) )  , and again that f − 1 is the con volution in verse of f , we obtain l ◮  h ◮ [ aγ ( u ) ⊗ a 1 r ]  = h γ ( v (5) ) aγ ( u ) f − 1  S ( v (2) ) , v (3)  γ  S ( v (1) )  ⊗ a v (4) 1 r i = h γ ( v (3) ) aγ ( u ) γ − 1 ( v (1) ) ⊗ a v (2) 1 r i , where v = l h . Since the last expressio n equals ( l h ) ◮ [ aγ ( u ) ⊗ a 1 r ], this finishes the pro of.  F or ea ch r ≥ 0, le t M r be E ⊗ A ⊗ r ⊗ , endow ed with the left H -mo dule structur e given b y ◮ . F or each r, s ≥ 0, let B r s : M r ⊗ k H ⊗ s k → M r +1 ⊗ k H ⊗ s k be the map defined b y B ( x ) = r X j =0 ( − 1) ( j +1) r h γ  h (3) 0 S ( h (1) 1 · · · h (1) s )  γ ( h (5) 1 ) · · · γ ( h (5) s ) ⊗ a j +1 ,r ⊗ a 0 U ( h (1) 0 , h (3) 1 s ) ⊗  a h (4) 1 s 1 j  h (2) 0 S ( h (2) 1 ··· h (2) s ) i ⊗ k h (6) 1 s , where x =  a 0 γ ( h 0 ) ⊗ a 1 r  ⊗ k h 1 s . F or each r , s ≥ 0, let ∂ r : H s ( H, M r ) → H s ( H, M r − 1 ) and D r : H s ( H, M r ) → H s ( H, M r +1 ) be the ma ps induced by d 0 r s and B r s , respec tively Prop osition 3.8. F or e ach s ≥ 0 , ^ H K s ( H, E ) =  H s ( H, M ∗ ) , ∂ ∗ , D ∗  is a mixe d c omplex and ther e is a c onver gent sp e ctr al se quenc e E 2 r s = HC r  ^ H K s ( H, E )  ⇒ HC K r + s ( E ) . 16 GRA CIELA CARBONI, JORGE A. GUCCIONE, AND JUAN J. GUCCIONE Pr o of. Consider the sp ectral sequence ( E v r s , δ v r s ) v ≥ 0 , asso ciated with th e filtration F 0  T ot ( BC ( X , d, D ))  ⊆ F 1  T ot ( BC ( X , d, D ))  ⊆ F 2  T ot ( BC ( X , d, D ))  ⊆ · · · of the complex T o t ( BC ( X , d, D )), given b y F i  T ot ( BC ( X , d, D )) n  = M j ≥ 0 F i − 2 j ( X n − 2 j ) u j , where F l ( X m ) = L 0 ≤ r ≤ l X r,m − r . An stra igh tfor ward computation shows that • E 0 r s = M j ≥ 0  M r − 2 j ⊗ k H ⊗ s  u j and δ 0 r s is M j ≥ 0 d 1 r − 2 j,s u j , • E 1 r s = M j ≥ 0 H s ( H, M r − 2 j ) u j and δ 1 r s is ∂ + D . F ro m this it is easy to see that ^ H K s ( H, E ) is a mixed co mplex and E 2 r s = HC r  ^ H K s ( H, E )  . In order to finish the pro of it suffices to no te that the filtr ation of T ot ( BC ( X , d, D )) int r oduced ab ov e is canonica lly b ounded, and so, by Theo r em 3 .2, the sp ectral sequence ( E v r s , δ v r s ) v ≥ 0 conv erg es to the cyclic homology of the K -algebra E .  Corollary 3.9. If H is sep ar able, t hen HC K n ( E ) = HC n  ^ H K 0 ( H, E )  . 4. Some decompositions of the mixed co m pl exes Let [ H , H ] b e the k -submo dule of H s panned by the set of all elements hl − l h with h, l ∈ H . It is ea sy to see that [ H , H ] is a coideal in H . Let ˘ H b e the quotient co algebra H / [ H , H ]. In this sec tio n we s tudy decomp ositions of the mixed complexes  E ⊗ E ⊗ ∗ ⊗ , b, B  ,  b X , b d, b D  and  X , d, D  induced by decomp ositions of ˘ H . F or h ∈ H , we let h denote the c la ss of h in ˘ H . Given a subco a lgebra C o f ˘ H and a rig h t ˘ H -como dule ( N , ρ ), w e set N C = { n ∈ N ρ ( n ) ∈ N ⊗ C } . It is well known that if ˘ H deco mposes a s a direct sum of a family ( C i ) i ∈ I of sub coalgebras, then N = L i ∈ I N C i . F or each n , the m o dule E ⊗ E ⊗ n ⊗ is an ˘ H -como dule via ρ n  a 0 γ ( h 0 ) ⊗ · · · ⊗ a n γ ( h n )  = h a 0 γ ( h (1) 0 ) ⊗ · · · ⊗ a n γ ( h (1) n ) i ⊗ k h (2) 0 · · · h (2) n , and the map ρ ∗ : E ⊗ E ⊗ ∗ ⊗ →  E ⊗ E ⊗ ∗ ⊗  ⊗ k ˘ H is a mo rphism of mixed complexes. This last fact implies that if C is a subcoa lg ebra of ˘ H , then b  E ⊗ E ⊗ n ⊗ C  ⊆ E ⊗ E ⊗ n − 1 ⊗ C and B  E ⊗ E ⊗ n ⊗ C  ⊆ E ⊗ E ⊗ n +1 ⊗ C . Let  E ⊗ E ⊗ ∗ ⊗ C , b C , B C  be the mixed sub complex of  E ⊗ E ⊗ ∗ ⊗ , b, B  , with mo dules E ⊗ E ⊗ n ⊗ C . W e let HH K,C ∗ ( E ), HC K,C ∗ ( E ), HP K,C ∗ ( E ) and HN K,C ∗ ( E ) denote its Hochschild, cyclic, p erio dic and negative homology , resp ectiv ely . Similarly , for eac h n ≥ 0, the mo dule b X n is an ˘ H -como dule v ia ρ n  a 0 γ ( h 0 ) ⊗ A γ ( h 1 s ) ⊗ a 1 ,n − s  = h a 0 γ ( h (1) 0 ) ⊗ A γ ( h (1) 1 s ) ⊗ a 1 ,n − s i ⊗ h (2) 0 . . . h (2) s , and the map ρ ∗ : b X ∗ → b X ∗ ⊗ ˘ H is a morphism of mixed complexes. Cons equen tly , if C is a subc oalgebra of ˘ H , then b d n ( b X C n ) ⊆ b X C n − 1 and b D n ( b X C n ) ⊆ b X C n +1 . CYCLIC HOMOLOGY OF CROSSED PRODUCTS 17 Let  b X C , b d C , b D C  be the mixed sub complex o f  b X , b d, b D  with mo dules b X C n . The homotopy equiv alent data intro duced in The o rem 2.4 induces by r e striction a ho- motopy equiv alent data b etw een  b X C , b d C , b D C  and  E ⊗ E ⊗ ∗ ⊗ C , b C , B C  . So, HH K,C ∗ ( E ), HC K,C ∗ ( E ), HP K,C ∗ ( E ) and H N K,C ∗ ( E ) are the Ho chsc hild, cyclic, pe ri- o dic a nd negative homology of  b X C , b d C , b D C  , resp ectiv e ly . Suppo se now the co cycle f is inv ertible. A direc t co mputatio n shows that the ˘ H -coactio n of ( X , d, D ), o btained by transp orting the one of ( b X , b d, b D ) through θ : ( b X , b d, b D ) → ( X , d, D ), is given b y [ a 0 γ ( h 0 ) ⊗ a 1 r ] ⊗ k h 1 s 7→  a 0 h (1) 0 ⊗ a 1 r  ⊗ k h (2) 1 s ⊗ h (2) 0 S ( h (1) 1 · · · h (1) s ) h (3) 1 · · · h (3) s . This implies that if if ˘ H is co commutativ e, then X C n = M r + s = n X C r s = M r + s = n E C ⊗ A ⊗ r ⊗ H ⊗ s . F or each s ubcoalg ebra C of ˘ H , we cons ide r the mixed sub complex  X C , d C , D C  of  X , d, D  with modules X C n . It is clea r that θ induces a n iso morphism θ C :  b X C , b d C , b D C  →  X C , d C , D C ) . So, HH K,C ∗ ( E ), HC K,C ∗ ( E ), HP K,C ∗ ( E ) and HN K,C ∗ ( E ) ar e the Ho chsc hild, cyclic, per iodic and negative homology o f  X C , d C , D C  , respec tively . By the discuss io n a t the b eginning of this subsectio n, if ˘ H decomp oses a s a dir e ct sum of a family ( C i ) i ∈ I of subcoa lgebras, then  E ⊗ E ⊗ ∗ ⊗ , b, B  = M i ∈ I  E ⊗ E ⊗ ∗ ⊗ C i , b C i , B C i   b X , b d, b D  = M i ∈ I  b X C i , b d C i , b D C i  and  X , d, D  = M i ∈ I  X C i , d C i , D C i  . In particular HH K ∗ ( E ) = L i ∈ I HH K,C i ∗ ( E ), etcetera. In the sequel w e use the notations int ro duced in Subsection 3.1 and 3.2. Lemma 4.1. Assu me that ˘ H is c o c ommutative and H is a flat k -mo dule. If C is a su b c o algebr a of ˘ H , t hen for e ach r, s ≥ 0 , e d  H K r ( A, E C ) ⊗ k H ⊗ n  ⊆ H K r ( A, E C ) ⊗ k H ⊗ n − 1 and e D  H K r ( A, E C ) ⊗ k H ⊗ n  ⊆ H K r ( A, E C ) ⊗ k H ⊗ n +1 . Pr o of. Left to the rea de r .  