Hochschild (Co)Homology of Differential Operator Rings 1

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1 Journal of Algebra 243, (2001 doi: /jabr , available online at on Hochschild (CoHomology of Differential Operator Rings 1 Jorge A. Guccione and Juan J. Guccione Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Pabellón 1 - Ciudad Universitaria, 1428 Buenos Aires, Argentina vander@mate.dm.uba.ar Communicated by Susan Montgomery Received July 19, 2000 We show that the Hochschild homology of a differential operator k-algebra E = A# f Ug is the homology of a deformation of the Chevalley Eilenberg complex of with coefficients in M A b. Moreover, when A is smooth and k is a characteristic zero field, we obtain a type of Hochschild Kostant Rosenberg theorem for these algebras. When A = k our complex reduces to the one obtained by C. Kassel (1988, Invent. Math. 91, for the homology of filtrated algebras whose associated graded algebras are symmetric algebras. In the last section we give similar results for the cohomology Academic Press INTRODUCTION Let k be a field and A an associative k-algebra with 1. An extension E A of A is a differential operator ring on A if there exists a k-lie algebra and a vector space embedding x x, of into E, such that for all x y a A: (1 xa a x = a x, where a a x is a derivation, (2 xy yx = x y + f x y, where [, ] is the bracket of and f A is a bilinear map, 1 Supported by UBACYT 01/TW79 and CONICET. We thank the referee for a substantial simplification in the proof of Theorem /01 $35.00 Copyright 2001 by Academic Press All rights of reproduction in any form reserved. 596

2 hochschild (cohomology 597 (3 for a given basis x i i I of, E is a free left A-module with the standard monomials in the x i s as a basis. This general construction was introduced in [Ch, Mc-R]. Several particular cases of this type of extensions have been considered previously in the literature. For instance: when is one dimensional, f is trivial and E is the Ore extension Ax δ, where δa =a x, when A = k, one obtains the algebras studied by Sridharan in [S], which are the quasi-commutative algebras E, whose associated graded algebra is a symmetric algebra. in [Mc, Sect. 2] this type of extension was studied under the hypothesis that A is commutative and x a a x is an action and in [B-G-R, Theorem 4.2] the case in which the cocycle is trivial was considered. In [B-C-M, D-T] the study of the crossed products A# f H of an algebra A by a Hopf algebra H was begun and in [M] it was proved that the differential operator rings on A are the crossed products of A by enveloping algebras of Lie algebras. In [G-G] we obtained a complex, simpler than the canonical one, giving the Hochschild homology of a general crossed product E = A# f H with coefficients in an arbitrary E-bimodule M. In the present paper we show that, for differential operator rings, a complex simpler than the one obtained in [G-G] also works, and we give some applications of this result. This paper is organized as follows: In Section 1 we recall the definition of differential operator rings following the Hopf algebra point of view of [B-C-M, D-T]. In Section 2 we recall a technical result, established in [G-G], that we need in order to carry out our computations. In Section 3 we get a resolution of a differential operator ring E = A# f U as an E-bimodule. This resolution is a mixture of the canonical Hochschild normalized resolution of A and the Chevalley Eilenberg resolution of. In Section 4 we study the Hochschild homology of E with coefficients in an arbitrary E-bimodule M. The main result is Theorem 4.1, where the promised complex, which is a deformation of the Chevalley Eilenberg complex of with coefficients in M A b, is obtained. Then we consider a natural filtration of this complex, and we derive from it the spectral sequence of [St] in a more explicit way than the original one. Then, we consider the case when A is a commutative smooth algebra. The result obtained by us under this condition is a common generalization of the Hochschild Kostant Rosenberg theorem and the computation given in [K] for the Hochschild homology of algebras whose associated graded algebras are symmetric algebras. Finally, in Section 5, we study the cohomology.

