Lie-Rinehart and Hochschild cohomology for algebras of differential operators
aa r X i v : . [ m a t h . K T ] J un LIE–RINEHART AND HOCHSCHILD COHOMOLOGY FORALGEBRAS OF DIFFERENTIAL OPERATORS
FRANCISCO KORDON AND THIERRY LAMBRE
Abstract.
Let (
S, L ) be a Lie–Rinehart algebra such that L is S -projective and let U be its universal enveloping algebra. In this paper we present a spectral sequencewhich converges to the Hochschild cohomology of U with values on a U -bimodule M and whose second page involves the Lie–Rinehart cohomology of the algebra and theHochschild cohomology of S with values on M . After giving a convenient descriptionof the involved algebraic structures we use the spectral sequence to compute explicitlythe Hochschild cohomology of the algebra of differential operators tangent to a centralarrangement of three lines. Introduction
The goal of this article is to apply homological algebra techniques for Lie–Rinehartalgebras to a problem of algebras of differential operators. We begin by describing aspectral sequence that converges to the Hochschild cohomology of the enveloping algebraof a Lie–Rinehart algebra. After that, we focus on the algebra of differential operators
Diff A associated to a central arrangement A of three lines. This is a graded associativealgebra that is at the same time the enveloping algebra of a Lie–Rinehart algebra: anexplicit calculation with the spectral sequence allows us to compute the Hilbert seriesof its Hochschild cohomology. We conclude by giving two other examples of algebras inwhich the spectral sequence proves useful.Let k be a field of characteristic zero and let A be a central hyperplane arrangementin a finite dimensional k -vector space V . Let S be the algebra of coordinates on V andlet Q ∈ S be a defining polynomial for A . The arrangement A is free if the Lie algebraDer A = { θ ∈ Der S : θ ( Q ) ∈ QS } of derivations of S tangent to A is a free S -module.It is not known what makes an arrangement free, but this condition is neverthelesssatisfied in many important examples; for instance, it is a theorem by H. Terao in [18]that reflection arrangements over C are free. We refer to P. Orlik and H. Terao’s book [15]for a general reference of hyperplane arrangements.The algebra Diff A of differential operators tangent to an arrangement A , first consid-ered by F. J. Calderón-Moreno in [5], is the algebra of differential operators on S whichpreserve the ideal QS of S and all its powers. We are interested in the Hochschildcohomology of Diff A when A is free. Date : June 3, 2020.
The first and simplest example of a free arrangement is that of a central line arrange-ment, that is, when V = k . Let l be the number of lines of such an arrangement: for l ≥
5, the Hochschild cohomology of
Diff A has been obtained as a Gerstenhaber algebraby the first author and M. Suárez-Álvarez in [9] starting from a projective resolution of Diff A as a bimodule over itself by means of explicit calculations that exploit a gradedalgebra structure on Diff A , but the calculations performed in this situation seem impossi-ble to emulate when l = 3 or l = 4. In this paper we are able to extend, in Corollaries 5.9and 5.10, some of these results to the most complicated case, which is when l = 3: Theorem A.
Let A be a central arrangement of three lines. The Hilbert series of HH • ( Diff A ) is h HH • ( Diff A ) ( t ) = 1 + 3 t + 6 t + 4 t . The first cohomology space HH ( Diff A ) is an abelian Lie algebra of dimension three. It is to prove Theorem A that Lie–Rinehart algebras come to into play: the pair( S, Der A ) is a Lie-Rinehart algebra. Recall that a Lie–Rinehart algebra ( S, L ) consistsof a commutative algebra S and a Lie algebra L with an S -module structure that actson S by derivations and which satisfies certain compatibility conditions analogous tothose satisfied by the pair ( S, Der S ). The universal enveloping algebra U of a Lie–Rinehart algebra ( S, L ) and the Lie–Rinehart cohomology H • S ( L, N ) = Ext • U ( S, N ) arean associative algebra and a cohomology theory that generalize the usual envelopingalgebra and the Lie algebra cohomology of the Lie algebra L by taking into accountits interaction with S —see the original paper [16] by G. Rinehart or the more modernexposition [8] by J. Huebschmann.If A is free, as remarked by L. Narváez Macarro in [12, Theorem 1.3.1], the envelopingalgebra of ( S, Der A ) is isomorphic to Diff A . To compute the Hochschild cohomologyin Theorem A above we employ a strategy that gives rise to a general method to approachthis kind of computations: we construct, in Corollary 3.3, a spectral sequence converg-ing to the Hochschild cohomology H • ( U, M ) of the enveloping algebra U with values onan U -bimodule M . For this sequence we need an U -module structure on H • ( S, M ), theHochschild cohomology of S with values on M . This U -module structure is constructedusing an injective resolution of M by U -bimodules and we see in Theorem 2.8 that it canbe computed explicitly from a projective resolution of S by S -bimodules. Moreover, theaction of each α ∈ L on H • ( S, M ), computed using projectives, by the endomorphism ∇ • α given in Remark 2.5 turns out suitable for computations. Theorem B.
Let ( S, L ) be a Lie–Rinehart pair such that L is an S -projective moduleand let M be an U -bimodule. There exist a U -module structure on H • ( S, M ) and afirst-quadrant spectral sequence E • converging to H • ( U, M ) with second page E p,q = H pS ( L, H q ( S, M )) . We give two other applications of Theorem B. First, in Subsection 6.1 we computethe Hochschild cohomology of a family of subalgebras of the Weyl algebra over a fieldof characteristic zero, that is, the algebras A h generated by elements x and y satisfying IE–RINEHART AND HOCHSCHILD COHOMOLOGY 3 the relation yx − xy = h for a given h ∈ k [ x ]. These algebras have been studied by G.Benkart, S. Lopes and M. Ondrus in the series of articles that start with [2] for a field ofarbitrary characteristic and, more recently, S. Lopes and A. Solotar in [11] have describedtheir Hochschild cohomology, with special emphasis on the Lie module structure of thesecond cohomology space over the first one, also in arbitrary characteristic. Some ofthe expressions we provide were nevertheless not found before and might be of interest.Second, in Subsection 6.2 we recover in a more direct and clear way a result by thesecond author and P. Le Meur in [10] that states that the enveloping algebra U of a Lie–Rinehart algebra ( S, L ) has Van den Bergh duality in dimension n + d if S has Van denBergh duality in dimension n and L is finitely generated and projective with constantrank d .Let us outline the organization of this article. In Section 1 we recall the definition ofLie–Rinehart pairs, their universal enveloping algebras and their cohomology theory. InSections 2 and 3 we describe the module structure on H • ( S, M ) and present the spectralsequence. After proving some useful lemmas regarding eulerian modules in Section 4we devote Section 5 to the computation of the Hochschild cohomology of the algebra ofdifferential operators of a central arrangement of three lines. Finally, in Section 6 weprovide the two other applications described above.We will denote the tensor product over the base field k simply by ⊗ or, sometimes,by | . Unless it is otherwise specified, all vector spaces and algebras will be over k . Givenan associative algebra A , the enveloping algebra A e is the vector space A ⊗ A endowedwith the product · defined by a ⊗ a · b ⊗ b = a b ⊗ b a , so that the category of A e -modules is equivalent to that of A -bimodules. The Hochschild cohomology of A withvalues on an A e -module M is defined as Ext • A e ( A, M ) and will be denoted by H • ( A, M )or, if M = A , by HH • ( A ). The book [22] by C. Weibel may serve as general referenceon this subject.The first author heartfully thanks his PhD advisor M. Suárez-Álvarez for his col-laboration, fruitful suggestions and overall help. We thank the Université ClermontAuvergne for hosting the first author in a postdoctoral position at the Laboratoirede Mathématiques Blaise Pascal during the year 2019-2020. Part of this work wasdone during the time the first author was supported by a full doctoral grant by CON-ICET and by the projects PIP-CONICET 12-20150100483, PICT 2015-0366 and UBA-CyT 20020170100613BA. 1. Lie–Rinehart algebras
We begin by recalling some basic facts about Lie-Rinehart algebras available in [16]and in [8]. Until Section 3 we assume k to be a field of arbitrary characteristic. IE–RINEHART AND HOCHSCHILD COHOMOLOGY 4
Definition 1.1.
Let S and ( L, [ − , − ]) be a commutative and a Lie algebra endowedwith a morphism of Lie algebras L → Der k ( S ) that we write α α S and a left S -module structure on L which we simply denote by juxtaposition. The pair ( S, L ) is a Lie–Rinehart algebra if the equalities ( sα ) S ( t ) = sα S ( t ) , [ α, sβ ] = s [ α, β ] + α S ( s ) β hold whenever s, t ∈ S and α, β ∈ L . Definition 1.2.
Let ( S, L ) be a Lie-Rinehart algebra. A Lie–Rinehart module —or ( S, L ) -module— is a vector space M that is at the same time an S -module and an L -Liemodule in such a way that ( sα ) · m = s · ( α · m ) , α · ( s · m ) = ( sα ) · m + α S ( s ) · m (1) for s ∈ S , α ∈ L and m ∈ M . Theorem 1.3.
Let ( S, L ) be a Lie-Rinehart algebra. ( i ) There exists an associative algebra U = U ( S, L ) , the universal enveloping algebraof ( S, L ) , endowed with a morphism of algebras i : S → U and a morphism ofLie algebras j : L → U that satisfies, for s ∈ S and α ∈ L , i ( s ) j ( α ) = j ( sα ) , j ( α ) i ( s ) − i ( s ) j ( α ) = i ( α S ( s )) (2) and universal with these properties. ( ii ) The category of U -modules is isomorphic to that of ( S, L ) -modules.Example . The obvious actions of S and L make of S an U -module. If g is a Liealgebra then ( k , g ) is a Lie–Rinehart algebra whose enveloping algebra is simply theusual enveloping algebra of g . If S = k [ x , . . . , x n ] then the full Lie algebra of derivations L = Der k S is a Lie–Rinehart algebra and its enveloping algebra is isomorphic to thealgebra of differential operators Diff ( S ) = A n , the n th Weyl algebra. Definition 1.5.
Let ( S, L ) be a Lie–Rinehart algebra with enveloping algebra U and let N be an U -module. The Lie–Rinehart cohomology of (
S, L ) with values on N is H • S ( L, N ) := Ext • U ( S, N ) . In many important situations, some of which will be illustrated in the examples below, L is a projective S -module, and in this case there is a well-known complex that computesthe Lie–Rinehart cohomology. Proposition 1.6.