Prop osition 4.2. Assume that ˘ H is c o c ommutative and H is a flat k -mo dule. L et C b e a su b c o algebr a of ˘ H and let ^ H K r ( A, E C ) =  H K r ( A, E C ) ⊗ k H ⊗ ∗ , e d C ∗ , e D C ∗  18 GRA CIELA CARBONI, JORGE A. GUCCIONE, AND JUAN J. GUCCIONE b e the submix e d c omplex of ^ H K r ( A, E ) with mo dules H K r ( A, E C ) ⊗ k H ⊗ n . Ther e is a c onver gent sp e ct ra l se qu enc e E 2 sr = HC s  ^ H K r ( A, E C )  ⇒ HC K,C r + s ( E ) . Pr o of. Left to the rea de r .  Lemma 4.3. Assume that ˘ H is c o c ommutative. If C is a sub c o algebr a of ˘ H , t hen M C n = E C ⊗ A ⊗ n ⊗ is an H -submo dule of M n for e ach n ≥ 0 . Mor e over ∂  H s ( H, M n )  ⊆ H s ( H, M n − 1 ) a n d D  H s ( H, M n )  ⊆ H s ( H, M n +1 ) . Pr o of. Left to the rea de r .  Prop osition 4.4. Assume that ˘ H is c o c ommu tative. L et C b e a sub c o algebr a of ˘ H and let ^ H K s ( H, E C ) =  H s ( H, M C ∗ ) , ∂ ∗ , D ∗  b e the submix e d c omplex of ^ H K s ( H, E ) with mo dules H s ( H, M C n ) . Ther e is a c on- ver gent sp e ctr al se quenc e E 2 r s = HC r  ^ H K s ( H, E C )  ⇒ HC K,C r + s ( E ) . Pr o of. Left to the rea de r .  Appendix A. This appendix is devoted to pro ve Prop ositions 2.1, 2.2 and 2 .5. Lemma A.1. We have σ n +1 = − σ 0 0 ,n +1   σ − 1 n +1   µ n + n X r =0 n − r X l =0 σ l r + l +1 ,n − r − l , Pr o of. By the definition of µ , σ − 1 and σ it suffices to prov e that σ l ( E ⊗ A ( E / A ) ⊗ n +1 A ⊗ A A ) = 0 for all l ≥ 1. Assume the result is false and let l ≥ 1 b e the minimal upp er index for which the ab o ve equalit y is wrong. Let x ∈ E ⊗ A ( E / A ) ⊗ n +1 A ⊗ A A . Then σ l ( x ) = − l − 1 X i =0 σ 0   d l − i   σ i ( x ) = − σ 0   d l   σ 0 ( x ) . But, b ecause σ 0 ( x ) ∈ E ⊗ A ( E / A ) ⊗ n +1 A ⊗ K , from the definition of d l it follows that d l   σ 0 ( x ) ∈ Im ( σ 0 ). Since σ 0   σ 0 = 0 , this implies that σ l ( x ) = 0, which con tradicts the assumption.  Lemma A.2. The c ont r acting homotopy σ satisfies σ   σ = 0 . Pr o of. By Lemma A.1 it will b e sufficient to see that σ 0   σ − 1   µ   σ 0   σ − 1   µ = 0 and σ l   σ l ′ = 0 for all l , l ′ ≥ 0. The first eq ualit y follows from the fact that µ   σ 0 = id and σ − 1   σ − 1 = 0. W e now prov e the last o ne. An inductive argument sho ws tha t there exists a map γ l such that σ l = σ 0   γ l   σ 0 for all l ≥ 1. So σ l ′   σ l = 0, since clearly σ 0   σ 0 = 0 .  R emark A.3 . The previous lemma implies that ψ n ( y ⊗ 1 ) = ( − 1) n σ   ψ ( y ) for all n ≥ 1. CYCLIC HOMOLOGY OF CROSSED PRODUCTS 19 Let L r s ⊆ U r s be the k -submo dules of E ⊗ A ( E / A ) ⊗ s A ⊗ A ⊗ r ⊗ E g enerated b y the simple tensors of the form 1 ⊗ A γ ( h 1 s ) ⊗ a 1 r ⊗ 1 a nd 1 ⊗ A γ ( h 1 s ) ⊗ a 1 r ⊗ γ ( h ) , resp ectiv e ly . Note that under the ide ntification X r s ≃ E ⊗ k H ⊗ s k ⊗ A ⊗ r ⊗ E , the subspaces and L r s and U r s of X r s corres p ond to k ⊗ k H ⊗ s k ⊗ A ⊗ r ⊗ k and k ⊗ k H ⊗ s k ⊗ A ⊗ r ⊗ H , resp ectiv e ly Lemma A.4. It is true that d l ( L r s ) ⊆ U r + l − 1 ,s − l , for e ach l ≥ 2 . Mor e over d 1 ( L r s ) ⊆ E L r,s − 1 + U r,s − 1 Pr o of. W e pro ceed by induction on l and r . F or l = 1 and r ≥ 0, the r esult follows immediately from