Hochschild cohomology of algebras of differential operators tangent to a central arrangement of lines
HHOCHSCHILD COHOMOLOGY OF ALGEBRAS OF DIFFERENTIALOPERATORS TANGENT TO A CENTRAL ARRANGEMENT OFLINES
FRANCISCO KORDON AND MARIANO SUÁREZ-ÁLVAREZ
Abstract.
Given a central arrangement of lines A in a 2-dimensional vector space V over a field of characteristic zero, we study the algebra D ( A ) of differential operatorson V which are logarithmic along A . Among other things we determine the Hochschildcohomology of D ( A ) as a Gerstenhaber algebra, establish a connection between thatcohomology and the de Rham cohomology of the complement M ( A ) of the arrangement,determine the isomorphism group of D ( A ) and classify the algebras of that form up toisomorphism. Let us fix a ground field k of characteristic zero, a vector space V and a centralarrangement of hyperplanes A in V . We let S be the algebra of polynomial functionsof V , fix a defining polynomial Q ∈ S for A , and consider, following K. Saito [13], theLie algebra Der ( A ) = { δ ∈ Der ( S ) : δ ( Q ) ∈ QS } of derivations of S logarithmic with respect to A , which is, geometrically speaking, theLie algebra of vector fields on V which are tangent to the hyperplanes of A . This Liealgebra is a very interesting invariant of the arrangement and has been the subject ofa lot of work — we refer to the book of P. Orlik and H. Terao [10] and the one byA. Dimca [4] for surveys on this subject. In particular, using this Lie algebra we candefine an important class of arrangements: we say that an arrangement A is free if Der ( A ) is free as a left S -module. For example, central arrangements of lines in theplane are free, as are, according to a beautiful result of Terao [16], the arrangements ofreflecting hyperplanes of a finite group generated by pseudo-reflectionNow, along with Der ( A ) we can consider also the associative algebra D ( A ) 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 : we call it the algebra ofdifferential operators tangent to the arrangement A . When A is free, this coincides withthe algebra of differential operators on S which preserve the ideal QS of S and all itspowers, studied for example by F. J. Calderón-Moreno [3] or by the second author in [15].The purpose of this paper is to study, from the point of view of non-commutative algebraand homological algebra, this algebra D ( A ) in the simplest case of free arrangement,that of central line arrangements. Date : July 25, 2018.2010
Mathematics Subject Classification.
Primary 16E40; Secondary 14N20.IMAS (CONICET) – Universidad de Buenos Aires. This work has been supported by the projects UBA-CYT 20020130100533BA, PIP-CONICET 112–201501– 00483CO, PICT 20150366 and MATHAMSUD-REPHOMOL. a r X i v : . [ m a t h . K T ] J u l LGEBRAS OF DIFFERENTIAL OPERATORS 2
Let us describe briefly our results. We thus assume in what follows that A is a centralarrangement of r +2 lines in a 2-dimensional vector space V , and for simplicity we supposethat A has at least five lines, so that r ≥
3. We let Q ∈ S be a defining polynomial for A ,that is, a square-free product of linear forms on V with the union of the hyperplanesof A as zero locus. As S is a subalgebra of D ( A ), we view Q as an element of the latter. Theorem A.
The algebra D ( A ) is a noetherian domain, it has global dimension and projective dimension as a bimodule over itself also equal to . The Hochschildcohomology HH • ( D ( A )) of D ( A ) has Hilbert series X i ≥ dim HH i ( D ( A )) · t i = 1 + ( r + 2) t + (2 r + 3) t + ( r + 2) t . The algebra D ( A ) has Hochschild homology and cyclic homology isomorphic to those of apolynomial algebra k [ X ] , and periodic homology and higher K -theory isomorphic to thatof the ground field k . It is a twisted Calabi–Yau algebra of dimension , the element Q of D ( A ) is normal, and the modular automorphism σ : D ( A ) → D ( A ) of D ( A ) is theunique one such that for all a ∈ D ( A ) one has Qa = σ ( a ) Qa.
These claims are contained in Propositions , , and . In Propositions and we describe completely the cup product and the Gerstenhaber Lie structureon HH • ( D ( A )) — we refer to their statements for the precise details, which are technical.The calculations needed in order to do these computations are annoyingly involved.We obtain a very concrete description of HH ( D ( A )) in Proposition : Theorem B.
Let Q = α · · · α r +2 be a factorization of the defining polynomial as aproduct of linear factors, so that α , . . . , α r +2 are linear polynomials on V whose zeroloci are the hyperplanes of A . ( i ) For each i ∈ { , . . . , r + 2 } there is a unique derivation ∂ i : D ( A ) → D ( A ) suchthat ∂ i ( f ) = 0 for all f ∈ S and ∂ i ( δ ) = δ ( α i ) /α i for all δ ∈ Der ( A ) . ( ii ) The set of classes of ∂ , . . . , ∂ r +2 in HH ( D ( A )) , which we view as the space ofouter derivations of the algebra D ( A )) , is a basis. The elements ∂ , . . . , ∂ r +2 are canonically determined and in a natural bijection withthe set of hyperplanes. We do not have a description along the same lines of the restof the cohomology. In Proposition , though, we do obtain the following piece ofinformation: Theorem C.
The subalgebra H of HH • ( D ( A )) generated by the component HH ( D ( A )) of degree is isomorphic to the de Rham cohomology of the complement M ( A ) of thearrangement. It is freely generated as a graded-commutative algebra by the r + 2 elements ∂ , . . . , ∂ r +2 of HH ( D ( A )) subject to the relations ∂ i ^ ∂ j + ∂ j ^ ∂ k + ∂ k ^ ∂ i = 0 , one for each choice of three pairwise distinct elements i , j , k of { , . . . , r + 2 } . Using our precise description of HH ( D ( A )) and the techniques of J. Alev andM. Chamarie [1], we arrive in Section at a description of the automorphism groupof the algebra D ( A ). Since the arrangement A is central, the Lie algebra Der ( A ) is a LGEBRAS OF DIFFERENTIAL OPERATORS 3 graded S -module, and that grading turns D ( A ) into a graded algebra: we will use thisstructure in the following result. Theorem D.
Let G be the subgroup of GL ( V ) of maps which preserve the arrangement A . ( i ) There is an action of G on a vector space W of dimension r + 2 such that thesemidirect product G (cid:110) W is isomorphic to the group Aut ( D ( A )) of algebraautomorphisms of D ( A ) which respect the grading. ( ii ) An element of D ( A ) is locally ad -nilpotent if and only if it belongs to S . Theset Exp ( A ) = { exp ad ( f ) : f ∈ S } of the automorphisms of D ( A ) obtained asexponentials of locally ad -nilpotent elements is a subgroup of the full group ofautomorphisms Aut ( D ( A )) . ( iii ) There is an action of
Aut ( D ( A )) on Exp ( A ) such that there is an isomorphismof groups Aut ( D ( A )) = Aut ( D ( A )) (cid:110) Exp ( A ) . This knowledge of the automorphism group of D ( A )) allows us to describe the set ofnormal elements of the algebra and its birational class: Theorem E.
The set of normal elements of D ( A ) is the saturated multiplicatively closedsubset generated by Q . The maximal normal localization of D ( A ) , which is therefore D ( A )[ Q ] , is isomorphic to the localization of the Weyl algebra Diff ( S )[ Q ] . Finally, using —as it is often done— normal elements, we are able to classify thealgebras under study up to isomorphism:
Theorem F.
Let A and A be two central arrangements of lines in V . The algebras D ( A ) and D ( A ) are isomorphic if and only if the arrangements A and A themselvesare linearly isomorphic. This means, essentially, that we can reconstruct the arrangement from the algebra D ( A )of its differential operators.We expect most of the above results to hold in the general case of a free arrangementof hyperplanes of arbitrary rank. As our computations here make clear, some technologyis needed in order to deal with more complicated cases. In future work, we will showhow to organize this computation using the language of Lie–Rinehart pairs [12] andtheir cohomology theory. On the other hand, one can interpret the second cohomologyspace HH ( D ( A )) as classifying infinitesimal deformations of the algebra D ( A ) and use HH ( D ( A )) and our description of the Gerstenhaber bracket to study the deformationtheory of D ( A ). This produces a somewhat concrete interpretation of the secondcohomology space in geometrical terms. As this involves quite a bit of calculation, wedefer the exposition of these results to a later paper.The contents of this paper are part of the doctoral thesis of the first author.The paper is organized as follows. We start in Section by giving a concrete realizationof the algebra D ( A ) as an iterated Ore extension of a polynomial ring and proving someuseful lemmas. In Section we construct a resolution for D ( A ) and in Sections and we present the computation of the Hochschild cohomology HH • ( D ( A )) and itsGerstenhaber algebra structure. Section gives the much easier determination of theHochschild homology, cyclic homology, periodic homology and K -theory of our algebra, LGEBRAS OF DIFFERENTIAL OPERATORS 4 followed by the proof, in Section , of the twisted Calabi–Yau property. Finally, in thelast section we determine the automorphism group of D ( A ) and classify the algebras ofthis form up to isomorphism. Some notations
We will use the symbols . and / to denote the left and right actions ofan algebra on a bimodule whenever this improves clarity. We will have a ground field k of characteristic zero. All vector spaces and algebras are implicitly defined over k , andunadorned ⊗ and hom are taken with respect to k . If M is a vector space, we will oftendenote by M an element of M about which we do not need to be specific.We refer to the book [10] for a general reference about hyperplane arrangements andtheir derivations, and to C. Weibel’s book [17] for generalities about homological algebraand, in particular, Hochschild, cyclic and periodic theories.1. The algebra of differential operators associated to a centralarrangement of lines
We fix once and for all a ground field k of characteristic zero and put S = k [ x, y ].We view S as a graded algebra as usual, with both x and y of degree 1, and for each p ≥ S p the homogeneous component of S of degree p .We write Der ( S ) the Lie algebra of derivations of S , which is a free left graded S -module, freely generated by the usual partial derivatives ∂ x , ∂ y : S → S , which arehomogeneous elements of Der ( S ) of degree −
1. On the other hand, we write D ( S )the associative algebra of regular differential operators on S , as defined, for example,in [9, §15.5]. As this is by definition a subalgebra of End k ( S ), there is a tautologicalstructure of left D ( S )-module on S .There is an injective morphism of algebras φ : S → D ( S ) such that φ ( s )( a ) = as for all s , a ∈ S which we will view as an identification; elements in its image are the differentialoperators of order zero. Since S is a regular algebra, the algebra D ( S ) is generated as asubalgebra of End k ( S ) by S and Der ( S ); see [9, Corollary 15.5.6]. A consequence of thisis that D ( S ) is generated as an algebra by x , y , ∂ x and ∂ y , and in fact these elementsgenerate it freely subject to the relations[ x, y ] = [ ∂ x , y ] = [ ∂ y , x ] = [ ∂ x , ∂ y ] = 0 , [ ∂ x , x ] = [ ∂ y , y ] = 1 . It follows easily from this that D ( S ) has a Z -grading with x and y in degree 1 and ∂ x and ∂ y in degree −
1, and that with respect to this grading, S is a graded D ( S )-module. We fix an integer r ≥ − A of r + 2 lines in theplane A . Up to a change of coordinates, we may assume that the line with equation x = 0 is one of the lines in A , so that the defining polynomial Q of the arrangementis of the form xF for some square-free homogeneous polynomial F ∈ S of degree r + 1which does not have x as a factor. Up to multiplying by a scalar, which does not changeanything substantial, we may assume that F = x ¯ F + y r +1 for some ¯ F ∈ S r .We let Der ( A ) be the Lie algebra of derivations of S that preserve the arrangement,as in [10, §4.1], so that Der ( A ) = { d ∈ Der ( S ) : d ( Q ) ∈ QS } . LGEBRAS OF DIFFERENTIAL OPERATORS 5
This a graded Lie subalgebra of
Der ( S ). The two derivations E = x∂ x + y∂ y , D = F ∂ y are elements of Der ( A ) of degrees 0 and r , and it follows immediately from Saito’s criterion[10, Theorem 4.19] that the set { E, D } is a basis of Der ( A ) as a graded S -module; thisis the content of Example 4.20 in that book.The algebra of differential operators tangent to the arrangement A is the subalge-bra D ( A ) of D ( S ) generated by S and Der ( A ). It follows immediately from the remarksabove that D ( A ) is generated by x , y , E and D , and a computation shows that thefollowing commutation relations hold in D ( A ):[ y, x ] = 0 , [ D, x ] = 0 , [ D, y ] = F, (1)[ E, x ] = x, [ E, y ] = y, [ E, D ] = rD.
Since these generators are homogeneous elements in D ( S ) —with E of degree 0, x and y of degree 1 and D of degree r — we see that the algebra D ( A ) is a graded subalgebraof D ( S ) and, by restricting the structure from D ( S ), that S is a graded D ( A )-module.The set of commutation relations given above is in fact a presentation of the alge-bra D ( A ). More precisely, we have: Lemma.
The algebra D ( A ) is isomorphic to the iterated Ore extension S [ D ][ E ] . It is anoetherian domain and the set { x i y j D k E l : i, j, k, l ≥ } is a k -basis for D ( A ) . Here we view D as a derivation of S , so that we way construct the Ore extension S [ D ],and view E as a derivation of this last algebra, so as to be able extend once more toobtain S [ D ][ E ]. Proof.
It is clear at this point that the obvious map π : S [ D ][ E ] → D ( A ) is a surjectivemorphism of algebras, so we need only prove that it is injective. To do that, let ussuppose that there exists a non-zero element L in S [ D ][ E ] whose image under themap π is zero, and suppose that L = P i,j ≥ f i,j D i E j , with coefficients f i,j ∈ S for all i , j ≥
0, almost all of which are zero. As L is non-zero, we may consider the number m = max { i + j : f i,j = 0 } .Let us now fix a point p = ( a, b ) ∈ A which is not on any line of the arrangement A ,so that aF ( a, b ) = 0, and let O p be the completion of S at the ideal ( x − a, y − b ) or,more concretely, the algebra of formal series in x − a and y − b . We view O p as a leftmodule over D ( S ) in the tautological way and, by restriction, as a left D ( A )-module.There exist formal series φ and ψ in O p such that E · φ = 1 , D · φ = 0 , E · ψ = 0 , D · ψ = x r . Indeed, we may choose φ = ln x to satisfy the first two conditions, and the last two onesare equivalent to the equations ∂ x ψ = − x r − yF , ∂ y ψ = x r F , which can be solved for ψ , as the usual well-known sufficient integrability condition fromelementary calculus holds. If now s , t ∈ N are such that s + t = m , a straightforward LGEBRAS OF DIFFERENTIAL OPERATORS 6 computation shows that L · φ s ψ t = s ! t ! x rt f s,t in O p , and this implies that f s,t = 0. Thiscontradicts the choice of m and this contradiction proves what we want. (cid:3) We will use the following two simple lemmas a few times:
Lemma.
Suppose that r ≥ . If α , β ∈ S are such that αF x + βF y = 0 , then α = β = 0 . The conclusion of this statement is false if r < Proof.