Suppose that L is S -projective and let Λ • S L denote the exterior al-gebra of L over S . The complex Hom S (Λ • S L, N ) with Chevalley–Eilenberg differentialscomputes H • S ( L, N ) .Example . For the Lie–Rinehart algebra ( k , g ) with g a Lie algebra, N is simply a g -Lie module and the complex Hom k (Λ • k L, N ) is the standard complex that computesthe Lie algebra cohomology H • ( g , N ). IE–RINEHART AND HOCHSCHILD COHOMOLOGY 5
Given a finite dimensional manifold M , we obtain a Lie–Rinehart algebra setting S = C ∞ ( M ), the algebra of smooth functions, and L = X ( M ), the Lie algebra ofvector fields on M . The enveloping algebra of this pair is isomorphic to the algebra ofglobally defined differential operators on the manifold —see [8, §1]. We can find in J.Nestruev’s [13, Proposition 11.32] that L is finitely generated and projective over S ; asthe complex Hom S (Λ • S L, S ) is the de Rham complex Ω • ( M ) of differential forms, thecohomology H • S ( L, S ) coincides with the de Rham cohomology of M . Example . A central hyperplane arrangement A in a finite dimensional vector space V is a finite set { H , . . . , H l } of subspaces of codimension 1. Let λ i : V → k be a linear formwith kernel H i for each i ∈ { , . . . , l } . We let S be the algebra of polynomial functionson V , fix a defining polynomial Q = λ · · · λ l ∈ S for A and consider the Lie algebraDer A := { θ ∈ Der k ( S ) : Q divides θ ( Q ) } of derivations tangent to the arrangement. The pair ( S, Der A ) is a Lie–Rinehart algebra,as one can readily check.An arrangement A is free , by definition, if Der A is a free S -module. In that case,as in [12, Theorem 1.3.1], the enveloping algebra of ( S, Der A ) is isomorphic to the algebra of differential operators tangent to the arrangement Diff A , that is, the algebra ofdifferential operators on S which preserve the ideal QS of S and all its powers. As seenin [5] or by M. Suárez-Álvarez in [19], it coincides with the associative algebra generatedinside the algebra End k ( S ) of linear endomorphisms of the vector space S by Der A andthe set of maps given by left multiplication by elements of S .For the Lie–Rinehart algebra ( S, L ) associated to a free hyperplane arrangement A ,the complex Hom S (Λ • S L, S ) is the complex of logarithmic forms Ω • ( A ), and its cohomol-ogy is isomorphic to the Orlik–Solomon algebra of A —here we refer to J. Wiens andS. Yuzvinsky’s [23]. When k = C , this algebra is, in turn, isomorphic to the cohomologyof the complement of the arrangement, as proved by P. Orlik and L. Solomon in [14].2. The U -module structure on H • ( S, M )Let (
S, L ) be a Lie–Rinehart algebra such that L is a projective S -module. Let U be its enveloping algebra and M be an U e -module. Since the inclusion of S in U is amorphism of algebras we can regard M as an S e -module and consider the Hochschildcohomology of S with values on M , denoted as before by H • ( S, M ). In this section wefirst construct an U -module structure on H • ( S, M ) from an U e -injective resolution of M ;afterwards, we construct S - and L -module structures on H • ( S, M ) from an S e -projectiveresolution of S ; finally, we show that these induce an U -module structure that coincideswith the one we have using injectives: this will allow us to compute the latter in practice.2.1. Using U e -injective modules. The second author and P. Le Meur introduce in[10, Lemma 3.2.1] the functor G = Hom S e ( S, − ) : U e Mod → U Mod , (3) IE–RINEHART AND HOCHSCHILD COHOMOLOGY 6 where for an U e -module M the left L -Lie module and left S -module structures onHom S e ( S, M ) are defined by the rules( α · ϕ )( s ) = ( α ⊗ · ϕ ( s ) − (1 ⊗ α ) · ϕ ( s ) − ϕ ( α S ( s )) , ( t · ϕ )( s ) = ( t ⊗ · ϕ ( s ) (4)for α ∈ L , ϕ ∈ Hom S e ( S, M ) and s, t ∈ S . Proposition 2.1.
Let M be an U e -module and let M → I • be an injective resolutionof M as an U e -module. The cohomology of the complex G ( I • ) = Hom S e ( S, I • ) is theHochschild cohomology H • ( S, M ) .Proof. Let I be an injective U e -module. The functor Hom S e ( − , I ) is naturally isomor-phic to Hom U e ( U e ⊗ S e − , I ), which is the composition of the exact functor Hom U e ( − , I )and U e ⊗ S e − . Now, the PBW-theorem in [16, §3] ensures that U is a projective S -module and, using Proposition IX.2.3 of H. Cartan and S. Eilenberg’s [6] , we obtainthat U e is S e -projective. As a consequence of this, the functor U e ⊗ S e − is exact andtherefore Hom S e ( − , I ) is exact as well. This implies that M → I • is in fact a resolutionof M by S e -injective modules, so that H • (Hom S e ( S, I • )) = Ext S e ( S, M ). (cid:3) From Proposition 2.1 and the functoriality of G = Hom S e ( S, − ) we can conclude thatif M → I • is an U e -injective resolution then the U -module structure on Hom S e ( S, I • )defined in (4) induces an U -module structure on H • ( S, M ): Corollary 2.2.
Let M be an U e -module and let M → I • be an U e -injective resolution.Let j ≥ , u ∈ U and denote the class in H j ( S, M ) of ϕ ∈ Hom S e ( S, I j ) by ¯ ϕ . Defining u · ¯ ϕ to be the class of u · ϕ as defined in (4) we obtain an U -module structure on H j ( S, M ) . Using S e -projective modules. In this subsection we define S - and L -modulestructures on H • ( S, M ) using projectives. To see that these structures are compatibleas in (1) we will show that the are equal to the ones in Subsection 2.1 using injectivesand conclude that they determine an U -module structure.2.2.1. The S -module structure. We start by letting P • → S be an S e -projective resolu-tion. For each i ≥ there is a left S -module structure on Hom S e ( P i , S ) given by ( s · φ )( p ) = sφ ( p ) for s ∈ S , φ ∈ Hom S e ( P i , S ) and p ∈ P i . (5)With this structure the differentials in the complex Hom S e ( P • , S ) become S -linear andtherefore the cohomology of this complex, which is canonically isomorphic to H • ( S, M ) ,inherits an S -module structure. It is straightforward to verify that this structure doesnot depend on the choice of the projective resolution. IE–RINEHART AND HOCHSCHILD COHOMOLOGY 7 δ -liftings. To give an L -Lie module structure on H • ( S, M ) using projectives wewill use the tools developed by M. Suárez-Álvarez in [20]. Let A be an algebra and δ : A → A a derivation. Given an A -module V , we say that a linear map f : V → V isa δ -operator if for every a ∈ A and v ∈ V we have f ( av ) = δ ( a ) v + af ( v ) . If, moreover, ε : P • → V is an A -projective resolution of V , a δ -lifting of f to P • isa family of δ -operators f • = ( f i : P i → P i , i ≥ such that the following diagramcommutes: · · · P P V · · · P P V f f f The construction in [20, §1] proceeds then as follows. Given an algebra A with aderivation δ , a δ -operator f : V → V and a projective resolution P • → V , a δ -lifting f • of f to P • is shown to always exist. This δ -lifting gives rise to an endomorphism f ♯ • of thecomplex Hom A ( P • , V ) defined for i ≥ and ϕ ∈ Hom A ( P i , V ) by f ♯i ( ϕ ) = f ◦ ϕ − ϕ ◦ f i .Moreover, f ♯ • induces an endomorphism ∇ • f of the cohomology Ext • A ( V, V ) which, conve-niently, does not depend neither on the choice of the δ -lifting or the projective resolution.We will now generalize this construction so that we can adapt it to our needs. Let usfirst recall two simple but fundamental results in the following Lemma. Lemma 2.3 ([20, §1.4,§1.6]) . Let V be a left A -module, let f : V → V be a δ -operatorand let ε : P • → V be a projective resolution. ( i ) There exists a δ -lifting f • of f to P • . ( ii ) If ε ′ : P ′• → V is another projective resolution, f • and f ′• are δ -liftings of f to ε and ε ′ and h • : P ′• → P • is an A -linear lifting of id V : V → V then f • h • − h • f ′• : P ′• → P • is an A -linear lifting of the zero map V → V . Proposition 2.4.