Theorem 1.4. Assume that s ≥ l > 1 , r = 0 and that the result for l ≥ 2 is tr ue for ev ery d j r ′ s ′ ’s with ar bitrary r ′ , s ′ and j < l . Let x = 1 ⊗ A γ ( h 1 s ) ⊗ 1. By the very definitio n o f d l , the ab ov e inclusio n of d 1 ( L r s ), and the inductiv e hypothes is d l ( x ) = − l − 1 X j =1 σ 0   d l − j   d j ( x ) ∈ σ 0   d l − 1 ( E L 0 ,s − 1 ) + l − 1 X j =1 σ 0   d l − j ( U j − 1 ,s − j ) = l − 1 X j =1 σ 0   d l − j ( U j − 1 ,s − j ) , where the last equalit y follows from the fact that Im ( σ 0 ) ⊆ ker( σ 0 ) and d l − 1 ( E L 0 ,s − 1 ) ⊆ Im ( σ 0 ) , by the definition of d l − 1 . Now, b y the inductiv e hypothesis , d l − j ( U j − 1 ,s − j ) ⊆ L l − 2 ,s − l E for l − j > 1 and d 1 ( U l − 2 ,s − l +1 ) ⊆ E U l − 2 ,s − l + L l − 2 ,s − l E . Thu s, b y the definition of σ 0 , we hav e d l ( x ) ∈ U l − 1 ,s − l . Suppose now that r > 0 and the result is true for all the d j r ′ s ′ ’s with arbitrary r ′ , s ′ and j < l , and for all the d l r ′ s ′ ’s with a rbitrary s ′ and r ′ < r . Let x = 1 ⊗ A γ ( h 1 s ) ⊗ a 1 r ⊗ 1 . Arguing as ab o ve w e see that d l ( x ) ≡ − σ 0   d l   d 0 ( x ) (mo d U r + l − 1 ,s − l ) . Finally , b y the definition of d 0 and the inductiv e hypothes is, σ 0   d l   d 0 ( x ) ∈ σ 0   d l ( AL r − 1 ,s + L r − 1 ,s A ) ⊆ σ 0 ( AU r + l − 2 ,s − l + U r + l − 2 ,s − l A ) ⊆ U r + l − 1 ,s − l , which finishes the pro of.  W e rec ursiv e ly define γ ( h 1 s ) ∗ a 1 r by • γ ( h 1 s ) ∗ a 1 r = a 1 r if s = 0 and γ ( h 1 s ) ∗ a 1 r = γ ( h 1 s ) if r = 0 , • If r, s ≥ 1, then γ ( h 1 s ) ∗ a 1 r = P r i =0 ( − 1) i γ ( h 1 ,s − 1 ) ∗ a h (1) s 1 i ⊗ γ ( h (2) s ) ⊗ a i +1 ,r . 20 GRA CIELA CARBONI, JORGE A. GUCCIONE, AND JUAN J. GUCCIONE Let V n be the k -submo dule o f B n ( E ) genera ted by the s imple tensor s 1 ⊗ x 1 n ⊗ 1 such that x i ∈ A ∪ H for 1 ≤ i ≤ n . Recall that H · Im ( f ) denotes the minimal k -submodule of A that includes Im ( f ) and is clos ed under the weak action o f H . W e will de no te by C n the E -subbimodule of E ⊗ E ⊗ n ⊗ E genera ted by all the simple tens ors 1 ⊗ x 1 ⊗ · · · ⊗ x n ⊗ 1 with some x i in H · Im ( f ). Prop osition A.5. The map φ satisfies φ (1 ⊗ A γ ( h 1 i ) ⊗ a 1 ,n − i ⊗ 1) ≡ 1 ⊗ γ ( h 1 i ) ∗ a 1 ,n − i ⊗ 1 mo dule F i − 1 ( B n ( E )) ∩ V n ∩ C n . Pr o of. W e pro ceed by induction on n . Let x = 1 ⊗ A γ ( h 1 i ) ⊗ a 1 ,n − i ⊗ 1. By item (2) of Theo rem 1.4, the fact tha t d l ( x ) ∈ U n − i + l − 1 ,i − l (b y Lemma A.4), and the inductiv e hypothesis ξ   φ   d l ( x ) ∈ F i − l +1 ( B n ( E )) ∩ V n ∩ C n for all l > 1, So, φ ( x ) ≡ ξ   φ   d 0 ( x ) + ξ   φ   d 1 ( x ) (mo d F i − 1 ( B n ( E )) ∩ V n ∩ C n ) . Moreov er, by the definition of d 0 and Theorem 1.4 ξ   φ   d 0 ( x ) = ( − 1) n ξ   φ (1 ⊗ A γ ( h 1 i ) ⊗ a 1 ,n − i ) , and ξ   φ   d 1 ( x ) = ( − 1) i ξ   φ (1 ⊗ A γ ( h 1 ,i − 1 ) ⊗ a h (1) i 1 ,n − i ⊗ γ ( h (2) i )) , since φ ( E L n − s − 1 ,s ) ⊆ E ⊗ E ⊗ n − 1 ⊗ K ⊆ ker( ξ ). The pr oof can be now easily finished using the inductiv e hypo thesis.  