Suppose that F , F and F are three distinct linear factors of F (here is wherewe need the hypothesis that r is at least 2) so that F = F F F F for some F ∈ S r − ; as F has degree at least 3, this is possible. We have F x ≡ F x F F F and F y ≡ F y F F F modulo F , so that ( αF x + βF y ) F F F ≡ mod F . Since F is square free, this tellsus that F divides αF x + βF y and, since both polynomials have the same degree and F = 0, that there exists a scalar λ such that αF x + βF y = λF . Of course, we can dothe same with the other two factors F and F . We can state this by saying that thematrix (cid:16) α x β x α y β y (cid:17) has the three vectors (cid:16) F x F y (cid:17) , (cid:16) F x F y (cid:17) and (cid:16) F x F y (cid:17) as eigenvectors. Since notwo of these are linearly dependent, because F is square-free, this implies that the matrixis in fact a scalar multiple of the identity, and there is a µ ∈ k such that α = µx and β = µy . The hypothesis is then that µ ( r + 1) F = µ ( xF x + yF y ) = 0, so that µ = 0. Thisproves the claim. (cid:3) If α , . . . , α r +1 ∈ S are such that F = Q r +1 i =1 α i , then the set of quotients { Fα , . . . , Fα r +1 } is a basis for S r .Proof. Suppose c , . . . , c r +1 ∈ k are scalars such that P r +1 i =1 c i Fα i = 0. If j ∈ { , . . . , r + 1 } ,we then have c j Fα j ≡ α j and, since F is square-free, this implies that in fact c j = 0. The set { Fα , . . . , Fα r +1 } is therefore linearly independent. Since dim S r = r + 1,this completes the proof. (cid:3) A projective resolution
We keep the situation of the previous section, and write from now on A insteadof D ( A ). Our immediate objective is to construct a projective resolution of A as an A -bimodule, and we do this by looking at A as a deformation of a commutative polynomialalgebra, which suggests that it should have a resolution resembling the usual Koszulcomplex. If U is a vector space and u ∈ U , there are derivations ∇ ux , ∇ uy : S → S ⊗ U ⊗ S of S into the S -bimodule S ⊗ U ⊗ S uniquely determined by the condition that ∇ ux ( x ) = 1 ⊗ u ⊗ , ∇ ux ( y ) = 0 , ∇ uy ( x ) = 0 , ∇ uy ( y ) = 1 ⊗ u ⊗ , and in fact we have, for every i , j ≥
0, that ∇ ux ( x i y j ) = X s + t +1= i x s ⊗ u ⊗ x t y j , ∇ uy ( x i y j ) = X s + t +1= j x i y s ⊗ u ⊗ y s . We consider the derivation ∇ = ∇ xx + ∇ yy : S → S ⊗ S ⊗ S ; it is the unique derivationsuch that ∇ ( α ) = 1 ⊗ α ⊗ α ∈ S . There is, on the other hand, a uniquemorphism of S -bimodules d : S ⊗ S ⊗ S → S ⊗ S such that d (1 ⊗ α ⊗
1) = α ⊗ − ⊗ α LGEBRAS OF DIFFERENTIAL OPERATORS 7 for all α ∈ S , and we have d ( ∇ ( f )) = f ⊗ − ⊗ f for all f ∈ S . To check this last equality, it is enough to notice that d ◦ ∇ : S → S ⊗ S is a derivation and, since S generates S as an algebra, that the equality holds when f ∈ S . Let V be the subspace of A spanned by x , y , D and E . This is a graded subspaceand its grading induces on the exterior algebra Λ • ( V ) an internal grading. If ω is anelement of an exterior power Λ p ( V ) of V , we write ( − ) ∧ ω the map of A -bimodules A ⊗ S ⊗ A → A ⊗ Λ p +1 V ⊗ A such that (1 ⊗ α ⊗ ∧ ω = 1 ⊗ α ∧ ω ⊗ α ∈ S . There is a chain complex P of free graded A -bimodules of the form A | Λ V | A A | Λ V | A A | Λ V | A A | V | A A | A d d d d (2)with A e -linear maps homogeneous of degree zero and such that d (1 | v |
1) = [ v, | , ∀ v ∈ V ; d (1 | x ∧ y |
1) = [ x, | y | − [ y, | x | d (1 | x ∧ E |
1) = [ x, | E | − [ E, | x |
1] + 1 | x | d (1 | y ∧ E |
1) = [ y, | E | − [ E, | y |
1] + 1 | y | d (1 | x ∧ D |
1) = [ x, | D | − [ D, | x | d (1 | y ∧ D |
1) = [ y, | D | − [ D, | y |
1] + ∇ ( F ); d (1 | D ∧ E |
1) = [ D, | E | − [ E, | D |
1] + r | D | d (1 | x ∧ y ∧ D |
1) = [ x, | y ∧ D | − [ y, | x ∧ D |
1] + [ D, | x ∧ y |
1] + ∇ ( F ) ∧ x ; d (1 | x ∧ y ∧ E |
1) = [ x, | y ∧ E | − [ y, | x ∧ E |
1] + [ E, | x ∧ y | − | x ∧ y | d (1 | x ∧ D ∧ E |
1) = [ x, | D ∧ E | − [ D, | x ∧ E |
1] + [ E, | x ∧ D | − ( r + 1) | x ∧ D | d (1 | y ∧ D ∧ E |
1) = [ y, | D ∧ E | − [ D, | y ∧ E |
1] + [ E, | y ∧ D | ∇ ( F ) ∧ E − ( r + 1) | y ∧ D | d (1 | x ∧ y ∧ D ∧ E |
1) = [ x, | y ∧ D ∧ E | − [ y, | x ∧ D ∧ E | D, | x ∧ y ∧ E | − [ E, | x ∧ y ∧ D | ∇ ( F ) ∧ x ∧ E + ( r + 2) | x ∧ y ∧ D | . That P is indeed a complex follows from a direct calculation. More interestingly, it isexact: Lemma.
The complex P is a projective resolution of A as an A -bimodule, with augmen-tation d : A | A → A such that d (1 |
1) = 1 .Proof.
For each p ∈ N we consider the subspace F p A = h x i y j D k E l : k + l ≤ p i of A . As aconsequence of Lemma 1.2, one sees that F A = ( F p A ) p ≥ is an exhaustive and increasingalgebra filtration on A and that the corresponding associated graded algebra gr ( A ) is LGEBRAS OF DIFFERENTIAL OPERATORS 8 isomorphic to the usual commutative polynomial ring k [ x, y, D, E ]. Since V is a subspaceof A , we can restrict the filtration of A to one on V , and the latter induces as usuala filtration on each exterior power Λ p V . In this way we obtain a filtration on eachcomponent of the complex P , which turns out to be compatible with its differentials,as can be checked by inspection. The complex gr ( P ) obtained from P by passing toassociated graded objects in each degree is isomorphic to the Koszul resolution of gr ( A )as a gr ( A )-bimodule and it is therefore acyclic over gr ( A ). A standard argument usingthe filtration of P concludes from this that the complex P itself acyclic over A . As itscomponents are manifestly free A -bimodules, this proves the lemma. (cid:3) One almost immediate application of having a bimodule projective resolution forour algebra is in computing its global dimension.
Proposition.
The global dimension of A is equal to . Of course, as A is noetherian, there is no need to distinguish between the left and theright global dimensions. Proof. If λ ∈ k let M λ be the left A -module which as a vector space is freely spannedby an element u λ and on which the action of A is such that x · u λ = y · u λ = D · u λ = 0and E · u λ = λu λ . It is easy to see that all 1-dimensional A -modules are of this form andthat M λ ∼ = M µ iff λ = µ , but we will not need this.The complex P ⊗ A M λ is a projective resolution of M λ as a left A -module, and thereforethe cohomology of hom A ( P ⊗ A M λ , M µ ) is canonically isomorphic to Ext • A ( M λ , M µ ).Identifying as usual hom A ( P ⊗ A M λ , M µ ) to M µ ⊗ Λ • V ∗ ⊗ M ∗ λ , we compute that thecomplex is M µ ⊗ M ∗ λ M µ ⊗ V ∗ ⊗ M ∗ λ M µ ⊗ Λ V ∗ ⊗ M ∗ λδ δ δ M µ ⊗ Λ V ∗ ⊗ M ∗ λ M µ ⊗ Λ V ∗ ⊗ M ∗ λδ with differentials given by δ (1) = ( µ − λ ) ⊗ ˆ E,δ ( a ⊗ ˆ x + b ⊗ ˆ y + c ⊗ ˆ D + d ⊗ ˆ D )= ( λ + 1 − µ ) a ⊗ ˆ x ∧ ˆ E + ( λ + 1 − µ ) b ⊗ ˆ y ∧ ˆ E + ( λ + r − µ ) c ⊗ ˆ D ∧ ˆ E,δ ( a ⊗ ˆ x ∧ ˆ y + b ⊗ ˆ x ∧ ˆ E + c ⊗ ˆ y ∧ ˆ E + d ⊗ ˆ x ∧ ˆ D + e ˆ y ∧ ˆ D + f ˆ D ∧ ˆ E )= ( µ − λ − a ⊗ ˆ x ∧ ˆ y ∧ ˆ E + ( µ − λ − r − d ⊗ ˆ x ∧ ˆ D ∧ ˆ E + ( µ − λ − r − e ⊗ ˆ y ∧ ˆ D ∧ ˆ E,δ ( a ⊗ ˆ x ∧ ˆ y ∧ ˆ D + b ⊗ ˆ x ∧ ˆ y ∧ ˆ E + c ⊗ ˆ x ∧ ˆ D ∧ ˆ E + d ⊗ ˆ y ∧ ˆ D ∧ ˆ E )= ( λ + r + 2 − µ ) a ⊗ ˆ x ∧ ˆ y ∧ ˆ D ∧ ˆ E. An easy computation shows that dim Ext pA ( M λ , M λ + r +2 ) = ( , if p = 3 or p = 4;0 , in any other case. LGEBRAS OF DIFFERENTIAL OPERATORS 9
In particular,
Ext A ( M λ , M λ + r +2 ) = 0 and therefore gldim A ≥
4. On the other hand, wehave constructed a projective resolution of A as an A -bimodule of length 4, so that theprojective dimension of A as a bimodule is pdim A e A ≤
4. Since gldim A ≤ pdim A e A ,the proposition follows from this. (cid:3) The Hochschild cohomology of D ( A ) We want to compute the Hochschild cohomology of the algebra A . Applying thefunctor hom A e ( − , A ) to the resolution P of we get, after standard identifications,the cochain complex A A ⊗ V ∗ A ⊗ Λ V ∗ A ⊗ Λ V ∗ A ⊗ Λ V ∗ d d s d s d s s which we denote simply by A ⊗ Λ V ∗ , with differentials such that d ( a ) = [ x, a ] ⊗ ˆ x + [ y, a ] ⊗ ˆ y + [ D, a ] ⊗ ˆ D + [ E, a ] ⊗ ˆ E ; d ( a ⊗ ˆ x ) = − [ y, a ] ⊗ ˆ x ∧ ˆ y + ( a − [ E, a ]) ⊗ ˆ x ∧ ˆ E − [ D, a ] ⊗ ˆ x ∧ ˆ D + ∇ ax ( F ) ⊗ ˆ y ∧ ˆ D ; d ( a ⊗ ˆ y ) = [ x, a ] ⊗ ˆ x ∧ ˆ y + ( a − [ E, a ]) ⊗ ˆ y ∧ ˆ E + ( ∇ ay ( F ) − [ D, a ]) ⊗ ˆ y ∧ ˆ D ; d ( a ⊗ ˆ D ) = [ x, a ] ⊗ ˆ x ∧ ˆ D + [ y, a ] ⊗ ˆ y ∧ ˆ D + ( ra − [ E, a ]) ⊗ ˆ D ∧ ˆ E ; d ( a ⊗ ˆ E ) = [ x, a ] ⊗ ˆ x ∧ ˆ E + [ y, a ] ⊗ ˆ y ∧ ˆ E + [ D, a ] ⊗ ˆ D ∧ ˆ E ; d ( a ⊗ ˆ x ∧ ˆ y ) = ([ D, a ] − ∇ ay ( F )) ⊗ ˆ x ∧ ˆ y ∧ ˆ D + ([ E, a ] − a ) ⊗ ˆ x ∧ ˆ y ∧ ˆ E ; d ( a ⊗ ˆ x ∧ ˆ E ) = − [ y, a ] ⊗ ˆ x ∧ ˆ y ∧ ˆ E − [ D, a ] ⊗ ˆ x ∧ ˆ D ∧ ˆ E + ∇ ax ( F ) ⊗ ˆ y ∧ ˆ D ∧ ˆ E ; d ( a ⊗ ˆ y ∧ ˆ E ) = [ x, a ] ⊗ ˆ x ∧ ˆ y ∧ ˆ E + ( ∇ ay ( F ) − [ D, a ]) ⊗ ˆ y ∧ ˆ D ∧ ˆ E ; d ( a ⊗ ˆ x ∧ ˆ D ) = − [ y, a ] ⊗ ˆ x ∧ ˆ y ∧ ˆ D + ([ E, a ] − ( r + 1) a ) ⊗ ˆ x ∧ ˆ D ∧ ˆ E ; d ( a ⊗ ˆ y ∧ ˆ D ) = [ x, a ] ⊗ ˆ x ∧ ˆ y ∧ ˆ D + ([ E, a ] − ( r + 1) a ) ⊗ ˆ y ∧ ˆ D ∧ ˆ E ; d ( a ⊗ ˆ D ∧ ˆ E ) = [ x, a ] ⊗ ˆ x ∧ ˆ D ∧ ˆ E + [ y, a ] ⊗ ˆ y ∧ ˆ D ∧ ˆ E ; d ( a ⊗ ˆ x ∧ ˆ y ∧ ˆ D ) = ( − [ E, a ] + ( r + 2) a ) ⊗ ˆ x ∧ ˆ y ∧ ˆ D ∧ ˆ E ; d ( a ⊗ ˆ x ∧ ˆ y ∧ ˆ E ) = ([ D, a ] − ∇ ay ( F )) ⊗ ˆ x ∧ ˆ y ∧ ˆ D ∧ ˆ E ; d ( a ⊗ ˆ x ∧ ˆ D ∧ ˆ E ) = − [ y, a ] ⊗ ˆ x ∧ ˆ y ∧ ˆ D ∧ ˆ E ; d ( a ⊗ ˆ y ∧ ˆ D ∧ ˆ E ) = [ x, a ] ⊗ ˆ x ∧ ˆ y ∧ ˆ D ∧ ˆ E. These differentials are homogeneous with respect to the natural internal grading on thecomplex A ⊗ Λ V ∗ coming from the grading of A . We denote γ : A ⊗ Λ V ∗ → A ⊗ Λ V ∗ the k -linear map whose restriction to each homogeneous component of the complex A ⊗ Λ V ∗ is simply the multiplication by the degree. There is a homotopy, drawn in the diagram (2)with dashed arrows, with s ( a ⊗ ˆ x + b ⊗ ˆ y + c ⊗ ˆ D + d ⊗ ˆ E ) = d, LGEBRAS OF DIFFERENTIAL OPERATORS 10 s ( a ⊗ ˆ x ∧ ˆ y + b ⊗ ˆ x ∧ ˆ E + c ⊗ ˆ y ∧ ˆ E + d ⊗ ˆ x ∧ ˆ D + e ⊗ ˆ y ∧ ˆ D + f ⊗ ˆ D ∧ ˆ E )= − b ⊗ ˆ x − c ⊗ ˆ y − f ⊗ ˆ D,s ( a ⊗ ˆ x ∧ ˆ y ∧ ˆ D + b ⊗ ˆ x ∧ ˆ y ∧ ˆ E + c ⊗ ˆ x ∧ ˆ D ∧ ˆ E + d ⊗ ˆ y ∧ ˆ D ∧ ˆ E )= b ⊗ ˆ x ∧ ˆ y + c ⊗ ˆ x ∧ ˆ D + d ⊗ ˆ y ∧ ˆ D,s ( a ⊗ ˆ x ∧ ˆ y ∧ ˆ D ∧ ˆ E ) = − a ⊗ ˆ x ∧ ˆ y ∧ ˆ D and such that d ◦ s + s ◦ d = γ : this tells us that γ induces the zero map on cohomol-ogy. Since our ground field k has characteristic zero, this implies that the inclusion( A ⊗ Λ V ∗ ) → A ⊗ Λ V ∗ of the component of degree zero of our complex A ⊗ Λ V ∗ is aquasi-isomorphism. r ≥ . Let us write the complex ( A ⊗ Λ V ∗ ) simply X and let us put T = k [ E ], which coincideswith A . The complex X has components X = A , X = A ⊗ ( k ˆ x ⊕ k ˆ y ) ⊕ A r ⊗ k ˆ D ⊕ A ⊗ k ˆ E, X = A ⊗ k ˆ x ∧ ˆ y ⊕ A ⊗ ( k ˆ x ∧ ˆ E ⊕ k ˆ y ∧ ˆ E ) ⊕ A r ⊗ k ˆ D ∧ ˆ E ⊕ A r +1 ⊗ ( k ˆ x ∧ ˆ D ⊕ k ˆ y ∧ ˆ D ) , X = A ⊗ ˆ x ∧ ˆ y ∧ ˆ E ⊕ A r +1 ⊗ ( k ˆ x ∧ ˆ D ∧ ˆ E ⊕ k ˆ y ∧ ˆ D ∧ ˆ E ) ⊕ A r +2 ⊗ k ˆ x ∧ ˆ y ∧ ˆ D, X = A r +2 ⊗ ˆ x ∧ ˆ y ∧ ˆ D ∧ ˆ E and, since r >
2, we have A = T, A = S T, A = S T,A r = ( S r ⊕ k D ) T, A r +1 = ( S r +1 ⊕ S D ) T, A r +2 = ( S r +2 ⊕ S D ) T. In fact, this is where our assumption that r ≥ r ≤