Let V and W be two A -modules, f : V → V and g : W → W two δ -operators and P • → V an A -projective resolution. Let f • = ( f i ) i ≥ be a δ -lifting of f to P • provided by Proposition 2.3. ( i ) There is an endomorphism ( f • , g ) = (( f i , g )) i ≥ of the complex of vector spaces Hom A ( P • , W ) such that if i ≥ and φ ∈ Hom A ( P i , W ) then ( f i , g )( φ ) = g ◦ φ − φ ◦ f i . (6)( ii ) The map ∇ • ( f,g ) : Ext • A ( V, W ) → Ext • A ( V, W ) induced by ( f • , g ) in cohomology isindependent of the choice of the projective resolution P • → S and the δ -lifting f • .Proof. Let i ≥ . As both g : W → W and f i : P i → P i are δ -operators and φ is A -linear, the difference ( f i , g )( φ ) = g ◦ φ − φ ◦ f i is A -linear. That ( f • , g ) is a morphismof complexes is an immediate consequence of the fact that so is f • . IE–RINEHART AND HOCHSCHILD COHOMOLOGY 8
For the second assertion we let ε ′ : P ′• → V be another A -projective resolution of V , f ′• be another δ -lifting of f to P • and ( f ′• , g ) be the graded endomorphism of Hom A ( P ′• , W ) in (6). We claim that if h : P ′• → P • is a morphism of complexes lifting the identity of S then the diagram Hom S e ( P • , W ) Hom S e ( P • , W )Hom S e ( P ′• , W ) Hom S e ( P ′• , W ) ( f • ,g ) h ∗• h ∗• ( f ′• ,g ) (7)commutes up to homotopy.Proposition 2.3 tells us that z • := f • h • − h • f ′• : P ′• → P • is an A -linear lifting of V → V and therefore z ∗• : Hom A ( P • , W ) → Hom A ( P ′• , W ) is homotopic to zero. Toprove the claim it is then enough to show that ( f ′ i , g ) ◦ h ∗ i − h ∗ i ◦ ( f i , g ) = z ∗ i for each i ≥ ,so that the zero-homotopic map z ∗• is the failure in the commutativity of the diagram (7).We have, for φ ∈ Hom S e ( P i , W ) , (cid:0) ( f ′ i , g ) ◦ h ∗ i − h ∗ i ◦ ( f i , g ) (cid:1) ( φ ) = ( f ′ i , g )( φ ◦ h i ) − h ∗ i (( f i , g )( φ ))= g ◦ ( φ ◦ h i ) − ( φ ◦ h i ) ◦ f ′ i − ( g ◦ φ ) ◦ h i + ( φ ◦ f i ) ◦ h i = φ ◦ f i ◦ h i − φ ◦ h i ◦ f ′ i = ( h ∗ i f ∗ i − f ′∗ i h ∗ i )( φ ) = z ∗ i ( φ ) . This proves the claim, and it follows at once that the endomorphisms that ( f • , g ) and ( f ′• , g ) induce on Ext • A ( V, W ) are equal. (cid:3) The L -Lie module structure. Let ( S, L ) be a Lie–Rinehart algebra, M be an U e -module and α ∈ L . To adapt the construction of Subsection 2.2.2 to our situationwe recall that α acts on S by the derivation α S : S → S and consider the followingassertions. (i) The map α eS = α S ⊗ ⊗ α S : S e → S e is a derivation. (ii) Viewing S as an S e -module via ( s ⊗ s ) · t := s ts , the derivation α S : S → S becomes an α eS -operator. (iii) The map α M : M → M such that α M ( m ) = ( α ⊗ · m − (1 ⊗ α ) · m satisfies α M (( s ⊗ t ) · m ) = α eS ( s ⊗ t ) · m + ( s ⊗ t ) · α M ( m ) for s, t ∈ S and m ∈ M , which is to say that, regarding M as an S e -module, α M is an α eS -operator.The first two claims can be proved with a straightforward calculation; for the thirdone, we let α , s , t and m as before and see that α M (( s ⊗ t ) · m ) = (( α ⊗ − ⊗ α )( s ⊗ t )) · m = ( αs ⊗ t − s ⊗ tα ) · m = (( α ( s ) + sα ) ⊗ t − s ⊗ ( αt − α ( t ))) · m IE–RINEHART AND HOCHSCHILD COHOMOLOGY 9 = α e ( s ⊗ t ) · m + ( sα ⊗ t − s ⊗ αt ) · m = α e ( s ⊗ t ) · m + s ⊗ t · α M ( m ) since α S ( s ) = sα − αs , as in (2).We may now specialize Proposition 2.4 to our situation. We take A = S e , δ = α eS : S e → S e , V = S,f = α S : S → S, W = M, g = α M : M → M and from this we obtain the maps α ♯ • := ( f ♯ • , g ) and ∇ • α := ∇ • ( f,g ) . More concretely: Remark . Let α ∈ L , M an U e -module and ε : P • → S an S e -projective resolu-tion. Let α • be an α eS -lifting of α S : S → S to P • , that is, a morphism of complexes α • = ( α q : P q → P q ) q ≥ such that ε ◦ α = α S ◦ ε and for each q ≥ , s , t ∈ S and p ∈ P q α q (( s ⊗ t ) · p ) = ( α S ( s ) ⊗ t + s ⊗ α S ( t )) · p + ( s ⊗ t ) · p. Denote by α ⊗ − ⊗ α : M → M the map such that m ( α ⊗ − ⊗ α ) · m . Theendomorphism α ♯ • of Hom S e ( P • , M ) is given for each q ≥ by α ♯q ( φ ) = ( α ⊗ − ⊗ α ) ◦ φ − φ ◦ α q , (8)and the map ∇ • α : H • ( S, M ) → H • ( S, M ) is the unique graded endomorphism such that ∇ qα ([ φ ]) = [ α ♯q ( φ )] , (9)where [-] denotes class in cohomology. Proposition 2.6.
Let
End ( H • ( S, M )) be the Lie algebra of linear endomorphisms of H • ( S, M ) with Lie structure given by the commutator. The map ∇ : L → End ( H • ( S, M )) defined by α
7→ ∇ • α is a morphism of Lie algebras.Proof. Let α, β ∈ L and call γ = [ α, β ] . Let α • , β • and γ • be α e , β e and γ eS -liftings,respectively. Observe that γ • is not necessarily the commutator of α • and β • . Let α ♯ • , β ♯ • and γ ♯ • be the endomorphisms of Hom S e ( P • , M ) defined as in (8) and consider theendomorphism θ • of Hom S e ( P • , M ) such that if i ≥ and φ ∈ Hom S e ( P i , M ) θ i ( φ ) = ( γ ⊗ − ⊗ γ ) ◦ φ − φ ◦ ( α i ◦ β i − β i ◦ α i ) . A straightforward calculation shows that the commutator α • ◦ β • − β • ◦ α • is a γ eS -liftingof γ and therefore Proposition 2.4 tells us that θ • and γ ♯ • induce the same endomorphismon cohomology. We claim that in fact θ • = α ♯ • ◦ β ♯ • − β ♯ • ◦ α ♯ • . Indeed, for i ≥ and φ ∈ Hom S e ( P i , M ) α ♯i ( β ♯i ( φ )) = ( α ⊗ − ⊗ α ) ◦ β ♯i ( φ ) − β ♯i ( φ ) ◦ α i = ( α ⊗ − ⊗ α ) ◦ (( β ⊗ − ⊗ β ) ◦ φ − φ ◦ β i ) − ( β ⊗ − ⊗ β ) ◦ φ ◦ α i − φ ◦ β i ◦ α i = ( αβ ⊗ − α ⊗ β − β ⊗ α + α ⊗ β ) ◦ φ − ( α ⊗ − ⊗ α ) ◦ φ ◦ α i − ( β ⊗ − ⊗ β ) ◦ φ ◦ α i − φ ◦ β i ◦ α i IE–RINEHART AND HOCHSCHILD COHOMOLOGY 10
These two expressions together with the equality αβ − βα = γ in U allow us to concludethat α ♯i ( β ♯i ( φ )) − β ♯i ( α ♯i ( φ )) = θ i ( φ ) , which proves the claim.We conclude in this way that H • ( γ ♯ • ) = H • ( θ • ) = H • ( α ♯ • ◦ β ♯ • − β ♯ • ◦ α ♯ • )= H • ( α ♯ • ) ◦ H • ( β ♯ • ) − H • ( β ♯ • ) ◦ H • ( α ♯ • ) , in virtue of the linearity of the functor H . This means that ∇ • γ = [ ∇ • α , ∇ • β ] . (cid:3) Example . It is easy to describe the endomorphism ∇ α of H ( S, U ) for a given α ∈ L .Let us choose a resolution P • of S with P = S e and augmentation ε : S e → S definedby ε ( s ⊗ t ) = st . As α eS is a α eS -operator and ε ◦ α eS = α S ◦ ε , we may choose an α eS -liftingwith α = α eS . According to the rule (8) we have α ♯ ( φ )(1 ⊗
1) = ( α ⊗ − ⊗ α ) · φ (1 ⊗ for all φ ∈ Hom S e ( P , M ) . (10)Identifying, as usual, each φ ∈ Hom S e ( S e , U ) with φ (1 ⊗ ∈ U , we can view H ( S, U ) as a subspace of U and then (10) tells us that ∇ α ( u ) = αu − uα for all u ∈ H ( S, U ) .2.3. Comparing the two actions.
We now prove that the S - and L -module structureson H • ( S, M ) constructed in Subsection 2.2 using projectives are equal to those inducedby the U -module structure in Subsection 2.1 using injectives. As a consequence, thisshows that the actions of S and L using projectives satisfy compatibility relations (1). Theorem 2.8.
Suppose L is S -projective. The S - and L -module structures on H • ( S, M ) determined by (4) using U e -injective modules are equal to those given in (5) and (9) using S e -projective modules.Proof. We will only prove that the L -module structures coincide —that the S -modulestructures are equal too is analogous and simpler. To begin with, we fix an U e -injectiveresolution η : M → I • , an S e -projective resolution ε : P • → S and α ∈ L . In (8), wegive endomorphisms of complexes α ♯ • of Hom S e ( P • , M ) and of Hom S e ( P • , I j ) for each j ≥ —we denote them the same way— which induce the map ∇ • α on their cohomologies H • ( S, M ) and H • ( S, I j ) . We first claim that the map η ∗ : Hom S e ( P • , M ) ∋ φ η ◦ φ ∈ Hom S e ( P • , I • ) satisfies, for each i ≥ and φ ∈ Hom S e ( P i , M ) , η ∗ ( α ♯i ( φ )) = α ♯i ( η ∗ ( φ )) . (11)Indeed, since η is a morphism of U e -modules it commutes with ⊗ α − α ⊗ and thus η ∗ ( α ♯i ( φ )) = η ◦ ( α ⊗ − ⊗ α ) ◦ φ − η ◦ φ ◦ α i = ( α ⊗ − ⊗ α ) ◦ η ◦ φ − η ◦ φ ◦ α i = α ♯i ( η ∗ ( φ )) . Let us see that, on the other hand, the map ε ∗ : Hom S e ( S, I • ) ∋ ϕ ϕ ◦ ε ∈ Hom S e ( P • , I • ) IE–RINEHART AND HOCHSCHILD COHOMOLOGY 11 satisfies that for each ϕ ∈ Hom S e ( S, I • ) ε ∗ ( α · ϕ ) = α ♯ ( ε ∗ ( ϕ )) . (12)Since α • is a lifting of α S : S → S to P • , we have that α ◦ ε = ε ◦ α and ε ∗ ( α · ϕ ) = ( α ⊗ − ⊗ α ) ◦ ϕ ◦ ε − ϕ ◦ α ◦ ε = ( α ⊗ − ⊗ α ) ◦ ϕ ◦ ε − ϕ ◦ ε ◦ α = α ♯ ( ε ∗ ( ϕ )) . As the morphisms of complexes ε ∗ and η ∗ are quasi-isomorphisms, the fact that they areequivariant with respect to the actions of α —as shown by (11) and (12)— allows us toconclude that the two actions of L on H • ( S, M ) coincide. (cid:3) The spectral sequence
Let ( S, L ) be a Lie–Rinehart algebra, let U be its enveloping algebra and let M bean U e -module. In this section we construct a spectral sequence which converges to theHochschild cohomology of U with values on M and whose second page involves the Lie–Rinehart cohomology of ( S, L ) and the Hochschild cohomology of S with values on M .Recall that in (3) we considered a functor G : U e Mod → U Mod defined on objects as G ( M ) = Hom S e ( S, M ) . We now consider the functor F : U Mod → U e Mod F ( N ) = U ⊗ S N (13)where we give to U ⊗ S N the U e -module structure in [7, (2.4)]. This structure is com-pletely determined by the rules ( v ⊗ · u ⊗ S n = vu ⊗ S n, (1 ⊗ α ) · u ⊗ S n = uα ⊗ S n − u ⊗ S α · n, (1 ⊗ s ) · u ⊗ S n = uα ⊗ S s · n for u, v ∈ U , n ∈ N and α ∈ L . With the functors G and F at hand, we can state thevery useful Proposition 3.4.1 of [10]. Proposition 3.1.