In the sequel we let J n denote the E -subbimodule of X n generated by all the simple tensors 1 ⊗ A x 1 ⊗ A · · · ⊗ A x s ⊗ a 1 ⊗ · · · ⊗ a r ⊗ 1 ( r + s = n ), with some a i in the image of the cocycle f . Lemma A.6. We have: (1) L et x = 1 ⊗ A γ ( h 1 i ) ⊗ a i +1 ,n . If i < n , then σ ( x ) = σ 0 ( x ) = ( − 1) n ⊗ A γ ( h 1 i ) ⊗ a i +1 ,n ⊗ 1 . (2) If z = 1 ⊗ A γ ( h 1 ,i − 1 ) ⊗ a i,n − 1 ⊗ a n γ ( h n ) , then σ l ( z ) ∈ U n − i + l +1 ,i − 1 − l for l ≥ 0 and σ l ( z ) ∈ J n for l ≥ 1 . (3) If z = 1 ⊗ A γ ( h 1 ,i − 1 ) ⊗ a i,n − 1 ⊗ γ ( h n ) , then σ l ( z ) = 0 for l ≥ 0 . (4) If z = 1 ⊗ A γ ( h 1 ,i − 1 ) ⊗ a i,n − 1 ⊗ a n γ ( h n ) and i < n , t hen σ ( z ) ≡ σ 0 ( z ) , mo dule L i − 2 l =0 ( U n − l,l ∩ J n ) . (5) If y = 1 ⊗ A γ ( h 1 ,n − 1 ) ⊗ a n γ ( h n ) , t hen σ ( y ) ≡ − σ 0   σ − 1   µ ( y ) + σ 0 ( y ) , mo dule L n − 2 l =0 ( U n − l,l ∩ J n ) . (6) If z = 1 ⊗ A γ ( h 1 ,n − 1 ) ⊗ γ ( h n ) , then σ ( z ) = − σ 0   σ − 1   µ ( z ) . (7) If z = 1 ⊗ A γ ( h 1 ,i − 1 ) ⊗ a i,n − 1 ⊗ γ ( h n ) and i < n , then σ ( z ) = 0 . CYCLIC HOMOLOGY OF CROSSED PRODUCTS 21 Pr o of. The first a ssertion impr o ves item (b) of the pro of of [G- G, Pr opositio n 1.2.2]. W e first claim that if l ≥ 1, then σ l ( x ) = 0. W e pro ceed b y induction on l . By the recursive definitio n o f σ l and the inductiv e hypothes is σ l ( x ) = − l − 1 X i =0 σ 0   d l − i   σ i ( x ) = − σ 0   d l   σ 0 ( x ) = ( − 1) n − 1 σ 0   d l ( x ⊗ 1) . In o rder to finish the pro o f of the claim it suffices to no te that σ 0   σ 0 = 0 and that, by the very definition, d l ( x ⊗ 1) ∈ Im ( σ 0 ). When i < n − 1 item (1) follows clear ly from the claim. When i = n − 1 it is necessar y to see also that σ l   σ − 1   µ ( x ) = 0, which is immediate, since σ − 1   µ ( x ) = 0 by the definitions of µ and σ − 1 . W e nex t prov e the first par t of item (2). B y definition this is clear for σ 0 . Ass ume the r esult is v alid for σ i with i < l . Then, b y Le mma A.4, σ l ( z ) = − l − 1 X j =0 σ 0   d l − j   σ j ( z ) ⊆ l − 1 X j =0 σ 0   d l − j ( U n − i + j +1 ,i − 1 − j ) ⊆ σ 0 ( E U n − i + l,i − 1 − l ) + σ 0 ( U n − i + l,i − 1 − l E ) = U n − i + l +1 ,i − 1 − l , as desire d. W e now prov e the second part. By Theor em 1.4, the r e c ursiv e definition of σ l and the definition of σ 0 , w e know t hat σ l ( z ) = − l − 1 X j =0 σ 0   d l − j   σ j ( z ) ≡ − σ 0   d 1   σ l − 1 ( z ) (mo d J n ) . Since σ 0   d 1   σ l − 1 ( z ) ∈ σ 0   d 1 ( U n − i + l,i − l ), in order to finish the pro of it suffices to see that σ 0   d 1 ( U n − i + l,i − l ) ⊆ J n , which is a direct consequence o f Theor em 1.4 and the definition of σ 0 . Item (3) follows immedia tely by induction on l . Items (4) and (5) follo w easily from the definition of σ , item (2) and Lemma A.1. Finally , items (6) and (7) follow from the definition of σ , item (3) and Lemma A .1.  