2, then these subspacesof A have different descriptions.The differentials in X can be computed to be given by δ ( a ) = xτ ( a ) ⊗ ˆ x + yτ ( a ) ⊗ ˆ y + Dτ r ( a ) ⊗ ˆ D,δ ( φa ⊗ ˆ x ) = − φyτ ( a ) ⊗ ˆ x ∧ ˆ y − ( F φ y a + φDτ r ( a )) ⊗ ˆ x ∧ ˆ D + ∇ φax ( F ) ⊗ ˆ y ∧ ˆ D,δ ( φa ⊗ ˆ y ) = φxτ ( a ) ⊗ ˆ x ∧ ˆ y + ( ∇ φay ( F ) − F φ y a − φDτ r ( a )) ⊗ ˆ y ∧ ˆ D,δ (( φ + λD ) a ⊗ ˆ D ) = ( φxτ ( a ) + λxDτ ( a )) ⊗ ˆ x ∧ ˆ D + ( φyτ ( a ) + λF ( τ ( a ) − a ) + λyDτ ( a )) ⊗ ˆ y ∧ ˆ D,δ ( a ⊗ ˆ E ) = xτ ( a ) ⊗ ˆ x ∧ ˆ E + yτ ( a ) ⊗ ˆ y ∧ ˆ E + Dτ r ( a ) ⊗ ˆ D ∧ ˆ E,δ ( φa ⊗ ˆ x ∧ ˆ y ) = ( F φ y a + φDτ r ( a ) − ∇ φay ( F )) ⊗ ˆ x ∧ ˆ y ∧ ˆ D,δ ( φa ⊗ ˆ x ∧ ˆ E ) = − φyτ ( a ) ⊗ ˆ x ∧ ˆ y ∧ ˆ E − ( F φ y a + φDτ r ( a )) ⊗ ˆ x ∧ ˆ D ∧ ˆ E + ∇ φax ( F ) ⊗ ˆ y ∧ ˆ D ∧ ˆ E, LGEBRAS OF DIFFERENTIAL OPERATORS 11 δ ( φa ⊗ ˆ y ∧ ˆ E ) = φxτ ( a ) ⊗ ˆ x ∧ ˆ y ∧ ˆ E + ( ∇ φay ( F ) − F φ y a − φDτ r ( a )) ⊗ ˆ y ∧ ˆ D ∧ ˆ E,δ (( φ + ψD ) a ⊗ ˆ x ∧ ˆ D ) = ( − φyτ ( a ) − ψF ( τ ( a ) − a ) − ψyDτ ( a )) ⊗ ˆ x ∧ ˆ y ∧ ˆ D,δ (( φ + ψD ) a ⊗ ˆ y ∧ ˆ D ) = ( φxτ ( a ) + ψxDτ ( a )) ⊗ ˆ x ∧ ˆ y ∧ ˆ D,δ (( φ + λD ) a ⊗ ˆ D ∧ ˆ E ) = ( φxτ ( a ) + λxDτ ( a )) ⊗ ˆ x ∧ ˆ D ∧ ˆ E + ( φyτ ( a ) + λyDτ ( a ) + λF ( τ ( a ) − a )) ⊗ ˆ y ∧ ˆ D ∧ ˆ E,δ (( φ + ψD ) a ⊗ ˆ x ∧ ˆ y ∧ ˆ D ) = 0 ,δ ( φa ⊗ ˆ x ∧ ˆ y ∧ ˆ E ) = ( F φ y a + φDτ r ( a ) − ∇ φay ( F )) ⊗ ˆ x ∧ ˆ y ∧ ˆ D ∧ ˆ E,δ (( φ + ψD ) a ⊗ ˆ x ∧ ˆ D ∧ ˆ E )= − ( φyτ ( a ) + ψyDτ ( a ) + ψF ( τ ( a ) − a )) ⊗ ˆ x ∧ ˆ y ∧ ˆ D ∧ ˆ E,δ (( φ + ψD ) a ⊗ ˆ y ∧ ˆ D ∧ ˆ E ) = ( φxτ ( a ) + ψxDτ ( a )) ⊗ ˆ x ∧ ˆ y ∧ ˆ D ∧ ˆ E. Here and below τ t : T → T is the k -linear map such that τ t ( E n ) = E n − ( E + t ) n for all n ∈ N , and φ and φ denote homogeneous elements of r of appropriate degrees and λ ascalar. We proceed to compute the cohomology of the complex X , starting with degreeszero and four, for which the computation is almost immediate. Indeed, since the kernel of τ and of τ r is k ⊆ T , it is clear that H ( X ) = ker δ = k . On the other hand, if ψ ∈ S and a ∈ T , we can write ψ = ψ x + ψ y for some ψ , ψ ∈ S and there is a b ∈ T suchthat τ ( b ) = a , so that δ ( − ψ Db ⊗ ˆ x ∧ ˆ D ∧ ˆ E + ψ Db ⊗ ˆ y ∧ ˆ D ∧ ˆ E ) = ( ψDa + S r +2 T ) ⊗ ˆ x ∧ ˆ y ∧ ˆ D ∧ ˆ E. Similarly, we have δ ( S r +1 T ⊗ ˆ x ∧ ˆ D ∧ ˆ E + S r +1 T ⊗ ˆ y ∧ ˆ D ∧ ˆ E ) = S r +2 T ⊗ ˆ x ∧ ˆ y ∧ ˆ D ∧ ˆ E .These two facts imply that the map δ is surjective, so that H ( X ) = 0. Let ω ∈ X be a 1-cocycle in X . There are then a , b , c , d , e , f ∈ T , k ∈ N and φ , . . . , φ k ∈ S r such that either k = 0 or φ k = 0, and ω = ( xa + yb ) ⊗ ˆ x + ( xc + yd ) ⊗ ˆ y + k X i =0 φ i E i + De ! ⊗ ˆ D + f ⊗ ˆ E. If ¯ e ∈ T is such that τ r (¯ e ) = e , then by replacing ω by ω − δ (¯ e ), which does not changethe cohomology class of ω , we can assume that e = 0. The formula for δ then showsthat ω is a coboundary iff it is equal to zero. The coefficient of ˆ x ∧ ˆ y in δ ( ω ) is x τ ( c ) + xy ( τ ( d ) − τ ( a )) − y τ ( b ) = 0 . We therefore have b , c , d − a ∈ k . The coefficient of ˆ D ∧ ˆ E , on the other hand, is Dτ r ( f ) = 0, so that also f ∈ k ; exactly the same information comes from the vanishingof the coefficients of ˆ x ∧ ˆ E and of ˆ y ∧ ˆ E . Since b ∈ k , the coefficient of ˆ x ∧ ˆ D is − F b − xDτ r ( a ) + k X i =0 φ i xτ ( E i ) = 0 . LGEBRAS OF DIFFERENTIAL OPERATORS 12
We see that τ r ( a ) = 0, so that a ∈ k , and that P ki =0 φ i xτ ( E i ) = F b . This implies that k ≤
1, that − φ x = F b and therefore, since x is not a factor of F by hypothesis, that φ = 0 and b = 0.Finally, using all the information we have so far, we can see that the vanishing of thecoefficient of ˆ y ∧ ˆ D in δ ( ω ) implies that F x xa + F y ( xc + yd ) = F d . Together with Euler’srelation F x x + F y y = ( r + 1) F this tells us that( cx + ( d − a ) y ) F y = ( d − ( r + 1) a ) F. (3)As F is square-free, it follows from this equality the polynomial cx + ( d − a ) y is zero sothat c = 0 and d = a and, finally, that a = 0. We conclude in this way that the set of1-cocycles φ ⊗ ˆ D + f ⊗ ˆ E, φ ∈ S r , f ∈ k is a complete, irredundant set of representatives for the elements of H ( X ). Let ω ∈ X be a 3-cocycle, so that ω = a ⊗ ˆ x ∧ ˆ y ∧ ˆ D + b ⊗ ˆ x ∧ ˆ y ∧ ˆ E + c ⊗ ˆ x ∧ ˆ D ∧ ˆ E + d ⊗ ˆ y ∧ ˆ D ∧ ˆ E for some a ∈ ( S r +2 ⊕ S D ) T , b ∈ S T , c , d ∈ ( S r +1 ⊕ S D ) T and δ ( ω ) = 0. For all φ ∈ S and e ∈ T we have δ ( φe ⊗ ˆ x ∧ ˆ E ) = − φyτ ( e ) ⊗ ˆ x ∧ ˆ y ∧ ˆ E + A r +1 ⊗ ˆ x ∧ ˆ D ∧ ˆ E + A r +1 ⊗ ˆ y ∧ ˆ D ∧ ˆ E and δ ( φe ⊗ ˆ y ∧ ˆ E ) = φxτ ( e ) ⊗ ˆ x ∧ ˆ y ∧ ˆ E + A r +1 ⊗ ˆ y ∧ ˆ D ∧ ˆ E, so that by adding to ω an element of δ ( S T ⊗ ˆ x ∧ ˆ E + S T ⊗ ˆ y ∧ ˆ E ), which does notchange the cohomology class of ω , we can suppose that b = 0. Similarly, for all φ ∈ S and all e ∈ T we have δ ( φe ⊗ ˆ x ∧ ˆ y ) = ( S r +2 T + φDτ r ( e )) ⊗ ˆ x ∧ ˆ y ∧ ˆ D, and for all φ ∈ S r +1 and all e ∈ T we have δ ( φe ⊗ ˆ x ∧ ˆ D ) = − φyτ ( e ) ⊗ ˆ x ∧ ˆ y ∧ ˆ D and δ ( φe ⊗ ˆ y ∧ ˆ D ) = φxτ ( e ) ⊗ ˆ x ∧ ˆ y ∧ ˆ D. Using this we see that, up to changing ω by adding to it a 3-coboundary, we can supposethat a = 0. Finally, for each φ ∈ S r and all e ∈ T we have δ ( φe ⊗ ˆ D ∧ ˆ E ) = φxτ ( e ) ⊗ ˆ x ∧ ˆ D ∧ ˆ E + A r +1 ⊗ ˆ y ∧ ˆ D ∧ ˆ E,δ ( De ⊗ ˆ D ∧ ˆ E ) = xDτ ( e ) ⊗ ˆ x ∧ ˆ D ∧ ˆ E + A r +1 ⊗ ˆ y ∧ ˆ D ∧ ˆ E Suppose that u = cx + ( d − a ) y is not zero. Differentiating in (3) with respect to y , we find that − raF y = uF yy . Since x does not divide F , we have F yy = 0, and then a = 0 and u divides F y : from (3)it follows then that u divides F , since the left hand side of that equality is non-zero, and this is absurdbecause F is square-free. LGEBRAS OF DIFFERENTIAL OPERATORS 13 and δ ( − y ⊗ ˆ x ∧ ˆ E + ¯ F E ⊗ ˆ D ∧ ˆ E ) = y r +1 ⊗ ˆ x ∧ ˆ D ∧ ˆ E + A r +1 ⊗ ˆ y ∧ ˆ D ∧ ˆ E, so we can also suppose that c ∈ y r +1 ET + yDT .There are l ≥ λ , . . . , λ l , µ , . . . , µ l ∈ k , φ , . . . , φ l ∈ S r +1 , ψ , . . . , ψ l ∈ S , ζ , . . . , ζ l ∈ S such that c = P li =1 λ i y r +1 E i + P li =0 µ i yDE i and d = P li =0 ( φ i + ψ i D ) E i .The vanishing of δ ( ω ) means precisely that l X i =1 λ i y r +2 τ ( E i )+ l X i =0 (cid:16) µ i y Dτ ( E i ) − µ i yF ( E +1) i − φ i xτ ( E i ) − ψ i xDτ ( E i ) (cid:17) = 0 . The left hand side of this equation is an element of S r +2 T ⊕ S DT . The componentin S DT is P li =0 ( µ i y − ψ i x ) Dτ ( E i ) = 0 and therefore µ i = ψ i = 0 for all i ∈ { , . . . , l } .On the other hand, the component in S r +2 T is l X i =1 λ i y r +2 τ ( E i ) − µ yF − l X i =0 φ i xτ ( E i ) = 0 . This implies that λ i y r +2 − φ i x = 0 if i ∈ { , . . . , l } , so that λ i = φ i = 0 for such i ,and then the equation reduces to λ y r +2 + µ yF − φ x = 0. Recalling from that F = y r +1 + x ¯ F , we deduce from this that λ = − µ and φ = µ y ¯ F . We conclude in thisway that every 3-cocycle is cohomologous to one of the form( µ yD − µ y r +1 E )ˆ x ∧ ˆ D ∧ ˆ E + ( φ + ψ D + µ y ¯ F E )ˆ y ∧ ˆ D ∧ ˆ E (4)with µ ∈ k , φ ∈ S r +1 and ψ ∈ S , and a direct computation shows that moreoverevery 3-cochain of this form is a 3-cocycle.Let now η be a 2-cochain η in X , so that η = A ⊗ k ˆ x ∧ ˆ y ⊕ A r +1 ⊗ ( k ˆ x ∧ ˆ D ⊕ k ˆ y ∧ ˆ D ) + u ⊗ ˆ x ∧ ˆ E + v ⊗ ˆ y ∧ ˆ E + w ⊗ ˆ D ∧ ˆ E with u , v ∈ A and w ∈ A r , and let us suppose that δ ( η ) is equal to the 3-cocycle (4).There are l ≥ α , . . . , α l , β , . . . , β l ∈ S , γ , . . . , γ l ∈ S r and ξ , . . . , ξ l ∈ k such that u = P li =0 α i E i , v = P li =0 β i E i and w = P li =0 ( γ i + ξ i D ) E i . The coefficient of ˆ x ∧ ˆ y ∧ ˆ E in δ ( η ) must be equal to zero, so that l X i =0 ( − α i y + β i x ) τ ( E i ) = 0 , and this implies that there are scalars ρ , . . . , ρ l ∈ k such that α i = ρ i x and β i = ρ i y forall i ∈ { , . . . , l } .Looking now at the coefficient of ˆ x ∧ ˆ D ∧ ˆ E in δ ( η ) and comparing with (4) we findthat l X i =0 (cid:16) − F α iy E i − α i Dτ r ( E i ) + γ i xτ ( E i ) + ξ i xDτ ( E i ) (cid:17) = µ yD − µ y r +1 E. (5) LGEBRAS OF DIFFERENTIAL OPERATORS 14
This is an equality of two elements of ST ⊕ SDT . Considering the components in DT ,we find that xD P li =1 ( − ρ i τ r ( E i ) + ξ i τ ( E i )) = µ yD , and this tells us that µ = 0 andthat l X i =1 (cid:16) − ρ i τ r ( E i ) + ξ i τ ( E i ) (cid:17) = 0 . (6)On the other hand, as the components in ST of the two sides of (5) are equal, we have − F α y + l X i =0 γ i xτ ( E i ) = 0 , so that γ i = 0 for all i ∈ { , . . . , l } and F α y + γ x = 0. As x does not divide F , wemust have α y = 0 and γ = 0; in particular, there is a ρ ∈ k such that α = ρ x .Finally, considering the coefficient of ˆ y ∧ ˆ D ∧ ˆ E of δ ( η ) and of (4) we see that l X i =0 (cid:16) ∇ α i E i x ( F ) + ∇ β i E i y ( F ) − F β iy E i − β i Dτ r ( E i )+ γ i yτ ( E i ) + ξ i yDτ ( E i ) − ξ i F ( E + 1) i (cid:17) = φ + ψ D, which at this point we can rewrite (using in the process the equality (6) above and thefact that ∇ xE i x ( F ) + ∇ yE i x ( F ) = F P rt =0 ( E + t ) i ) as ρ xF x + β F y − F β y + ξ − l X i =1 ρ i r X t =1 ( E + t ) i − ξ i ( E + 1) i !! = φ + ψ D. It follows at once that ψ = 0 and that, in fact, ρ xF x + β F y − F β y + ξ − l X i =1 ρ i r X t =1 t i − ξ i !! = φ . The polynomial φ is then in the linear span of xF x , xF y , yF y and F inside S r +1 . Euler’srelation implies that already the first three polynomials span this subspace, and we have δ ( x ⊗ ˆ x ∧ ˆ E ) = xF x ⊗ ˆ y ∧ ˆ D ∧ ˆ E,δ ( x ⊗ ˆ y ∧ ˆ E ) = xF y ⊗ ˆ y ∧ ˆ D ∧ ˆ E,δ ( y ⊗ ˆ y ∧ ˆ E − D ⊗ ˆ D ˆ E ) = yF y ⊗ ˆ y ∧ ˆ D ∧ ˆ E. (7)We conclude in this way that the only 3-coboundaries among the cocycles of the form (4)are the linear combinations of the right hand sides of the equalities (7); these threecocycles are, moreover, linearly independent. This means that there is an isomorphism H ( X ) ∼ = k ω ⊕ S D ⊗ ˆ y ∧ ˆ D ∧ ˆ E ⊕ S r +1 h xF x , xF y , yF y i ⊗ ˆ y ∧ ˆ D ∧ ˆ E, with ω = ( yD − y r +1 E ) ⊗ ˆ x ∧ ˆ D ∧ ˆ E + y ¯ F E ⊗ ˆ y ∧ ˆ D ∧ ˆ E, LGEBRAS OF DIFFERENTIAL OPERATORS 15 and that, in particular, dim H ( X ) = r + 2, since the denominator appearing in the righthand side of this isomorphism is a 3-dimensional vector space —this follows at once fromLemma . We consider now a 2-cocycle ω ∈ X and a ∈ S T , b , c ∈ S T , d , e ∈ S r +1 T ⊕ S DT and f ∈ S r T ⊕ DT such that ω = a ⊗ ˆ x ∧ ˆ y + b ⊗ ˆ x ∧ ˆ E + c ⊗ ˆ y ∧ ˆ E + d ⊗ ˆ x ∧ ˆ D + e ⊗ ˆ y ∧ ˆ D + f ⊗ ˆ E ∧ ˆ D. Adding to ω an element of δ ( T ⊗ ˆ E ), we can assume that f ∈ S r T ; adding an elementof δ ( S T ⊗ ˆ x ⊕ S T ⊗ ˆ y ), we can suppose that a = 0; finally, adding an elementof δ (( S r T ⊕ DT ) ⊗ ˆ D ) we can suppose that d ∈ y r +1 T ⊕ yDT . In this situation,there are an integer l ≥ α , . . . , α l , β , . . . , β l ∈ S , λ , . . . , λ l , µ , . . . , µ l ∈ k , φ , . . . , φ l ∈ S r +1 , ψ , . . . , ψ l ∈ S and ξ , . . . , ξ l ∈ S r such that b = P li =0 α i E i , c = P li =0 β i E i , d = P li =0 ( λ i y r +1 + µ i yD ) E i , e = P li =0 ( φ i + ψ i D ) E i and f = P li =0 ξ i E i .As δ ( − y ⊗ ˆ x + ¯ F E ⊗ ˆ D ) = y r +1 ⊗ ˆ x ∧ ˆ D + S r +1 ⊗ ˆ y ∧ ˆ D, we can assume that λ = 0.The coefficient of ˆ x ∧ ˆ y ∧ ˆ E in δ ( ω ) is P li =0 ( − α i y + β i x ) τ ( E i ) = 0, and this implies thatthere are scalars ρ , . . . , ρ l ∈ k such that α i = ρ i x and β i = ρ i y for each i ∈ { , . . . , l } .The coefficient of ˆ x ∧ ˆ D ∧ ˆ E in δ ( ω ) is l X i =0 (cid:0) − F α iy E i − α i Dτ r ( E i ) + ξ i xτ ( E i ) (cid:1) = 0 . (8)It follows that P li =0 α i Dτ r ( E i ) = 0, so that α = · · · = α l = 0; as a consequence ofthis, we have that ρ = · · · = ρ l = 0 and β = · · · = β l = 0. The equality (8) also tellsus that − F α y + P li =0 ξ i xτ ( E i ) = 0, and from this we see that ξ = · · · = ξ l = 0 and − F α y − ξ x = 0, so that α y = 0 and ξ = 0, since x does not divide F . In particular,there is a ρ ∈ k such that α = ρ x .The coefficient of ˆ y ∧ ˆ D ∧ ˆ E in δ ( ω ) is l X i =0 (cid:16) ∇ α i E i x ( F ) + ∇ β i E i y ( F ) − F β iy E i − β i Dτ r ( E i ) + ξ i yτ ( E i ) (cid:17) = ρ xF x + β F y − β y F = (cid:0) ρ − ( r + 1) − β y (cid:1) xF x + (cid:0) β x x + (1 − (1 + r ) − ) β y y ) F y = 0 , and our Lemma implies then that β = 0 and ρ = 0. Finally, we consider thecoefficient of ˆ x ∧ ˆ y ∧ ˆ D : l X i =0 (cid:16) − λ i y r +2 τ ( E i )+ µ i yF ( E +1) i − µ i y Dτ ( E i )+ φ i xτ ( E i )+ ψ i xDτ ( E i ) (cid:17) = 0 . Looking at the terms involving D in this equation, we see that l X i =0 ( − µ i y + ψ i x ) Dτ ( E i )) = 0 , LGEBRAS OF DIFFERENTIAL OPERATORS 16 so µ = · · · = µ l = 0 and ψ = · · · = ψ l = 0. The terms not involving D add up to µ yF + l X i =0 ( − λ i y r +2 + φ i x ) τ ( E i ) = 0 , so that λ = · · · = λ l = 0, φ = · · · = φ l = 0 and µ yF + λ y r +1 − φ x = 0, which impliesthat λ = − µ and φ = µ y ¯ F .After all this, we see that every 2-cocycle in our complex is cohomologous to one ofthe form( µ yD − µ y r +1 E )ˆ x ∧ ˆ D + ( φ + ψ D + µ y ¯ F E )ˆ y ∧ ˆ D + ξ ˆ D ∧ ˆ E (9)with µ ∈ k , φ ∈ S r +1 , ψ ∈ S and ξ ∈ S r . Computing we find that all elements ofthis form are in fact 2-cocycles.Let us now suppose that the cocycle (9), which we call again ω , is a coboundary, sothat there exist k ≥ α , . . . , α k , β , . . . , β k ∈ S , σ , . . . , σ k ∈ S r , ζ , . . . , ζ k ∈ k and u ∈ T such that if η = k X i =0 α i E i ˆ x + k X i =0 β i E i ˆ y + k X i =0 ( σ i + ζ i D ) E i ˆ D + u ˆ E, we have δ ( η ) = ω . The coefficient of ˆ D ∧ ˆ E in δ ( η ) is Dτ r ( u ) so, comparing with (9),we see that we must have ξ = 0 and u ∈ k ; it follows from this that the coefficients ofˆ E ∧ ˆ E and of ˆ y ∧ ˆ E in δ ( η ) vanish. On the other hand, the coefficient of ˆ x ∧ ˆ y in δ ( η )is P ki =0 ( − α i y + β i x ) τ ( E i ): as this has to be zero, we see that there exist ρ , . . . , ρ k ∈ k such that α i = ρ i x and β i = ρ i y for each i ∈ { , . . . , k } .The coefficient of ˆ x ∧ ˆ D in δ ( η ) is k X i =0 (cid:0) − F α iy E i − α i Dτ r ( E i ) + σ i xτ ( E i ) + ζ i xDτ ( E i ) (cid:1) = µ yD − µ y r +1 E. (10)This means, first, that P ki =1 (cid:0) − ρ i xDτ r ( E i )+ ζ i xDτ ( E i ) (cid:1) = µ yD and this is only possibleif µ = 0 and k X i =1 (cid:0) − ρ i τ r ( E i ) + ζ i τ ( E i ) (cid:1) = 0 . (11)Second, the equality (10) implies that k X i =0 (cid:0) − F α iy E i + σ i xτ ( E i ) (cid:1) = − F α y + k X i =1 σ i xτ ( E i ) = 0 , so that σ = · · · = σ k = 0 and F α y + σ x = 0, which tells us that σ = 0 and α y = 0;there is then a ρ ∈ k such that α = ρ x .Finally, the coefficient of ˆ y ∧ ˆ D in δ ( η ) is k X i =0 (cid:0) ∇ α i E i x ( F ) + ∇ β i E i y ( F ) − F β iy E i − β i Dτ r ( E i ) + σ i yτ ( E i ) − ζ i F ( E + 1) i + ζ i yDτ ( E i ) (cid:1) = φ + ψ D. LGEBRAS OF DIFFERENTIAL OPERATORS 17
Looking only at the terms which are in S DT , we see that yD k X i =1 ( − ρ i τ r ( E i ) + ζ i τ ( E i )) = ψ D and, in view of (11), it follows from this that ψ = 0. The terms in S r +1 T , on the otherhand, are ρ xF x + β F y + F − β y − ζ + k X i =0 (cid:16) ρ i r X t =1 ( E + t ) i − ζ i ( E + 1) i (cid:17)! = φ , and proceeding as before we see that φ is in the linear span of xF x , xF y and yF y .Computing, we find that δ ( x ⊗ ˆ x ) = xF x ⊗ ˆ y ∧ ˆ D,δ ( x ⊗ ˆ y ) = xF y ⊗ ˆ y ∧ ˆ D,δ ( y ⊗ ˆ y − D ˆ D ) = yF y ⊗ ˆ y ∧ ˆ D. We thus conclude that there is an isomorphism H ( X ) ∼ = k ω ⊕ S r +1 h xF x , xF y , yF y i ⊗ ˆ y ∧ ˆ D ⊕ S D ⊗ ˆ y ∧ ˆ D ⊕ S r ⊗ ˆ D ∧ ˆ E, with ω = ( yD − y r +1 E ) ⊗ ˆ x ∧ ˆ D + y ¯ F E ⊗ ˆ y ∧ ˆ D , and that, in particular, the dimensionof H ( X ) is 2 r + 3. We can summarize our findings as follows:
Proposition.
Suppose that r ≥ . For all p ≥ we have HH p ( A ) = 0 . There areisomorphisms HH ( A ) ∼ = k ,HH ( A ) ∼ = S r ⊗ ˆ D ⊕ k ⊗ ˆ E,HH ( A ) ∼ = k ω ⊕ S r +1 h xF x , xF y , yF y i ⊗ ˆ y ∧ ˆ D ⊕ S D ⊗ ˆ y ∧ ˆ D ⊕ S r ⊗ ˆ D ∧ ˆ E,HH ( A ) ∼ = k ω ⊕ S r +1 h xF x , xF y , yF y i ⊗ ˆ y ∧ ˆ D ∧ ˆ E ⊕ S D ⊗ ˆ y ∧ ˆ D ∧ ˆ E, with ω = ( yD − y r +1 E ) ⊗ ˆ x ∧ ˆ D + y ¯ F E ⊗ ˆ y ∧ ˆ D,ω = ( yD − y r +1 E ) ⊗ ˆ x ∧ ˆ D ∧ ˆ E + y ¯ F E ⊗ ˆ y ∧ ˆ D ∧ ˆ E. The Hilbert series of the Hochschild cohomology of A is h HH • ( A ) ( t ) = 1 + ( r + 2) t + (2 r + 3) t + ( r + 2) t = (1 + t )(1 + ( r + 1) t + ( r + 2) t ) . (cid:3) In fact, in each of the isomorphisms appearing in the statement of the proposition wehave given a set of representing cocycles. This will be important in what follows, whenwe compute the Gerstenhaber algebra structure on the cohomology of A . LGEBRAS OF DIFFERENTIAL OPERATORS 18
We have chosen a system of coordinates in the vector space containing the arrange-ment A in such a way that one of the lines is given by the equation x = 0. This wasuseful in picking a basis for the S -module of derivations Der ( A ) and, as a consequence,obtaining a presentation of the algebra A amenable to the computations we wanted tocarry out, but the unnaturality of our choice is reflected in the rather unpleasant form ofthe representatives that we have found for cohomology classes —a consequence of thecombination of the truth of Hermann Weyl’s dictum that the introduction of coordinatesis an act of violence together with that of the everyday observation that violence doesnot lead to anything good. In the next section we will be able to obtain a more naturaldescription. In Proposition we considered only line arrangements with r ≥
3, that is, with atleast 5 lines. As we explained in , without the restriction the method of calculationthat we followed has to be modified, and it turns out that this is not only a technicaldifference: the actual results are different. Let us describe what happens, starting withthe factorizable cases: • If there are no lines, so that r = −
2, the arrangement is empty and D ( A ) is thesecond Weyl algebra k [ x, y, ∂ x , ∂ y ]. • If there is one line, then D ( A ) is k [ x, y, x∂ x , ∂ y ] and this is isomorphic to U ( s ) ⊗ A ,with U ( s ) the enveloping algebra of the non-abelian 2-dimensional Lie algebra s and A the first Weyl algebra. • If there are two lines, so that r = 0, then D ( A ) is k [ x, y, x∂ x , y∂ y ], which isisomorphic to U ( s ) ⊗ U ( s ).The Hochschild cohomology of the Weyl algebras is well-known —for example, from [14]—as is that of U ( s ). Using this and Künneth’s formula we find that when − ≤ r ≤ i ∈ N . dim HH i ( D ( A )) = r + 2 i ! . Finally, we have the cases of three and four lines. Up to isomorphism of arrangements, onecan assume that the defining polynomials are Q = xy ( x − y ) and Q = xy ( x − y )( x − λy )for some λ ∈ k \ { , } , respectively. One can compute the cohomology of D ( A ) inthese cases along the lines of what we did above, but the computation is surprisinglymuch more involved. We have done the computation using an alternative, much moreefficient approach —using a spectral sequence that computes in general the Hochschildcohomology of the enveloping algebra of a Lie–Rinehart pair— on which we will reportin an upcoming paper. Let us for now simply summarize the result: when r is 2 or 3, theHilbert series of HH • ( A ) is h HH • ( A ) ( t ) = 1 + ( r + 2) t + (2 r + 4) t + ( r + 3) t . This differs from the general case of Proposition in the coefficients of t and t .For our immediate purposes, we remark that in all cases HH ( D ( A )) has dimensionequal to the number of lines in the arrangement A , and that its concrete description isthe same in all cases. LGEBRAS OF DIFFERENTIAL OPERATORS 19 The Gerstenhaber algebra structure on HH • ( D ( A )) Let B A be the usual bar resolution for A as an A -bimodule. There is a morphismof complexes φ : P → B A over the identity map of A such that φ = φ K + φ N with φ K , φ N : P → B A maps of A -bimodules such that φ K (1 | v ∧ · · · ∧ v p |
1) = X π ∈ S p ( − ε ( π ) | v π (1) | · · · | v π ( p ) | , whenever p ≥ v , . . . , v p ∈ V , with the sum running over permutations of degree p ,and φ N (1 |
1) = 0; φ N (1 | v |
1) = 0 , ∀ v ∈ V ; φ N (1 | x ∧ y |
1) = φ N (1 | x ∧ E |
1) = φ N (1 | y ∧ E |
1) = φ N (1 | x ∧ D | φ N (1 | D ∧ E |
1) = 0; φ N (1 | y ∧ D |
1) = q (1) | ¯ q (2) | q (3) | − F | | | φ N (1 | x ∧ y ∧ E |
1) = φ N (1 | x ∧ D ∧ E |
1) = 0; φ N (1 | x ∧ y ∧ D |
1) = q (1) | ¯ q (2) | q (3) | x | − q (1) | ¯ q (2) | x | q (3) | q (1) | x | ¯ q (2) | q (3) | − F | x | | | − F | | | x | φ N (1 | y ∧ D ∧ E |
1) = q (1) | ¯ q (2) | q (3) | E | − q (1) | ¯ q (2) | E | q (3) | q (1) | E | ¯ q (2) | q (3) | − F | E | | | − F | | | E | . Here q (1) | ¯ q (2) | q (3) denotes the element ∇ ( F ) ∈ S ⊗ S ⊗ S , with an omitted sum.On the other hand, there is a morphism of complexes of A -bimodules ψ : B A → P over the identity map of A such that ψ (1 |
1) = 1 | ,ψ (1 | w |
1) = w (1) | w (2) | w (3) , for all standard monomials w ; ψ (1 | yD | y |
1) = − y | y ∧ D | − q (1) | q (2) ∧ y | q (3) ; ψ (1 | y r +1 E | y |
1) = − y r +1 | y ∧ E | ψ (1 | E | w |
1) = − w (1) | w (2) ∧ E | w (3) for all standard monomials w ; ψ (1 | v | w |
1) = − | w ∧ v | , if v, w ∈ { x, y, D, E } and vw is not standard; ψ (1 | w | x |
1) = − w (1) | x ∧ w (2) | w (3) for all standard monomials w ;and ψ (1 | u | v |
1) = 0whenever u and v are standard monomials of A such that the concatenation uv is alsoa standard monomial. This morphism ψ can be taken —and we will take it— to benormalized, so that it vanishes on elementary tensors of B A with a scalar factor. LGEBRAS OF DIFFERENTIAL OPERATORS 20
We need the comparison morphisms that we have just described in order to computethe Gerstenhaber bracket on HH • ( A ), but we start with a more immediate application:obtaining a natural basis of the first cohomology space HH ( A ). Proposition. ( i ) If α is a non-zero element of S that divides Q , then there exists aunique derivation ∂ α : A → A such that ∂ α ( f ) = 0 for all f ∈ S and ∂ α ( δ ) = δ ( α ) α for all δ ∈ Der ( A ) . ( ii ) If Q = α . . . α r +1 is a factorization of Q as a product of elements of S , thenthe cohomology classes of the r + 2 derivations ∂ α , . . . , ∂ α r +1 of A freely spanthe vector space HH ( A ) . Here we are viewing HH ( A ) as the vector space of outer derivations of A , as usual.It should be noticed that the derivation ∂ α associated to a linear factor of Q does notchange if we replace α by one of its non-zero scalar multiples: this means that the basisof HH ( A ) is really indexed by the lines of the arrangement A . Proof. ( i ) Let us fix a non-zero element α in S dividing Q . There is at most onederivation ∂ α : A → A as in the statement of the proposition simply because thealgebra A is generated by the set S ∪ Der ( A ). In order to prove that there is sucha derivation, we need only recall from [10, Proposition 4.8] that δ ( α ) ∈ αS for all δ ∈ Der ( A ) and check that the candidate derivation respects the relations (1) of thatpresent the algebra A .( ii ) We need to pass from the description of HH ( A ) as the space of outer derivationsto its description in terms of the complex X that was used to compute it: we do thiswith the comparison morphism φ : P → B A over the identity map that we describedin . If δ : A → A is a derivation of A and ˜ δ : A ⊗ A ⊗ A → A is the map such that˜ δ ( a ⊗ b ⊗ c ) = aδ ( b ) c for all a , b , c ∈ A , which is a 1-cocycle on B A , the composition¯ δ ◦ φ : A ⊗ V ⊗ A → A is a 1-cocycle in the complex hom A e ( P , A ) whose cohomologyclass corresponds to δ in the usual description of HH ( A ) as the space of outer derivationsof A . In the notation that we used in , this cohomology class is that of δ ( x ) ⊗ ˆ x + δ ( y ) ⊗ ˆ y + δ ( D ) ⊗ ˆ D + δ ( E ) ⊗ ˆ E ∈ A ⊗ ˆ V .