3 598 guccione and guccione 1. PRELIMINARIES Let A be a k-algebra and H a Hopf algebra. A weak action of H on A is a bilinear map h a a h from H A to A such that, for h H a b A, (1 ab h = h a h1 b h2, (2 1 h = ɛh1, (3 a 1 = a. By an action of H on A we mean a weak action such that (4 a 1 h = a hl for all h l H a A. Let A be a k-algebra and H a Hopf algebra with a weak action on A. Given a k-linear map f H H A we let A# f H denote the k-algebra (in general non-associative and without 1 whose underlying vector space is A H and whose multiplication is given by a hb l = ab h1 f ( h 2 l 1 h 3 l 2 hl for all a b A h l H. The element a h of A# f H will usually be written a#h to remind us H is weakly acting on A. The algebra A# f H is called a crossed product if it is associative with 1#1 as identity element. In [B-C-M] was proved that this happens if and only if f and the weak action satisfy the following conditions (1 (normality of f for all h H we have f h1=f1h=ɛh1 A, (2 (cocycle condition for all h l m H we have 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 hlm hl (3 (twisted module condition for all h l H a A we have ( a l 1 h 1 ( f h 2 l 2 = f ( h 1 l 1 a h2 l 2 hl hl From now on, we assume that H is the enveloping algebra U of a Lie algebra. In this case, item (1 of the definition of weak action implies that ab x = a x b + ab x for x. So, a weak action determines a linear map δ Der k A by δxa =a x. Moreover if h a a h is an action, then δ is a homomorphism of Lie algebras. Reciprocally given a linear map δ Der k A, there exists a (generality non-unique weak action of U on A such that δxa =a x. When δ is a homomorphism of Lie algebras, there is a unique action of U on A such that δxa =a x. For a proof of these facts see [B-C-M]. Next we show that each normal cocycle f U U A is convolution invertible, giving a formula for f 1.

4 hochschild (cohomology 599 Remark 1.1. Each normal cocycle f U U A is convolution invertible. Moreover, for each h U and each family x 1 x r of elements of, we have f 1 1h=f 1 h 1 =ɛh1 A and f 1 x 1 x r h r = 1 l l=1 1 p 1 p l p 1 + +p l =r f ( x τ1 x τp1 h 1 τ Sh p1 p l h f ( x τp1 +1 x τp1 +p 2 h 2 f ( x τp1 + +p l 1 +1 x τr h 1 where Sh p1 p l denotes the multishuffles associated to p 1 p l. That is, { ( ( i i+1 } Sh p1 p l = τ r τ 1 + p j < <τ p j for 0 i<l j=1 j=1 This fact can be proved by a direct computation. 2. A METHOD FOR CONSTRUCTING RESOLUTIONS Let k be a commutative ring with 1 and E a k-algebra. In this section we recall a result that we will use in Section 3. For the proof we remit to [G-G]. Let 3 µ 2 d12 0 d22 0 Y 2 X02 X 12 2 µ 1 d11 0 d21 0 Y 1 X01 X 11 1 µ 0 d10 0 d20 0 Y 0 X00 X 10 be a diagram of E-bimodules and morphisms of E-bimodules verifying: (1 The column and the rows are chain complexes, (2 Each X rs is isomorphic to a free E-bimodule E X rs E, (3 Each row is contractible as a complex of left E-modules, with a chain contracting homotopy σ0s 0 Y s X 0s and σ 0 r+1s X rs X r+1s r 0.