Let V ′ n be the k -submo dule of E ⊗ E ⊗ n ⊗ E gener ated b y the simple tenso r s 1 ⊗ x 1 n ⊗ 1 such tha t #( { j : x j / ∈ A ∪ H} ) ≤ 1 (Note that V n ⊆ V ′ n ). Prop osition A. 7. L et R i = F i ( B n ( E )) \ F i − 1 ( B n ( E )) . The fol lowing e qu ali ties hold: (1) ψ (1 ⊗ γ ( h 1 i ) ⊗ a i +1 ,n ⊗ 1) = 1 ⊗ A γ ( h 1 i ) ⊗ a i +1 ,n ⊗ 1 . (2) If x = 1 ⊗ x 1 n ⊗ 1 ∈ R i ∩ V n and ther e exists 1 ≤ j ≤ i such t hat x j ∈ A , then ψ ( x ) = 0 . (3) If x = 1 ⊗ γ ( h 1 ,i − 1 ) ⊗ a i γ ( h i ) ⊗ a i +1 ,n ⊗ 1 , then ψ ( x ) ≡ 1 ⊗ A γ ( h 1 ,i − 1 ) ⊗ A a i γ ( h i ) ⊗ a i +1 ,n ⊗ 1 + 1 ⊗ A γ ( h 1 ,i − 1 ) ⊗ a i ⊗ a h (1) i i +1 ,n ⊗ γ ( h (2) i ) , mo dule L i − 2 l =0 ( U n − l,l ∩ J n ) . (4) If x = 1 ⊗ γ ( h 1 ,j − 1 ) ⊗ a j γ ( h j ) ⊗ γ ( h j +1 ,i ) ⊗ a i +1 ,n ⊗ 1 with j < i , then ψ ( x ) ≡ 1 ⊗ A γ ( h 1 ,j − 1 ) ⊗ A a j γ ( h j ) ⊗ A γ ( h j +1 ,i ) ⊗ a i +1 ,n ⊗ 1 , mo dule L i − 2 l =0 ( U n − l,l ∩ J n ) . 22 GRA CIELA CARBONI, JORGE A. GUCCIONE, AND JUAN J. GUCCIONE (5) If x = 1 ⊗ γ ( h 1 ,i − 1 ) ⊗ a i,j − 1 ⊗ a j γ ( h j ) ⊗ a j +1 ,n ⊗ 1 with j > i , then ψ ( x ) ≡ 1 ⊗ A γ ( h 1 ,i − 1 ) ⊗ a ij ⊗ a h (1) j j +1 ,n ⊗ γ ( h (2) j ) , mo dule L i − 2 l =0 ( U n − l,l ∩ J n ) . (6) If x = 1 ⊗ x 1 n ⊗ 1 ∈ R i ∩ V ′ n and ther e exists 1 ≤ j 1 < j 2 ≤ i su ch that x j 1 ∈ A and x j 2 ∈ H , then ψ ( x ) = 0 . Pr o of. 1) W e pro ceed b y induction on n . The case n = 0 is trivial. Suppose n > 0 and the result is v a lid for n − 1. Ass ume first th at i < n . By Remark A.3 and the inductive h yp othesis, ψ (1 ⊗ γ ( h 1 i ) ⊗ a i +1 ,n ⊗ 1) = ( − 1) n σ   ψ (1 ⊗ γ ( h 1 i ) ⊗ a i +1 ,n ) = ( − 1 ) n σ (1 ⊗ A γ ( h 1 i ) ⊗ a i +1 ,n ) , and the r esult fo llows from item (1) of Lemma A.6 . Assume now that i = n . B y Remark A.3, the inductiv e hypothes is a nd item (6) of Lemma A.6, ψ (1 ⊗ γ ( h 1 n ) ⊗ 1) = ( − 1 ) n σ   ψ (1 ⊗ γ ( h 1 n )) = ( − 1 ) n +1 σ 0   σ − 1   µ (1 ⊗ A γ ( h 1 ,n − 1 ) ⊗ γ ( h n )) . The result follows now imm ediately from the definitions of µ , σ − 1 and σ 0 . 2) W e pro ceed by induction on n . Assume first that ther e e x ist j 1 < j 2 < n such that x j 1 ∈ A a nd x j 2 ∈ H . By Remark A.3 and the inductiv e hypothesis, ψ ( x ) = ( − 1) n σ   ψ (1 ⊗ x 1 n ) = ( − 1) n σ (0) = 0 . Assume no w that x 1 n = γ ( h 1 ,i − 1 ) ⊗ a i,n − 1 ⊗ γ ( h n ). By Remark A.3 and item (1), ψ ( x ) = ( − 1) n σ   ψ (1 ⊗ x 1 n ) = ( − 1) n σ (1 ⊗ A γ ( h 1 ,i − 1 ) ⊗ a i,n − 1 ⊗ γ ( h n )) , and the result follows from item (7) of Lemma A.6. 3) W e pro ceed b y induction o n n . Assume first that i < n . Let y = 1 ⊗ A γ ( h 1 ,i − 1 ) ⊗ A a i γ ( h i ) ⊗ a i +1 ,n , z = 1 ⊗ A γ ( h 1 ,i − 1 ) ⊗ a i ⊗ a h (1) i i +1 ,n − 1 ⊗ γ ( h (2) i ) a n . By Remark A.3 and the inductiv e hypothesis , ψ ( x ) = ( − 1) n σ   ψ (1 ⊗ γ ( h 1 ,i − 1 ) ⊗ a i γ ( h i ) ⊗ a i +1 ,n ) ≡ ( − 1) n σ ( y + z ) , mo dule σ  L i − 2 l =0 ( U n − 1 − l,l ∩ J n − 1 ) A  . So , by it ems (1) and (4) of Lemma A.6, ψ ( x ) ≡ ( − 1 ) n σ 0 ( y + z ) , mo dule L i − 2 l =0 ( U n − l,l ∩ J n ) + σ 0  L i − 2 l =0 ( U n − 1 − l,l ∩ J n − 1 ) A  . Using the definition of σ 0 we obtain immediately the desired expr ession for ψ ( x ). Ass ume now that i = n . Let y = 1 ⊗ γ ( h 1 ,n − 1 ) ⊗ a n γ ( h n ) and z = 1 ⊗ A γ ( h 1 ,n − 1 ) ⊗ a n γ ( h n ) . By Remark A.3, item (1) of the presen t prop osition and item (5) of L e mma A.6, ψ ( x ) = ( − 1) n σ   ψ ( y ) = ( − 1 ) n σ ( z ) ≡ ( − 1) n +1 σ 0   σ − 1   µ ( z ) + ( − 1) n σ 0 ( z ) , mo dule L n − 2 l =0 ( U n − l,l ∩ J n ). The established formula for ψ ( x ) follows now eas ily from the definitions of µ , σ − 1 and σ 0 . 