Using this, we can now prove the second part of the proposition. We can suppose withoutloss of generality that α = x , and then the class of δ α in HH ( A ) is that of1 ⊗ ˆ E. On the other hand, for each i ∈ { , . . . , r + 1 } , computing we find that the class of ∂ α i is α iy Fα i ⊗ ˆ D + 1 ⊗ ˆ E. It follows easily from the second part of Lemma that these r + 2 classes span HH ( A )and, since the dimension of this space is exactly r + 2, do so freely. (cid:3) LGEBRAS OF DIFFERENTIAL OPERATORS 21
The cup product.4.3.
We describe the associative algebra structure on HH • ( A ) given by the cup product. Proposition.
The cup product on HH • ( A ) is such that S r ⊗ ˆ D ^ S r ⊗ ˆ D = 0; φ ˆ D ^ ˆ E = φ ˆ D ∧ ˆ E, ∀ φ ∈ S r ; S r ⊗ ˆ D ^ HH ( A ) = 0;1 ⊗ ˆ E ^ ω = ω ;1 ⊗ ˆ E ^ κ ⊗ ˆ y ∧ ˆ D = κ ⊗ ˆ y ∧ ˆ E ∧ ˆ D, ∀ κ ∈ S r +1 / h xF x , xF y , yF y i ;1 ⊗ ˆ E ^ ψD ⊗ ˆ y ∧ ˆ D = ψD ⊗ ˆ y ∧ ˆ D ∧ ˆ E, ∀ ψ ∈ S ;1 ⊗ ˆ E ^ S r ⊗ ˆ D ∧ ˆ E = 0 . These equalities completely describe the multiplicative structure on HH • ( A ). Proof.
There is a morphism of complexes of A -bimodules ∆ : P → P ⊗ A P that lifts thecanonical isomorphism A → A ⊗ A A such that ∆ = ∆ K + ∆ N , with • ∆ K : P → P ⊗ A P the map of A -bimodules such that for whenever p ≥ v , . . . , v p ∈ V we have∆ K (1 | v ∧ · · · ∧ v p |
1) = X ( − ε | v i ∧ · · · ∧ v i r | ⊗ | v j ∧ · · · ∧ v j s | , with the sum taken over all decompositions r + s = p with r , s ≥
0, andall permutations ( i , . . . , i r , j , . . . , j s ) of (1 , . . . , p ) such that i < · · · < i r and j < · · · < j s , and where ε is the signature of the permutations, • and ∆ N : P → P ⊗ A P the map of A -bimodules such that∆ N (1 |
1) = 0;∆ N (1 | v |
1) = 0 , ∀ v ∈ V ;∆ N (1 | v ∧ w |
1) = 0 , if v , w ∈ { x, y, D, E } , v = w and { v, w } 6 = { y, D } ;∆ N (1 | y ∧ D |
1) = f (1) | f (2) | f (3) ⊗ | f (4) | f (5) ;∆ N (1 | x ∧ y ∧ D |
1) = ∆ N (1 | x ∧ y ∧ E |
1) = ∆ N (1 | x ∧ D ∧ E |
1) = 0;∆ N (1 | y ∧ D ∧ E |
1) = − f (1) | f (2) ∧ E | f (3) ⊗ | f (4) | f (5) + f (1) | f (2) | f (3) ⊗ | f (4) ∧ E | f (5) . Here we have written f (1) | f (2) | f (3) | f (4) | f (5) the image of F under the composition S S ⊗ S ⊗ S S ⊗ S ⊗ S ⊗ S ⊗ S, ∇ id S ⊗ id S ⊗∇ with an omitted sum, à la Sweedler.We leave the verification that this does define a morphism of complexes to the reader.One can compute the cup product on HH • ( A ) using this diagonal morphism ∆. Indeed,we view HH • ( A ) as the cohomology of the complex hom A e ( P , A ), and if φ and ψ area p - and a q -cocycle in that complex, the cup product of their cohomology classes is LGEBRAS OF DIFFERENTIAL OPERATORS 22 represented by the composition P p + q P p ⊗ A P q A ⊗ A A = A, ∆ p,q φ ⊗ ψ with ∆ p,q the component P p + q → P p ⊗ P q of the morphism ∆. The multiplication tablegiven in the statement of the composition can be computed in this way, item by item. (cid:3) ( i ) For all i , j , k ∈ { , . . . , r + 1 } we have ∂ α i ^ ∂ α j + ∂ α j ^ ∂ α k + ∂ α k ^ ∂ α i = 0 (12) and HH ( A ) ^ HH ( A ) = S r ⊗ ˆ D ∧ ˆ E . ( ii ) The subalgebra H of HH • ( A ) generated by HH ( A ) is the graded-commutativealgebra freely generated by its elements ∂ α , . . . , ∂ α r +1 of degree subject to the (cid:0) r +23 (cid:1) relations (12) . This subalgebra H is isomorphic to the de Rham cohomology of the complement ofthe arrangement of lines A . This follows from a direct computation of this cohomologyor, in fact, from the solution of Arnold’s conjecture by Brieskorn; this is discussed indetail in [10, Section 5.4]. Proof.
Using Proposition and the description given in the proof of Proposition for the derivations ∂ α i we compute immediately that ∂ α i ^ ∂ α j = − (cid:12)(cid:12)(cid:12)(cid:12) α ix α jx α iy α jy (cid:12)(cid:12)(cid:12)(cid:12) Qα i α j for all i , j ∈ { , . . . , r + 1 } . Using this, we see that for all i , j , k ∈ { , . . . , r + 1 } we have ∂ α i ^ ∂ α j + ∂ α j ^ ∂ α k + ∂ α k ^ ∂ α i = − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α i α j α k α ix α jx α kx α iy α jy α ky (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Qα i α j α k = 0 , as the determinant vanishes. This proves the first claim of ( i ). The second one followsimmediately from the description of the cup product of Proposition .( ii ) Let F = L n ≥ F n be the free graded-commutative algebra generated by r + 2generators w , . . . , w r +1 of degree 1 subject to the relations w i w j + w j w k + w k w i = 0, onefor each choice of i , j , k ∈ { , . . . , r +1 } . We have F n = 0 if n ≥
3: if i , j , k ∈ { , . . . , r +1 } we have w i w j w k = ( w i w j + w j w k + w k w i ) w k = 0, because of graded-commutativity. Onthe other hand, we have dim F ≤ r + 1. To see this, we notice that F is spanned byproducts w i w j with 1 ≤ i < j ≤ r + 1. If i + 1 < j then w i w j = − w i +1 w j − w i +1 w i : itfollows from this that the set of monomials { w i w i +1 : 0 ≤ i ≤ r } already spans F .The first part of the proposition implies that there is a surjective morphism of gradedalgebras f : F → H such that f ( w i ) = ∂ α i for all i ∈ { , . . . , r + 1 } , and this map is alsoinjective because the dimension of the component of degree 2 of H , which is S r ⊗ ˆ D ∧ ˆ E ,is r + 1. (cid:3) Proposition describes meaningfully a part of the associative algebra HH • ( A ),the subalgebra H generated by HH ( A ), in terms of the geometry of the arrangement A .It is not clear how to make sense of the complete algebra. We can make the following LGEBRAS OF DIFFERENTIAL OPERATORS 23 observation, though. Let us write HH ( A ) = k ω ⊕ ( S r +1 / h xF x , xF y , yF y i ⊕ S D ) ⊗ ˆ y ∧ ˆ D, which is a complement of H in HH ( A ), and let Q = α . . . α r +1 be a factorizationof Q as a product of linear factors. If δ : A → A is derivation of A , then our descriptionof HH ( A ) implies that there exist scalars δ , . . . , δ r +1 ∈ k and an element u ∈ A suchthat δ = P r +1 i =0 δ i ∂ α u + ad ( u ), and it follows easily from Proposition that the map ζ ∈ HH ( A ) δ ^ ζ ∈ HH ( A )is either zero or an isomorphism, provided P r +1 i =0 δ i is zero or not. The Gerstenhaber bracket.4.6.
Using the comparison morphisms of , we can now compute the Gerstenhaberbracket. As usual, this is very laborious.
Proposition. In HH • ( A ) we have [0 , • ] n [ HH ( A ) , HH • ( A )] = 0 , [1 , n [ HH ( A ) , HH ( A )] = 0 , [1 , [ HH ( A ) , S r ⊗ ˆ D ∧ ˆ E ] = 0 , [ u ⊗ ˆ D + λ ⊗ ˆ E, ( v + wD ) ⊗ ˆ y ∧ ˆ D ] = uw ⊗ ˆ y ∧ ˆ D, [ u ⊗ ˆ D + λ ⊗ ˆ E, ω ] = (( µ − λ ) yF x + µy ¯ F − y ¯ u ) ⊗ ˆ y ∧ ˆ D, [1 , [ u ⊗ ˆ D + λ ⊗ ˆ E, ( v + wD ) ⊗ ˆ y ∧ ˆ D ∧ ˆ E ] = uw ⊗ ˆ y ∧ ˆ D ∧ ˆ E, [ u ⊗ ˆ D + λ ⊗ ˆ E, ω ] = (( µ − λ ) yF x + µy ¯ F − y ¯ u ) ⊗ ˆ y ∧ ˆ D ∧ ˆ E, [2 , [ S r ⊗ ˆ D ∧ ˆ E, S r ⊗ ˆ D ∧ ˆ E ] = 0 , [ u ⊗ ˆ D ∧ ˆ E, ( v + wD ) ⊗ ˆ y ∧ ˆ D ] = uw ⊗ ˆ y ∧ ˆ D ∧ ˆ E, [ u ⊗ ˆ D ∧ ˆ E, ω ] = ( µyF x + µy ¯ F − y ¯ u ) ⊗ ˆ y ∧ ˆ D ∧ ˆ E, [( S r +1 + S D ) ⊗ ˆ y ∧ ˆ D, ( S r +1 + S D ) ⊗ ˆ y ∧ ˆ D ] = 0 , [( S r +1 + S D ) ⊗ ˆ y ∧ ˆ D, ω ] = 0 , [ ω , ω ] = 0 . Here u ∈ S r , λ ∈ k , v ∈ S r +1 , w ∈ S and µ ∈ k and ¯ u ∈ S r − are such that u = λy r + x ¯ u .Proof. Let us first recall from [6] how one can compute the Gerstenhaber bracket inthe standard complex hom A e ( B A, A ). If f : A ⊗ q → A is a q -cochain in the standardcomplex hom A e ( B A, A ), which we identify as usual with hom ( A ⊗• , A ), and p ≥ q , wedenote w p ( f ) : A ⊗ p → A p − q +1 the p -cochain in the same complex such that w p ( f )( a ⊗ · · · ⊗ a p )= p − q +1 X i =1 ( − ( q − i − a ⊗ · · · ⊗ a i − ⊗ f ( a i ⊗ · · · ⊗ a i + q − ) ⊗ a i + q ⊗ · · · ⊗ a p . LGEBRAS OF DIFFERENTIAL OPERATORS 24
If now α and β are a p - and a q -cocycle in the standard complex, the Gerstenhabercomposition (cid:5) (which is usually written simply ◦ ) of α and β is the ( p + q − α (cid:5) β = α ◦ w p + q − ( β )and the Gerstenhaber bracket is the graded commutator for this composition, so that[ α, β ] = α (cid:5) β − ( − ( p − q − β (cid:5) α. Next, if α and β are now a p - and a q -cochain in the complex hom A e ( P , A ), we can liftthem to a p -cochain ˜ α = α ◦ ψ p and a q -cochain ˜ β = β ◦ ψ q in the standard complex hom A e ( B A, A ), and the Gerstenhaber bracket of the classes of α and β is then representedby the ( p + q − α, ˜ β ] ◦ φ p + q − . This is the computation we have to do in orderto compute brackets in HH • ( A ), except that in some favorable circumstances we cantake advantage of the compatibility of the bracket with the product to cut down thework. We do this in several steps. • Since the morphism ψ is normalized and HH ( A ) is spanned by 1 ∈ k , it followsimmediately that[ HH ( A ) , HH • ( A )] = 0 . • The Gerstenhaber bracket on HH ( A ) is induced by the commutator of derivations.From Proposition we have a basis of HH ( A ) whose elements are classes ofcertain derivations, and it is immediate to check that those derivations commute,so that[ HH ( A ) , HH ( A )] = 0 . (13) • We know that the subspace S r ⊗ ˆ D ∧ ˆ E of HH ( A ) is HH ( A ) ^ HH ( A ). Since HH • ( A ) is a Gerstenhaber algebra and we now that (13) holds, it follows that[ HH ( A ) , S r ⊗ ˆ D ∧ ˆ E ] = 0 . For exactly the same reasons we also have that[ S r ⊗ ˆ D ∧ ˆ E, S r ⊗ ˆ D ∧ ˆ E ] = 0 . • Let α = u ⊗ ˆ D + λ ⊗ ˆ E , with u ∈ S r and λ ∈ k . If β = ( v + wD ) ⊗ ˆ y ∧ ˆ D , with v ∈ S r +1 and w ∈ S , one can compute that ( ˜ α (cid:5) ˜ β ) ◦ φ = uw ⊗ ˆ y ∧ ˆ D and that( ˜ β (cid:5) ˜ α ) ◦ φ = 0: it follows from this that[ α, ( v + wD ) ⊗ ˆ y ∧ ˆ D ] = uw ⊗ ˆ y ∧ ˆ D. On the other hand, we have (˜ ω (cid:5) ˜ α ) ◦ φ = 0 and[ ˜ α, ˜ ω ] ◦ φ = ( ˜ α (cid:5) ˜ ω ) ◦ φ = ( yu − λy r +1 ) ⊗ ˆ x ∧ ˆ D + λy ¯ F ⊗ ˆ y ∧ ˆ D = (cid:0) ( µ − λ ) yF x + µy ¯ F − y ¯ u (cid:1) ⊗ ˆ y ∧ ˆ D − δ (cid:0) (( µ − λ ) ¯ F − y ¯ u ) E ⊗ ˆ D + ( λ − µ ) y ⊗ ˆ x (cid:1) with ¯ u ∈ S r − and µ ∈ k chosen so that u = µy r + x ¯ u .Finally, if v ∈ S r +1 and w ∈ S , using the compatibility of the bracket and theproduct and what we know so far we see that[ α, ( v + wD ) ⊗ ˆ y ∧ ˆ D ∧ ˆ E ] = [ α, ⊗ E ^ ( v + wD ) ⊗ ˆ y ∧ ˆ D ] LGEBRAS OF DIFFERENTIAL OPERATORS 25 = 1 ⊗ E ^ [ α, ( v + wD ) ⊗ ˆ y ∧ ˆ D ]= 1 ⊗ E ^ uw ⊗ ˆ y ∧ ˆ D = uw ⊗ ˆ y ∧ ˆ D ∧ ˆ E and, similarly, that[ α, ω ] = [ α, ω ^ ⊗ ˆ E ] = [ α, ω ] ^ ⊗ ˆ E + ω ^ [ α, ⊗ ˆ E ]= (cid:0) ( µ − λ ) yF x + µy ¯ F − y ¯ u (cid:1) ⊗ ˆ y ∧ ˆ D ∧ ˆ E. • Let u ∈ S r . If v ∈ S r +1 and w ∈ S , we have[ u ⊗ ˆ D ∧ ˆ E, ( v + wD ) ⊗ ˆ y ∧ ˆ D ] = [ u ⊗ ˆ D ^ ⊗ ˆ E, ( v + wD ) ⊗ ˆ y ∧ ˆ D ]= [ u ⊗ ˆ D, ( v + wD ) ⊗ ˆ y ∧ ˆ D ] ^ ⊗ ˆ E + u ⊗ ˆ D ^ [1 ⊗ ˆ E, ( v + wD ) ⊗ ˆ y ∧ ˆ D ]= uw ⊗ ˆ y ∧ ˆ D ^ ⊗ ˆ E = uw ⊗ ˆ y ∧ ˆ D ∧ ˆ E. Similarly,[ u ⊗ ˆ D ∧ ˆ E, ω ] = [ u ⊗ ˆ D ^ ⊗ ˆ E, ω ]= [ u ⊗ ˆ D, ω ] ^ ⊗ ˆ E + u ⊗ ˆ D ^ [1 ⊗ ˆ E, ω ]= (cid:0) µyF x + µy ¯ F − y ¯ u (cid:1) ⊗ ˆ y ∧ ˆ D ∧ ˆ E. if u = µy r + x ¯ u with µ ∈ k and ¯ u ∈ S r − . • Let now α = ( v + wD ) ⊗ ˆ y ∧ ˆ D and β = ( s + tD ) ⊗ ˆ y ∧ ˆ D , with v , s ∈ S r +1 and w , t ∈ S . We claim that ( ˜ α (cid:5) ˜ β ) ◦ φ = 0, so that, by symmetry, we have[ ˜ α, ˜ β ] ◦ φ = 0. To verify our claim, we compute:1 | x ∧ y ∧ E | φ k [ x, y, E ] ⊗ w ( ˜ β ) | x ∧ D ∧ E | φ k [ x, D, E ] ⊗ w ( ˜ β ) | x ∧ y ∧ D | φ | x | y | D | − | x | D | y | | D | x | y | − | D | y | x | | y | D | x | − | y | x | D | S ⊗ w ( ˜ β ) | ( s + tD ) | x | − | x | ( s + tD ) | ψ s (1) | x ∧ s (2) | s (3) − t (1) | x ∧ t (2) | t (3) D − t | x ∧ D | α | y ∧ D ∧ E | φ | y | D | E | − | y | E | D | | E | y | D | − | E | D | y | | D | E | y | − | D | y | E | k [ x, y, E ] ⊗ w ( ˜ β ) | ( s + tD ) | E | − | E | ( s + tD ) | ψ s (1) | s (2) ∧ E | s (3) + t (1) | t (2) ∧ E | t (3) D + t | D ∧ E | α . LGEBRAS OF DIFFERENTIAL OPERATORS 26 • Let again α = ( v + wD ) ⊗ ˆ y ∧ ˆ D , with v ∈ S r +1 and w ∈ S , and let us computethat (˜ ω (cid:5) ˜ α ) ◦ φ = − w ( yD − y r +1 E ) ⊗ ˆ x ∧ ˆ y ∧ ˆ D .1 | x ∧ y ∧ z | φ k [ x, y, E ] ⊗ w (˜ α ) | x ∧ D ∧ E | φ k [ x, D, E ] ⊗ w (˜ α ) | x ∧ y ∧ D | φ | x | y | D | − | x | D | y | | D | x | y | − | D | y | x | | y | D | x | − | y | x | D | S ⊗ w (˜ α ) | ( v + wD ) | x | | x | ( v + wD ) | ψ v (1) | x ∧ v (2) | v (3) − w (1) | x ∧ w (2) | w (3) D − w | x ∧ D | ω w ( yD − y r +1 E )1 | y ∧ D ∧ E | φ | y | D | E | − | y | E | D | | E | y | D | − | E | D | y | | D | E | y | − | D | y | E | k [ x, y, E ] ⊗ w (˜ α ) | ( v + wD ) | E | − | E | ( v + wD ) | ψ v (1) | v (2) ∧ E | v (3) + w (1) | w (2) ∧ E | w (3) D + w | D ∧ E | ω . Similarly, we have that ( ˜ α (cid:5) ˜ ω ) ◦ φ = y ( v + wD ) ⊗ ˆ x ∧ ˆ y ∧ ˆ D :1 | x ∧ y ∧ z | φ k [ x, y, E ] ⊗ w (˜ ω ) | x ∧ D ∧ E | φ | x | D | E | − | x | E | D | | E | x | D | − | E | D | x | | D | E | x | − | D | x | E | w (˜ ω ) | E | ( yD − y r +1 E ) | | ( yD − y r +1 ) | E | ψ | y ∧ E | D − y | D ∧ E | r X i =0 y i | y ∧ E | y r − iα | x ∧ y ∧ D | φ | x | y | D | − | x | D | y | | D | x | y | − | D | y | x | | y | D | x | − | y | x | D | S ⊗ w (˜ ω ) | x | y ¯ F E | x | − | ( yD − y r +1 E ) | y |
1+ 1 | y ¯ F E | x | | y | ( yD − y r +1 E ) | ψ y | y ∧ D | − y r +1 | y ∧ E | − ( y ¯ F E ) (1) | x ∧ ( y ¯ F E ) (2) | ( y ¯ F E ) (3)LGEBRAS OF DIFFERENTIAL OPERATORS 27 α y ( v + wD )1 | y ∧ D ∧ E | φ | y | D | E | − | y | E | D | | E | y | D | − | E | D | y | | D | E | y | − | D | y | E | k [ x, y, E ] ⊗ w (˜ ω ) | E | y ¯ F E | | y ¯ F E | E | ψ ( y ¯ F E ) (1) | ( y ¯ F E ) (2) ∧ E | ( y ¯ F E ) (3) α . It follows from this that[˜ ω , ˜ α ] ◦ φ = − w ( yD − y r +1 E ) ⊗ ˆ x ∧ ˆ y ∧ ˆ D + y ( v + wD ) ⊗ ˆ x ∧ ˆ y ∧ ˆ D = ( yv + y r +1 E ) ⊗ ˆ x ∧ ˆ y ∧ ˆ D and, as we say in , this is a coboundary. • The one computation that remains is that of the bracket of ω with itself, whichis represented by the 3-cocycle[˜ ω , ˜ ω ] ◦ φ = 2(˜ ω (cid:5) ˜ ω ) ◦ φ = 2 y ¯ F E ⊗ ˆ x ∧ ˆ y ∧ ˆ D, (14)as can be seen from the following calculation:1 | x ∧ y ∧ z | φ k [ x, y, E ] ⊗ w (˜ ω ) | x ∧ D ∧ E | φ | x | D | E | − | x | E | D | | E | x | D | − | E | D | x | | D | E | x | − | D | x | E | w (˜ ω ) | E | ( yD − y r +1 E ) | | ( yD − y r +1 ) | E | ψ | y ∧ E | D − y | D ∧ E | r X i =0 y i | y ∧ E | y r − iω | x ∧ y ∧ D | φ | x | y | D | − | x | D | y | | D | x | y | − | D | y | x | | y | D | x | − | y | x | D | S ⊗ w (˜ ω ) | x | y ¯ F E | x | − | ( yD − y r +1 E ) | y |
1+ 1 | y ¯ F E | x | | y | ( yD − y r +1 E ) | ψ y | y ∧ D | − y r +1 | y ∧ E | − ( y ¯ F E ) (1) | x ∧ ( y ¯ F E ) (2) | ( y ¯ F E ) (3) ω y ¯ F E | y ∧ D ∧ E | φ | y | D | E | − | y | E | D | | E | y | D | − | E | D | y | | D | E | y | − | D | y | E | k [ x, y, E ] ⊗ w (˜ ω ) | E | y ¯ F E | | y ¯ F E | E | ψ ( y ¯ F E ) (1) | ( y ¯ F E ) (2) ∧ E | ( y ¯ F E ) (3) ω . Now the 3-cocycle (14) is a coboundary, again by what we saw in , so thatthe class of ω has bracket-square zero.This completes the proof of the proposition. (cid:3) Hochschild homology, cyclic homology and K -theory For completeness, we determine the rest of the ‘usual’ homological invariants of ouralgebra A . Recall that our ground field k is of characteristic zero. Proposition.
The inclusion T = k [ E ] → A induces an isomorphism in Hochschildhomology and in cyclic homology. In particular, there are isomorphisms of vector spaces HH i ( A ) ∼ = ( T, if i = 0 or i = 1 ; , if i ≥ ; HC i ( A ) ∼ = ( T, if i = 0 ; HC i ( k ) , if i > .On the other hand, the inclusion k → A induces an isomorphism in periodic cyclichomology and in higher K -theory.Proof. As we know, the algebra A is N -graded and for each n ∈ N its homogeneouscomponent A n of degree n is the eigenspace corresponding to the eigenvalue n of thederivation ad ( E ) : A → A . On one hand, this grading of A induces as usual an N -gradingon the Hochschild homology HH • ( A ) of A ; on the other, the derivation ad ( E ) inducesa linear map L ad ( E ) : HH • ( A ) → HH • ( A ) as in [8, §4.1.4] and, in fact, for all n ∈ N the homogeneous component HH • ( A ) n of degree n for that grading coincides with theeigenspace corresponding to the eigenvalue n of L ad ( E ) . As the derivation ad ( E ) is inner,it follows from [8, Proposition 4.1.5] that the map L ad ( E ) is actually the zero map andthis tells us in our situation that HH • ( A ) n = 0 for all n = 0. Of course, this means that HH • ( A ) = HH • ( A ) and, since A is non-negatively graded, it is immediate that the 0thhomogeneous component HH • ( A ) coincides with the Hochschild homology HH • ( A )of A and that the map HH • ( A ) → HH • ( A ) induced by the inclusion A , → A is anisomorphism. Now, in the notation of [8, Theorem 4.1.13], this tells us that ≈ HH • ( A ) = 0so that by that theorem we also have ≈ HC • ( A ) = 0: this means precisely that the inclusion A , → A induces an isomorphism HC • ( A ) → HC • ( A ) in cyclic homology. Together withthe well-known computation of the Hochschild homology of a polynomial ring and thatof the cyclic homology of symmetric algebras [8, Theorem 3.2.5], this proves the firstclaim of the statement.In the proof of the lemma of we constructed an increasing filtration F on thealgebra A with F − A = 0 and such that the corresponding graded algebra is thecommutative polynomial ring gr A = k [ x, y, D, E ] with generators x and y in degree 0and D and E in degree 1. In particular, both gr A and its subalgebra gr A of degree 0 havefinite global dimension. It follows from a theorem of D. Quillen [11, p. 117, Theorem 7]that the inclusion k [ x, y ] = F A → A induces an isomorphism K i ( k [ x, y ]) → K i ( A ) in K -theory for all i ≥
0. Similarly, the theorem of J. Block [2, Theorem 3.4] tells us
LGEBRAS OF DIFFERENTIAL OPERATORS 29 that that inclusion induces an isomorphism HP • ( k [ x, y ]) → HP • ( A ) in periodic cyclichomology. As the inclusion k → k [ x, y ] induces an isomorphism in K -theory and inperiodic cyclic homology, we see that the second claim of the proposition holds. (cid:3) The Calabi–Yau property
The enveloping algebra A e of A is a bimodule over itself, with left and right actions . and / given by ‘outer’ and ‘inner’ multiplication, respectively, so that if a ⊗ b , c ⊗ d and e ⊗ f are elementary tensors in A e , we have a ⊗ b . c ⊗ d / e ⊗ f = ace ⊗ f db. From this bimodule structure we obtain a duality functor hom A e ( − , A e ) : A e Mod → Mod A e . On the other hand, using the anti-automorphism τ : A e → A e such that τ ( a ⊗ b ) = b ⊗ a for all a , b ∈ A , we can turn a right A e -module M into a left A e -module, with action u . m = m / τ ( u ) for all u ∈ A e and all m ∈ M . In this way, we obtain an isomorphism ofcategories τ ∗ : Mod A e → A e Mod . We denote ( − ) ∨ : A e Mod → A e Mod the composition τ ∗ ◦ hom A e ( − , A e ).Let now W be a finite dimensional vector space, let W ∗ be the vector space dual to W ,and view A ⊗ W ⊗ A and A ⊗ W ∗ ⊗ A as left A e -modules using the usual ‘exterior’ action.There is a unique k -linear mapΦ : A ⊗ W ∗ ⊗ A → ( A ⊗ W ⊗ A ) ∨ such that Φ( a ⊗ φ ⊗ b )(1 ⊗ w ⊗
1) = φ ( w ) b ⊗ a and it is an isomorphism of left A e -modules:we will view it in all that follows as an identification. The algebra A is twisted Calabi-Yau of dimension with modularautomorphism σ : A → A such that σ ( x ) = x, σ ( y ) = y, σ ( D ) = D + F y , σ ( E ) = E + r + 2 . Let us recall from [7] that this means that A has a resolution of finite length byfinitely generated projective A -bimodules, that Ext iA e ( A, A e ) = 0 if i = 4 and that Ext A e ( A, A e ) ∼ = A σ , the A -bimodule obtained from A by twisting its right action usingthe automorphism σ , so that a . x / b = axσ ( b ) for all a , b ∈ A and all x ∈ A σ . Proof.