5 600 guccione and guccione We define E-bimodule morphisms drs l X rs X r+l 1s l r 0 and 1 l s, recursively by σ 0 0s 1 s µ s x if r = 0 and l = 1, drs l x = l 1 j=1 σ0 l 1s l dl j j 1s j dj 0sx if r = 0 and 1 <l s, l i j=0 σ0 r+l 1s l dl j r+j 1s j dj rsx if r>0, for x = 1 x 1 with x X rs. Theorem 2.1. Let µ Y 0 E be a morphism of E-bimodules such that E µ Y 0 1 Y1 2 Y2 3 Y3 4 Y4 5 Y5 6 Y6 7 is a complex that is contractible as a complex of left E-modules. Then where E µ X 0 d 1 X1 d 2 X2 d 3 X3 d 4 X4 d 5 X5 d 6 X6 d 7 µ = µ µ 0 X n = X rs and d n = is a relative projective resolution of E as an E-bimodule. r+l>0 s drs l l=0 3. A RESOLUTION FOR A DIFFERENTIAL OPERATOR RING Let E = A# f U be a crossed product. In this section we obtain an E-bimodule resolution X d of E, that is simpler than the canonical of Hochschild. Then an explicit expression of the boundary maps of this resolution is given. To begin, we fix some notations: (1 For each k-algebra B and each r, we write B = B/k B r = B B (r times and B r = B B (r times. Moreover, for b B, we also let b denote the class of b in B. (2 Given a 0 a r A r+1 and 0 i<j r, we write a ij = a i a j. (3 For each Lie k-algebra and each s, we write s = (s times. (4 Given x = x 1 x s s and 1 i s, we write xî = x 1 ˆx i x s. (5 Given x = x 1 x s s and 1 i<j s, we write xîĵ = x 1 ˆx i ˆx j x s.

6 3.1. The Complex Y X hochschild (cohomology 601 Let be the direct sum of two copies y x x and z x x of, endowed with the bracket given by y x y x = y x x and y x z x = z x z x = z x x. Note that is the semi-direct sum arising from the adjoint action of g on itself. Let π U U be the algebra map defined by πy x =πz x =x. Let be the exterior algebra generated by. That is, the algebra generated by the elements e x x and the relations e λx+x = λe x + e x and e 2 x = 0 λ k x x. Let us consider the action of U on determined by e y x x = e x x and e z x x = 0. The enveloping algebra U of acts weakly on A via a e u = a πu e + a e u a A e and u U. Moreover, the map f U U A, defined by f u v =fπuπv 1, is a normal 2-cocycle which satisfies the twisted module condition. Theorem Let Y be the graded algebra generated by A, the degree zero elements y x z x x, the degree one elements e x x, and the relations y λx+x = λ yx + y x y x a = a x + ay x e x y x = y x e x + e x x z λx+x = λz x + z x z x a = a x + az x e x z x = z x e x e λx+x = λe x + e x e x a = ae x e 2 x = 0 y x y x = y x y x + y x x + f x x f x x z x y x = y x z x + z x x + f x x f x x z x z x = z x z x + z x x + f x x f x x Let x i i I be a basis of with indexes running on an ordered set I. For each i I let us write y i = y xi z i = z xi, and e i = e xi. Then each Y s is a free A-module with basis ( y m 1 e δ 1 z n 1 y m l e δ l z n l l 0i1 < < I m j n j 0δ j 01 m j + δ j + n j > 0δ 1 + +δ l = s Proof. Let ϑ Y A # f U be the homomorphism of algebras defined by ϑa =a 1#1 for all a A and ϑy x =1 1#y x, ϑz x =1 1#z x and ϑe x =1 e x #1 for all x. Because of the Poincaré Birkhoff Witt theorem, ( ϑ y m 1 e δ 1 z n 1 y m l e δ l z n l ( l 0i1 < < I m j n j 0 and δ j 01 is a basis of A # f U as an A-module. The theorem follows immediately from this fact.

7 602 guccione and guccione Remark Note that E is a subalgebra of Y by embedding a A to a and x to y x. This gives rise to a structure of the left E-module on Y s. Similarly we consider Y as a right E-module via the embedding of E in Y that sends a A to a and x to z x. Theorem Let µ Y 0 E be the algebra map defined by µ a = a for a A and µ y i = µ z i =x i for i I. There is a unique derivation Y Y 1 such that 1 e i=z i y i for i I. Moreover, the chain complex of E-bimodules E µ Y 0 1 Y 1 2 Y 2 3 Y 3 4 Y 4 5 Y 5 6 Y 6 7 is contractible as a complex of k-modules. A chain contracting homotopy is given by σ 0 a#x m 1 x m l =az m 1 z m l and ( σ s+1 ay m 1 e δ 1 z n 1 y m l e δ l z n l = m 1 +n 1 z m j 1+n j 1 i j 1 yi h j e ij j<α 0 h<m j az z m j+n j h 1 i j y m j+1 i j+1 e δ j+1 i j+1 z n j+1 i j+1 y m l e δ l z n l where α = mink δ k = 1 (in particular δ 1 = =δ α 1 = 0. Proof. We must check that µ σ 0 = id σ 0 µ + 1 σ 1 = id and s+1 σ s+1 + σ s s = id for all s>0. It is immediate that and µ σ 0 ( a#x m 1 σ 0 µ ( ay m 1 z n 1 M mδn v x m l = µ ( az m 1 z m l m = a#x 1 x m l y m l z n ( l m = σ0 a#x 1 +n 1 x m l+n l m = az 1 +n 1 z m l+n l Let us compute s+1 σ s+1 for s 0 and σ s s for s>0. To abbreviate we write = y m u e δ u z n u y m v e δ v z n v for 1 u<v l We have Z m+n v = z m u+n u z m v+n v for 1 u<v l δ h = δ 1 + +δ h for 1 h l s+1 σ s+1ami mδn 1l = s ( az m+n u<α u 1 y h e iu = u<α az m+n ( u 1 y h iu z m u+n u h y h+1 u<α v α Mi mδn u+1l 1 δ v δ v az m+n u 1 yi h u e iu M mδn +1l

8 hochschild (cohomology 603 M mδn +1v 1 y m v + u<α M mδn z n v+1 Mi mδn v+1l 1 δ v δ v az m+n u 1 yi h u e iu v α where α = mink δ k = 1. Since we obtain u<α s+1 σ s+1am mδn l On the other hand +1v 1 y m v+1 z n v Mi mδn v+1l az m+n ( u 1 y h iu z m u+n u h y h+1 = az m+n ( m u 1 z u +n u u<α = az m+n α 1 M mδn l ami mδn 1l = az m+n α 1 Miαl mδn u<α v α M mδn +1v 1 y m v + u<α M mδn y m u z n u M mδn +1l + am mδn l M mδn +1l 1 δ v δ v az m+n u 1 yi h u e iu z n v+1 Mi mδn v+1l 1 δ v δ v az m+n u 1 yi h u e iu v α +1v 1 y m v+1 z n v Mi mδn v+1l ( σ s s ammδn l =σ s 1 δv 1 δ v ami mδn 1v 1 v α ( y m v z n v+1 y m v+1 z n v M mδn +1l Since ( σ s ami mδn ( m 1α 1 y α z n α+1 y m α+1 = az m+n α 1 y m α e iα z n α M mδn +1l z n α Mi mδn ( m u+1α 1 y α z n α+1 y m α+1 +1l M mδn az m+n u 1 yi h u e iu u<α z n α +1l M mδn

9 604 guccione and guccione and, for v>α, we obtain ( σ s ami mδn ( m 1v 1 y v z n v+1 y m v+1 z n v +1l M mδn = u<α az m+n u 1 yi h u e iu Mi mδn ( m u+1v 1 y v z n v+1 y m v+1 z n v +1l M mδn σ s s ammδn l =az m+n α 1 Mi mδn αl + 1 δ v δ v az m+n u 1 yi h u e iu u<α v α M mδn +1v 1 y m v u<α M mδn z n v+1 Mi mδn v+1l 1 δ v δ v az m+n u 1 yi h u e iu v α +1v 1 y m v+1 z n v Mi mδn v+1l The result follows immediately from these facts The Resolution X d Let Y s = E s U s 0 and X rs = E s A r E r s 0. The groups X rs are E-bimodules in an obvious way and the groups Y s are E-bimodules via the left canonical action and the right action a 0 v x wa#u = ( a 0 a w 1 v 1 ( f w 2 u 1 v 2 uvw ( v 3 x w 3 u 2 where x = x 1 x s. Let us consider the diagram. 3 µ 2 d12 0 d22 0 Y 2 X02 X 12 2 µ 1 d11 0 d21 0 Y 1 X01 X 11 1 µ 0 d10 0 d20 0 Y 0 X00 X 10