4) W e pr o ceed b y induction on n . When i < n the same a rgumen t that in item (3) works. Assume now that j < i − 1 and i = n . Let y = 1 ⊗ γ ( h 1 ,j − 1 ) ⊗ a j γ ( h j ) ⊗ γ ( h j +1 ,n ) , z = 1 ⊗ A γ ( h 1 ,j − 1 ) ⊗ A a j γ ( h j ) ⊗ A γ ( h j +1 ,n − 1 ) ⊗ γ ( h n ) . CYCLIC HOMOLOGY OF CROSSED PRODUCTS 23 By Remark A.3 and the inductiv e hypothesis , ψ ( x ) = ( − 1 ) n σ   ψ ( y ) ≡ ( − 1 ) n σ ( z ) , mo dule σ  L n − 3 l =0 ( U n − 1 − l,l ∩ J n − 1 ) E  . So , by items (4) and (6) of Lemma A.6, ψ ( x ) ≡ ( − 1) n +1 σ 0   σ − 1   µ ( z ) , mo dule L n − 4 l =0 ( U n − l,l ∩ J n ) + σ 0  L n − 3 l =0 ( U n − 1 − l,l ∩ J n − 1 ) E  . The formula for ψ ( x ) follows now easily from the definitio ns o f µ , σ − 1 and σ 0 . Assume finally that j = i − 1 and i = n . Let y = 1 ⊗ A γ ( h 1 ,n − 2 ) ⊗ A a n − 1 γ ( h n − 1 ) ⊗ γ ( h n ) , z = 1 ⊗ A γ ( h 1 ,n − 2 ) ⊗ a n − 1 ⊗ γ ( h n − 1 ) γ ( h n ) . By Remark A.3 and item (3), ψ ( x ) = ( − 1) n σ   ψ (1 ⊗ γ ( h 1 ,n − 2 ) ⊗ a n − 1 γ ( h n − 1 ) ⊗ γ ( h n )) ≡ ( − 1) n σ ( y + z ) , mo dule σ  L n − 3 l =0 ( U n − 1 − l,l ∩ J n − 1 ) E  . So , b y the fact that σ 0 ( z ) ∈ U 2 ,n − 2 ∩ J n , and items (4) and (6) of Lemma A.6, ψ ( x ) ≡ ( − 1) n +1 σ 0   σ − 1   µ ( y ) , mo dule L n − 2 l =0 ( U n − l,l ∩ J n ) + σ 0  L n − 3 l =0 ( U n − 1 − l,l ∩ J n − 1 ) E  . The formula for ψ ( x ) follows now easily from t he definitions of µ , σ − 1 and σ 0 . 5) W e pro ceed b y induction o n n . Let y = 1 ⊗ γ ( h 1 ,i − 1 ) ⊗ a i,j − 1 ⊗ a j γ ( h j ) ⊗ a j +1 ,n , z = 1 ⊗ A γ ( h 1 ,i − 1 ) ⊗ a ij ⊗ a h (1) j j +1 ,n − 1 ⊗ γ ( h (2) j ) a n . By Remark A.3 and item (1) or the inductiv e hypo thesis (dep ending on j = n or j < n ), ψ ( x ) = ( − 1 ) n σ   ψ ( y ) ≡ ( − 1 ) n σ ( z ) , mo dule σ  L i − 2 l =0 ( U n − l − 1 ,l ∩ J n − 1 ) A  . Thus, b y item (4) of Lemma A.6, ψ ( x ) = ( − 1) n σ 0   ψ ( y ) ≡ ( − 1 ) n σ 0 ( z ) , mo dule L i − 2 l =0 ( U n − l,l ∩ J n ) + σ 0  L i − 2 l =0 ( U n − l − 1 ,l ∩ J n − 1 ) A  . The r esult is obtained now immediately using the definition of σ 0 . 6) W e pro ceed by inductio n on n . By Remark A.3 and item (2) or the inductiv e hypothesis (depending on x n / ∈ A ∪ H or x n ∈ A ∪ H ), ψ ( x ) = ( − 1) n σ   ψ (1 ⊗ x 1 n ) = ( − 1) n σ (0) = 0 , as desired.  Lemma A.8. L et R i = F i ( B n ( E )) \ F i − 1 ( B n ( E )) . The fol lowing e qualities hold: (1) φ   ψ  1 ⊗ γ ( h 1 i ) ⊗ a 1 ,n − i ⊗ 1  ≡ 1 ⊗ γ ( h 1 i ) ∗ a 1 ,n − i ⊗ 1 mo dule F i − 1 ( B n ( E )) ∩ V n . (2) If x = 1 ⊗ x 1 n ⊗ 1 ∈ R i ∩ V n and ther e exists 1 ≤ j ≤ i such that x i ∈ A , then φ   ψ ( x ) = 0 . (3) If x = 1 ⊗ γ ( h 1 ,i − 1 ) ⊗ a i γ ( h i ) ⊗ a i +1 ,n ⊗ 1 , then φ   ψ ( x ) ≡ a h (1) 1 ,i − 1 i ⊗  γ ( h (2) 1 ,i − 1 ) ⊗ γ ( h i )  ∗ a i +1 ,n ⊗ 1 + 1 ⊗ γ ( h 1 ,i − 1 ) ∗  a i ⊗ a h (1) i i +1 ,n  ⊗ γ ( h (2) i ) , mo dule F i − 1 ( B n ( E )) ∩ AV n + F i − 2 ( B n ( E )) ∩ V n H . 