A direct computation shows that there is indeed an automorphism σ of A as inthe statement of the proposition. We already know that A has a resolution P of length 4by finitely generated free A -bimodules, so we need only compute Ext • A e ( A, A e ), and thisis the cohomology of the complex P ∨ obtained by applying the functor described in to P . Using the identifications introduced there, this complex P ∨ is A ⊗ A A ⊗ V ∗ ⊗ A A ⊗ Λ V ∗ ⊗ A A ⊗ Λ V ∗ ⊗ A A ⊗ Λ V ∗ ⊗ A d ∨ d ∨ d ∨ d ∨ with left A e -linear differentials such that d ∨ (1 ⊗
1) = − [ x, ⊗ ˆ x ⊗ − [ y, ⊗ ˆ y ⊗ − [ D, ⊗ ˆ D ⊗ − [ E, ⊗ ˆ E ⊗ LGEBRAS OF DIFFERENTIAL OPERATORS 30 d ∨ (1 ⊗ ˆ x ⊗
1) = [ y, ⊗ ˆ x ∧ ˆ y ⊗
1] + [ D, ⊗ ˆ x ∧ ˆ D ⊗
1] + [ E, ⊗ ˆ x ∧ ˆ E ⊗ ⊗ ˆ x ∧ ˆ E ⊗ ∇ ˆ y ∧ ˆ Dx ( F ); d ∨ (1 ⊗ ˆ y ⊗
1) = − [ x, ⊗ ˆ x ∧ ˆ y ⊗
1] + [ D, ⊗ ˆ y ∧ ˆ D ⊗
1] + [ E, ⊗ ˆ y ∧ ˆ E ⊗ ⊗ ˆ y ∧ ˆ E ⊗ ∇ ˆ y ∧ ˆ Dy ( F ); d ∨ (1 ⊗ ˆ D ⊗
1) = − [ x, ⊗ ˆ x ∧ ˆ D ⊗ − [ y, ⊗ ˆ y ∧ ˆ D ⊗
1] + [ E, ⊗ ˆ D ∧ ˆ E ⊗ r ⊗ ˆ D ∧ ˆ E ⊗ d ∨ (1 ⊗ ˆ E ⊗
1) = − [ x, ⊗ ˆ x ∧ ˆ E ⊗ − [ y, ⊗ ˆ y ∧ ˆ E ⊗ − [ D, ⊗ ˆ D ∧ ˆ E ⊗ d ∨ (1 ⊗ ˆ x ∧ ˆ y ⊗
1) = − [ D, ⊗ ˆ x ∧ ˆ y ∧ ˆ D ⊗ − ˜ ∇ ˆ x ∧ ˆ y ∧ ˆ Dy ( F ) − [ E, ⊗ ˆ x ∧ ˆ y ∧ ˆ E ⊗ − ⊗ ˆ x ∧ ˆ y ∧ ˆ E ⊗ d ∨ (1 ⊗ ˆ x ∧ ˆ D ⊗
1) = [ y, ⊗ ˆ x ∧ ˆ y ∧ ˆ D ⊗ − [ E, ⊗ ˆ x ∧ ˆ D ∧ ˆ E ⊗ − ( r + 1) ⊗ ˆ x ∧ ˆ D ∧ ˆ E ⊗ d ∨ (1 ⊗ ˆ x ∧ ˆ E ⊗
1) = [ y, ⊗ ˆ x ∧ ˆ y ∧ ˆ E ⊗
1] + [ D, ⊗ ˆ x ∧ ˆ D ∧ ˆ E ⊗ ∇ ˆ y ∧ ˆ D ∧ ˆ Ex ( F ); d ∨ (1 ⊗ ˆ y ∧ ˆ D ⊗
1) = − [ x, ⊗ ˆ x ∧ ˆ y ∧ ˆ D ⊗ − [ E, ⊗ ˆ y ∧ ˆ D ∧ ˆ E ⊗ − ( r + 1) ⊗ ˆ y ∧ ˆ D ∧ ˆ E ⊗ d ∨ (1 ⊗ ˆ y ∧ ˆ E ⊗
1) = − [ x, ⊗ ˆ x ∧ ˆ y ∧ ˆ E ⊗
1] + [ D, ⊗ ˆ y ∧ ˆ D ∧ ˆ E ⊗ ∇ ˆ y ∧ ˆ D ∧ ˆ Ey ( F ); d ∨ (1 ⊗ ˆ D ∧ ˆ E ⊗
1) = − [ x, ⊗ ˆ x ∧ ˆ D ∧ ˆ E ⊗ − [ y, ⊗ ˆ y ∧ ˆ D ∧ ˆ E ⊗ d ∨ (1 ⊗ ˆ x ∧ ˆ y ∧ ˆ D ⊗
1) = [ E, ⊗ ˆ x ∧ ˆ y ∧ ˆ D ∧ ˆ E ⊗ r + 2) ⊗ ˆ x ∧ ˆ y ∧ ˆ D ∧ ˆ E ⊗ d ∨ (1 ⊗ ˆ x ∧ ˆ y ∧ ˆ E ⊗
1) = − [ D, ⊗ ˆ x ∧ ˆ y ∧ ˆ D ∧ ˆ E ⊗ − ˜ ∇ ˆ x ∧ ˆ y ∧ ˆ D ∧ ˆ Ey ( F ); d ∨ (1 ⊗ ˆ x ∧ ˆ D ∧ ˆ E ⊗
1) = [ y, ⊗ ˆ x ∧ ˆ y ∧ ˆ D ∧ ˆ E ⊗ d ∨ (1 ⊗ ˆ y ∧ ˆ D ∧ ˆ E ⊗
1) = − [ x, ⊗ ˆ x ∧ ˆ y ∧ ˆ D ∧ ˆ E ⊗ , where each ˜ ∇ ux is the image of ∇ ux under the map a ⊗ u ⊗ b b ⊗ u ⊗ a , and the samewith each ˜ ∇ uy .Let us now identify P ⊗ A A σ with P as vector spaces, remembering that the bimodulestructure on P with this identification is given by a . x / b = axσ ( b ) for all a , b ∈ A andall x ∈ P . There is a morphism of complexes of A -bimodules ψ : P ∨ → P ⊗ A A σ suchthat ψ (1 ⊗ ˆ x ∧ ˆ y ∧ ˆ D ∧ ˆ E ⊗
1) = 1 ⊗ ψ (1 ⊗ ˆ y ∧ ˆ D ∧ ˆ E ⊗
1) = − ⊗ x ⊗ ψ (1 ⊗ ˆ x ∧ ˆ D ∧ ˆ E ⊗
1) = 1 ⊗ y ⊗ ψ (1 ⊗ ˆ x ∧ ˆ y ∧ ˆ E ⊗
1) = − ⊗ D ⊗ − ξ ; ψ (1 ⊗ ˆ x ∧ ˆ y ∧ ˆ D ⊗
1) = 1 ⊗ E ⊗ LGEBRAS OF DIFFERENTIAL OPERATORS 31 ψ (1 ⊗ ˆ D ∧ ˆ E ⊗
1) = − ⊗ x ∧ y ⊗ ψ (1 ⊗ ˆ x ∧ ˆ D ⊗
1) = 1 ⊗ y ∧ E ⊗ ψ (1 ⊗ ˆ y ∧ ˆ D ⊗
1) = − ⊗ x ∧ E ⊗ ψ (1 ⊗ ˆ y ∧ ˆ E ⊗
1) = 1 ⊗ x ∧ D ⊗ x ∧ ξ ; ψ (1 ⊗ ˆ x ∧ ˆ E ⊗
1) = − ⊗ y ∧ D ⊗ ζ ; ψ (1 ⊗ ˆ x ∧ ˆ y ⊗
1) = − ⊗ D ∧ E ⊗ − ξ ∧ E ; ψ (1 ⊗ ˆ E ⊗
1) = 1 ⊗ x ∧ y ∧ D ⊗ ψ (1 ⊗ ˆ D ⊗
1) = − ⊗ x ∧ y ∧ E ⊗ ψ (1 ⊗ ˆ y ⊗
1) = 1 ⊗ x ∧ D ∧ E ⊗ x ∧ ξ ∧ E ; ψ (1 ⊗ ˆ x ⊗
1) = − ⊗ y ∧ D ∧ E ⊗ ζ ∧ E ; ψ (1 ⊗
1) = 1 ⊗ x ∧ y ∧ D ∧ E ⊗ , where ξ ∈ A ⊗ V ⊗ A and ζ ∈ A ⊗ Λ V ⊗ A are chosen so that d ( ξ ) = ˜ ∇ y ( F ) − | F y , d ( ζ ) = ξy − yξ − | y | F y − ˜ ∇ xx ( F ) + ∇ ( F ) . That there are elements which satisfy these two conditions follows immediately from theexactness of the Koszul resolution of S as an S -bimodule —indeed, the right hand sidesof the two conditions are cycles in that complex— but we can exhibit a specific choice: ifwe write F = P a + b = r +1 c a x a y b , with c , . . . , c r − ∈ k , then we can pick ξ = X a + b = r +1 s + t +1= b − ( t + 1) c a y s | y | x a y t , ζ = X a + b = r +1 s + t +1= bs + t +1= a c a x s y s | x ∧ y | x t y t . That these formulas for ψ do indeed define a morphism of complexes follows from a directcomputation and it is easy to see that it is in fact an isomorphism, as for an appropriateordering of the bases of the bimodules involved the matrices for the components of ψ are upper triangular. Of course, it therefore induces an isomorphism in cohomologyand, since A σ is A -projective on the left, we conclude that there are isomorphisms of A -bimodules H i ( P ∨ ) ∼ = H i ( P ⊗ A A σ ) ∼ = ( A σ if i = 4;0 if i > (cid:3) LGEBRAS OF DIFFERENTIAL OPERATORS 32 Automorphisms, isomorphisms and normal elements
Our next objective is to compute the group of automorphisms of the algebra A .We start by describing some graded automorphisms of A . Later we will see that theseare, in fact, all the graded automorphisms of our algebra, and that together with theexponentials of locally ad -nilpotent elements they generate the whole group Aut ( A ). Lemma. If (cid:0) a bc d (cid:1) ∈ GL ( k ) and e ∈ k × are such that ad − bc ) e Q ( ax + by, cx + dy ) = Q ( x, y ) , and v ∈ k and φ ∈ S r , then there is a homogeneous algebra automorphism θ : A → A such that θ ( x ) = ax + by, θ ( y ) = cx + dy, θ ( E ) = E + v and θ ( D ) = ( φ − ebFax + by E + eD, if b = 0 ; φ + eD, if not. (15) Proof.
This is proved by a straightforward calculation. It should be noted that thequotient appearing in the formula (15) is always a polynomial. (cid:3)
Recall that a higher derivation of A is a sequence d = ( d i ) i ≥ of linear maps A → A such that d = id A and for all a , b ∈ A and all i ≥ higher Leibniz identity d i ( ab ) = X s + t = i d s ( a ) d t ( b ) . It is clear that if d = ( d i ) i ≥ is a higher derivation and m ≥
0, then the sequence d [ m ] = ( d [ m ] i ) i ≥ with d [ m ] i = ( d i/m , if i is divisible by m ;0 , if notis also a higher derivation. On the other hand, if d = ( d i ) i ≥ and d = ( d i ) i ≥ are higherderivations of A , we can construct a new higher derivation ( d i ) i ≥ , which we denote d ◦ d , putting d i = P s + t = i d s ◦ d t for all i ≥
0. Finally, if δ : A → A is a derivation of A ,then the sequence ( i ! δ i ) i ≥ is a higher derivation, which we denote by exp ( δ ); notice thatthis makes sense because our ground field k has characteristic zero.We let D ( A ) be the associative subalgebra of End k ( A ) generated by Der ( A ), and saythat two higher derivations d = ( d i ) i ≥ and d = ( d i ) i ≥ of A are equivalent, and write d ∼ d , if for all i ≥ d i − d i is in the subalgebra of End k ( A ) generated by D ( A )and d , . . . , d i − ; one can check that this is indeed an equivalence relation on the set ofhigher derivations. We recall the following very useful lemma from [1]:
Lemma. If d = ( d i ) i ≥ is a higher derivation of A , then d i ∈ D ( A ) for all i ≥ . LGEBRAS OF DIFFERENTIAL OPERATORS 33
Proof.
The result is an easy consequence of the fact that if d is a higher derivation of A and j ≥ , then there exists a higherderivation d = ( d i ) i ≥ such that d ∼ d , d i = 0 if < i < j , and d j isan element of Der ( A ) . (16)To prove that this holds, let d = ( d i ) i ≥ and suppose there is an j ≥ d i = 0 if 1 < i < j . The higher Leibniz identity implies that d j is an element of Der ( A ),and then we can consider the higher derivation exp ( − d j ) [ j ] . We let d = ( d i ) i ≥ be thecomposition exp ( − d j ) [ j ] ◦ d . It is immediate that d ∼ d and a simple computation showsthat d i = 0 if 1 < i < j + 1. The claim (16) follows inductively from this. (cid:3) An element of A commutes with x and with y if and only if it belongs to S .Proof. The sufficiency of the condition is clear. To prove the necessity, let e ∈ A besuch that [ x, e ] = [ y, e ] = 0. There are an integer m ≥ φ , . . . , φ m in the subalgebra generated by x , y and D in A such that e = P mi =0 φ i E i , and wehave 0 = [ x, e l ] = P mi =0 φ i τ ( E i ): this tells us that φ i = 0 if i >
0, and that e = φ .In particular, there are an integer n ≥ ψ , . . . , ψ n in S such that e = P ni =0 ψ i D i . If i ≥ D i , y ] ≡ iF D i − mod L i − j =0 SD j , so that0 = [ e, y ] = n X i =0 ψ i [ D i , y ] ≡ nψ n F D n − mod n − M j =0 SD i . Proceeding by descending induction we see from this that ψ i = 0 if i >
0, so that e = ψ ∈ S . (cid:3) If θ : A → A is an automorphism of A such that for all i ≥ and all a ∈ A i we have θ ( a ) ∈ a + L j>i A j , then here exists an f ∈ S , uniquely determined upto the addition of a constant, such that θ ( x ) = x, θ ( y ) = y, θ ( D ) = D − F f y , θ ( E ) = E − [ E, f ] . Conversely, every f ∈ S determines in this way an automorphism of A satisfying thatcondition.Proof. Let θ : A → A be an automorphism of A as in the statement. For each j ≥ θ j : A → A of degree j such that for each i ≥ a ∈ A i the element θ j ( a ) is the ( i + j )th homogeneous component of θ ( a ). We have that for all a ∈ A we have θ j ( a ) = 0 for j ≥ θ ( a ) = P j ≥ θ i ( a ) and, moreover, the sequence( θ j ) j ≥ is a higher derivation of A . In particular, it follows from Lemma that θ i ∈ D ( A ) for all i ≥ . (17)We know, from Proposition , that Der ( A ) = S r ˆ D ⊕ k ˆ E ⊕ InnDer ( A ). If u is anirreducible factor of xF , then ( φ ˆ D )( uA ), ˆ E ( uA ) and [ a, uA ] are all contained in uA forall φ ∈ S r and all a ∈ A , and therefore (17) implies that that θ ( uA ) ⊆ uA . As ourargument also applies to the inverse automorphism θ − , we have θ − ( uA ) ⊆ uA and,therefore, θ ( uA ) = uA . Since all units of A are in k , we see that θ ( u ) = u . Since of xF has two linearly independent linear factors, we can conclude that θ ( x ) = x and θ ( y ) = y . LGEBRAS OF DIFFERENTIAL OPERATORS 34
Let θ ( E ) = E + e + · · · + e l with e i ∈ A i for each i ∈ { , . . . , l } . We have x = θ ( x ) = [ θ ( E ) , θ ( x )] = [ E, x ] + [ e , x ] + · · · + [ e l , x ]and, by looking at homogeneous components, we see that [ e i , x ] = 0 for all i ∈ { , . . . , l } Similarly, [ e i , y ] = 0 for such i , and therefore Lemma tells us that e , . . . , e l ∈ S .Suppose now that θ ( D ) = D + d r +1 + · · · + d l with d j ∈ A j for each j ∈ { r + 1 , . . . , l } .Considering the equality [ θ ( E ) , θ ( D )] = rθ ( D ) we see that d r + i = i F e iy for each i ∈ { , . . . , l } . Putting f = − P li =1 1 i e i , we obtain the first part of the lemma. Thesecond part follows from a direct verification. (cid:3) The automorphisms described in Proposition are precisely the exponentialsof the inner derivations corresponding to locally ad -nilpotent elements of A . This is aconsequence of the following result: Proposition.
An element of A is locally ad -nilpotent if and only if it belongs to S . If f ∈ S , then the automorphism exp ad ( f ) maps x , y , D and E to x , y , D − F f y and E − [ E, f ] , respectively.Proof. Suppose that e ∈ A is a locally ad -nilpotent element. The kernel ker ad ( e ) is afactorially closed subalgebra of A , so that whenever a , b ∈ A and ad ( e )( ab ) = 0 we have ad ( e )( a ) = 0 or ad ( e )( b ) = 0; see [5] for the proof of this in the commutative case, whichadapts to ours.Since [ x i y j D k E l , x ] = − x i +1 y j D k τ ( E l ) for all i , j , k , l ≥
0, we have [
A, x ] ⊆ xA andfrom this we see immediately that [ A, xA ] ⊆ xA . This implies that there is a sequence( u k ) k ≥ in A such that ad ( e ) k ( x ) = xu k for all k ≥
0. Since e is locally ad -nilpotent,we can consider the integer k = max { k ∈ N : ad ( e ) k ( x ) = 0 } , and then we have0 = xu k ∈ ker ad ( e ). As ker ad ( e ) is factorially closed, we see that ad ( e )( x ) = 0. Inother words, the element e commutes with x .There are an integer m ≥ φ , . . . , φ m in the subalgebra generatedby x , y and D in A such that e = P mi =0 φ i E i , and we have 0 = [ x, e ] = P mi =0 φ i τ ( E i ):this tells us that φ i = 0 if i >
0, and that e = φ . In particular, there are an integer n ≥ ψ , . . . , ψ n in S such that e = P ni =0 ψ i D i .An induction shows that [ D i , F ] ∈ F A for all i ≥
0, and using this we see that[ e, F ] = P ni =0 ψ i [ D i , F ] ∈ F A , from which it follows that in fact [ e, F A ] ⊆ F A . Thereis therefore a sequence ( v i ) i ≥ of elements of A such that ad ( e ) i ( F ) = F v i for all i ≥ ad ( e ) allows us to consider the integer i = max { i ∈ N : ad ( e ) i ( F ) = 0 } , and then 0 = F v i ∈ ker ad ( e ). If ax + by is any of the factors of F , we have b = 0 and ax + by ∈ ker ad ( e ): clearly, this implies that y commutes with e .In view of Lemma , we see that e ∈ S : this proves the necessity of the conditionfor local ad -nilpotency given in the lemma. Its sufficiency is a direct consequence ofthe fact that the graded algebra associated to the filtration on A described in iscommutative. Finally, the truth of the last sentence of the proposition can be verified byan easy computation. (cid:3) LGEBRAS OF DIFFERENTIAL OPERATORS 35
We write
Aut ( A ) the set all automorphisms of A described in Lemma , and Exp ( A ) the set of all automorphisms of A described in Proposition ; they are subgroupsof the full group of automorphisms Aut ( A ). Theorem G.
The group
Aut ( A ) is the semidirect product Aut ( A ) (cid:110) Exp ( A ) , corre-sponding to the action of Aut ( A ) on Exp ( A ) given by θ · exp ad ( f ) = exp ad ( θ − ( f )) for all θ ∈ Aut ( A ) and f ∈ S . The subgroup Aut ( A ) is precisely the set of auto-morphisms of A preserving the grading and Exp ( A ) is the set of exponentials of locallynilpotent inner derivations of A . Notice that the action described in this statement makes sense, as θ ( S ) = S whenever θ belongs to Aut ( A ). Proof.
Let θ : A → A be an automorphism and let us write θ ( E ) = e + · · · + e l , θ ( x ) = x + · · · + x l , θ ( y ) = e + · · · + y l , θ ( D ) = d + · · · + d l with e i , x i , y i , d i ∈ A i foreach i ∈ { , . . . , l } . Since θ is an automorphism, we have[ θ ( E ) , θ ( x )] = θ ( x ) , [ θ ( E ) , θ ( y )] = θ ( y ) , [ θ ( E ) , θ ( D )] = rθ ( D ) . (18)Looking at the degree zero parts of these equalities, and remembering that A is acommutative ring, wee see x = y = d = 0. As θ ( x ) = 0, we can consider the number s = min { i ∈ N : x i = 0 } and we have s >
0. Looking that the component of degree s of the first equality in (18), we see that [ e , x s ] = x s . This means that the restriction ad ( e ) : A s → A s has a nonzero fixed vector. Now A s as a right k [ E ]-module is freewith basis { x i y j D k : i + j + rk = s } , the map ad ( e ) is right k [ E ]-linear, and coincideswith right multiplication by − τ s ( e ) on A s . Clearly, the existence of nonzero fixed vectorimplies that − τ s ( e ) = 1, so that e = uE + v for some u ∈ k × and v ∈ k with su = 1.Putting now s = min { i ∈ N : y i = 0 } and s = min { u ∈ N : d i = 0 } and looking atthe components in the least possible degree in the second and third equations of (18), wefind that s u = 1 and s u = r . In particular, s = s and s = rs .Suppose for a moment that s >
1. As θ ( x ), θ ( y ) and θ ( D ) are in the ideal ( A s )generated by A s , the composition q : A → A of θ with the quotient map A → A/ ( A s ) isa surjection such that q ( A ) = A/ ( A s ). This is impossible, as A is a commutative ringand A/ ( A s ) is not: we therefore have s = 1 and, as a consequence, u = 1.There exist a , b , c , d ∈ k [ E ] such that x = xa + yb and y = xc + yd . The fourelements θ ( E ), θ ( x ), θ ( y ) and θ ( D ) generate A and, as θ ( D ) is in L i ≥ r A i , the elements x and y are in the subalgebra generated by the first three. It follows at once that x , y ∈ x k [ E ] + y k [ E ] and, therefore, that (cid:0) a bc d (cid:1) ∈ GL ( k [ E ]).Let us write f ∈ k [ E ] ~f ∈ k [ E ] the unique algebra morphism such that ~E = E + 1.We have [ θ ( x ) , θ ( y )] = 0 and in degree 2 this tells us that x ( a~c − ~ac ) + xy T + y ( b ~d − ~bd ) = 0 , so that a~c = ~ac, b ~d = ~cd. (19)Suppose that a is not constant. As the characteristic of k is zero (and possibly afterreplacing k by an algebraic extension, which does not change anything) there is then a LGEBRAS OF DIFFERENTIAL OPERATORS 36 ξ ∈ k such that a ( ξ ) = 0 and ~a ( ξ ) = a ( ξ + 1) = 0, and the first equality in (19) impliesthat c ( ξ ) = 0. The determinant of (cid:0) a bc d (cid:1) is thus divisible by E − ξ , and this is impossible.Similarly, we find that all of b , c , d must be constant.Since d r ∈ A r , there exist k ≥ φ , . . . , φ k ∈ S r and h ∈ k [ E ] such that d r = P ki =0 φ i E i + Dh . The component of degree r + 1 of [ θ ( D ) , θ ( x )] is0 = [ d r , x ] = − k X i =0 ( ax + by ) φ i τ ( E i ) − ( ax + by ) Dτ ( h ) + bF~h. We thus see that h is constant, that φ i = 0 if i ≥
2, and that( ax + by ) φ + bhF = 0 . If b = 0, then φ = 0, and if instead b = 0, then either h = 0 and we see that ax + by divides F and that φ = − bhF/ ( ax + by ), or h = 0 and φ = 0. In any case, we see that d r = ( φ − hbFax + by E + hD, if b = 0; φ + hD, if not.Finally, the component of degree r + 1 of the equality [ θ ( D ) , θ ( y )] = θ ( F ) tells us that F ( ax + by, cx + dy ) = ( ad − bc ) h xFax + by . It follows now from Lemma that there is a graded automorphism θ : A → A suchthat θ ( x ) = ax + by , θ ( y ) = cx + dy , θ ( E ) = E + v and θ ( D ) = d r . The composition θ − ◦ θ satisfies the hypothesis of Proposition , and then there exists an f ∈ S suchthat θ = θ ◦ exp ad ( f ). This shows that Aut ( A ) = Aut ( A ) · Exp ( A ). Moreover, if θ is agraded automorphism, then so is exp ad ( f ) = θ − ◦ θ and, since it maps E to E − [ E, f ],this is possible if and only if f ∈ k , that is, if and only if exp ad ( f ) = id A ; this provesthe last claim of the theorem.Finally, computing the action of both sides of the equation on the generators of A , wesee that exp ad ( f ) ◦ θ = θ ◦ exp ad ( θ − ( f ))for all f ∈ S and all θ ∈ Aut ( A ), and this tells us that Aut ( A ) is indeed a semidirectproduct Aut ( A ) (cid:110) Exp ( A ). (cid:3) As usual, we say that an element u of A is normal if uA = Au . Such an element,since it is not a zero-divisor, determines an automorphism θ u : A → A uniquely by thecondition that ua = θ u ( a ) u for all u ∈ A . Proposition.
Let Q = α · · · α r +1 be a factorization of Q as a product of linear factors.The set of non-zero normal elements of A is N ( A ) = { λα i · · · α i r +1 r +1 : λ ∈ k × , i , . . . , i r +1 ∈ N } . This set is the saturated multiplicatively closed subset of A or of S generated by Q .Proof. A direct computation shows that each of the factors α , . . . , α r +1 of Q is normalin A , so the set N ( A ) is contained in the set of normal elements of A , for the latteris multiplicatively closed. The set N ( A ) is multiplicatively closed and it is saturatedbecause S is closed under divisors in A , and it is clear that as a saturated multiplicatively LGEBRAS OF DIFFERENTIAL OPERATORS 37 closed it is generated by Q . To conclude the proof, we have to show that every non-zeronormal element of A belongs to N ( A ).Let u be a normal element in A and let θ u : A → A be the associated automorphism, sothat ua = θ u ( a ) u for all a ∈ A . There are k , l ∈ N with k ≤ l and elements u k , . . . u l ∈ A such that u = u k + · · · + u l , u i ∈ A i if k ≤ i ≤ l , and u k = 0 = u l . Similarly, there are s , t ∈ N with s ≤ t and elements e s , . . . , e t ∈ A such that θ u ( E ) = e s + · · · + e t , e i ∈ A i if s ≤ i ≤ t , and e s = 0 = e t . As we have u k E + · · · + u l E = uE = θ u ( E ) u = e s u k + · · · + e t u l with u k E , u l E , e s u k and e t u l all non-zero, looking at the homogeneous components ofboth sides we see that s = t = 0. This means that θ u ( E ) = f ( E ) ∈ k [ E ], and thereforethe above equality is really of the form u k E + · · · + u l E = f ( E ) u k + · · · + f ( E ) u l . It follows from this that u i E = f ( E ) u i = u i f ( E + i ) for all i ∈ { k, . . . , l } and thereforethat E = f ( E + k ) and that E = f ( E + l ). Since our ground field has characteristic zero,this is only possible if k = l : the element u is homogeneous of degree l .Now, since ua = θ u ( a ) u for all a ∈ A , the homogeneity of u implies immediately that θ u is a homogeneous map. There are n ∈ N and φ , . . . , φ n in the subalgebra of A generated by x , y and D , such that φ n = 0 and u = P ni =0 φ i E i . As θ u ( x ) has degree 1, itbelongs to S and we have θ u ( x ) n X i =0 φ i E i = θ u ( x ) u = ux = n X i =0 φ i E i x = x X i =0 φ i ( E + 1) i . Considering only the terms that have E n as a factor we see that θ u ( x ) = x , and then theequality tells us that in fact P ni =0 φ i E i = P i =0 φ i ( E + 1) i . Looking now at the termswhich have E n − as a factor here we see that moreover n = 0, so that u ∈ k [ x, y, D ].There exist then m ∈ N and ψ , . . . , ψ m ∈ S such that ψ m = 0 and u = P mi =0 ψ i D i . As θ u ( y ) has degree 1, it belongs to S and we have θ u ( y ) m X i =0 ψ i D i = θ u ( y ) u = uy = m X i =0 ψ i D i y = m X i =0 yψ i D i + m X i =0 ψ i [ D i , y ] . Comparing the terms that have D m as a factor we conclude that also θ u ( y ) = y .As θ u fixes x and y , the element u commutes with x and y , and Lemma allows usto conclude that u is in S l . Moreover, we know that all homogeneous automorphismsof A are those described in Lemma , so there exist φ ∈ S r and e ∈ k × such that θ u ( D ) = φ + eD . We then have that uD = θ u ( D ) u = ( φ + eD ) u = φu + euD + eu y F and this implies that e = 1 and φu + u y F = 0. Suppose now that α is a linear factorof u and let k ∈ N and v ∈ S be such that u = α k v and v is not divisible by α . The lastequality becomes φα k v + kα k − α y vF + α k v y F = 0 and implies that α divides α y F : thismeans that α is a non-zero multiple of x or a linear factor of F . As u can be factored asa product of linear factors, we can therefore conclude that u belongs to the set describedin the statement of the proposition. (cid:3) LGEBRAS OF DIFFERENTIAL OPERATORS 38
There is a close connection between normal elements, the first Hochschild cohomologyspace that we computed in Section 3 and the modular automorphisms of A . Proposition.
Let Q = α · · · α r +1 be a factorization of Q as a product of linear factors. ( i ) Every linear combination of the derivations ∂ α , . . . , ∂ α r +1 : A → A described inProposition is locally nilpotent. ( ii ) If u = λα i · · · α i r +1 r +1 , with λ ∈ k × and i , . . . , i r +1 ∈ N , is a normal elementof A , then the automorphism θ u : A → A associated to u is θ u = exp − r +1 X j =0 i j ∂ α j . This automorphism is such that θ u ( f ) = f for all f ∈ S and θ u ( δ ) = δ + δ ( u ) u for all δ ∈ Der ( A ) . ( iii ) The modular automorphism σ : A → A described in Proposition coincideswith the automorphism θ Q associated to the normal element Q . Another immediate application of the determination of the set of normal elementsis the classification under isomorphisms of our algebras.
Proposition.
Let A and A be two central arrangements of lines in A . The algebras D ( A ) and D ( A ) are isomorphic if and only if the arrangements A and A are isomorphic.Proof. The sufficiency of the condition being obvious, we prove only its necessity. Wewill denote with primes the objects associated to the arrangement A , so that for example A = D ( A ) and so on. Moreover, in view of the sufficiency of the condition we cansuppose without loss of generality that both arrangements A and A contain the linewith equation x = 0.Let us suppose that there is an isomorphism of algebras φ : A → A . Since φ maps locally ad -nilpotent elements to locally ad -nilpotent elements, it follows fromProposition that φ ( S ) = S and therefore that φ restricts to an isomorphism ofalgebras φ : S → S . On the other hand, φ also maps normal elements to normalelements, so that φ restricts to a monoid homomorphism φ : N ( A ) → N ( A ). Let Q = α · · · α r +1 and Q = α · · · α r +1 be the factorizations of Q and of Q as products oflinear factors. The invertible elements of the monoid N ( A ) are the units of k and thequotient N ( A ) / k × is the free abelian monoid generated by (the classes of) α , . . . , α r +1 and, of course, a similar statement holds for the other arrangement. Since φ induces anisomorphism N ( A ) / k × → N ( A ) / k × we see, first, that r = r and, second, that thereare a permutation π of the set { , . . . , r + 1 } and a function λ : { , . . . , r + 1 } → k × suchthat φ ( α i ) = λ ( i ) α π ( i ) for all i ∈ { , . . . , r + 1 } . As there are at least two lines in eacharrangement, this implies that the restriction φ | S : S → S restricts to an isomorphism ofvector spaces φ : S → S , so that φ | S is linear, and that φ ( Q ) = Q . It is clear that thisimplies that the arrangements A and A are isomorphic. (cid:3) A simple and final observation that we can make at this point is that our algebra A and the full algebra D ( S ) of regular differentials operators of S are birational, that is, LGEBRAS OF DIFFERENTIAL OPERATORS 39 that they have the same fields of quotients. In fact, the two algebras become isomorphicalready after localization at a single element:
Proposition.
The inclusion A → D ( S ) induces after localization at Q an isomorphism A [ Q ] → D ( S )[ Q ] and, in particular, A and D ( S ) have isomorphic fields of fractions. That both localizations actually exist follows from the usual characterization of quotientrings; see, for example, [9, Chapter 2].
Proof.
Clearly the map A [ Q ] → D ( S )[ Q ] induced by the inclusion is injective, and it issurjective since S is contained in its image as are ∂ y = F D and ∂ x = x E − yQ D . (cid:3) References [1] J. Alev and M. Chamarie,
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Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad deBuenos Aires. Ciudad Universitaria, Pabellón I (1428) Ciudad de Buenos Aires, Argentina.
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