The functor F is left adjoint to G . Theorem 3.2.
Assume L is S -projective and let N and M be a left U -module and an U e -module. There is a first-quadrant spectral sequence E • converging to Ext • U e ( F ( N ) , M ) with second page E p,q = Ext pU ( N, H q ( S, M )) . Proof.
Let Q • → N be an U -projective resolution of N and let M → I • be an U e -injective resolution. Consider the double complex X • , • = Hom U ( Q • , G ( I • )) IE–RINEHART AND HOCHSCHILD COHOMOLOGY 12 and denote its total complex by Z • . There are two spectral sequences for this doublecomplex: we will use the first one to compute H • ( Z ) and the second one will be the onewe are looking for. From the first filtration on Z • with ˜ F q Z p = M r + s = ps ≥ q X r,s we obtain a first spectral sequence converging to H ( Z • ) . Its zeroth page ˜ E is ˜ E p,q = Hom U ( Q p , G ( I q )) and its differential comes from the one on Q • . We claim that for each s ≥ , the functor Hom U ( − , G ( I s )) is exact. Indeed, by the adjunction of Proposition 3.1 it is naturallyisomorphic to Hom U e ( F ( − ) , I s ) , which is the composition of the functors F = U ⊗ S ( − ) and Hom U e ( − , I s ) and these are exact because U is left projective over S and I s is U e -injective. The first page ˜ E of the spectral sequence is therefore given by ˜ E p,q = Hom U ( N, G ( I q )) ∼ = Hom U e ( F ( N ) , I q ) if p = 0 ; if p = 0 and its differential is induced by that of I • . Now, as the complex Hom U e ( F ( N ) , I • ) computes Ext • U e ( F ( N ) , M ) using injectives, we obtain that the second page is ˜ E p,q = Ext qU e ( F ( N ) , M ) if p = 0 ; if p = 0 .This spectral sequence thus degenerates at its the second page, so that we see that H • ( Z ) is isomorphic to Ext • U e ( F ( N ) , M ) .The second filtration on Z • is given by F p Z q = M r + s = qr ≥ p X r,s and determines a second spectral sequence E • that also converges to H ( Z • ) . Its differ-ential on E is induced by the one on I • ; as Q p is U -projective for each p ≥ , the cohomol-ogy of Hom U ( Q p , G ( I • )) is given in its q th place precisely by E p,q = Hom U ( Q p , H q ( S, M )) —recall that, according to Proposition 2.1, the cohomology of G ( I • ) is H • ( S, M ) . Sincethe differentials in E are induced by those of Q • , for each q ≥ the cohomology of therow E • ,q is E p,q = Ext pU ( N, H q ( S, M )) . The spectral sequence E • is therefore the onewe were looking for. (cid:3) Specializing Theorem 3.2 to the case in which N = S we obtain the following corollary,which is in fact the result we are mainly interested in. Corollary 3.3. If L is S -projective then for each U e -module M there is a first-quadrantspectral sequence E • converging to H • ( U, M ) with second page E p,q = H pS ( L, H q ( S, M )) . IE–RINEHART AND HOCHSCHILD COHOMOLOGY 13
The following examples illustrate what happens when we take M = U in the twoextreme situations. Example . Suppose first that L = 0 . The enveloping algebra U is just S and Λ • S L = S ,so the resolution U ⊗ Λ • S L of S is simply Q • = U ⊗ S S . The double complex X • , • is there-fore Hom S ( S, Hom S e ( S, I • )) , which is isomorphic to Hom S e ( S, I • ) and the cohomologyof the complex Z • in the proof is HH • ( S ) , the Hochschild cohomology of S . Example . If S = k and L = g is a Lie algebra then H • ( S, U ) = Ext • k e ( k , U ) is just U ,the second page of our spectral sequence is H • ( g , U ) and we recover from Corollary 3.3the well-known fact that the Hochschild cohomology of the enveloping algebra of aLie algebra equals its Lie cohomology with values on U with the adjoint action, asin [6, XIII.5.1]. 4. Eulerian modules
We assume from now on that k is a field of characteristic zero. In this section wepay attention to a particular but rather frequent situation in which some calculations toattain the second page of the spectral sequence in Corollary 3.3 can be significantlyshortened. Let S = k [ x , . . . , x n ] . The usual graded algebra structure on S , suchthat | x i | = 1 if ≤ i ≤ n , induces a grading on the Lie algebra Der S that makeseach partial derivative ∂ i have degree − . Let L be a Lie subalgebra of Der S that isalso an S -submodule of Der S freely generated by homogeneous derivations α , . . . , α l ,where α = e = x ∂ + · · · + x n ∂ n is the eulerian derivation. The pair ( S, L ) isa Lie-Rinehart algebra and, since L is free, its enveloping algebra U admits the set { α n . . . α n l l : n , . . . , n l ≥ } as an S -module basis of U thanks to the PBW-theoremin [16, §3]. The graded structures on S and Der S induce on L and on U a graded Liealgebra and a graded associative algebra structures. Definition 4.1. A Z -graded left U -module N = L i ∈ Z N i is eulerian if the action of e on N satisfies e · n = in if n ∈ N i . The Lie–Rinehart cohomology H S ( L, N ) . Recall from Proposition 1.6 that theLie-Rinehart cohomology of ( S, L ) with values on an U -module N is the cohomology ofthe complex C • S ( L, N ) = Hom S (Λ • S L, N ) with differentials d r : C rS ( L, N ) → C r +1 S ( L, N ) determined by ( d r f )( α i ∧ · · · ∧ α i r +1 ) = r +1 X j =1 ( − j +1 α i j · f ( α i ∧ · · · ∧ ˇ α i j ∧ · · · ∧ α i r +1 )+ X ≤ j Let N be an eulerian U -module. The inclusion of the component ofdegree zero C • S ( L, N ) ֒ → C • S ( L, N ) is a quasi-isomorphism.Proof. Let γ • : C • S ( L, N ) → C • S ( L, N ) be the linear map whose restriction to each homo-geneous component of the complex C • S ( L, N ) is the multiplication by degree. A straight-forward calculation shows that the homotopy s = ( s r : C rS ( L, N ) → C r − S ( L, N )) r ≥ given by ( s r f )( α i ∧ · · · ∧ α i r ) = f ( e ∧ α i ∧ · · · ∧ α i r ) satisfies s ◦ d + d ◦ s = γ . Weobtain from this that γ induces the zero map in cohomology and then, as the fieldhas characteristic zero, each of the cohomologies of the subcomplexes of nonzero degreeare trivial. (cid:3) Corollary 4.3. If N is an eulerian U -module then the subspace T i ≥ ker( α i : N → N ) of N is isomorphic to H S ( L, N ) . The Hochschild cohomology H • ( S, M ) . To compute the Hochschild cohomologyof S we use the Koszul resolution of S available in [22, §4.5]. Lemma 4.4. Let W be the subspace of S with basis ( x , . . . , x n ) . The complex P • = S e ⊗ Λ • W with differentials b • : P • → P •− defined for s, t ∈ S and ≤ i < · · · < i r ≤ n by b r ( s | t ⊗ x i ∧ · · · ∧ x i r ) = r X j =1 ( − j +1 ( sx i j | t − s | x i j t ) ⊗ x i ∧ · · · ∧ ˇ x i j ∧ · · · ∧ x i r and augmentation ε : S e → S given by ε ( s | t ) = st is a resolution of S by free S e -modules.Here the symbol | denotes the tensor product inside S e and ˇ x i j means that x i j is omitted. If M is an graded U e -module M , the cohomology of the complex Hom S e ( P • , M ) is H • ( S, M ) . The graded algebra S induces a grading on this complex which is preservedby the differentials and therefore H • ( S, M ) inherits a graded structure. We denote by H • ( S, M ) i the i th homogeneous component of H • ( S, M ) for each i ∈ Z . Proposition 4.5. If M = L i ∈ Z M i is a graded U e -module such that ( e ⊗ − ⊗ e ) · m = im for all m ∈ M i then for each q ∈ Z the q th Hochschild cohomology space H q ( S, M ) is aneulerian U -module.Proof. That H q ( S, U ) i is a graded U -module for each i can be seen from (4). Follow-ing Remark 2.5 we denote by e S : S → S the action of e on S and by e eS the derivation e S ⊗ ⊗ e S : S e → S e . We let, for q ≥ , e q : P q → P q be the e eS -operator such that e q (1 | ⊗ x i ∧ · · · ∧ x i q ) = q | ⊗ x i ∧ · · · ∧ x i q (14) IE–RINEHART AND HOCHSCHILD COHOMOLOGY 15 if ≤ i < · · · < i q ≤ n . A small calculation allows us to deduce from (14) that thecollection of maps ( e q ) q ≥ is a e eS -lifting of e S to P • . Let now φ ∈ Hom S e ( P q , M ) be anhomogeneous map of degree i and write m i ,...,i q := φ (1 | ⊗ x i ∧ . . . x i q ) ∈ M i + q . Ourhypothesis on M allows us to see that e ♯q ( φ )(1 | ⊗ x i ∧ · · · ∧ x i q )= ( e ⊗ − ⊗ e ) · m i ,...,i q − φ ◦ e q (1 | ⊗ x i ∧ · · · ∧ x i q )= ( i + q ) m i ,...,i q − qm i ,...,i q = im i ,...,i q and therefore ∇ qe ([ φ ]) = i [ φ ] . (cid:3) The algebra of differential operators tangent to a centralarrangement of three lines In this section we describe the example that motivated us to construct the spectralsequence of Corollary 3.3: it