math.KT 2010-03-17 0

Cohomology ring of differential operator rings

We compute the multiplicative structure in the Hocshchild cohomology ring of a differential operators ring and the cap product of Hochschild cohomology on the Hochschild homology.

Authors: Graciela Carboni, Jorge A. Guccione, Juan J. Guccione

COHOMOLOGY RING OF DIFFERENTIAL OPERA TOR RINGS GRACI ELA CARBONI, JOR GE A. GUCCIONE, AND JUAN J. GUCCIONE Abstract. W e compute the multiplicative structure in the Ho cshc hild coho- mology ring of a differen tial op erators ri ng and the cap pro duct of Ho chsc hi ld cohomology on the Ho ch sc hild homology . Introduction Let k be a field and A a n a sso ciative k -a lgebra with 1. An extension E / A of A is a differ ential op er ator ring on A if ther e exists a Lie k -algebra g and a k -vector space embedding x 7→ x , of g in to E , such that for a ll x, y ∈ g , a ∈ A : (1) xa − ax = a x , where a 7→ a x is a deriv a tio n, (2) xy − y x = [ x, y ] g + f ( x, y ), where [ − , − ] g is the br ack et of g and f : g × g → A is a k -bilinea r map, (3) for a giv en basis ( x i ) i ∈ I of g , the algebra E is a free left A -mo dule with the standard monomials in the x i ’s as a basis. This general co nstruction was introduce d in [Ch] and [Mc-R]. Several particula r cases of this type of extens io ns ha ve b een consider ed prev iously in the literature. F or instance: - when g is one dimensional and f is trivial, E is the Ore extension A [ x, δ ], where δ ( a ) = a x , - when A = k , o ne obtain the alg e bras studied by Sridhara n in [S], whic h are the quasi- comm utative alg ebras E , who se asso ciated graded algebr a is a symmetric algebra, - in [Mc, § 2] this type of extensio ns w as studied under the hyp o thesis that A is comm utative and ( x, a ) 7→ a x is an action, and in [B-G-R, Theorem 4.2] the case in whic h the cocycle is trivial was c o nsidered. In [B-C-M] and [D-T] the study of the cr ossed pro ducts A # f H of a k -alg ebra A b y a Hopf k -a lg ebra H was begun, a nd in [M] was pr oved that the differ en tial op erator r ings on A are the crossed pro ducts o f A by en veloping algebra s of Lie algebras . In [G-G1] complexes , simpler than the canonic a l ones, g iving the Ho chsc hild homology a nd cohomolo gy of a differential ope r ator ring E with co efficients in an E - bimodule M , were obtained. In this pap er we con tin ue this inv estigation by studying the Hocshchild cohomolog y ring of E and the cap pro duct H p ( E , M ) × HH q ( E ) → H p − q ( E , M ) ( q ≤ p ) , 2000 Mathematics Subje ct Class ific ation. Pri mary 16E40; Secondary 16S32. Key wor ds and phr ases. Differential op erator rings; Hochsc hild (co)homology; cup product; cap pro duct. Supported UBACYT 095. UBACYT 095 and PIP 112-200801-0090 0 (CONICET). UBACYT 095 and PIP 112-200801-0090 0 (CONICET). 1 2 GRA CIELA CARBONI, JOR GE A. GUCCIONE, AND JUAN J. GUCCIONE in terms of the ab ov e mentioned complexes. Moreov er we g eneralize the r esults of [G-G1] by considering the (co)homolog y of E rela tiv e to a subalgebr a K of A which is stable under the a ction of g (which we a lso call the Ho chsc hild (co)homology of the K -algebra E ), a nd we seized the oppo rtunit y to fix some minor mistakes a nd to simplify some pro ofs in [G-G1]. The pap er is o rganized in the follo wing wa y: In Section 1 w e obtain a pr o jective resolution ( X ∗ , d ∗ ) of the E -bimo dule E , relative to the family of all epimorphism of E -bimo dules which split as ( E , K )-bimo dule maps. In Section 2 w e determine and study comparison maps be t w een ( X ∗ , d ∗ ) and the relative to K normalized Ho c hschild r esolution ( E ⊗ K E ⊗ ∗ K ⊗ K E , b ′ ∗ ) of E . In Sections 3 a nd 4 we apply the ab ov e results in order to obtain complexes ( X K ∗ ( M ) , d ∗ ) a nd ( X ∗ K ( M ) , d ∗ ), simpler that the ca no nical ones, giving Ho chsc hid homolog y and cohomolo gy of the K -a lg ebra E with coefficie n ts in an E -bimo dule M , resp e ctiv ely . The main r esults are Theorems 3.4 and 4.4, in whic h we obtain morphisms X ∗ K ( E ) ⊗ X ∗ K ( E ) → X ∗ K ( E ) and X K ∗ ( M ) ⊗ X ∗ K ( E ) → X K ∗ ( M ) , inducing the cup and ca p pro duct, resp ectively . Finally in Section 5 , assuming that A is a symmetric algebra, w e obtain fu rther simplifications . 1. Preliminaries Let k b e a fie ld. In this pap er all the algebras are over k . Let A be an algebra and H a Hopf a lgebra. W e are going use the Sweedler notation ∆( h ) = P ( h ) h (1) ⊗ k h (2) for the comultiplication ∆ of H . A we ak action o f H on A is a k -bilinea r map ( h, a ) 7→ a h , from H × A to A , such that (1) ( ab ) h = P ( h ) a h (1) b h (2) , (2) 1 h = ǫ ( h )1, (3) a 1 = a , for h ∈ H , a, b ∈ A . By an action of H o n A we mean a w eak action suc h that ( a l ) h = a hl for all h, l ∈ H , a ∈ A. Let A be an a lgebra and let H be a Ho pf algebra acting weakly on A . Given a k -linea r map f : H ⊗ k H → A we le t A # f H denote the a lgebra (in general no n asso ciative a nd without 1) whose underlying vector spa c e is A ⊗ k H and whose m ultiplication is given by ( a ⊗ k h )( b ⊗ k l ) = X ( h )( l ) ab h (1) f ( h (2) , l (1) ) ⊗ k h (3) l (2) , for a ll a, b ∈ A , h, l ∈ H . The element a ⊗ k h of A # f H will usually be written a # h . The alge br a A # f H is ca lled a cr osse d pr o duct if it is as so ciative with 1#1 as identit y element . In [B- C- M] it was prov en that this happ ens if and only if the map f and the weak action of H o n A s a tisfy the follo wing conditions (1) (Nor ma lit y of f ) for all h ∈ H w e ha v e f ( h, 1) = f (1 , h ) = ǫ ( h )1 A , (2) (Co cycle condition) for all h, l , m ∈ H we have X ( h )( l )( m ) f  l (1) , m (1)  h (1) f  h (2) , l (2) m (2)  = X ( h )( l ) f  h (1) , l (1)  f  h (2) l (2) , m  , (3) (Twisted mo dule condition) for all h, l ∈ H and a ∈ A we hav e X ( h )( l )  a l (1)  h (1) f  h (2) , l (2)  = X ( h )( l ) f  h (1) , l (1)  a h (2) l (2) . COHOMOLOGY RING OF DIFFERENTIAL OPERA TOR RINGS 3 F rom no w on w e assume that H is the enveloping alg e bra U ( g ) of a Lie a lgebra g . In this case, item (1) of the definition of weak action implies that ( ab ) x = a x b + ab x for each x ∈ g and a, b ∈ A . So, a w eak action determines a k - linear map δ : g → Der k ( A ) by δ ( x )( a ) = a x . Moreov er if ( h, a ) 7→ a h is an action, then δ is a homomorphism of Lie a lgebras. Co n versely , given a k -linea r map δ : g → Der k ( A ), there exists a (generality non- unique) w eak action of U ( g ) on A suc h that δ ( x )( a ) = a x . When δ is a homomorphism of Lie alg e br as, there is a unique a c tion of U ( g ) on A suc h that δ ( x )( a ) = a x . F or a pr oo f of thes e facts see [B-C-M]. It is eas y to see that each nor mal co cyc le f : U ( g ) ⊗ k U ( g ) → A is con volution inv ertible. F or a pro of see [G-G1, Remark 1.1]. Next we r ecall so me results and no tations from [G-G1] that we will need later . Let K b e a suba lg ebra of A which is stable under the w eak action of g (that is λ x ∈ K for all λ ∈ K and x ∈ g ) and le t E = A # f U ( g ) b e a cr ossed pro duct. W e are going to mo dify the sign of some boundary maps in order to obtain simple expressions for the comparison maps. T o b egin, we fix some notations: (1) The unadorned tensor pro duct ⊗ means the tensor product ⊗ K ov er K , (2) F or B = A or B = E a nd eac h r ∈ N , we wr ite B = B /K , B r = B ⊗ · · · ⊗ B ( r times) and B r = B ⊗ · · · ⊗ B ( r times). Moreov er, for b ∈ B we also le t b denote the class of b in B . (3) F or each Lie alg ebra g and s ∈ N , we write g ∧ s = g ∧ · · · ∧ g ( s times). (4) Thr o ughout this pap er w e will write a 1 r for a 1 ⊗ · · · ⊗ a r ∈ A r and x 1 s for x 1 ∧ · · · ∧ x s ∈ g ∧ s (5) F or a 1 r and 0 ≤ i < j ≤ r , we wr ite a ij = a i ⊗ · · · ⊗ a j . (6) F or x 1 s and 1 ≤ i ≤ s , we wr ite x 1 b ıs = x 1 ∧ · · · ∧ b x i ∧ · · · ∧ x s . (7) F or x 1 s and 1 ≤ i < j ≤ s , we w r ite x 1 b ı b s = x 1 ∧ · · · ∧ b x i ∧ · · · ∧ b x j ∧ · · · ∧ x s . Let Y ∗ be the gra ded algebra generated by A and the elemen ts y x , z x ( x ∈ g ) in degree zero, the elemen ts e x ( x ∈ g ) in degree one, and the relations y λx + x ′ = λy x + y x ′ , z λx + x ′ = λz x + z x ′ , e λx + x ′ = λe x + e x ′ , y x a = a x + ay x , z x a = a x + az x , e x a = ae x , e x ′ y x = y x e x ′ + e [ x ′ ,x ] g , e x ′ z x = z x e x ′ , e 2 x = 0 , y x ′ y x = y x y x ′ + y [ x ′ ,x ] g + f ( x ′ , x ) − f ( x, x ′ ) , z x ′ y x = y x z x ′ + z [ x ′ ,x ] g + f ( x ′ , x ) − f ( x, x ′ ) , z x ′ z x = z x z x ′ + z [ x ′ ,x ] g + f ( x ′ , x ) − f ( x, x ′ ) , where [ x ′ , x ] g denotes the Lie brack et of x ′ and x in g . Note that E is a subalg ebra of Y ∗ via the embedding that tak es a ∈ A to a and 1# x to y x for all x ∈ g . This gives rise to an structure of left E -mo dule on Y ∗ . Similarly we consider Y ∗ as a right E -mo dule via the em bedding of E in Y ∗ that tak es a ∈ A to a and 1# x to z x for all x ∈ g . Let ( g i ) i ∈ I be a basis of g with indexes running on an ordered set I . F or each i ∈ I let us write y i = y g i , z i = z g i , e i = e g i and ρ i = z i − y i . 4 GRA CIELA CARBONI, JOR GE A. GUCCIONE, AND JUAN J. GUCCIONE Theorem 1.1. Each Y s is a fr e e left E -mo dule with b asis ρ m 1 i 1 e δ 1 i 1 · · · ρ m l i l e δ l i l  l ≥ 0 , i 1 < · · · < i l ∈ I , m j ≥ 0 , δ j ∈ { 0 , 1 } m j + δ j > 0 , δ 1 + · · · + δ l = s  . Pr o of. Let e g b e the dir ect sum of tw o copies { y x : x ∈ g } and { z x : x ∈ g } of g , endow ed with the bra c ket given by [ y x ′ , y x ] e g = y [ x ′ ,x ] g and [ z x ′ , z x ] e g = [ z x ′ , y x ] e g = z [ x ′ ,x ] g . Note that e g is the semi-dir ect sum aris ing from the a djoin t action o f g on itself. Let π : U ( e g ) → U ( g ) b e the algebra map defined by π ( y x ) = π ( z x ) = x . Let Λ( g ) b e the exterior algebra generated by g . Tha t is, the alg ebra gener a ted by the elements e x , with x ∈ g , a nd the relations e λx + x ′ = λe x + e x ′ and e 2 x = 0, with λ ∈ k and x, x ′ ∈ g . Let us consider the action of U ( e g ) on Λ( g ) determined b y e y x x ′ = e [ x,x ′ ] g and e z x x ′ = 0 . The env eloping algebra U ( e g ) of e g acts weakly on A ⊗ k Λ( g ) via ( a ⊗ k e ) u = X ( u ) a π ( u (1) ) ⊗ k e u (2) ( a ∈ A , e ∈ Λ( g ) and u ∈ U ( e g )). Moreov er, the map e f : U ( e g ) × U ( e g ) / / A ⊗ k Λ( g ) , defined b y e f ( u, v ) = f ( π ( u ) , π ( v )) ⊗ k 1, is a norma l 2-c o cy cle which satisfies the t wisted mo dule condition. L e t η : Y ′ ∗ / / ( A ⊗ k Λ( g ))# e f U ( e g ) be the homomorphism o f a lg ebras defined by η ( a ) = ( a ⊗ k 1)#1 for all a ∈ A and η ( y x ) = (1 ⊗ k 1)# y x , η ( z x ) = (1 ⊗ k 1)# z x and η ( e x ) = (1 ⊗ k e x )#1 fo r all x ∈ g . Because of the P oincar´ e-Birkhoff-Witt theorem, η  y n 1 j 1 · · · y n h j h ρ m 1 i 1 · · · ρ m l i l   h, l ≥ 0, j 1 < · · · < j h , i 1 < · · · < i l ∈ I and m j , n j ≥ 0  , is a ba s is of ( A ⊗ k Λ( g ))# e f U ( e g ) as a left A ⊗ k Λ( g )-mo dule. The theorem follows easily from this fact.  R emark 1 .2 . A similar argument shows that each Y s is a free right E -mo dule with the same basis. Theorem 1.3. L et e µ : Y 0 → E b e the algebr a map define d by e µ ( a ) = a for a ∈ A and e µ ( y x ) = e µ ( z x ) = 1# x for x ∈ g . T her e is a unique derivation ∂ ∗ : Y ∗ → Y ∗− 1 such t hat ∂ ( e x ) = z x − y x for x ∈ g . Mor e over, the chain c omplex 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 c ontr ac tible as a c omplex of ( E , A ) -bimo dules. A chain c ontra cting homotopy σ − 1 0 : E / / Y 0 , σ − 1 s +1 : Y s / / Y s +1 ( s ≥ 0) , is given by σ − 1 (1) = 1 , σ − 1  ρ m 1 i 1 e δ 1 i 1 · · · ρ m l i l e δ l i l  = ( ρ m 1 − 1 i 1 e i 1 ρ m 2 i 2 e δ 2 i 2 · · · ρ m l i l e δ l i l if δ 1 = 0 , 0 if δ 1 = 1 , wher e we assume that i 1 < · · · < i l , δ 1 + · · · + δ l = s and m l + δ l > 0 . Pr o of. A direct computation shows that - e µ   σ − 1 (1) = e µ (1) = 1, COHOMOLOGY RING OF DIFFERENTIAL OPERA TOR RINGS 5 - σ − 1   e µ (1) = σ − 1 (1) = 1 and ∂   σ − 1 (1) = ∂ (0 ) = 0, - If x = ρ m 1 i 1 x ′ , where m 1 > 0 and x ′ = ρ m 2 i 2 · · · ρ m l i l with i 1 < · · · < i l , then σ − 1   e µ ( x ) = σ − 1 (0) = 0 and ∂   σ − 1 ( x ) = ∂ ( ρ m 1 − 1 i 1 e i 1 x ′ ) = x , - Le t x = ρ m 1 i 1 e δ 1 i 1 x ′ , wher e m l + δ l > 0 and x ′ = ρ m 2 i 2 e δ 2 i 2 · · · ρ m l i l e δ l i l with i 1 < · · · < i l and δ 1 + · · · + δ l = s > 0. If δ 1 = 0 , then σ − 1   ∂ ( x ) = σ − 1  ρ m 1 i 1 ∂ ( x ′ )  = ρ m 1 − 1 i 1 e i 1 ∂ ( x ′ ) , ∂   σ − 1 ( x ) = ∂  ρ m 1 − 1 i 1 e i 1 x ′  = x − ρ m 1 − 1 i 1 e i 1 ∂ ( x ′ ) , and if δ 1 = 1 , then σ − 1   ∂ ( x ) = σ − 1  ρ m 1 +1 i 1 x ′ − ρ m 1 i 1 e i 1 ∂ ( x ′ )  = x , ∂   σ − 1 ( x ) = ∂ (0) = 0 . The result follows immediately from all these facts.  F or each s ≥ 0 w e consider E ⊗ k g ∧ s as a right K -mo dule via ( c ⊗ k x ) λ = c λ ⊗ k x . F or r, s ≥ 0, let X r s = ( E ⊗ k g ∧ s ) ⊗ A r ⊗ E . The g r oups X r s are E - bimodules in an obvious wa y . Let us consider the diagram of E -bimo dules and E -bimo dule maps . . . ∂ 3   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 µ ∗ : X 0 ∗ → Y ∗ and d 0 ∗∗ : X ∗∗ → X ∗− 1 , ∗ , are defined b y: µ (1 ⊗ k x 1 s ⊗ 1) = e x 1 . . . e x s , d 0 (1 ⊗ k x 1 s ⊗ a 1 r ⊗ 1) = ( − 1) s a 1 ⊗ k x 1 s ⊗ a 2 r ⊗ 1 + r − 1 X i =1 ( − 1) i + s ⊗ k x 1 s ⊗ a 1 ,i − 1 ⊗ a i a i +1 ⊗ a i +1 ,r ⊗ 1 + ( − 1) r + s ⊗ k x 1 s ⊗ a 1 ,r − 1 ⊗ a r , Each horizontal complex in this diagram is contractible a s a c omplex of ( E , K )- bimo dules. A c hain con tracting homotop y is the family σ 0 0 s : Y s / / X 0 s , σ 0 r +1 ,s : X r s / / X r +1 ,s ( r ≥ 0) , of ( E , K )-bimo dule maps, defined by σ 0 ( e x 1 · · · e x s z x s +1 · · · z x n ) = X j a j ⊗ k x 1 s ⊗ 1# w j , where P j a j # w j = (1 # x s +1 ) · · · (1# x n ), and σ 0 (1 ⊗ k x 1 s ⊗ a 1 r ⊗ a r +1 # w ) = ( − 1) r + s +1 ⊗ k x 1 s ⊗ a 1 ,r +1 ⊗ 1# w ( r ≥ 0) . (In order to see that the σ 0 ’s are r ight K -linear it is necessary to use that K is stable under the action of g ). Moreov er, eac h X r s is a pro jectiv e E -bimo dule relativ e to 6 GRA CIELA CARBONI, JOR GE A. GUCCIONE, AND JUAN J. GUCCIONE the family of all epimorphism o f E -bimo dules which split as ( E , K )-bimo dule ma ps . W e define E -bimo dule maps d l r s : X r s / / X r + l − 1 ,s − l ( r ≥ 0 and 1 ≤ l ≤ s ), recursively by: d l ( y ) =          − σ 0   ∂   µ ( y ) if l = 1 and r = 0, − σ 0   d 1   d 0 ( y ) if l = 1 and r > 0, − P l − 1 j =1 σ 0   d l − j   d j ( y ) if l > 1 and r = 0 , − P l − 1 j =0 σ 0   d l − j   d j ( y ) if l > 1 and r > 0 , where y = 1 ⊗ k x 1 s ⊗ a 1 r ⊗ 1. Theorem 1.4. The c omplex (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 X 5 d 5 o o . . . , d 6 o o wher e µ (1 ⊗ 1) = 1 , X n = M r + s = n X r s and d n = X r + s = n r + l> 0 s X l =0 d l r s , is a pr oje ctive r esolut ion of t he E -bimo dule E , r elative to the family of al l epimor- phism of E -bimo du les which split as ( E , K ) -bimo dule maps. Mor e over an explicit c ont r acting homotop y σ 0 : E / / X 0 , σ n +1 : X n / / X n +1 ( n ≥ 0) , of (1) , as a c omplex of ( E , K ) -bimo dules, is gi ven by σ 0 = σ 0   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 − l − r , wher e 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 (0 < l ≤ s, r ≥ 0) ar e r e cursively define d by σ l = − l − 1 X j =0 σ 0   d l − j   σ j . Pr o of. By [G-G2, Cor o llary A.2].  The boundar y maps of the pro jective resolution of E that w e just found ar e defined recursively . Next w e g ive closed formulas fo r them. COHOMOLOGY RING OF DIFFERENTIAL OPERA TOR RINGS 7 Theorem 1.5. F or x i , x j ∈ g , we put b f ij = f ( x i , x j ) − f ( x j , x i ) . We ha ve: d 1 (1 ⊗ k x 1 s ⊗ a 1 r ⊗ 1) = s X i =1 ( − 1) i +1 # x i ⊗ k x 1 b ıs ⊗ a 1 r ⊗ 1 + s X i =1 ( − 1) i ⊗ k x 1 b ıs ⊗ a 1 r ⊗ 1# x i + s X i =1 1 ≤ h ≤ r ( − 1) i ⊗ k x 1 b ıs ⊗ a 1 ,h − 1 ⊗ a x i h ⊗ a h +1 ,r ⊗ 1 + X 1 ≤ i 0, we can assume that l = 0 . In this case the assertion follows immediately from the definition of σ 0 . (3) By the definition of σ 0 and Theorem 1.5, σ 0   d 1   σ 0 ( e i 1 · · · e i n ) = σ 0   d 1 (1 ⊗ k g i 1 ∧ · · · ∧ g i n ⊗ 1) = 0 and σ 0   d 2   σ 0 ( e i 1 · · · e i n ) = σ 0   d 2 (1 ⊗ k g i 1 ∧ · · · ∧ g i n ⊗ 1) = 0 . Item (3) follows now eas ily by induction on l , since, by the recursive definition of σ l and Theorem 1.5, σ 1 = − σ 0   d 1   σ 0 and σ l = − σ 0   d 1   σ l − 1 − σ 0   d 2   σ l − 2 for l ≥ 2. (4) It is similar to the pro of of item (3). (5) Since e i a = ae i for all i ∈ I and a ∈ A , σ − 1   µ  a ⊗ k g i 1 ∧ · · · ∧ g i n ⊗ a ′  = σ − 1  ae i 1 · · · e i n a ′  = σ − 1  aa ′ e i 1 · · · e i n  = 0 , where the last equalit y follo ws from the definition of σ − 1 . COHOMOLOGY RING OF DIFFERENTIAL OPERA TOR RINGS 9 (6) W e ha ve σ − 1   µ  1 ⊗ k g i 1 ∧ · · · ∧ g i n ⊗ 1# g i n +1  = σ − 1  e i 1 · · · e i n z i n +1  = σ − 1  e i 1 · · · e i n ( y i n +1 + ρ i n +1 )  = σ − 1  y i n +1 e i 1 · · · e i n  + σ − 1  e i 1 · · · e i n ρ i n +1  , where z i n +1 , y i n +1 and ρ i n +1 are a s in Theorem 1.1. So, in order to finish the pr o of it suffices to note that σ − 1  y i n +1 e i 1 · · · e i n  = 0 and σ − 1  e i 1 · · · e i n ρ i n +1  = ( ( − 1) n e i 1 · · · e i n +1 if i n < i n +1 , 0 otherwise, which follows immediately from the fact that e i j ρ i n +1 = ρ i n +1 e i j + e [ x i j ,x i n +1 ] g for all j such that i j > i n +1 , and from the definition of σ − 1 .  Theorem 2.3. L et ( g i ) i ∈ I b e the b asis of g c onsider e d in The or em 1.1. Assume that c 1 n = c 1 ⊗ · · · ⊗ c n ∈ E n is a simple tensor with c j ∈ A ∪ { 1# g i : i ∈ I } for al l j ∈ { 1 , . . . , n } . If ther e exist 0 ≤ s ≤ n and i 1 < · · · < i s in I , such that c j = 1 # g i j for 1 ≤ j ≤ s and c j ∈ A for s < j ≤ n , t hen ϑ (1 ⊗ c 1 n ⊗ 1) = 1 ⊗ k g i 1 ∧ · · · ∧ g i s ⊗ c s +1 ,n ⊗ 1 . Otherwise, ϑ (1 ⊗ c 1 n ⊗ 1) = 0 . Pr o of. F or all n ≥ 0 w e define P n by c 1 n ∈ P n if there is i 1 < · · · < i s in I such that c j = 1 # g i j for j ≤ s and c j ∈ A for j > s . W e now pro ceed by induction on n . The case n = 0 is immediate. Assume that the res ult is v alid for ϑ n . By item (1) of Lemma 2.2 and the recursive definition of ϑ n , w e ha ve σ   ϑ ( c ′ 0 n ⊗ 1) = σ   σ   ϑ   b ′ ( c ′ 0 n ⊗ 1) = 0 , and so ϑ (1 ⊗ c 1 ,n +1 ⊗ 1) = ( − 1) n +1 σ   ϑ (1 ⊗ c 1 ,n +1 ) . Assume that c j ∈ A ∪ { 1# g i : i ∈ I } for all j ∈ { 1 , . . . , n + 1 } . In order to finis h the pro of it suffices to sho w that - If c 1 ,n +1 / ∈ P n +1 , then σ   ϑ (1 ⊗ c 1 ,n +1 ) = 0, - If c 1 ,n +1 = 1 # g i 1 ⊗ · · · ⊗ 1# g i s ⊗ a s +1 ,n +1 ∈ P n +1 , then σ   ϑ (1 ⊗ c 1 ,n +1 ) = ( − 1) n +1 ⊗ k g i 1 ∧ · · · ∧ g i s ⊗ a s +1 ,n +1 ⊗ 1 . If c 1 n / ∈ P n , then ϑ (1 ⊗ c 1 ,n +1 ) = 0 by the inductive hypothesis. It rema ins to consider the case c 1 n ∈ P n . W e divide this in to three subca ses. 1) If c 1 n = 1 # g i 1 ⊗ · · · ⊗ 1# g i s ⊗ a s +1 ,n and c n +1 = a n +1 ∈ A , then σ   ϑ (1 ⊗ c 1 ,n +1 ) = σ  1 ⊗ k g i 1 ∧ · · · ∧ g i s ⊗ a s +1 ,n +1  = σ 0  1 ⊗ k g i 1 ∧ · · · ∧ g i s ⊗ a s +1 ,n +1  = ( − 1) n +1 ⊗ k g i 1 ∧ · · · ∧ g i s ⊗ a s +1 ,n +1 ⊗ 1 , by the inductiv e hypothesis, items (4 ) and (5) o f Lemma 2.2, and the definitio ns of σ and σ 0 . 2) If c 1 n = 1 # g i 1 ⊗ · · · ⊗ 1# g i s ⊗ a s +1 ,n with s < n and c n +1 = 1# g i n +1 , then σ   ϑ (1 ⊗ c 1 ,n +1 ) = σ  1 ⊗ k g i 1 ∧ · · · ∧ g i s ⊗ a s +1 ,n ⊗ 1# g i n +1  = 0 , by the inductive h ypothesis , the definition of σ and item (2) o f Lemma 2.2. 10 GRA CIELA CARBONI, JOR GE A. GUCCIONE, AND JUAN J. GUCCIONE 3) If c 1 n = 1 # g i 1 ⊗ · · · ⊗ 1# g i n and c n +1 = 1 # g i n +1 , then σ   ϑ (1 ⊗ c 1 ,n +1 ) = σ  1 ⊗ k g i 1 ∧ · · · ∧ g i n ⊗ 1# g i n +1  = − σ 0   σ − 1   µ  1 ⊗ k g i 1 ∧ · · · ∧ g i n ⊗ 1# g i n +1  = ( ( − 1) n +1 ⊗ k g i 1 ∧ · · · ∧ g i n +1 ⊗ 1 if c 1 ,n +1 ∈ P n +1 , 0 otherwise, by the inductiv e h ypo thesis, items (2), (3) and (6) of Lemma 2 .2, and the definitions of σ and σ 0 .  3. The Hochschild cohomology Let E = A # f U ( g ) and M an E -bimo dule. In this section we obtain a cochain complex ( X ∗ K ( M ) , d ∗ ), simpler than the canonica l one, giving the Ho chsc hild coho - mology of the K -alg e br a E with co efficients in M . When K = k our result reduce to the one obtained in [G-G1, Sec tion 5]. Then, we obtain an express ion that gives the cup pr o duct of the Hochsc hild cohomology of E in terms of ( X ∗ K ( E ) , d ∗ ). As usual, given c ∈ E and m ∈ M , we let [ m, c ] denote the commutator mc − cm . 3.1. The compl ex ( X ∗ K ( M ) , d ∗ ) . F or r, s ≥ 0, let X r s K ( M ) = Ho m K e ( A r ⊗ k g ∧ s , M ) , where A r ⊗ k g ∧ s is co ns idered as a K -bimo dule via the canonical actions o n A r . W e define the mor phis m d r s l : X r + l − 1 ,s − l K ( M ) / / X r s K ( M ) (with 0 ≤ l ≤ min(2 , s ) a nd r + l > 0), by: d 0 ( ϕ )( a 1 r ⊗ k x 1 s ) = a 1 ϕ ( a 2 r ⊗ k x 1 s ) + r − 1 X i =1 ( − 1) i ϕ ( a 1 ,i − 1 ⊗ a i a i +1 ⊗ a i +2 ,r ⊗ k x 1 s ) + ( − 1) r ϕ ( a 1 ,r − 1 ⊗ k x 1 s ) a r , d 1 ( ϕ )( a 1 r ⊗ k x 1 s ) = s X i =1 ( − 1) i + r  ϕ ( a 1 r ⊗ k x 1 b ıs ) , 1# x i  + s X i =1 1 ≤ h ≤ r ( − 1) i + r ϕ ( a 1 ,h − 1 ⊗ a x i h ⊗ a h +1 ,r ⊗ k x 1 b ıs ) + X 1 ≤ i 0 min( s, 2) X l =0 d r s l . Note that if f ( g ⊗ k g ) ⊆ K , then ( X ∗ K ( M ) , d ∗ ) is the to tal complex of the double complex  X ∗∗ K ( M ) , d ∗∗ 0 , d ∗∗ 1  . Theorem 3.1. The Ho chschild c ohomo lo gy H ∗ K ( E , M ) , of the K -algebr a E with c o efficients in M , is the c ohomolo gy of ( X ∗ K ( M ) , d ∗ ) . Pr o of. It is an immediate consequence of the ab ov e discussion.  3.2. The comparis on maps. The ma ps θ ∗ and ϑ ∗ , introduced in Section 2, induce quasi-isomo rphisms θ ∗ :  Hom K e  E ∗ , M  , b ∗  / / ( X ∗ K ( M ) , d ∗ ) and ϑ ∗ : ( X ∗ K ( M ) , d ∗ ) / /  Hom K e  E ∗ , M  , b ∗  which are inv erse one o f each other up to homotopy . Prop osition 3.2. We have θ ( ψ )( a 1 r ⊗ k x 1 s ) = X τ ∈ S s ( − 1) r s sg( τ ) ψ   1# x τ (1) ⊗ · · · ⊗ 1# x τ ( s )  ∗ a 1 r  Pr o of. This follows immedia tely from Prop osition 2.1.  In the sequel w e consider that X r s K ⊆ X r + s K in the canonical wa y . Theorem 3.3. L et ( g i ) i ∈ I b e the b asis of g c onsider e d in The or em 1.1 and let ϕ ∈ X r s K . Assume that c 1 ,r + s = c 1 ⊗ · · · ⊗ c r + s ∈ E r + s is a simple ten sor with c j ∈ A ∪ { 1# g i : i ∈ I } for al l j ∈ { 1 , . . . , r + s } . If c j = 1# g i j with i 1 < · · · < i s in I for 1 ≤ j ≤ s and c j ∈ A for s < j ≤ r + s , then ϑ ( ϕ )( c 1 ,r + s ) = ( − 1) r s ϕ ( c s +1 ,r + s ⊗ k g i 1 ∧ · · · ∧ g i s ) . Otherwise, ϑ ( ϕ )( c 1 ,r + s ) = 0 . Pr o of. This follows immedia tely from Theorem 2.3.  As usual, in the fo llowing subsection we will write HH ∗ K ( E ) ins tea d of H ∗ K ( E , E ). 3.3. The cup pro duct. Recall that the cup pr o duct of HH ∗ K ( E ) is given in terms of  Hom K e  E ∗ , E  , b ∗  , b y ( ψ ⌣ ψ ′ )( c 1 ,m + n ) = ψ ( c 1 m ) ψ ′ ( c m +1 ,m + n ) , where ψ ∈ Ho m K e ( E m , E ) and ψ ′ ∈ Hom K e ( E n , E ). In this subsection w e compute the cup pro duct in ter ms of the small complex ( X ∗ K ( E ) , d ∗ ). Giv en ϕ ∈ X r s K ( E ) and ϕ ′ ∈ X r ′ s ′ K ( E ) we define ϕ • ϕ ′ ∈ X r + r ′ ,s + s ′ K ( E ) by ( ϕ • ϕ ′ )( a 1 r ′′ ⊗ k x 1 s ′′ ) = X 1 ≤ j 1 < ··· 0), by: d 0 ( m ⊗ a 1 r ⊗ k x 1 s ) = ma 1 ⊗ a 2 r ⊗ k x 1 s + r − 1 X i =1 ( − 1) i m ⊗ a 1 ,i − 1 ⊗ a i a i +1 ⊗ a i +2 ,r ⊗ k x 1 s + ( − 1) r a r m ⊗ a 1 ,r − 1 ⊗ k x 1 s , d 1 ( m ⊗ a 1 r ⊗ k x 1 s ) = s X i =1 ( − 1) i + r [(1# x i ) , m ] ⊗ a 1 r ⊗ k x 1 b ıs + s X i =1 1 ≤ h ≤ r ( − 1) i + r m ⊗ a 1 ,h − 1 ⊗ a x i h ⊗ a h +1 ,r ⊗ k x 1 b ıs COHOMOLOGY RING OF DIFFERENTIAL OPERA TOR RINGS 13 + X 1 ≤ i 0 min( s, 2) X l =0 d l r s . Note that if f ( g ⊗ k g ) ⊆ K , then ( X K ∗ ( M ) , d ∗ ) is the to tal complex of the double complex  X K ∗∗ ( M ) , d 0 ∗∗ , d 1 ∗∗  . Theorem 4.1. The Ho chschild homo lo gy H K ∗ ( E , M ) , of the K -algebr a E with c o- efficients in M , is the homol o gy of ( X K ∗ ( M ) , d ∗ ) . Pr o of. It is an immediate consequence of the ab ov e discussion.  4.2. The comparis on maps. The ma ps θ ∗ and ϑ ∗ , introduced in Section 2, induce quasi-isomo rphisms θ ∗ : ( X K ∗ ( M ) , d ∗ ) / /  M ⊗ E ∗ [ M ⊗ E ∗ ,K ] , b ∗  and ϑ ∗ :  M ⊗ E ∗ [ M ⊗ E ∗ ,K ] , b ∗  / / ( X K ∗ ( M ) , d ∗ ) which are inv erse one o f each other up to homotopy . Prop osition 4.2. We have θ ( m ⊗ a 1 r ⊗ k x 1 s ) = X τ ∈ S s ( − 1) r s sg( τ ) m ⊗  1# x τ (1) ⊗ · · · ⊗ 1# x τ ( s )  ∗ a 1 r Pr o of. This follows immedia tely from Prop osition 2.1.  Theorem 4.3. L et ( g i ) i ∈ I b e the b asis of g c onsider e d in The or em 1.1. Assume that c 1 n = c 1 ⊗ · · · ⊗ c n ∈ E n is a simple tensor with c j ∈ A ∪ { 1# g i : i ∈ I } for al l j ∈ { 1 , . . . , n } . If ther e exist 0 ≤ s ≤ n and i 1 < · · · < i s in I , such that c j = 1 # g i j for 1 ≤ j ≤ s and c j ∈ A for s < j ≤ n , t hen ϑ ( m ⊗ c 1 n ) = ( − 1) s ( n − s ) m ⊗ c s +1 ,n ⊗ k g i 1 ∧ · · · ∧ g i s . Otherwise, ϑ ( m ⊗ c 1 n ) = 0 . Pr o of. This follows immedia tely from Theorem 2.3.  14 GRA CIELA CARBONI, JOR GE A. GUCCIONE, AND JUAN J. GUCCIONE 4.3. The cap pro duct. Reca ll that the cap pr o duct H K p ( E , M ) × HH q K ( E ) → H K p − q ( E , M ) ( q ≤ p ) , is defined in terms of  M ⊗ E ∗ [ M ⊗ E ∗ ,K ] , b ∗  and  Hom K e  E ∗ , E  , b ∗  , b y m ⊗ c 1 p a ψ = mψ ( c 1 q ) ⊗ c q +1 ,p , where ψ ∈ Hom K e ( E q , E ). In this s ubsection we compute the cup pro duct in terms of the small complexes ( X K ∗ ( M ) , d ∗ ) and ( X ∗ K ( E ) , d ∗ ). Given m ⊗ a 1 r ⊗ k x 1 s ∈ X K r s ( M ) and ϕ ′ ∈ X r ′ s ′ K ( E ) with r ≥ r ′ and s ≥ s ′ , we define ( m ⊗ a 1 r ⊗ k x 1 s ) • ϕ ′ ∈ X K r − r ′ ,s − s ′ ( M ) by ( m ⊗ a 1 r ⊗ k x 1 s ) • ϕ ′ = X 1 ≤ j 1 < ··· r or s ′ > s , then ϑ  θ ( m ⊗ a 1 r ⊗ k x 1 s ) a ϑ ( ϕ ′ )  = 0 , and, if r ′ ≤ r and s ′ ≤ s , then ϑ  θ ( m ⊗ a 1 r ⊗ k x 1 s ) a ϑ ( ϕ ′ )  = X 1 ≤ j 1 < ··· 0), be the E -bimo dule morphis ms defined by δ 0 (1 ⊗ x 1 s ⊗ v 1 r ⊗ 1) = r X i =1 ( − 1) i + s  v i ⊗ x 1 s ⊗ v 1 b ır ⊗ 1 − 1 ⊗ x 1 s ⊗ v 1 b ır ⊗ v i  , δ 1 (1 ⊗ x 1 s ⊗ v 1 r ⊗ 1) = s X i =1 ( − 1) i +1 # x i ⊗ x 1 b ıs ⊗ v 1 r ⊗ 1 + s X i =1 ( − 1) i ⊗ x 1 b ıs ⊗ v 1 r ⊗ 1# x i + s X i =1 1 ≤ h ≤ r ( − 1) i ⊗ x 1 b ıs ⊗ v 1 ,h − 1 ∧ v x i h ∧ v h +1 ,r ⊗ 1 + X 1 ≤ i 0 s X l =0 δ l r s , is a pr oje ctive r esolution of the E -bimo dule E . Mor e over, the family of maps Γ ∗ : Z ∗ → X ∗ , given by Γ(1 ⊗ x 1 s ⊗ v 1 r ⊗ 1) = X σ ∈ S r sg( σ ) ⊗ x 1 s ⊗ v σ (1) ⊗ · · · ⊗ v σ ( r ) ⊗ 1 , 16 GRA CIELA CARBONI, JOR GE A. GUCCIONE, AND JUAN J. GUCCIONE defines an morphism of E -bimo dule c omplexes fr om ( Z ∗ , δ ∗ ) t o ( X ∗ , d ∗ ) . Pr o of. It is clear that each Z n is a pro jective E -bimo dule a nd a direct c omputation shows that Γ ∗ is a morphism of complexes. Let G 0 ∗ ⊆ G 1 ∗ ⊆ G 2 ∗ ⊆ G 3 ∗ ⊆ . . . and F 0 ∗ ⊆ F 1 ∗ ⊆ F 2 ∗ ⊆ F 3 ∗ ⊆ . . . be the filtr a tion of Z ∗ and X ∗ resp ectively , defined b y G i n = M r + s = n s ≤ i Z r s and F i n = M r + s = n s ≤ i X r s . In or der to see that Γ ∗ is a quasi- isomorphism it is sufficient to show that it induces a quasi-iso morphism betw een the gra ded complexes asso ciated with the filtrations int ro duced above. In other words that the maps Γ ∗ s : ( Z ∗ s , δ 0 ∗ s ) / / ( X ∗ s , d 0 ∗ s ) ( s ≥ 0) , defined b y Γ(1 ⊗ x 1 s ⊗ v 1 r ⊗ 1) = X σ ∈ S r sg( σ ) ⊗ x 1 s ⊗ v σ (1) ⊗ · · · ⊗ v σ ( r ) ⊗ 1 , are quasi-iso morphisms, which follo ws easily from Propo sition 2 .1.  5.1. Ho c hsc hi ld cohomology. Let M b e an E -bimo dule. F or r, s ≥ 0, let Z r s ( M ) = Ho m k ( V r ⊗ g ∧ s , M ) . W e define the mor phis m δ r s l : Z r + l − 1 ,s − l ( M ) / / Z r s ( M ) (with 0 ≤ l ≤ min(2 , s ) a nd r + l > 0) by: δ 0 ( ϕ )( v 1 r ⊗ x 1 s ) = r X i =1 ( − 1) i [ v i , ϕ ( v 1 b ır ⊗ x 1 s )] , δ 1 ( ϕ )( v 1 r ⊗ x 1 s ) = s X i =1 ( − 1) i + r  ϕ ( v 1 r ⊗ x 1 b ıs ) , 1# x i  + s X i =1 1 ≤ h ≤ r ( − 1) i + r ϕ ( v 1 ,h − 1 ∧ v x i h ∧ v h +1 ,r ⊗ x 1 b ıs ) + X 1 ≤ i 0 min( s, 2) X l =0 δ r s l . Note that if f ( g ⊗ g ) ⊆ k , then ( Z ∗ ( M ) , δ ∗ ) is the total complex of the double complex  Z ∗∗ ( M ) , δ ∗∗ 0 , δ ∗∗ 1  . Theorem 5. 2. The Ho chschild c oh omolo gy H ∗ ( E , M ) , of E with c o efficients in M , is t he c ohomolo gy of ( Z ∗ ( M ) , δ ∗ ) . The map Γ : ( Z ∗ , δ ∗ ) → ( X ∗ , d ∗ ) induces a quasi-isomorphism Γ ∗ : ( X ∗ k ( M ) , d ∗ ) / / ( Z ∗ ( M ) , δ ∗ ) . Prop osition 5.3. We have Γ( ϕ )( v 1 r ⊗ x 1 s ) = X σ ∈ S r sg( σ ) ϕ ( v σ (1) ⊗ · · · ⊗ v σ ( r ) ⊗ x 1 s ) . Pr o of. This follows immedia tely from Theorem 5.1.  5.2. The cup pro duct. In this subsection w e compute the cup product of HH ∗ ( E ) in terms of the complex ( Z ∗ ( E ) , δ ∗ ). Given φ ∈ Z r s ( E ) and φ ′ ∈ Z r ′ s ′ ( E ), we define φ ⋆ φ ′ ∈ Z r + r ′ ,s + s ′ ( E ) b y ( φ ⋆ φ ′ )( v 1 r ′′ ⊗ x 1 s ′′ ) = X 1 ≤ i 1 < ··· 0) by: δ 0 ( m ⊗ v 1 r ⊗ x 1 s ) = r X i =1 ( − 1) i [ m, v i ] ⊗ v 1 b ır ⊗ x 1 s , δ 1 ( m ⊗ v 1 r ⊗ x 1 s ) = s X i =1 ( − 1) i + r [1# x i , m ] ⊗ v 1 r ⊗ x 1 b ıs + s X i =1 1 ≤ h ≤ r ( − 1) i + r m ⊗ v 1 ,h − 1 ∧ v x i h ∧ v h +1 ,r ⊗ x 1 b ıs + X 1 ≤ i 0 min( s, 2) X l =0 δ l r s . Note that if f ( g ⊗ g ) ⊆ k , then ( Z ∗ ( M ) , δ ∗ ) is the total complex of the double complex  Z ∗∗ ( M ) , δ 0 ∗∗ , δ 1 ∗∗  . Theorem 5.5. The Ho chsc hild homolo gy H ∗ ( E , M ) , of E with c o efficients in M , is t he homolo gy of ( Z ∗ ( M ) , δ ∗ ) . COHOMOLOGY RING OF DIFFERENTIAL OPERA TOR RINGS 19 Pr o of. It is an immediate consequence of the ab ov e discussion.  The map Γ : ( Z ∗ , δ ∗ ) → ( X ∗ , d ∗ ) induces a quasi-isomorphism Γ ∗ : ( Z ∗ ( M ) , δ ∗ ) / / ( X k ∗ ( M ) , d ∗ ) . Prop osition 5.6. We have Γ( m ⊗ v 1 r ⊗ x 1 s ) = X σ ∈ S r sg( σ ) m ⊗ v σ (1) ⊗ · · · ⊗ v σ ( r ) ⊗ x 1 s . Pr o of. This follows immedia tely from Theorem 5.1.  5.4. The cap pro duct. In this subsection we compute the cap pr o duct of H p ( E , M ) × HH q ( E ) → H p − q ( E , M ) ( q ≤ p ) , in terms of the complexes ( Z ∗ ( M ) , δ ∗ ) and ( Z ∗ ( E ) , δ ∗ ). Giv en m ⊗ v 1 s ⊗ x 1 s ∈ Z r s ( M ) and φ ′ ∈ Z r ′ s ′ ( E ) with r ≥ r ′ and s ≥ s ′ , we define ( m ⊗ v 1 r ⊗ x 1 s ) ⋆ φ ′ ∈ Z r − r ′ ,s − s ′ ( M ) by ( m ⊗ v 1 r ⊗ x 1 s ) ⋆ φ ′ = X 1 ≤ i 1 < ···

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