10 hochschild (cohomology 605 where Y Y 1 µ X 0 Y and d 0 X X 1, are defined by s s a#v x w= 1 i af ( v 1 x 1 i #v 2 x 2 i xî w s 1 i vx i wx i µ s a 0 #v x a 1 #w = v d 0 rs a 0#v x a 1r+1 #w = v af ( x 1 i w 1 v 1 #v 2 xî x 2 1 i+j a#v x i x j xîĵ w + a 0 a v1 1 #v 2 x w a 0 a v1 1 #v 2 x a 2r+1 #w r 1 i a 0 #v x a 1i 1 a i a i+1 a i+1r+1 #w i w 2 where a 1r+1 = a 1 a r+1 and x = x 1 x s. It is immediate that the µ s s and the drs 0 s are E-bimodule maps. In the proof of Theorem we will see that the s s are also. Each horizontal complex X s is the tensor product E U A A A b A E, where E U is a right A- module via the canonical inclusion of A in E. Hence, the family σ0s 0 Y s X 0s σ 0 r+1s X rs X r+1s r 0, of left E-module maps, defined by σ 0 r+1s a 0#v x a 1r+1 #w= 1 r+1 a 0 #v x a 1r+1 1#w is a contracting homotopy of Y s µ s X0s d 0 1s X 1s d 0 2s X 2s d 0 2s X 3s d 0 3s X 4s d 0 4s r 1 Moreover each X rs is a projective relative E-bimodule. We define E-bimodule maps d l rs X rs X r+l 1s l r 0 and 1 l s recursively by σ 0 0s 1 s µ s y if r = 0 and l = 1, drs l y = l 1 j=1 σ0 l 1s l dl j j 1s j dj 0sy if r = 0 and 1 <l s, l 1 j=0 σ0 r+l 1s l dl j r+j 1s j dj rsy if r>0, where y = 1#1 x 1 x s a 1r 1#1 X rs.

11 606 guccione and guccione Theorem The complex E µ X 0 d 1 X1 d 2 X2 d 3 X3 d 4 X4 d 5 X5 d 6 X6 d 7 where µa 0 #v a 1 #w = vw a 0 a v1 1 f ( v 2 w 1 #v 3 w 2, X n = X rs and d n = r+l>0 is a relative projective resolution of the E-bimodule E. s drs l l=0 Proof. Let µ Y 0 E and Y be as in Theorem and let µ Y 0 E be the E-bimodule map defined by µa v w = af ( v 1 w 1 #v 2 w 2 vw Let ϑ Y Y be the isomorphism of E-bimodule complexes, determined by ϑ s x 1 x s =e x1 e xs. Since µ = µ ϑ 0,we obtain from Theorem that the complex of E-bimodules E µ Y 0 1 Y1 2 Y2 3 Y3 4 Y4 5 Y5 6 Y6 7 is contractible as a complex of k-modules. Hence, the result follows immediately from Theorem 2.1. The boundary maps of the relative projective resolution of E that we just found are defined recursively. Next we compute these morphisms. Theorem 3.3. d 1 rs a 0#1 x a 1r+1 #1= For x i x j, weput ˆ f ij = f x i x j f x j x i. We have s 1 i+r+1 a 0 #x i xî a 1r+1 #1 + + s 1 i+r a 0 #1 xî a 1r+1 #x i s 1 h r+1 + d 2 rs a 0#1 x a 1r+1 #1= 0 h r 1 i+r a 0 #1 xî a 1h 1 a x i h a h+1r+1#1 1 i+j+r a 0 #1 x i x j xîĵ a 1r+1#1 1 i+j+h a 0 #1 xîĵ a 1h ˆ f ij a hr+1 #1 and d l rs = 0 for all l 3, where a 1r+1 = a 1 a r+1 and x = x 1 x s.

12 hochschild (cohomology 607 Proof. To unify the expressions in the proof, we put d0s 0 = µ sd 1s 1 = s and d 1s 2 = 0. First, we compute the maps d1 rs.forr = 1 the assertion is trivial. Suppose r 0 and the result is valid for d 1 r s with 1 r <r. Since, for all r s 0, 1 then σ rs b 0 #v x b 1r 1 1#w =0 d 1 rs a 0#1 x a 1r 1#1 = σ 0 r s 1 d1 r 1s d0 rs a 0#1 x a 1r 1#1 = 1 r+1 σ 0 r s 1 d1 r 1s a 0#1 x a 1r #1 Hence, the formula for drs 1 a 0#1 x a 1r 1#1 follows immediately by induction on r. Now, let us compute drs 2. Suppose r 0 and the result is valid for d 2 r s with 0 r <r. Using (1 twice, we get d 2 rs a 0#1 x a 1r 1#1 = σ 0 r+1s 2 ( dr 1s 2 d0 rs + d1 r s 1 rs d1 a0 #1 x a 1r 1#1 ( r 1 = σ 0 r+1s 2 1 i+j+h+r+1 a 0 #1 xîĵ a 1h fˆ ij a h+1r #1 σ 0 r+1s 2 = h=0 σ 0 r+1s 2 = l=0 h=0 ( d 1 r s 1 ( s 1 j+r a 0 #1 xĵ a 1r 1#1 1#x j j=1 r 1 1 i+j+h a 0 #1 xîĵ a 1h fˆ ij a h+1r 1#1 ( 1 i+j a 0 #1 xîĵ a 1r fˆ ij #1 r 1 i+j+h a 0 #1 xîĵ a 1h fˆ ij a h+1r 1#1 To prove that drs l = 0 for all l>2, it is sufficient to check that σ 0 r+2s 4 d2 r+1s 2 d2 rs a 0#1 x a 1r 1#1 =0 σ 0 r+1s 3 d1 r+1s 2 d2 rs a 0#1 x a 1r 1#1 =0 σ 0 r+1s 3 d2 r s 1 d1 rs a 0#1 x a 1r 1#1 =0 Next, we give an explicit formula for the comparison map between X d and the canonical normalized Hochschild resolution E E E b. Using this map it is easy to obtain explicit quasi-isomorphisms from the complex obtained in the following section for the Hochschild homology into the canonical one, and similarly for the cohomology.

13 608 guccione and guccione Remark 3.4. There is a map of complexes θ X d E E E b, given by θ r+s 1 E x 1 x s a 1 a r 1 E = τ s sgτ1 E 1#x τ1 1#x τs a 1 #1 a r #1 1 E where denotes the shuffle product defined by e 1 e s e s+1 e n = σ sn s-shuffles sgσe σ1 e σn 4. THE HOCHSCHILD HOMOLOGY Let E = A# f U and M be an E-bimodule. We use Theorem in order to construct a complex X E M, simpler than the canonical one, giving the Hochschild homology of E with coefficients in M. The Complex X E M. Let r s l 0 with l min2s and r + l>0. We define the morphism d rs l M A r s M A r+l 1 s l,by r 1 d rs 0 m a 1r x =ma 1 a 2r x + 1 i m a 1i 1 a i a i+1 a i+2r x + 1 r a r m a 1r 1 x s d rs 1 m a 1r x = 1 i+r 1#x i m m1#x i a 1r xî d 2 rs m a 1r x = + s 1 h r + 0 h r 1 i+r m a 1h 1 a x i h a h+1r xî 1 i+j+r m a 1r x i x j xîĵ 1 i+j+h m a 1h ˆ f ij a h+1r xîĵ where a 1r = a 1 a r x = x 1 x s, and ˆ f ij = f x i x j f x j x i. Theorem 4.1. The Hochschild homology H A M, ofe with coefficients in M, is the homology of X E M = X 0 d 1 X 1 d 2 X 2 d 3 X 3 d 4 X 4 d 5 X 5 d 6 X 6 d 7

14 hochschild (cohomology 609 where X n = M A r s and d n = r+l>0 mins2 l=0 d rs l Proof. It follows from the fact that X E M is the complex obtained by taking the tensor product M E e X d, where X d is the complex of Theorem 3.2.1, and using the identifications ϑ rs M A τ s M E e E s A r E given by ϑ rs m a 1r x =m 1#1 x a 1r 1#1. Note that when f takes its values in k, then X E M is the total complex of the double complex M A d 0 d Stefan s Spectral Sequence Next we show that the complex X E M has a natural filtration, which gives a more explicit version of the homology spectral sequence obtained in [St.] For each x, we have the morphism x M A b M A b, defined by x r m a 1r =1#xm m1#x a 1r + 1 h r m a 1h 1 a x h a h+1r. Proposition For each x x the endomorphisms of H A M induced by x x x x and by x x coincide. Consequently H A M is a right U-module. Proof. By a standard argument it is sufficient to prove it for H 0 A M. In this case, the assertion can be easily checked, using that 1#x 1#x = f x x#1 + 1#x x for all x x. The chain complex X E M has the filtration F 0 F 1 F 2 where F i X n = M A r s r 0 0 s i Using this fact and Proposition 4.2.1, we obtain the following: Corollary There is a converging spectral sequence E 2 rs = H s H r A M H r+s E M Given an A-bimodule M we let A M denote the k-submodule of M generated by the commutators am ma a A and m M. M Corollary If A is separable, then H E M =H (. AM

15 610 guccione and guccione 4.3. Smooth Algebras Let A be a commutative ring and let M be a symmetric A-bimodule. In the famous paper [H-K-R] was proved that if A is a commutative smooth k- algebra, then H n A M =M A n A/k, where n A/k denotes the A-module of differential n-forms of A. Next, we generalize this result by computing the Hochschild homology of a differential operator ring E = A# f U with coefficients in an E-bimodule M which is symmetric as an A-bimodule, under the hypothesis that k and A is a commutative smooth k-algebra. Let us assume that k A is a commutative ring, and M is symmetric as an A-bimodule. For each r s l 0 with 1 l min2s, we define the morphism d rs l M A A/k r s M A r+l 1 s l,by A/k d rs 1 m A da 1 da r x s = 1 i+r 1#x i m m1#x i A da 1 da r xî + s 1 h r + 1 i+r m da 1 da h 1 da x i h da h+1 da r xî 1 i+j+r m A da 1 da r x i x j xîĵ d rs 2 m A da 1 da r x = 1 i+j m A dfˆ ij da 1 da r xîĵ where x = x 1 x s and ˆ f ij = f x i x j x j x i. Consider the complex where X E M = X 0 X n = d 1 X1 d 2 X2 d 3 X3 d 4 X4 M A r A/k s and d n = r+l>0 d 5 mins2 l=1 d rs l Let ϑ n X n X n be the map ϑ n m a 1r x =1/r!m A da 1 da r x. It is easy to check that ϑ X E M X E M is a morphism of complexes, which is a quasi-isomorphism when A is smooth. Hence, in this case, the Hochschild homology of E with coefficients in M is the homology of X E M. A filtrated algebra E is called quasi-commutative if its associated graded algebra is commutative. In [S] was proved that if E is a quasi-commutative

16 hochschild (cohomology 611 algebra whose associated graded algebra is a polynomial ring, then E is isomorphic to a differential polynomial ring k# f U, with a finite dimensional Lie algebra. The Hochschild homology of this type of algebras was computed in [K, Theorem 3]. Next, we generalize this result. Remark Now, we assume that acts trivially on A. Consider the symmetric algebra S = S A, endowed with the Poisson bracket defined by a x =a b =0 and x y =fx y fy x+x y a A x y. It is easy to check that X E E is isomorphic to the canonical complex S/k δ introduced by Brylinski and Koszul in [B, Ko], respectively. In fact an isomorphism S/k δ X E E is given by r+s Pda 1 da r dx 1 dx s = 1 s ηpda 1 da r x 1 x s, where P S a 1 a r A x 1 x s and η S E is the symmetrization ηay 1 y n = a n! σ n y σ1 y σn Compatibility with the Canonical Decomposition Let us assume that k Ais a commutative ring, M is symmetric as an A-bimodule, and the cocycle f takes its values in k. In [G-S] was obtained a decomposition of the canonical Hochschild complex M A b. It is easy to check that the maps d 0 and d 1 are compatible with this decomposition. Since d 2 is the zero map, we obtain a decomposition of X E M, and then a decomposition of H E M. 5. THE HOCHSCHILD COHOMOLOGY Let E = A# f U and M be an E-bimodule. Using again Theorem 3.5 we construct a complex X E M, simpler than the canonical one, giving the Hochschild cohomology of E with coefficients in M. The Complex X E M. Let r s l 0 with l min2s and r + l>0. We define the morphism d rs l Hom k A r+l 1 s l M Hom k A r s M, by r 1 d 0 rs ϕa 1r x =a 1 ϕa 2r x+ 1 i ϕa 1i 1 a i a i+1 a i+2r x + 1 r ϕa 1r 1 xa r s d 1 rs ϕa 1r x = 1 i+r ϕa 1r xî1#x i 1#x i ϕa 1r xî

17 612 guccione and guccione + s 1 h r + d 2 rs ϕa 1r x = 0 h r 1 i+r ϕa 1h 1 a x i h a h+1r xî 1 i+j+r ϕa 1r x i x j xîĵ 1 i+j+h ϕa 1h ˆ f ij a h+1r xîĵ where a 1r = a 1 a r x = x 1 x s, and fˆ ij = f x i x j f x j x i. Applying the functor Hom E e M to the complex X d of Theorem 3.2.1, and using the identifications ϑ rs Hom k A r s M Hom E ee s A r E M given by ϑ rs ϕ1#1 x a 1r 1#1 =ϕa 1r x, we obtain the complex where X E M = X 0 X n = d 1 X 1 d 2 X 2 d 3 X 3 d 4 X 4 Hom k A r s M and d n = d 5 r+l>0 mins2 l=0 d rs l Note that when f takes its values in k, then X E M is the total complex of the double complex Hom k A M d 0 d 1. Theorem 5.1. The Hochschild cohomology H E M, ofe with coefficients in M, is the homology of X E M. Proof. It is an immediate consequence of the above discussion Stefan s Spectral Sequence Next we show that the complex X E M has a natural filtration, which gives a more explicit version of the cohomology spectral sequence obtained in [St]. For each x, we have the map x Hom k A Mb Hom k A Mb defined by r x ϕa 1r = 1#xϕa 1r ϕa 1r 1#x r h=1 ϕa 1h 1 a x h a h+1r. Proposition For each x x the endomorphisms of H A M induced by x x x x and by x x coincide. Consequently H A M is a left U-module.

18 hochschild (cohomology 613 Proof. It is similar to the proof of Proposition The cochain complex X E M has the filtration F 0 F 1 where F i X n = Hom k A r s M r 0s i From this fact and Proposition 5.2.1, we obtain the following: Corollary There is a converging spectral sequence E rs 2 = Hs H r A M H r+s E M Given an A-bimodule M we let M A denote the k-submodule of M consisting of the elements m verifying am = ma for all a A. Corollary If A is separable, then H E M =H M A Smooth Algebras Let us assume that k A is a commutative ring, and M is symmetric as an A-bimodule. For each r s l 0 with 1 l min2s, we define the morphism d rs l Hom A r+l 1 A/k s l M Hom A A/k r s M, by d 1 rs ϕda 1 da r x = s 1 i+r ϕda 1 da r xî 1#x i + s 1 h r + d 2 rs ϕda 1 da r x = 1 i+r ϕda 1 d h 1 da x i h d h+1 d r xî 1 i+j+r ϕda 1 da r x i x j xîĵ 1 i+j ϕd ˆ f ij da 1 da r xîĵ where x = x 1 x s ˆ f ij = f x i x j f x j x i and ϕda 1 da r xî, 1#x i = ϕda 1 da r xî1#x i 1#x i ϕda 1 da r xî. Consider the complex where X E M = X 0 X n = d 1 X 1 d 2 X 2 d 3 X 3 d 4 X 4 Hom k r A/k s M and d n = d 5 r+l>0 mins2 l=1 d rs l

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