is the algebra of differential operators Diff A tangent to acentral arrangement of lines A , whose Hochschild cohomology was studied by the firstauthor and M. Suárez-Álvarez in [9]. We will regard Diff A as the enveloping algebra ofa Lie–Rinehart algebra and compute the second page E p,q = H pS ( L, H q ( S, U )) of thespectral sequence of Corollary 3.3 for a central line arrangement of three lines. Afterstudying the Lie-Rinehart cohomology in a generic situation, we will compute whatwe need of H • ( S, U ) and the action of U to obtain the second page and, finally, theHochschild cohomology HH • ( U ) in Corollary 5.9.Let S = k [ x, y ] and write the defining polynomial of the arrangement Q = xF with F = y ( tx + y ) , for some t ∈ k . H. Saito’s criterion [17, Theorem 1.8.ii] allows us to seethat the two derivations E = x∂ x + y∂ y , D = F ∂ y form an S -basis of Der A . In [9] there is a convenient presentation of U = Diff A . It isgenerated by the symbols x , y , D and E subject to the relations [ y, x ] = 0 , [ D, x ] = 0 , [ D, y ] = F, [ E, x ] = x, [ E, y ] = y, [ E, D ] = D, where the bracket [ a, b ] between two elements stands for the commutator ab − ba . More-over, the set { x i y i D i E i : i , . . . , i ≥ } is a basis of U as a vector space.As in Section 4, we view S as a graded algebra, with both x and y of degree , andfor each i ≥ we write S i the homogeneous component of S of degree i . This gradinginduces one in L := Der A and also on U : Proposition 5.1. There is a grading on the algebra U with | x | = | y | = | D | = 1 and | E | = 0 . Given i ≥ the i th homogeneous component U i of U is the right k [ E ] -modulegenerated by the set { x r y s D t : r + s + t = i } . IE–RINEHART AND HOCHSCHILD COHOMOLOGY 16 For convenience, we denote by ψ ′ the image of ψ ∈ k [ E ] under the linear map k [ E ] → k [ E ] such that E n E n − ( E + 1) n for every n ≥ . Recall that ⊗ or | denotethe tensor product over k and that we may sometimes omit it to alleviate notation.5.1. The Lie-Rinehart cohomology H • S ( L, N ) . We let V L be the subspace of L withbasis ( D, E ) and V ∗ L be its dual space, and denote the dual basis by ( ˆ D, ˆ E ) . Let N bean eulerian U -module. The Lie-Rinehart cohomology H • S ( L, N ) of ( S, L ) with valueson N is the cohomology of the complex C • S ( L, N ) , which is isomorphic via standardidentifications to the complex N ⊗ Λ • V ∗ L given by N N ⊗ V ∗ L N ⊗ Λ V ∗ Ld d with differentials d ( n ) = D · n ⊗ ˆ D + E · n ⊗ ˆ E ; d ( n ⊗ ˆ D + m ⊗ ˆ E ) = ( D · m − E · n + n ) ⊗ ˆ D ∧ ˆ E. Proposition 5.2. Let N = L i ∈ Z N i be an eulerian U -module and ∇ D : N → N bethe restriction of the action of D . There are isomorphisms of vector spaces H pS ( L, N ) ∼ = ker ∇ D , if p = 0 ; coker ∇ D ⊗ k ˆ D ⊕ ker ∇ D ⊗ k ˆ E, if p = 1 ; coker ∇ D ⊗ k ˆ D ∧ ˆ E, if p = 2 and H pS ( L, N ) = 0 for every other p ∈ Z . We notice that the cohomology H • S ( L, N ) depends only on the map N → N givenby multiplication by D . Proof. Thanks to Proposition 4.2, we need only compute the cohomology of the sub-complex of N ⊗ Λ • V ∗ L of degree zero. This subcomplex is N N ⊗ k ˆ D ⊕ N ⊗ k ˆ E N ⊗ k ˆ D ∧ ˆ E d d with differentials given by d ( n ) = D · n ⊗ ˆ D and d ( n ⊗ ˆ D + m ⊗ ˆ E ) = D · m ⊗ ˆ D ∧ ˆ E. The claim in the proposition follows immediately from the these expressions. (cid:3) In Proposition 4.5 we saw that the U -modules H • ( S, U ) are eulerian and, as a conse-quence of this, to get H • S ( L, H • ( S, U )) we may use the following strategy: to computethe homogeneous components of degree and of H • ( S, U ) and then to describe themap ∇ • D : H • ( S, U ) → H • ( S, U ) given by the action of D . IE–RINEHART AND HOCHSCHILD COHOMOLOGY 17 The Hochschild cohomology H • ( S, U ) . Let W be the subspace of S with basis ( x, y ) . Applying Hom S e ( − , U ) to the Koszul resolution in Lemma 4.4 and using standardidentifications we obtain the complex U U ⊗ Hom( W, k ) U ⊗ Hom(Λ W, k ) δ δ (15)with differentials δ ( u ) = [ x, u ]ˆ x + [ y, u ]ˆ yδ ( a ˆ x + b ˆ y ) = ([ x, b ] − [ y, a ]) ˆ x ∧ ˆ y, where (ˆ x, ˆ y ) is the dual basis of ( x, y ) and ˆ x ∧ ˆ y is the linear morphism Λ W → k thatsends x ∧ y to one. The cohomology of the complex (15) is H • ( S, U ) . Proposition 5.3. There are isomorphisms of graded vector spaces H ( S, U ) ∼ = S and H ( S, U ) ∼ = k [ D ] ⊗ k [ E ] ⊗ k (ˆ x ∧ ˆ y ) . Proof. Evidently, H ( S, U ) , the subset of U of elements that commute with x and y ,contains S : let us prove that they are equal. Given u ∈ H ( S, U ) , there exist v , . . . , v m in the subalgebra of U generated by x , y and D such that u = P mi =0 v i E i . The condition u, x ] implies that P mi =0 v i ( E i ) ′ and therefore that v i = 0 for every i > , so thatthere exist f , . . . , f n ∈ S such that u = P ni =0 f i D i . An inductive argument using that u, y ] = n X i =0 f i [ D i , y ] ≡ nf n D n − mod n − M i =0 SD i allows us to see that f i = 0 if i > and therefore to conclude that u ∈ S .We compute H ( S, U ) directly from the complex (15). Denote by S ≥ the space ofpolynomials with no constant term. We claim that S ≥ D k k [ E ] is contained in the imageof δ for every k ≥ . Indeed, if f, g ∈ S and ψ ∈ k [ E ] then δ ( gϕ ˆ x + f ψ ˆ y ) = ( xf ψ ′ − ygϕ ′ )ˆ x ∧ ˆ y, so that our claim is true if k = 0 . Assume now that k > and that for every j < k theinclusion S ≥ D j k [ E ] ⊂ Im δ holds. Given f ∈ S and ψ ∈ k [ E ] , we have that δ ( f D k ψ ˆ y ) = xf D k ψ ′ ˆ x ∧ ˆ y and δ ( f D k ψ ˆ x ) = ( − f [ y, D k ] ψ − f D k yψ ′ )ˆ x ∧ ˆ y = ( − f [ y, D k ]( ψ − ψ ′ ) − f yD k ψ ′ )ˆ x ∧ ˆ y ≡ − f yD k ψ ′ ˆ x ∧ ˆ y mod Im δ , which proves the claim. We easily see, on the other hand, that the intersection of k [ D ] k [ E ] with Im δ is trivial, so that H ( S, U ) ∼ = k [ D ] k [ E ]ˆ x ∧ ˆ y , as we wanted. (cid:3) IE–RINEHART AND HOCHSCHILD COHOMOLOGY 18 The computation of H ( S, U ) is significantly more involved than the one just above.As we are after the Lie–Rinehart cohomology H • S ( L, H ( S, U )) , thanks to Proposition 5.2we need only compute the homogeneous components of H ( S, U ) of degree and . Proposition 5.4. The graded vector space H ( S, U ) satisfies dim H ( S, U ) = 5 and dim H ( S, U ) = 8 . Moreover, H ( S, U ) is generated by the classes of the cocycles ofthe complex (15) η = ( − yE + D )ˆ x + tyE ˆ y, η = y ˆ x, η = x ˆ y, η = y ˆ y, η = D ˆ y, and H ( S, U ) is generated by the classes of the cocycles ζ = ( D − yDE + y ( E − E ))ˆ x + (2 tyDE + tF E + ty ( E − E ))ˆ y,ζ = ( − y E + yD )ˆ x + ty E ˆ y, ζ = y ˆ x, ζ = x ˆ y,ζ = xy ˆ y, ζ = xD ˆ y, ζ = yD ˆ y, ζ = D ˆ y. Proof. The homogeneous component of degree zero of the complex (15) is U U ˆ x ⊕ U ˆ y U ˆ x ∧ ˆ y δ δ with U = k [ E ] , U = S k [ E ] ⊕ D k [ E ] , U = S k [ E ] ⊕ S D k [ E ] ⊕ D k [ E ] (16)and differentials given by δ ( φ ) = xφ ′ ˆ x + yφ ′ ˆ y,δ (( xϕ + yϕ + Dϕ )ˆ x ) = (cid:16) − xyϕ ′ − y ϕ ′ − yDϕ ′ − F ( ϕ ′ − ϕ ) (cid:17) ˆ x ∧ ˆ y,δ (( xψ + yψ + Dψ )ˆ y ) = ( x ψ ′ + xyψ ′ + xDψ ′ )ˆ x ∧ ˆ y, where φ , ϕ ’s and ψ ’s denote elements of k [ E ] .Let a, b ∈ U and let ω = a ˆ x + b ˆ y be a -cocycle. Up to adding a coboundary we maysuppose that the component of a in x k [ E ] is zero: we may therefore write a = yϕ + Dϕ , b = xψ + yψ + Dψ , (17)with Greek letters in k [ E ] . The coboundary δ ( ω ) belongs to U ˆ x ∧ ˆ y , which decomposesas in (16). The vanishing of the component in D k [ E ] does not give any information,that of the one in S D k [ E ] tells us that ϕ ′ = ψ ′ = 0 and, finally, that of S k [ E ] tellsus that x ψ ′ + xyψ ′ = y ϕ ′ − F ϕ ′ . (18)Let us put λ := ϕ . Looking at the component on y k [ E ] of equation (18) and keepingin mind that F = y + txy we see that ϕ ′ = λ and, using this, that xψ ′ + yψ ′ = − λty .There exist then µ ∈ k and f ∈ S such that ϕ = − λE + µ, xψ + yψ = λtyE + f. IE–RINEHART AND HOCHSCHILD COHOMOLOGY 19 As a cocycle ω = a ˆ x + v ˆ y satisfying (17) is a coboundary only if it is zero, we concludethat H ( S, U ) ∼ = k η ⊕ k y ˆ x ⊕ ( S ⊕ k D )ˆ y, with η = ( − yE + D )ˆ x + tyE ˆ y .We now compute H ( S, U ) . The component of degree of the complex (15) is U U ˆ x ⊕ U ˆ y U ˆ x ∧ ˆ y δ δ with U = S k [ E ] ⊕ S D k [ E ] ⊕ S D k [ E ] ⊕ D k [ E ] and differentials δ ( xφ + yφ + Dρ )= ( x φ ′ + xyφ ′ + xDρ ′ )ˆ x + ( xyφ ′ + y φ ′ + yDρ ′ + F ( ρ ′ − ρ )ˆ y,δ (cid:16)(cid:16)X x i y j ϕ ij + xDϕ + yDϕ + D ϕ (cid:17) ˆ x (cid:17) = − X x i y j +1 ϕ ′ ij − xyDϕ ′ − xF ( ϕ ′ − ϕ ) − y Dϕ ′ − yF ( ϕ ′ − ϕ ) − yD ϕ ′ − F D ( ϕ ′ − ϕ ) − F F y ( ϕ ′ − ϕ ) ,δ (cid:16)(cid:16)X x i y j ψ ij + xDψ + yDψ + D ψ (cid:17) ˆ y (cid:17) = X x i +1 y j ψ ′ ij + x Dψ ′ + xyDψ ′ + xD ψ ′ . In all the sums that appear here the indices i and j are such that i + j = 2 and we haveomitted the factor ˆ x ∧ ˆ y for δ . Again, all Greek letters lie in k [ E ] .Let us put, once again, ω = a ˆ x + b ˆ y , this time with a and b in U . Up to coboundaries,we write, with the same conventions as before, a = y ϕ + yDϕ + D ϕ, b = X x i y j ψ ij + xDψ + yDψ + D ψ. Let us examine the condition δ ( ω ) = 0 component by component according to ourdescription of U in (16) above.In D k [ E ] there is no condition at all. In S D k [ E ] we have xD ψ ′ − yD ϕ ′ = 0 , sothat ψ and ϕ are scalars. In S D k [ E ] the condition reads x Dψ ′ + xyDψ ′ = y Dϕ ′ + 2 F D ( ϕ ′ − ϕ ) . (19)Writing F = y + txy and looking at the terms that are in y k [ E ] we find ϕ ′ − ϕ , andthen ϕ = − ϕE + λ for some λ ∈ k . What remains of (19) implies that xψ ′ + yψ ′ = − tyϕ and therefore there exists h ∈ S such that xDψ + yDψ = 2 ϕtyDE + hD. Finally, we look at S k [ E ] : we have X x i +1 y j ψ ′ ij = y ϕ ′ + yF ( ϕ ′ − ϕ ) − F F y ϕ. In particular, using that F y = 2 y + tx and looking at the terms in y k [ E ] , we find that ϕ ′ + ( ϕ ′ − ϕ ) + 2( ϕ ′ − ϕ ) , or, rearranging, ϕ ′ = − ϕE + λ . “Integrating”, wesee there exists µ ∈ k such that ϕ = ϕ ( E − E ) − λE + µ. IE–RINEHART AND HOCHSCHILD COHOMOLOGY 20 Now, as F F y = 2 y + 3 txy + t x y , we must have X x i y j ψ ′ ij = ty ( ϕ ′ − ϕ ) − (3 ty + t xy ) ϕ, and, integrating yet another time, we get P x i y j ψ ij = φ ( tF E + ty ( E − E )) + λty E ,We conclude in this way that every -cocycle of degree is cohomologous to one ofthe form ω = ϕζ + λζ + f ˆ y + hD ˆ y + ψD ˆ y + µy ˆ x (20)where ζ and ζ are the cocycles in the statement, ϕ , λ , ψ , µ ∈ k , h ∈ S and f ∈ S .It is easy to see from the expression we have for δ that such a cocycle is a coboundaryif and only if it is a scalar multiple of F ˆ y . The upshot of all this is that H ( S, U ) ∼ = h ζ , ζ i ⊕ k y ˆ x ⊕ (cid:16) S / ( F ) ⊕ S D ⊕ k D (cid:17) ˆ y, as we wanted. (cid:3) The action of U on H • ( S, U ) . As we have already computed in Propositions 5.3and 5.4 the homogeneous components of degrees and of the Hochschild cohomol-ogy H q ( S, U ) for each q , Proposition 5.2 tells us that in order to compute the sec-ond page E • ,q = H • S ( L, H q ( S, U )) it remains only to find the kernel and the cokernelof ∇ qD : H q ( S, U ) → H q ( S, U ) . Proposition 5.5. ( i ) The kernel of ∇ D : H ( S, U ) → H ( S, U ) is k and itscokernel is S , the subspace of S with basis ( x, y ) . ( ii ) The kernel of ∇ D : H ( S, U ) → H ( S, U ) is k D ˆ x ∧ ˆ y and its cokernel is zero. ( iii ) The map ∇ D : H ( S, U ) → H ( S, U ) is a monomorphism and its cokernel isgenerated by the classes of the cocycles ζ , ζ , and ζ given in Proposition 5.4.Proof. Recall that H • ( S, U ) is computed from the Koszul resolution P • of Lemma 4.4,where W is the vector space spanned by x and y . To describe the action of D on H • ( S, U ) we need a lifting of D S : S → S to an P • . We obtain one by letting, for each q ∈ { , , } , D q : P q → P q be the D eS -operator such that D (1 | 1) = 0 ,D (1 | ⊗ y ) = (1 | y + y | tx | ⊗ y + t | y ⊗ x, D (1 | ⊗ x ) = 0 ,D (1 | ⊗ x ∧ y ) = (1 | y + y | tx | ⊗ x ∧ y, as a straightforward calculation shows. From the description of D we see that therestriction to S → S of the map ∇ D : S → S is zero, thus proving assertion ( i ).We recall from Proposition 5.3 that the homogeneous components of degree and of H ( S, U ) are D k [ E ]ˆ x ∧ ˆ y and D k [ E ]ˆ x ∧ ˆ y , respectively. Let us compute the kerneland the cokernel of ∇ D : H ( S, U ) → H ( S, U ) . We have D ♯ ( D ϕ ˆ x ∧ ˆ y ) = (cid:16) [ D, D ϕ ] − D ϕ ˆ x ∧ ˆ y ( D (1 | ⊗ x ∧ y )) (cid:17) ˆ x ∧ ˆ y IE–RINEHART AND HOCHSCHILD COHOMOLOGY 21 and, as in the second term there never appears a higher power of D than D , D ♯ ( D ϕ ˆ x ∧ ˆ y ) ≡ D ϕ ′ ˆ x ∧ ˆ y mod Im δ . The claim in the second item follows from this.For ( iii ) we give explicit formulas for the evaluation of ∇ D : H ( S, U ) → H ( S, U ) and, at the same time, compute its cokernel. Suppose that ω is a representative of aclass in H ( S, U ) chosen as in (20). As D ♯ ( D ˆ y ) = ( − F y D − F )ˆ y , we see that up toadding to ω an element in the image of ∇ D we may suppose that h = h x , for some h ∈ k .Let α , β and γ in k and define φ = αy ˆ x + ( βx + γy )ˆ y . Since φ ( D (1 | ⊗ y )) is equalto γxF x − αyF x − βxF y , we have D ♯ ( φ ) = ([ D, αy ] − φ ( D (1 | x | x + ([ D, βx + γy ] − φ ( D (1 | y | y = αF ˆ x + ( γyF y + αyF x + βxF y )ˆ y. In view of this, it is easy to see that we may choose α , β and γ in such a waythat ω + D ♯ ( φ ) , which is a cocycle of the form (20), has µ = 0 and f = 0 since { yF x , xF y , F } spans S .Let us see that the -cocycle ζ belongs to the image of ∇ D . Using the -cocycle η = ( − yE + D )ˆ x + tyE ˆ y we get D ♯ ( η )(1 | ⊗ x ) = [ D, − yE + D ] = − F E + yD, and D ♯ ( η )(1 | ⊗ y ) = [ D, tyE ] − η ( D (1 | ⊗ y ))= tF E − tyD − t ( − yE + D ) y − ( tx + y ) tyE − tyEy = − tyD + ty + t ( y + txy ) , which belongs to S + k yD . We already know that the elements of ( S + k yD ) ˆ y arecoboundaries: it follows that D ♯ ( η ) ≡ ( − F E + yD )ˆ x modulo coboundaries. Now, thedifference between D ♯ ( η ) and ζ is cohomologous to txyE ˆ x + ty E ˆ y , which is in turnequal to δ ( − tyE ) . As a consequence of this, we have that ∇ D ( η ) is equal to ζ incohomology.We conclude from the preceding calculation that coker (cid:0) ∇ D : H ( S, U ) → H ( S, U ) (cid:1) is generated by the classes of ζ , xD ˆ y , and D ˆ y . Since these classes are linearly indepen-dent, the dimension of this cokernel is . Finally, we can use the dimension theorem tosee that ∇ D : H ( S, U ) → H ( S, U ) is a monomorphism. (cid:3) The second page. We have already made all the computations required for thesecond page of the spectral sequence. IE–RINEHART AND HOCHSCHILD COHOMOLOGY 22 Proposition 5.6. The second page of the spectral sequence E • of Corollary 3.3 converg-ing to the Hochschild cohomology HH • ( U ) has dimensions dim E p,q = q p (21) Proof. Let q ≥ and recall that the q th row E • ,q is equal to the Lie-Rinehart cohomology H • S ( L, H q ( S, U )) . Thanks to Proposition 4.5, H q ( S, U ) is an eulerian U -module and wemay use Proposition 5.2, which asserts that to obtain H • S ( L, H q ( S, U )) we need onlythe nullity and rank of ∇ qD : H q ( S, U ) → H q ( S, U ) . This information is provided byProposition 5.5. (cid:3) Corollary 5.7. The dimension of HH ( U ) is or .Proof. The differential in the second page (21) could be non-zero, since neither thedomain nor the codomain of the map d , : E , → E , are. As dim E , = 1 , thedifferential d , is either zero or a monomorphism. If it is zero, the sequence degeneratesand using Corollary 3.3 we obtain that dim HH ( U ) = 4 ; if not, we have dim HH ( U ) = 3 . (cid:3) It follows from Corollary 5.7 that to see whether the sequence degenerates or not it isenough to compute the dimension of HH ( U ) : this provided in the next proposition. Proposition 5.8. The dimension of HH ( U ) is at least .Proof. The Hochschild cohomology of the algebra of differential operators U on an ar-rangement of more than five lines is computed in [9] from a complex that we may stilluse. This complex is given by U ⊗ Λ • V ∗ U , where V U is the subspace of U spanned by x , y , D and E , or more graphically U U ⊗ V ∗ U U ⊗ Λ V ∗ U U ⊗ Λ V ∗ U U ⊗ Λ V ∗ U , d d d d with differentials such that d ( u ⊗ ˆ x ∧ ˆ y ) = ([ D, u ] − ∇ uy ( F )) ⊗ ˆ x ∧ ˆ y ∧ ˆ D + ([ E, u ] − u ) ⊗ ˆ x ∧ ˆ y ∧ ˆ E ; d ( u ⊗ ˆ x ∧ ˆ E ) = − [ y, u ] ⊗ ˆ x ∧ ˆ y ∧ ˆ E − [ D, u ] ⊗ ˆ x ∧ ˆ D ∧ ˆ E + tuy ⊗ ˆ y ∧ ˆ D ∧ ˆ E ; d ( u ⊗ ˆ y ∧ ˆ E ) = [ x, u ] ⊗ ˆ x ∧ ˆ y ∧ ˆ E + (( tx + 2 y ) u − [ y, u ] − [ D, u ]) ⊗ ˆ y ∧ ˆ D ∧ ˆ E ; d ( u ⊗ ˆ x ∧ ˆ D ) = − [ y, u ] ⊗ ˆ x ∧ ˆ y ∧ ˆ D + ([ E, u ] − u ) ⊗ ˆ x ∧ ˆ D ∧ ˆ E ; d ( u ⊗ ˆ y ∧ ˆ D ) = [ x, u ] ⊗ ˆ x ∧ ˆ y ∧ ˆ D + ([ E, u ] − u ) ⊗ ˆ y ∧ ˆ D ∧ ˆ E ; IE–RINEHART AND HOCHSCHILD COHOMOLOGY 23 