24 GRA CIELA CARBONI, JORGE A. GUCCIONE, AND JUAN J. GUCCIONE (4) If x = 1 ⊗ γ ( h 1 ,j − 1 ) ⊗ a j γ ( h j ) ⊗ γ ( h j +1 ,i ) ⊗ a i +1 ,n ⊗ 1 with j < i , then φ   ψ ( x ) ≡ a h (1) 1 ,j − 1 j ⊗  γ ( h (2) 1 ,j − 1 ) ⊗ γ ( h j i )  ∗ a i +1 ,n ⊗ 1 , mo dule F i − 1 ( B n ( E )) ∩ AV n + F i − 2 ( B n ( E )) ∩ V n H . (5) If x = 1 ⊗ x 1 n ⊗ 1 ∈ R i ∩ V ′ n and ther e exists 1 ≤ j ≤ i such t hat x j ∈ A , then φ   ψ ( x ) ∈ F i − 1 ( B n ( E )) ∩ V n H . Pr o of. Item (1) follo ws from item (1) o f Pro position A.7 a nd P ropo s ition A.5, and item (2) follows from item (2) of Pro position A.7. W e nex t prove item (3 ). By item (3) of Prop osition A.7 , φ   ψ ( x ) ≡ φ  a h (1) 1 ,i − 1 i ⊗ A γ ( h (2) 1 ,i − 1 ) ⊗ A γ ( h i ) ⊗ a i +1 ,n ⊗ 1  + φ  1 ⊗ A γ ( h 1 ,i − 1 ) ⊗ a i ⊗ a h (1) i i +1 ,n ⊗ γ ( h (2) i )  , mo dule φ  L i − 2 l =0 U n − l,l  . So , by Prop osition A.5 φ   ψ ( x ) ≡ a h (1) 1 ,i − 1 i ⊗  γ ( h (2) 1 ,i − 1 ) ⊗ γ ( h i )  ∗ a i +1 ,n ⊗ 1 + 1 ⊗ γ ( h 1 ,i − 1 ) ∗  a i ⊗ a h (1) i i +1 ,n  ⊗ γ ( h (2) i ) , mo dule F i − 1 ( B n ( E )) ∩ AV n + F i − 2 ( B n ( E )) ∩ V n H . W e leave the task to pr o ve items (4) and (5) to the reader.  Prop osition A. 9. L et R i = F i ( B n ( E )) \ F i − 1 ( B n ( E )) . If x = 1 ⊗ x 1 n ⊗ 1 ∈ R i ∩ V ′ n , then ω ( x ) ∈ F i ( B n +1 ( E )) ∩ V n +1 . Pr o of. W e first claim that if x = 1 ⊗ x 1 n ⊗ 1 ∈ R i ∩ V n , then ω ( x ) = 0. F or n = 1 this is immediate, since ω 1 = 0 by definition. Assume that n > 1 and the claim holds for n − 1. Then, ω ( x ) = ξ  φ   ψ ( x ) − ( − 1) n ω (1 ⊗ x 1 n )  = ξ   φ   ψ ( x ) = 0 , where the last eq ualit y follows from the fa c ts that φ   ψ ( x ) ∈ V n (b y items (1) a nd (2) of Lemma A.8 ) and V n ⊆ ker( ξ ). W e now prov e the prop osition by induction on n . This is trivial for n = 1 since w 1 = 0. Assume that n > 1 and the prop osition is true for n − 1. Let x = 1 ⊗ x 1 n ⊗ 1 ∈ R i ∩ V ′ n . Since ω ( x ) = ξ  φ   ψ ( x ) − ( − 1 ) n ω (1 ⊗ x 1 n )  , and, b y items (3), (4) and (5) of Lemma A.8, ξ   φ   ψ ( x ) ∈ F i ( B n +1 ( E )) ∩ V n +1 , in order to finish the proof it suffices to chec k that ξ   ω (1 ⊗ x 1 n ) ∈ F i ( B n +1 ( E )) ∩ V n +1 . By the inductiv e hypo thesis and the claim, • If x n ∈ A , then ω (1 ⊗ x 1 n ) ∈ F i ( B n ( E )) ∩ V n A , • If x n ∈ H , then ω (1 ⊗ x 1 n ) ∈ F i − 1 ( B n ( E )) ∩ V n H , • If x n / ∈ A ∪ H , then ω (1 ⊗ x 1 n ) = 0. In all these cases the required inclusion is true.  Pro ofs of Propos itions 2.1, 2.2 and 2.5. They follow immediately from P ropo - sitions A.5, A.9 and A.7, respectively .  CYCLIC HOMOLOGY OF CROSSED PRODUCTS 25 References [A-K] R. Akbarp our and M. Khalkhali, Hopf Algebr a Equivariant Cyclic Homolo gy and Cy clic Homolo gy of Cr osse d Pr o duct Algebr as , J. Reine Angew. Math., vol 559, (2003) 137–152. [B-C-M] R. J. Blattner, M. Cohen and S. 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E-mail addr ess : gcarb o@dm.uba .ar Dep ar t amento de Ma tem ´ atica, F acul t ad de Ciencias Exact as y Na tura les, P abell ´ on 1 - Ciud ad Un iversit aria, (1428) Buen os Aires, Argentina. E-mail addr ess : vande r@dm.uba .ar Dep ar t amento de Ma tem ´ atica, F acul t ad de Ciencias Exact as y Na tura les, P abell ´ on 1 - Ciud ad Un iversit aria, (1428) Buen os Aires, Argentina. E-mail addr ess : jjguc ci@dm.ub a.ar