d ( u ⊗ ˆ D ∧ ˆ E ) = [ x, u ] ⊗ ˆ x ∧ ˆ D ∧ ˆ E + [ y, u ] ⊗ ˆ y ∧ ˆ D ∧ ˆ E ; d ( u ⊗ ˆ x ∧ ˆ y ∧ ˆ D ) = ( − [ E, u ] + 3 u ) ⊗ ˆ x ∧ ˆ y ∧ ˆ D ∧ ˆ Ed ( u ⊗ ˆ x ∧ ˆ y ∧ ˆ E ) = ([ D, u ] − ( tx + 2 y ) u + [ y, u ]) ⊗ ˆ x ∧ ˆ y ∧ ˆ D ∧ ˆ E ; d ( u ⊗ ˆ x ∧ ˆ D ∧ ˆ E ) = − [ y, u ] ⊗ ˆ x ∧ ˆ y ∧ ˆ D ∧ ˆ E ; d ( u ⊗ ˆ y ∧ ˆ D ∧ ˆ E ) = [ x, u ] ⊗ ˆ x ∧ ˆ y ∧ ˆ D ∧ ˆ E. As V U is a homogeneous subspace of U , the grading of U induces on the exterioralgebra Λ • V U an internal grading. There is as well a natural internal grading on thecomplex U ⊗ Λ • V ∗ U coming from the grading of U , with respect to which the differentialsare homogeneous. Moreover, the inclusion X • = ( U ⊗ Λ • V ∗ U ) ֒ → U ⊗ Λ • V ∗ U of thecomponent of degree zero of the complex U ⊗ Λ • V ∗ U is a quasi-isomorphism: we will usethe complex X • again to compute HH ( U ) .We borrow from our previous calculations the following four cochains in X : ω = D ˆ x ∧ ˆ y ∧ ˆ E + (cid:16) D E − yDE + F ( E − E + E ) / (cid:17) ⊗ ˆ y ∧ ˆ D ∧ ˆ E + (cid:16) − tD E + 2 tyDE + tf E (cid:17) ⊗ ˆ y ∧ ˆ D ∧ ˆ E,ω = (cid:16) D − yDE + y ( E − E ) (cid:17) ⊗ ˆ x ∧ ˆ D ∧ ˆ E + (cid:16) tyDE + tF E + ty ( E − E ) (cid:17) ⊗ ˆ y ∧ ˆ D ∧ ˆ E,ω = D ⊗ ˆ y ∧ ˆ D ∧ ˆ E,ω = xD ⊗ ˆ y ∧ ˆ D ∧ ˆ E. It is straightforward to see that these cochains are in fact cocycles. We will now showthat the classes of these cocycles are linearly independent, so that dim HH ( U ) ≥ . Wetake a linear combination ω = P i =1 λ i ω i with λ , . . . , λ ∈ k and suppose that thereexists a cochain ξ in X such that d ( ξ ) = ω . Since the component of ω in ˆ x ∧ ˆ y ∧ ˆ D iszero, we may write ξ = u ⊗ ˆ x ∧ ˆ E + v ⊗ ˆ y ∧ ˆ E + w ⊗ ˆ D ∧ ˆ E, with u , v and w in U , and there exist then α i , β i , γ i ∈ k [ E ] with ≤ i ≤ such that u = xα + yα + Dα , v = xβ + yβ + Dβ , w = xγ + yγ + Dγ . We now examine each component of the equality d ( ξ ) = ω . In ˆ x ∧ ˆ y ∧ ˆ D there is nothingto see. In ˆ x ∧ ˆ y ∧ ˆ E we have − [ y, u ] + [ x, v ] = λ D , or − xyα ′ − y α ′ − yDα ′ − F ( α ′ − α ) + x β ′ + xyβ ′ + xDβ ′ = λ D . This is an equality in U , which we may decompose as L i + j + k =2 x i y j D k k [ E ] . Lookingat D k [ E ] we get λ = 0 , from yD k [ E ] , x k [ E ] and xD k [ E ] we obtain α ′ = β ′ = β ′ = 0 and xy k [ E ] and y k [ E ] tell us that α = α ′ and β ′ = α ′ − tα . IE–RINEHART AND HOCHSCHILD COHOMOLOGY 24 In ˆ x ∧ ˆ D ∧ ˆ E , equation d ( ξ ) = ω reads − xDα ′ − F α − yDα ′ + x γ ′ + xyγ ′ + xDγ ′ = λ ( D − yDE + y ( E − E )) . The component in D k [ E ] of this equality is λ . From xD k [ E ] and yD k [ E ] weobtain γ ′ = α ′ and α ′ = 0 and from x k [ E ] , xy k [ E ] and y k [ E ] we get γ ′ = 0 , γ ′ = tα and α = 0 . In particular, that α = 0 implies that α = 0 and that β ′ = α ′ = γ ′ (22)We finally look at the component in ˆ y ∧ ˆ D ∧ ˆ E of d ( ξ ) = ω , which is tuy + ( tx + 2 y ) v − [ y, v ] − [ D, v ] + [ y, w ] = λ D + λ xD. This is an equality in U = L i + j + k =2 x i y j D k k [ E ] . In D k [ E ] we have λ , and in yD k [ E ]2 β yD − yDβ ′ + yDγ ′ = 0 , which in the light of (22) implies β = 0 . With this at hand we see that in xD k [ E ] itonly remains λ .We have seen at this point that the only coboundary among the cocycles of theform ω = P i =1 λ i ω i is ω = 0 . This shows that the classes of ω , . . . , ω are linearlyindependent, thus finishing the proof. (cid:3) Corollary 5.9. Let A be a central arrangement of three lines. The Hilbert seriesof HH • ( Diff A ) is h HH • ( Diff A ) ( t ) = 1 + 3 t + 6 t + 4 t . Proof. Proposition 5.8 implies at once that the spectral sequence degenerates at E .The dimensions in the statement are a consequence of the convergence of the sequencein Corollary 3.3 and the information in Proposition 5.6. (cid:3) As a consequence of the information we have gathered so far we can easily describethe Lie algebra structure on HH ( Diff A ) . Let us, again, call U = Diff A and recall that HH ( U ) is isomorphic to the space OutDer U of outer derivations of U , that is, thequotient of the derivations of U modulo inner derivations, and that the commutator ofderivations induces a Lie algebra structure on OutDer U . We know from [9, Proposi-tion 4.2] that if f ∈ S divides xF then there is a derivation ∂ f : U → U such that ∂ f ( x ) = ∂ f ( y ) = 0 , ∂ f ( D ) = Ff ∂ y f and ∂ f ( E ) = 1 . Let then f = x , f = y and f = tx + y and put ∂ i := ∂ f i for ≤ i ≤ . Corollary 5.10. Let A be a central arrangement of three lines. The Lie algebra of outerderivations of Diff A together with the commutator is an abelian Lie algebra of dimensionthree generated by the classes of the derivations ∂ , ∂ and ∂ . IE–RINEHART AND HOCHSCHILD COHOMOLOGY 25 Proof. We claim that the classes of ∂ , ∂ and ∂ are linearly independent in OutDer( U ) .Indeed, let u ∈ U and λ , λ , λ ∈ k be such that X λ i ∂ i ( v ) = [ u, v ] for every v ∈ U . (23)Evaluating (23) on each s ∈ S the left side vanishes and therefore Proposition 5.3 tellsus that u ∈ S . Write u = P j ≥ u j with u j ∈ S j . Evaluating now (23) on E we obtain P i λ i = − P j ju j . In each homogeneous component S j with j = 0 we have ju j = 0 andtherefore u ∈ S = k and P i λ i = 0 . This equation and the one we get evaluating (23)on D , that is, λ ( tx + y ) + λ y = 0 ∈ S , finally tell us that λ = λ = λ = 0 .The classes of ∂ , ∂ and ∂ span OutDer U because, thanks to Corollary 5.9, itsdimension is three. The composition ∂ i ◦ ∂ j : U → U is evidently equal to zero forany ≤ i, j ≤ , as a straightforward calculation shows, and therefore the Lie algebrastructure in OutDer U vanishes. (cid:3) It is possible also to use the spectral sequence of Corollary 3.3 to obtain HH • ( Diff A ) for arrangements with any l ≥ , but we will not perform this calculation here. Theresult is h HH • ( Diff A ) ( t ) = ( lt + 2 lt + ( l + 1) t , if l = 3 , ; lt + (2 l − t + lt , if l ≥ .This shows that the case in which l is or is genuinely different to that in which l ≥ .If l ≤ , the algebra Diff A is not very interesting, since it is isomorphic to algebras withwell-known Hochschild cohomology —see [9, §3.8].6. Other applications The Hochschild cohomology of a family of subalgebras of the Weyl al-gebra. Let k be a field of characteristic zero, fix a nonzero h ∈ k [ x ] and consider thealgebra A h with presentation k h x, y i ( yx − xy − h ) . Setting h = 1 the algebra A h is the Weyl algebra A that already appeared in Ex-ample 1.4, when h = x it is the universal enveloping algebra of the two-dimensionalnon-abelian Lie algebra and if h = x , it is the Jordan plane studied in [1].We let S = k [ x ] and consider the Lie algebra L freely generated by y = h ddx as an S -submodule of Der S . It is straightforward to see that ( S, L ) is a Lie–Rinehart algebrawhose enveloping algebra U is isomorphic to A h . We will use the spectral sequence ofCorollary 3.3 to compute the Hochschild cohomology HH • ( A h ) of A h : we will describeexplicitly the second page and find that the spectral sequence degenerates at that page. IE–RINEHART AND HOCHSCHILD COHOMOLOGY 26 The Hochschild cohomology H • ( S, U ) . The augmented Koszul complex P • : 0 S e S e S δ ε with δ ( s ⊗ t ) = sx ⊗ t − s ⊗ xt and augmentation ε ( s ⊗ t ) = st is an S e -projectiveresolution of S and therefore the Hochschild cohomology H • ( S, U ) is, after identify-ing Hom S e ( S e , U ) with U , the cohomology of the complex U δ −→ U with differential δ ( u ) = [ x, u ] . Proposition 6.1. There are isomorphisms of vector spaces H ( S, U ) ∼ = S, H ( S, U ) ∼ = U/hU, H q ( S, U ) = 0 if q ≥ .Proof. The isomorphisms in the statement come, of course, of the computation of thecohomology of U δ −→ U . Let us first deal with ker δ . Writing u = P ri =0 f i y i with f , . . . , f r ∈ S and r the greatest index such that f r = 0 , we have that δ ( u ) = rf r hy r − + v u (24)for some v u ∈ L r − i =0 Sy i . If δ ( u ) = 0 then its principal symbol rf r hy r − must be equalto zero and then, because the field has characteristic zero, either r = 0 or f r = 0 . Thissecond possibility contradicts our assumptions, and therefore r = 0 and u ∈ S . That,reversely, S is contained in the kernel of δ is evident.The second claim of the statement follows from the fact that the image of δ is theright ideal generated by h , that is, hU . For this, we can see that hSy i belongs to theimage of δ for every i ≥ with a straightforward inductive argument using (24). (cid:3) The action of U on H • ( S, U ) . As S acts just by left multiplication, to determinethe action of U it is enough to explicit that of y : for this have at hand Remark 2.5. Proposition 6.2. Under the isomorphisms H ( S, U ) ∼ = U and H ( S, U ) ∼ = U/hU ofProposition 6.1, the action of L on H • ( S, U ) is determined by ∇ y ( s ) = hs ′ for s ∈ S ; ∇ y (¯ u ) = − h ′ u for u ∈ U ,where the overline denotes class modulo hU .Proof. We use Example 2.7 to see that y acts on H ( S, U ) = S as in the statement. Todescribe its action on H ( S, U ) we need a lifting y • = ( y , y ) of y S : S → S to P • . Letus define y ( s ⊗ t ) = hs ′ ⊗ ⊗ ht, y ( s ⊗ t ) = hs ′ ⊗ ⊗ ht ′ + s ∆( h ) t, where ∆ : S → S e is the unique derivation of S such that ∆( x ) = 1 ⊗ , that is, it is theonly linear map such that δ ( x j ) = P s + t = j +1 x s ⊗ x t if j ≥ . We readily see that y and IE–RINEHART AND HOCHSCHILD COHOMOLOGY 27 y are y eS -operators and that the diagram S e S e SS e S e S δ εδ y εy y commutes, and thus the pair ( y , y ) is in fact one of the y eS -liftings we were looking for.We now compute y ♯ : Hom S e ( S e , U ) → Hom S e ( S e , U ) following (8), and for that welet φ ∈ Hom S e ( S e , U ) . Bearing in mind the isomorphism Hom S e ( S e , U ) ∼ = U induced bythe evaluation in ⊗ , we need only compute y ♯ ( φ )(1 ⊗ 1) = [ y, φ (1 ⊗ − φ (∆( h )) . Assuming without losing generality that φ (1 ⊗ 1) = f y i and h = x j for f ∈ S and i , j ≥ , we obtain y ♯ ( φ )(1 ⊗ 1) = [ y, f y i ] − X s + t = j +1 x s f y i x t = hf ′ y i − X s + t = j +1 x s f ( x t y i + [ y i , x t ]) ≡ − jx j − f mod hU, since [ y i , x t ] ∈ hU for all i, t ≥ . Taking class in cohomology and identifying φ with φ (1 ⊗ , we get ∇ y ( f y i ) = − h ′ f y i , and the stated result follows from this. (cid:3) The Lie-Rinehart cohomology. Let us now compute H • S ( L, H i ( S, U )) for each i ∈ Z . Using the complex in Proposition 1.6 to compute Lie–Rinehart cohomologyof S , we see that this is the cohomology of the complex H i ( S, U ) H i ( S, U ) . ∇ iy Proposition 6.3. Let d = gcd( h, h ′ ) and let I be the ideal of S/ ( h ) generated by theclass of h/d . There are isomorphisms of vector spaces H S ( L, H ( S, U )) ∼ = k , H S ( L, H ( S, U )) ∼ = S/ ( h ) ,H S ( L, H ( S, U )) ∼ = I [ y ] , H S ( L, H ( S, U )) ∼ = S/ ( d )[ y ] . and H pS ( L, H q ( S, U )) = 0 if p, q ≥ .Proof. We make use of the explicit description of ∇ iy in Proposition 6.2. For i = 0 , thisamounts to the cohomology of S y −→ S , and we readily see that the kernel of this map is k and its image, hS .Consider now the case in which i = 1 and recall that H ( S, U ) is isomorphic to U/hU .As U/hU is the quotient of the free noncommutative algebra in x and y by the relations xy − yx = h and h = 0 , we may identify H ( S, U ) with S ( h ) [ y ] . IE–RINEHART AND HOCHSCHILD COHOMOLOGY 28 For each f ∈ S we write ˜ f its class in S/ ( h ) . This way, given v ∈ S ( h ) [ y ] , there are f , . . . , f r ∈ k [ x ] such that v = P ri =0 ˜ f i y i and, as our findings on ∇ y of Proposition 6.2allow us to see, ∇ y ( v ) = − r X i =0 g h ′ f i y i . It is immediate that the cokernel of ∇ y is S ( h,h ′ ) [ y ] . To compute its kernel, let ussuppose that ∇ y ( u ) = 0 . For each i ∈ { , . . . , r } we have that h divides h ′ f i andtherefore, if d denotes the greatest common divisor of h and h ′ , we have that h/d divides f i . Denoting by I the ideal of S ( h ) generated by the class of h/d , we conclude that H S ( L, H ( S, U )) is isomorphic to the space of polynomials I [ y ] with coefficients in I . (cid:3) The conclusion of this calculation is the following description of the Hochschild coho-mology of U —this time we write A h instead of U . Proposition 6.4. There are isomorphisms of vector spaces HH i ( A h ) ∼ = k if i = 0 ; S/ ( h ) ⊕ I [ y ] if i = 1 ; S ( d ) [ y ] if i = 2 ; otherwise,where d stands for the greatest common divisor of h and its derivative h ′ and I is theideal of S ( h ) generated by the class of h/d .Proof. The spectral sequence E • of Corollary 3.3 converges to HH • ( A h ) and its secondpage, given by E p,q = H pS ( L, H q ( S, U )) , is completely computed in Proposition 6.3. Thesequence degenerates because E p,q = 0 if p, q ≥ . (cid:3) The first cohomology space had already been obtained, in other words, in [3, Theo-rem 5.7. (iii) ] and the second one in [11, Corollary 3.11]. However, the spectral sequenceargument provides conceptual simplifications and, as a consequence of that, this compu-tation is significantly shorter.6.2. The Van den Bergh duality property for U . An appropriate specialization ofCorollary 3.3 allows us to recover one of the main results of [10], which we recall afterthe following preliminary definition. Let n ≥ . An algebra A has Van den Bergh duality of dimension n if A has a resolution of finite length by finitely generated projective A -bimodules and there exists an invertible A -bimodule D such that there is an isomorphismof A -bimodules Ext iA e ( A, A ⊗ A ) = ( if i = n ; D if i = n . IE–RINEHART AND HOCHSCHILD COHOMOLOGY 29 The Van den Bergh duality property for an algebra A is important because, as canbe seen in [21], it relates the Hochschild cohomology of A with its homology in a wayanalogue to Poincaré duality: indeed, for each A -bimodule M it produces a canonicalisomorphism H i ( A, M ) → H n − i ( A, D ⊗ A M ) .Let us consider the left U -module structure on Λ dS L ∨ ⊗ S D discussed in the first sectionof [10]. If F is the functor from left U -modules to U e -modules in (13) then evidently F (Λ dS L ∨ ⊗ S D ) becomes an U e -module. Theorem 6.5. Let ( S, L ) be a Lie–Rinehart algebra such that S has Van den Berghduality in dimension n and L is finitely generated and projective with constant rank d as an S -module and let L ∨ = Hom S ( L, S ) . The enveloping algebra U of the algebra hasVan den Bergh duality in dimension n + d and there is an isomorphism of U e -modules Ext n + dU e ( U, U e ) ∼ = F (Λ dS L ∨ ⊗ S D ) . Lemma 6.6. Let A be an algebra and T and P two A -modules such that T admits aprojective resolution by finitely generated A -modules and P is flat. There is an isomor-phism Ext • A ( T, P ) ∼ = Ext • A ( T, A ) ⊗ A P. Proof of Lemma 6.6. Let Q • be such a resolution of T . For each i ≥ , the evidentmap from Hom A ( Q i , A ) ⊗ A P to Hom A ( Q i , P ) is an isomorphism because Q i is finitelygenerated and projective. As P is flat, the cohomology of the complex Hom A ( Q • , A ) ⊗ A P is isomorphic to Ext • A ( T, A ) ⊗ A P . (cid:3) Proof of Theorem 6.5. The homological smoothness of U follows from Lemma 5.1.2 of[10], whose proof does not depend on this theorem.Let us write D for the dualizing bimodule Ext nS e ( S, S e ) . We take, specializing Corol-lary 3.3, M = U e to obtain a spectral sequence E • such that E p,q = H pS ( L, H q ( S, U e )) = ⇒ H p + q ( U, U e ) . Let us first deal with H q ( S, U e ) . As we observed in the proof of Proposition 2.1, the U e -module U e is S e -projective and, since S has Van den Bergh duality, it admits a resolutionby finitely generated projective S e -modules. We may therefore use Lemma 6.6 to seethat H q ( S, U e ) ∼ = H q ( S, S e ) ⊗ S e U e , which is zero if q = n and isomorphic to D ⊗ S e U e if q = n . As a consequence of this, ourspectral sequence E • degenerates at its second page and thus H p + n ( U, U e ) is isomorphicto H pS ( L, D ⊗ S e U e ) for each p ∈ Z .The Chevalley–Eilenberg complex from Proposition 1.6 is an U -projective resolutionof S by finitely generated modules. On the other hand, the dualizing module D is S -projective because it is invertible —see Chapter 6 in the book [4] by F. Anderson and IE–RINEHART AND HOCHSCHILD COHOMOLOGY 30 K. Fuller. We conclude that the U -module D ⊗ S e U e is projective, and we can applyLemma 6.6 we obtain an isomorphism H • S ( L, D ⊗ S e U e ) ∼ = H • S ( L, U ) ⊗ U ( D ⊗ S e U e ) . 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(1997), no. 4, 1653–1662, DOI 10.1090/S0002-9947-97-01894-1. Laboratoire de Mathématiques Blaise Pascal, UMR6620 CNRS, Université ClermontAuvergne, Campus des Cézeaux, 3 place Vasarely, 63178 Aubière cedex, France E-mail address , F. Kordon: [email protected] E-mail address , Th. Lambre:, Th. Lambre: