Lobachevskii Journal of Mathematics http://ljm.ksu.ru Vol. 14, 2004, 85–123

© S. Skryabin

Serge Skryabin
DEGREE ONE COHOMOLOGY FOR THE LIE ALGEBRAS OF DERIVATIONS
(submitted by M. Arslanov)

Abstract. Let R be a commutative ring and W a Lie algebra of its derivations which is an R -submodule in the full derivation algebra DerR . We consider a class of W -modules generalizing the natural representations of the Lie algebras of vector fields in tensor fields of arbitrary type. The main result consists in the determination of the cohomology of those modules in degree 1. Its applications include a description of derivations and the universal central extension for the Lie algebra W .

Introduction

This paper is concerned with the Lie algebras of derivations of a commutative ring R which are closed under the natural R -module structure on all the derivations. The basic motivating examples are the Lie algebras of all vector fields of respective smoothness class on C manifolds, real analytic ones or Stein spaces. Lie algebras of this type appear also in the classification of simple Lie algebras of finite dimension over fields with nonzero characteristic [15, 25]. Whereas the Lie algebras in the classes just mentioned were studied often separately, a natural generality of results describing their properties can be obtained in the settings of an arbitrary commutative ring R . This point of view in regard to the Lie algebra isomorphisms and the structure of ideals was emphasized in Grabowski’s paper [7]. The same questions received a further treatment in [9, 13, 22].

Another group of problems centers around the representation theory. When dealing with representations it is reasonable to make certain assumptions on the ring R and an R -module Lie algebra of derivations W Der R which express algebraically the idea of absence of singularities (although Lie algebras associated with singular analytic spaces or algebraic varieties were a subject of specific interest in some recent work [e.g. 2, 10, 21]). Under these assumptions there is a class of W -modules which are glued, in a sense, from a family of finite dimensional irreducible modules of any possible type over the general linear Lie algebras gln(Rm) , where the parameter m runs through the maximal ideals of R . They generalize the natural representations of the Lie algebras of vector fields by Lie derivatives in the sections of vector bundles associated with the representations of the general linear group GLn .

Earlier I gave a description of submodules and intertwining operators for those modules [24]. The purpose of the present paper is to determine the Chevalley-Eilenberg cohomology groups in degree 1. By theorem 5.5 these groups vanish except for modules of several exceptional types. Theorem 5.6 gives the solution in exceptional cases.

Two traditional applications of cohomology are derivations and central extensions. We show that, under our assumptions, the derivation algebra Der W is isomorphic with the normalizer of W in Der R . This generalizes the description of derivations obtained by Takens [26] for the Lie algebra of vector fields on a smooth manifold, by Grabowski [8] in the real analytic and Stein cases, and by Jacobson [12] and Ree [18] for the Lie algebras of Witt type in positive characteristic.

All central extensions of W are trivial, according to theorem 7.1, when the rank of W as a projective R -module is greater than 1. If the rank equals 1 then the kernel of the universal central extension is canonically isomorphic with the cohomology group H1 (Ω) of the de Rham complex relative to W . This generalizes the classical construction of the Virasoro algebra as the universal central extension of the Lie algebra of C vector fields on a circle. In this case Ω is the ordinary de Rham complex, and dim H1(Ω) = 1 . Another interesting example considered by Wagemann recently [27] is the Lie algebra Vect 1,0Σ of complexified C vector fields of type (1, 0) on a Riemann surface Σ . The relative de Rham complex is here the complex 0 Ω0,0 Ω1,0 0 of C differential forms of type (, 0) with respect to the -differential. As it comes by taking the global sections from a fine resolution of the sheaf O¯ of antiholomorphic functions on Σ , its cohomology in degree 1 is isomorphic with H1 (Σ,O¯) . Therefore the kernel of the universal central extension has dimension equal to the genus of Σ . A nontrivial central extension is known also for the Zassenhaus algebras in positive characteristic [5].

By now the most comprehensive results on cohomology of the Lie algebras we consider have been achieved in the cases of formal and C vector fields (see [6]). The cohomology with coefficients in tensor fields of certain types was computed in all dimensions. However, the technique of glueing used here is not available in other situations. It should be mentioned that the Lie algebras and modules in those results were understood as topological objects, and the cohomology computed was that of continuous cochains. On the contrary, we deal with arbitrary linear cocycles. In fact, we don’t even need a ground field and work over the ring of integers. Nevertheless, we show that every 1-cocycle in our settings is a differential operator of order at most 3. Therefore the continuous and algebraic cohomologies for the Lie algebras of vector fields coincide in degree 1. Another feature of our approach is that there is no big difference between local and global aspects. All constructions are done without the resort to glueing of cocycles defined locally.

1. The category of representations

At the beginning we fix notations and recall the definition of a certain representation category from [24, section 6]. Let R be a commutative, associative and unital ring, W Der R a Lie -algebra of derivations such that RW = W , that is, W is an R -submodule in Der R . Put Ω1 = Hom R(W,R) and define df Ω1 for each f R by the rule df(D) = Df , D W . The assumptions throughout the whole paper are as follows:

(1.1) 2 is invertible in R ,

(1.2) W is R -projective of constant finite rank n > 0 ,

(1.3) Ω1 = R dR .

We furnish the R -module g = Ω1 RW with an R -bilinear Lie product setting

[θ D,θ D] = θ,Dθ Dθ,Dθ D

for elements θ,θ Ω1 and D,D W , where , stands for the natural pairing Ω1 × W R . There is an isomorphism σW of g onto the Lie algebra glRW of all R -linear transformations of W defined by the rule

σW (θ D)(D) = θ,DD.

Definition 1.1. Denote by C1 the category whose objects are additive groups M together with a system of operators fM , ρM (D) , σM (T) defined for each f R , D W , T g so that the following properties are satisfied:

(1.4) the operators fM give M the structure of an R -module ,

(1.5) the operators ρM(D) give M the structure of a W -module ,

(1.6) [ρM(D),fM] = (Df)M ,

(1.7) [σM(T),fM] = 0 ,

(1.8) ρM (fD) = fM ρM(D) + σM(df D) ,

(1.9) σM (fT) = fM σM(T) .

The morphisms in C1 are the maps that commute with the actions of R and W . Denote by C0 the full subcategory of C1 consisting of objects M with σM = 0 .

The category C1 is closed under several operations. If M,N are its objects then the R -modules M RN and Hom R(M,N) are in a natural way objects of C1 too. We agree to write M N suppressing the subscript in the tensor product. The corresponding operators are given by

ρMN(D)(u v) =ρM(D)u v + u ρN(D)v, σMN(T)(u v) =σM(T)u v + u σN(T)v, ρHom R(M,N)(D)ξ =ρM(D) ξ ξ ρN(D) σHom R(M,N)(T)ξ =σM(T) ξ ξ σN(T),

where u M , v N , ξ Hom R(M,N) . These are well defined in view of the compatibility conditions (1.6), (1.7). In particular, the r -fold tensor power rM of the underlying R -module of M is an object of C1 . The same is valid for the symmetric power SrM and the exterior power rM of the R -module M as these are factors of rM by subgroups stable under all operators involved. Both W and g operate in the tensor, symmetric and exterior algebras of the R -module M via derivations.

We regard R as an object of C0 letting ρR be the natural action of W on R and σR = 0 . Similarly, W together with the adjoint representation ρW and σW defined earlier is an object of C1 . Hence Ω1 and g are objects of C1 too. Most of the natural R -linear maps that we will happen to deal with are in fact morphisms in C1 . For instance, so is σM : g End RM for any M C1 . That σM is a W -equivariant map is asserted in Lemma 1.1 below. Another example is the contraction γ : Ω1 g R defined by the rule γ(θ D) = θ,D for θ Ω1 and D W .

Note that (1.2) enables one to define the trace function tr: End RW R as follows. The R -module nW is projective of rank 1. Therefore End R nWR . There is a natural representation of the Lie algebra glRW in nW via R -linear transformations. Each element TW glRW acts as a multiplication by a certain element of R called the trace of TW . Now γ(T) = tr σW (T) for T g . Similarly, γ(T) is the trace of the R -linear endomorphism σΩ1(T) .

It may be helpful to think of a particular example in which R is the ring of C functions and W the Lie algebra of C vector fields on a smooth manifold. In this case Ω1 is the module of linear differential forms. Tensor fields of any possible type constitute an object of the category C1 according to the constructions above. The representation ρ is given by Lie derivatives, whereas σ involves certain contractions of tensors. This generalizes to real analytic manifolds and Stein spaces.

Lemma 1.1. Let M be an object of C1 . Then the operators σM(T) with T g define a representation of g and [ρM(D),σM(T)] = σM(ρg(D)T) for all D W , T g . Every morphism in C1 is a g -module homomorphism.

Proof. In view of (1.3) every element of g is a sum of certain dg D with g R and D W , so it suffices to consider only such elements. If T = dg D then σM (T) = ρ M(gD) g M ρM(D) , and

[ρM(D),σM(T)] = ρM([D,gD]) g M ρM([D,D]) (Dg) M ρM(D) = σM(ρg(D)T)

since ρg(D)T = dg [D,D] + d(Dg) D . Now for T = df D with f R and D W we express σM(T) from (1.8) and get

[σM(T),σM(T)] = σ M(ρg(fD)T f ρ g(D)T) = σ M(σg(T)T).

One checks that

σg(T)T = σ Ω1(T)(dg) T + dg σ W (T)D = dg,Ddf D dg df,DD = [T,T].

The final conclusion of the Lemma follows again from (1.8).

If M C1 and m is a maximal ideal of R then MmM is a module for the Lie algebra gmg over the field Rm . We may identify gmg with the Lie algebra gl(WmW) of all linear transformations of the vector space WmW . In a sense M can be regarded as being glued from a family of representations of general linear Lie algebras parametrized by the maximal ideals of R . This is indeed a correct point of view provided M is R -projective. Of particular importance are the following assumptions on an object Q C1 :

(1.10) Q is a finitely generated projective R -module ,

(1.11) for each maximal ideal m of R the quotient QmQ is an absolutely irreducible module for the Lie algebra gmg .

Lemma 1.2. Suppose that Q C1 satisfies (1.10), (1.11) . Then the associative R -algebra End RQ is generated by the endomorphisms σQ(T) with T g .

Proof. Put A = End RQ , and let B be its subalgebra (containing the identity endomorphism) generated by all σQ (T) . For each maximal ideal m of R the image of B in AmA End Rm(QmQ) is the associative subalgebra generated by all endomorphisms of an absolutely irreducible representation in QmQ . It is therefore the whole AmA , i.e., B + mA = A . Since Q is a finitely generated projective R -module, A is finitely generated over R too. The global version of Nakayama’s Lemma [1, II, § 3, Proposition 11] yields B = A .

It turns out that for application to the central extensions in section 7 more general objects of C1 have to be dealt with. Hence we are led to the following

Definition 1.2. Suppose that Q C1 is an object satisfying (1.10), (1.11) . We say that an object M C1 is of type Q if there is a homomorphism of associative R -algebras End RQ End RM which takes σQ(T) to σM (T) for each T g .

If M is of type Q , then so is every subobject M M as well. In fact, the image of End RQ in End RM is the subalgebra, say AM , generated by the endomorphisms σM(T) with T g . Each σM(T) is the restriction of σM(T) . Therefore M is stable under AM , and the resulting homomorphism of R -algebras AM End RM takes σM (T) to σM(T) . Similarly, along with M , every its factor object is of type Q . A particular example of an object satisfying (1.10), (1.11) is R itself. An object M C1 is of type R if and only if σM(T) = 0 for all T g , that is, M C0 .

Lemma 1.3. Suppose that Q is an object of C1 satisfying (1.10), (1.11) . Then the functor M0M0 Q is an equivalence between C0 and the full subcategory of C1 consisting of objects of type Q .

Proof. Put A = End RQ . Note that Q is a projective generator in the category of R -modules. By Morita theory the functor M0M0 Q is an equivalence between the categories of R -modules and A -modules with the inverse equivalence M Hom A(Q,M) . We will check that it induces an equivalence between the categories in question.

Suppose that M C1 is of type Q . Then the homomorphism A End RM afforded by the definition 1.2 makes M into an A -module. Let M0 = Hom A(Q,M) . Then M0 H where H = Hom R(Q,M) is an object of C1 . In view of Lemma 1.2 η H is in M0 if and only if η commutes with the action of g . Hence M0 is the kernel of the morphism φ : H Hom R(g,H) in C1 defined by the rule

φ(η)(T) = σH(T)η = σM(T) η η σQ(T)

for η H , T g . Thus M0 C1 and, since the induced action of g in M0 is trivial, in fact M0 C0 . The canonical map M0 Q M is a morphism in C1 . It is bijective by Morita theory.

Conversely, suppose that M0 C0 . Then M = M0 Q is an object of C1 and σM(T) = id σQ(T) for all T g . The assignment ξ id ξ defines a homomorphism of R -algebras A End RM which takes σQ(T) to σM (T) . Thus M is of type Q . The canonical map M0 Hom A(Q,M) is a bijective morphism in C0 .

Lemma 1.4. Let Q,Q be two objects of C1 satisfying (1.10), (1.11) . Suppose that M,M C 1 are objects of type Q and Q respectively, so that MM0 Q and M M 0 Q for some M0,M0 C 0 . If the gmg -modules QmQ and Q mQ are not isomorphic for every maximal ideal m of R then Mor C1(M,M) = 0 . If Q = Q then Mor C1(M,M) Mor C0(M0,M0) .

Proof. Suppose there is a nonzero morphism M M in C1 . Its image N is a factor object of M and a subobject of M in C1 . It is therefore of type Q and Q simultaneously. Put A = End RQ , A = End RQ , and let B be the subalgebra of the associative R -algebra End RN generated by all endomorphisms σN(T) with T g . There is a homomorphism of R -algebras A B which takes σQ(T) to σN (T) for each T g . Clearly it is surjective. Since A is finitely generated over R , so is B as well. Furthermore, B0 because N0 . By Nakayama’s Lemma there exists a maximal ideal m of R such that BmB . Fix such an ideal.

The factor algebra BmB is a homomorphic image of a simple associative algebra AmA End Rm(QmQ) . It follows that BmBAmA . By symmetry we have BmBAmA as well. Up to isomorphism, QmQ is a unique simple module for AmA . Similarly, AmA has a unique simple module QmQ . If we let AmA operate in QmQ via the algebra isomorphism φ : AmA → ∼ AmA constructed above, there has to be an isomorphism of AmA -modules ι : QmQ → ∼ QmQ . Denote by σm : gmg AmA the reduction modulo m of the map σQ : g A and by σm : gmg AmA the reduction of σQ : g A . Then σm = φ σm by the construction. Hence ι is an isomorphism of gmg -modules. This proves the first statement of the Lemma. The second one is a general fact that a category equivalence is bijective on morphisms.

Next we are going to introduce certain operators on objects M C1 . Whereas by the definition these operators are R -linear endomorphisms of M , Lemma 1.5 shows that they can be expressed in terms of the endomorphisms ρM (D) , D W , solely. It is worth keeping in mind that the χM defined below is an R -multilinear function of its arguments. For θ,θ Ω1 and D,D W put

χM(θ,θ,D,D) =(σ M(θ D) θ,DM)σM(θ D) +(σM(θ D) θ,D M)σM(θ D).

Lemma 1.5. Let M C1 . Then for all f,g R and D,D W we have

ρM(fgD)ρM(D) + ρ M(fD)ρM(gD) + ρM(gD)ρM(fD) ρ M(D)ρM(fgD) = χ M(df,dg,D,D).

This is checked straightforwardly using relations (1.6)–(1.9) [24, Lemma 6.1].

We need a modification of the category C1 in which the role of Ω1 is transferred to the R -module of Kähler differentials which we denote as Ω ˜1 . Recall that it is defined together with a derivation d : R Ω ˜1 , universal in the class of derivations with values in R -modules (a -linear map Δ : R M with M an arbitrary R -module is a derivation if Δ(fg) = fΔg + gΔf for all f, g R ). The universality property gives a unique R -linear map Ω ˜1 Ω1 rendering commutative the diagram

         R @
    d~~~~    @@@@d
   ~~~           @@@
^ 1-------------// 1
Ω               Ω

This map is surjective in view of (1.3). It induces an R -bilinear pairing Ω ˜1 × W R . The same formula as in the case of g defines now a Lie multiplication on g˜ = Ω ˜1 W , the tensor product being over R .

Definition 1.3. Denote by C˜1 the category whose objects are additive groups M together with a system of operators fM , ρM (D) , σM (T) defined for each f R , D W , T g˜ subject to the conditions (1.4) (1.9) . The morphisms in C˜1 are the maps that commute with the actions of R and W .

The epimorphism of R -modules Ω ˜1 Ω1 induces a surjective homomorphism of Lie algebras g ˜ g over R . The category C1 can be identified therefore with the full subcategory in C ˜ 1 whose objects M satisfy σM(T) = 0 for all T in the kernel of g˜ g . All natural operations in C1 have their analogues in C˜1 . If M C˜1 then its R -linear transformations χM (θ,θ,D,D) make sense for elements θ,θ Ω ˜1 and D, D W . Lemmas 1.1 and 1.5 generalize to C˜1 as they are formal consequences of (1.4)–(1.9).

Lemma 1.6. The R -module Ω ˜1 can be in a unique way furnished with additional operators which make it an object of C ˜ 1 and the universal derivation d : R Ω ˜1 a W -equivariant map. Furthermore,

σΩ ˜1(θ D)θ = θ,Dθforθ,θ Ω ˜1andD W.

The canonical map Ω ˜1 Ω1 is an epimorphism in C˜1 and its kernel J an object of the subcategory C0 .

Proof. Put M = Ω ˜1 . Given D W , make the direct sum of additive groups E = M M into an R -module setting

f (θ,θ) =(fθ + (Df)θ,fθ)forf Randθ,θ M.

The projection π : E M onto the second summand is a homomorphism of R -modules. One checks that the assignment f(dDf,df) defines a derivation R E . By the universality property of Kähler differentials there is an R -linear map φ : M E sending df to (dDf,df) . The composite πφ has to be the identity endomorphism of M . Therefore φ(θ) =(ρ(D)θ,θ) for some operator ρ(D) on M . One has ρ(D)(df) = dDf and

ρ(D)(fθ) = f ρ(D)θ + (Df)θ

for all f R , D W , θ M . These two identities determine ρ(D) uniquely as M = R dR . It follows ρ([D,D]) = [ρ(D),ρ(D)] for D,D W , i.e., ρ is a Lie algebra representation. Next, ρ(fD) fM ρ(D) is an R -linear transformation of M . Indeed, this is a consequence of the identity (1.6) which we have verified above. One computes

(ρ(fD) fM ρ(D))(dg) = d(f Dg) f d(Dg) = (Dg) df = dg,Ddf = σ(df D)(dg)

where f,g R , D W and σ = σM is defined in the statement of the Lemma. Clearly σ satisfies (1.7) and (1.9). Since M = R dR , (1.8) holds too. The map Ω ˜1 Ω1 is a W -equivariant homomorphism of R -modules, i.e., a morphism in the category C˜1 . If θ J then θ ,D = 0 for all D W . It follows from the definition of σ that σ(T)θ = 0 for all T g˜ . Thus J C0 .

2. The differential order of a cocycle

A 1 -cocycle φ : W M with coefficients in a W -module M is a -linear map satisfying

φ([D,D]) = ρ M(D)φ(D) ρ M(D)φ(D)

for all D,D W . In this section we will prove that, when M C1 , every 1-cocycle is a differential operator of order at most 3. In general, given two R -modules M and N , a -linear map ξ : N M and an element f R , define

δfξ = ξ fN fM ξ

where fM and fN are multiplication operators on M and N . We call ξ a differential operator of order r if δf1δfr+1ξ = 0 for all f1 ,,fr+1 R . Denote by Diff r(N,M) the group of all differential operators of order r . For f, g R one has

δfδgξ = δgδfξandδfgξ = (δfξ) gN + fM (δgξ).

It follows that the map R × × R(rtimes) Hom R(N,M) given by the rule (f1,,fr)δf1δfrξ is symmetric and is a derivation in each of its arguments whenever ξ Diff r(N,M) . By the universality property of Kähler differentials it induces a symmetric R -multilinear map Ω ˜1 × ×Ω ˜1(rtimes) Hom R(N,M) , hence also an R -linear map from the r -th symmetric power SrΩ ˜1 to Hom R(N,M) . There is therefore an R -linear map

ξ : SrΩ ˜1 N M

such that

ξ(df 1dfr u) = (δf1δfrξ)(u)

for f1,,fr R and u N . It is called the r -th order symbol of ξ . The notation ξ that we use is somewhat ambiguous as every differential operator of order r is also a differential operator of order r + 1 . If P is a third R -module and η : P N a -linear map then δf (ξ η) = δfξ η + ξ δfη . It follows by induction that ξ η is a differential operator of order r + s when ξ Diff r(N,M) and η Diff s(P,N) . Its symbol can be computed as

(ξ η)(θ 1θr+s u) = ξ(θ i1θir η(θ ir+1θir+s u))

for θ1,,θr+s Ω ˜1 and u P , where the sum ranges over all permutations of indices 1, ,r + s such that i1 < < ir and ir+1 < < ir+s .

The exact sequence 0 J Ω ˜1 Ω1 0 in our settings induces a surjective homomorphism of symmetric algebras SΩ ˜1 SΩ1 whose kernel is the ideal of SΩ ˜1 generated by J . If the symbol of a differential operator ξ Diff r(N,M) vanishes on all elements in the image of the canonical map (J Sr1Ω ˜1) N SrΩ ˜1 N then it induces an R -linear map Sr Ω1 N M which we still call the symbol of ξ and denote as ξ .

Lemma 2.1. Let M C1 . Then every 1-coboundary φ : W M is a differential operator of order 1 .

Proof. There is m M such that φ(D) = ρM(D)m for all D W . By (1.8) (δfφ)(D) = σM(df D)m for f R , D W , whence δf φ is an R -linear map according to (1.9).

Lemma 2.2. Suppose that φ : W M is a 1 -cocycle where M is an object of C1 . If φ is a differential operator of order 2 then its symbol φ induces a morphism S2Ω1 W M in C1 . If φ is a differential operator of order 3 and either rk RW > 1 or 3 is invertible in R then φ induces a morphism S3Ω1 W M .

Proof. Assume φ Diff 3(W,M) . Its symbol φ : S3Ω ˜1 W M is an R -module homomorphism. We will show that φ is a W -module homomorphism as well, hence a morphism in C ˜ 1 . The Lie algebra W operates on the -linear maps ξ : W M in a natural way, and D δfξ = δDfξ + δfDξ for f R , D W . Now Dφ is the coboundary of φ(D) M , whence δfδg(Dφ) = 0 for all f,g R by Lemma 2.1. We deduce

D δf1δf2δf3φ = δDf1δf2δf3φ + δf1δDf2δf3φ + δf1δf2δDf3φ

for f1,f2,f3 R and D W . Evaluating at D W yields

ρM(D)φ(ω D) φ(ω [D,D]) = φ(Dω D)

where ω = df1 df2 df3 . Since Ω ˜1 = R dR , the equality holds actually for all ω S3Ω ˜1 and gives the W -invariance of φ .

Next we want to show that φ factors through S3Ω1 W . The kernel k of the canonical homomorphism g˜ g annihilates M and W since both modules are in C1 . As φ is g ˜ -equivariant by the C˜1 version of Lemma 1.1, it must vanish on the R -submodule K S3Ω ˜1 W spanned by the tensors ω D with ω k S3Ω ˜1 and D W . We will show that k S3Ω ˜1 = J S2Ω ˜1 , which means that K is the kernel of the canonical epimorphism S3 Ω ˜1 W S3Ω1 W , as required. Recall that g˜ operates in the symmetric algebra SΩ ˜1 via derivations. Given D W , let iD denote the derivation of SΩ ˜1 such that iDf = 0 for f R and iD θ = θ,D for θ Ω ˜1 . The composite θiD of iD with the multiplication by θ Ω ˜1 is again a derivation. Hence θ D acts in the symmetric algebra as θiD , both derivations having the same values on elements of R and Ω ˜1 . Now k is spanned by the tensors θ D with θ J , D W . Therefore k S3Ω ˜1 = JN , where N S2Ω ˜1 is the R -submodule spanned by the elements iDω with D W , ω S3Ω ˜1 . It remains to verify that N = S2Ω ˜1 .

If P Ω ˜1 is a finitely generated R -submodule then so is the span P2 S2Ω ˜1 of all θθ with θ, θ P . By Nakayama’s Lemma the inclusion P2 N holds if and only if P2 N + mP2 for all maximal ideals m of R . It suffices to prove these inclusions only for those P that project onto the whole Ω1 since the latter is finitely generated over R . Fix P and m . Let θ P . If rkRW > 1 then Ω1 mΩ1(WmW) has dimension > 1 over Rm . We can find D W and θ P such that θ,D 0 , θ ,D 1 modulo m . Then θ2 i D(θ2θ) modulo mP2 , so that θ2 N + mP2 . In view of (1.1) the squares θ2 span the whole P2 . Suppose now rk RW = 1 but 3R = R . If θ mP we can find D W such that θ, D 1 modulo m . Then 3θ2 i D(θ3) modulo mP2 , whence again θ2 N + mP2 . If θ mP , take θ P such that θ mP . Since the squares of θ , θ + θ , θ θ are all in N + mP2 , so is θ2 as well. That completes the proof. The case when φ is a differential operator of order 2 is treated similarly.

Lemma 2.3. The -linear span X of all endomorphisms χW (θ,θ,D,D) with θ,θ Ω1 and D, D W coincides with A = End RW .

Proof. Obviously, X is an R -submodule in A . Since A is finitely generated over R , it suffices to show that A = X + mA for every maximal ideal m of R . Note that

σW (θ D) σW (θ D) = θ,Dσ W (θ D)

and therefore χW (θ,θ,D,D) = 4θ,DσW (θ D) where we take θ,θ Ω1 and D,D W . If θ, Dm then we can find f R such that 4θ,Df 1 modulo m . Multiplying the previous equality by f , we deduce immediately that σW (θ D) X + mA . Suppose that θ,D m but DmW . Then θ,Dm for some θ Ω1 . Since σW (θ D) and σW ((θ + θ) D) are both in X + mA , so is σW (θ D) as well. Suppose finally that D mW . Pick out D W such that DmW . Then σW (θ D) and σW (θ (D + D)) are both in X + mA , whence so is σW (θ D) . We proved that σW (g) X + mA , which gives the assertion because A = σW (g) .

Proposition 2.4. Let M C1 . Then every 1 -cocycle φ : W M is a differential operator of order 3 . If rk RW > 1 then every 1 -cocycle is actually a differential operator of order 2 .

Proof. If D W then Dφ is a coboundary. By Lemma 2.1

Dφ = ρM(D) φ φ ρW (D) Diff 1(W,M).

In view of (1.6) ρM(D) Diff 1(M,M) and similarly ρW (D) Diff 1(W,W) . Hence

ρM(D)ρM(D)φ φρ W (D)ρW (D) = ρ M(D)(ρM(D)φ φρ W (D)) +(ρM(D)φ φρW (D))ρW (D)

is a differential operator of order 2 for all D,D W . The linear combination given in Lemma 1.5 yields

χM(θ,θ,D,D) φ φ χ W (θ,θ,D,D) Diff 2(W,M) ()

for all θ,θ dR Ω1 and D, D W . The inclusion holds actually for all θ,θ Ω1 since χM(fθ,θ,D,D) = χ M(θ,θ,D,fD) , where f R , and similarly for χW . We get

(δfφ) χW (θ,θ,D,D) = φ f W χW (θ,θ,D,D) f M φ χW (θ,θ,D,D) = fM (χM(θ,θ,D,D) φ φ χ W (θ,θ,D,D)) χM(fθ,θ,D,D) φ + φ χ W (fθ,θ,D,D) Diff 2(W,M).

It follows (δf1δf4φ) χ = 0 for all f1,,f4 R and all χ in the subgroup X End RW described in Lemma 2.3. Since 1W X , we deduce δf1δf4φ = 0 , i.e., φ is a differential operator of order 3 .

Suppose rk RW > 1 further on. Consider the symbol φ : S3Ω1 W M . The map φ is a differential operator of order 2 if and only if φ = 0 . Put

N = {ω S3Ω1ω D Ker φforallD W}.

Applying δf1δf2δf3 to () , we get

χM(θ,θ,D,D) δ f1δf2δf3φ = δf1δf2δf3φ χW (θ,θ,D,D),i.e., χM(θ,θ,D,D)(φ(θ 1θ2θ3 D)) = φ(θ 1θ2θ3 χW (θ,θ,D,D)D)

for all θ,θ,θ 1,θ2,θ3 dR and D, D,D W . In view of (1.3) this holds actually for all θ ’s in Ω1 . In other words,

χM(θ,θ,D,D) φ = φ (id χ W (θ,θ,D,D)). ()

It follows that all endomorphisms id χW (θ,θ,D,D) , hence by Lemma 2.3 all endomorphisms in id End RW , leave the kernel of φ stable. The endomorphism id σW (θ D) decomposes as

S3Ω1 Wiθ −−−→ S3Ω1 jD −−→ S3Ω1 W

where the maps iθ , jD are given by

iθ(ω D) = θ,Dω,j D(ω) = ω D,ω S3Ω1,D W.

As Ker φ is stable under jD iθ for every D W and θ Ω1 by the above, we see that iθ(Ker φ) N for every θ . On the other hand, φ is a morphism in C1 by Lemma 2.2. In particular, φ commutes with the actions of g . Therefore

χM(θ,θ,D,D) φ = φ χ S3Ω1W (θ,θ,D,D).

Comparing this with () , we see that the endomorphisms

χS3Ω1W (θ,θ,D,D) id χ W (θ,θ,D,D)

have images in Ker φ .

Now we are ready to prove that N = S3Ω1 , and so φ = 0 . By Nakayama’s Lemma it suffices to show that S3 Ω1 = N + m S3Ω1 for all maximal ideals m of R . Fix m . Let θ, θ Ω1 . We will check that θ2θ N + m S3Ω1 , whence our assertion. Since the vector space WmW and its dual Ω1mΩ1 have dimension > 1 over Rm , we can find D1 W , D1 mW , such that θ,D 1 0 , and then find D2 W , θ1 , θ2 Ω1 such that θi,Dj δij modulo m for i,j = 1, 2 . Put

a =(χS3Ω1W (θ,θ,D1,D1)idχW (θ,θ,D1,D1))(θ12θD 2) S3Ω1W.

As we noted above, a Ker φ . Now compute a . Using Lemma 2.5 below, we get a = ω1 D1 + ω2 D2 where ω1 turns out to be irrelevant, while

ω2 = χS3Ω1(θ,θ,D1,D1)(θ12θ) 4θ2θmodm S3Ω1.

It follows iθ2(a) ω2 4θ2θ modulo m S3Ω1 . Since iθ2 (a) N , we get the conclusion about θ2θ .

Lemma 2.5. Suppose that θ Ω1 , D W and A, B C1 . Then the endomorphism χAB(θ,θ,D,D) is equal to

χA(θ,θ,D,D) 1B + 4σA(θ,D) σB(θ,D) + 1A χB(θ,θ,D,D).

Denote by iD the derivation of the symmetric algebra SΩ1 such that iDf = 0 for f R and iDθ = θ,D for θ Ω1 . If A = SΩ1 and ω A then

χA(θ,θ,D,D)ω = 2θ2 i D2ω.

Proof. One checks (i) straightforwardly using the definitions of operators. Under hypotheses of (ii), σA(θ,D)ω = θ iDω . The conclusion of (ii) follows from the computation

σA(θ,D)2ω = θ2 i D2ω + θ,Dθ i Dω = θ2 i D2ω + θ,Dσ A(θ,D)ω.

3. First order prolongations

If M is an object of the category C0 then the standard cochain complex C(W,M) of -multilinear alternating maps W × × W M contains a subcomplex CR(W,M) whose elements are R -multilinear maps. The cohomology HR(W,M) of the latter is one of the ingredients in the cohomology H (W,M) of the ambient complex and can not be simplified any further in the general settings. Since W is a finitely generated projective R -module, CR (W,M)M Ω where Ω is the exterior algebra of the R -module Ω1 . In particular, CR(W,R)Ω generalizes the classical de Rham complex. If now M C1 then a part of H (W,M) is related to the cohomology of R -multilinear cochains for a certain extension of W which we describe below.

Consider, more generally, a pair ^W , π where ^W is an additive group endowed with structures of a Lie algebra over and a module over R , and π : ^ W W a map which is surjective and is a homomorphism of both structures simultaneously. Assume, moreover, that

(3.1) [D ˜,fD˜] = f[D˜,D˜] +(π(D˜)f)D˜ , for all f R and D˜,D˜ ^ W .

Such algebraic structures under different names were considered by many people [11, 14, 17, 19, 23]. In particular, the pair W , idW satisfies these conditions. Suppose M is an additive group endowed with an R -module and a ^W -module structures. Denote by ρ˜M(D˜) and fM the operators on M corresponding to elements D˜ ^ W and f R . If the identities

(3.2) [ρ ˜M(D˜),fM] =(π(D˜)f)M,

(3.3) ρ ˜ M(fD˜) = fM ρ˜M(D˜)

are fulfilled for M then the R -multilinear alternating maps ^W × × ^ W M form a cochain complex with respect to the standard differential (see the references above). Denote by HR( ^W,M) its cohomology.

The kernel k of π is an ideal of ^W on which the Lie multiplication is R -bilinear in view of (3.1). Similarly, (3.2), (3.3) show that the induced action k × M M is an R -bilinear operation. Denote by HR(k,M) the cohomology of the standard complex of R -multilinear alternating maps k × × k M . The Lie algebra ^W operates naturally in that cohomology group. As the action of k is trivial, there is the induced action of the factor algebra W ^ Wk . In particular, the k -invariants in M form an R -submodule Mk which has the induced structure of a W -module. As (3.2), (3.3) carry over to the induced representation of W in Mk , the R -linear cohomology HR(W,Mk) is defined too.

There is an analogue of the Hochschild-Serre spectral sequence relating the cohomology groups just described (see [16] for the construction in the settings of Lie algebroids). We will need only an exact sequence associated with the initial terms of this spectral sequence.

Lemma 3.1. There is an exact sequence

0 HR1(W,Mk) H R1( ^W,M) H R1(k,M)W H R2(W,Mk).

Proof. The map on the left is obtained by taking the composites of the R -linear 1 -cocycles W Mk with π . It is injective since a 1 -coboundary ^W M factors through W only if it is the differential of a 0-cochain lying in Mk . Restricting the R -linear 1 -cocycles ^W M to k gives the next map. Its images are W -invariant classes in HR1(k,M) since ^W acts in HR1( ^W,M) trivially. If a cocycle ^W M vanishes on k then it has values in Mk and factors through W , which shows the exactness at the second term.

Suppose now we are given an R -linear 1 -cocycle φ : k M whose cohomology class is W -invariant. Then η(D˜) = D˜φ is a coboundary for every D˜ ^ W . We get thus a map η : ^ W BR1(k,M) with values in the group of coboundaries k M . If f R , D ˜ ^ W , T k then

η(fD˜)(T) = ρ˜M(fD˜)φ(T) φ([fD˜,T]) = f ρM(D˜)φ(T) φ(f[D˜,T]) = f η(D˜)(T).

Hence η is R -linear. Obviously η(T) is the coboundary of φ(T) for every T k . We can extend φ to an R -linear map φ ˜ : ^ W M with the property that η(D˜) is the coboundary of φ˜(D˜) for every D˜ ^ W . In fact, the exact sequence

0 k ^ W W 0

splits as a sequence of R -modules in view of (1.2). In other words, ^W = k c with c an R -submodule. Since c is R -projective and the differential M BR1(k,M) is an epimorphism of R -modules, the restriction c BR1(k,M) of η can be lifted to an R -linear map φ : c M . Taking φ˜ to be φ on k and φ on c fulfills our requirement. Let now ψ : ^ W × ^ W M be the coboundary of φ˜ . Then ψ vanishes when one of its arguments is in k . It induces therefore a 2 -cocycle ψ¯: W × W M which takes values in Mk . The last map in the statement of the Lemma takes the cohomology class of φ to that of ψ¯ . One can check that it is well defined and gives the exactness in the same way as for ordinary Lie algebras.

Now take ^W = g × W to be the direct product of underlying additive groups. Note that W operates on g by means of ρg as a Lie algebra of derivations. Furnish ^W with the semidirect product of Lie algebra structures and a certain R -module structure. Explicit formulas are

[(T,D),(T,D)] =([T,T] + ρ g(D)T ρ g(D)T,[D,D]), f (T,D) = (fT df D,fD),

where f R , T, T g and D, D W . The projection π onto the second factor is clearly a homomorphism of both structures. The same definitions are in effect when g is replaced by g˜ . The kernel of π is g in the former case and g˜ in the latter. Identity (3.1) is a bit cumbersome, but we propose a more sophisticated argument in just a moment.

Definition 3.1. We call g × W (respectively g˜ × W ) with the Lie algebra and R -module structures just described the first order prolongation of W with kernel g (respectively g ˜ ).

Lemma 3.2.Suppose that ^W is the first order prolongation of W with kernel g or g˜ . Then C1 (respectively C ˜ 1 ) is isomorphic to the category whose objects are additive groups M together with an R -module and a ^W -module structures satisfying identities (3.2), (3.3) and morphisms are the maps compatible with both structures.

Proof. If we fix an R -module structure on an additive group M then the rule

ρ˜M(T,D) = σM(T) + ρM(D),T g,D W,

establishes a one-to-one correspondence between the families of operators ρ ˜ M(D˜) on M defined for each D ˜ ^ W and the families of operators ρM(D) , σM (T) defined for each D W and T g . Property (3.2) translates to the pair of (1.6) and (1.7), property (3.3) to (1.8) and (1.9). If ρ˜M is a Lie algebra representation then so is its restriction ρM to W . Conversely, if M C˜1 then it is immediate from Lemma 1.1 that ρ˜M is a Lie algebra representation. The same Lemma shows also that the morphisms in C˜1 are precisely the maps that are homomorphisms of R -module and ^W -module structures.

Now we can verify (3.1). Suppose that M is an object of C1 (respectively C ˜ 1 ) such that the induced representation ρ˜M is faithful. For instance, we can take M = W in case when the kernel is g . Since W is a faithful R -module by (1.2), property (3.2) implies that ker ρ˜M ker π = g . However g acts in W faithfully. In case of kernel g˜ we can take M = Ω ˜1 . Again (1.2) ensures the faithfulness of σM and ρ˜M . It remains to observe that, in view of (3.2) and (3.3), the elements at both sides of (3.1) act in M as

fM [ρ˜M(D˜),ρ˜M(D˜)] +(π(D˜)f) M ρ˜M(D˜).

Thus we can apply Lemma 3.1 to the situation where M is an object of C1 or even C ˜ 1 and ^W is the first order prolongation of W with kernel g˜ . In this case k = g˜ . Consider now the canonical embedding ι : W ^ W such that ι(D) = (0,D) for D W . Then ι is a differential operator of order 1. In fact (δfι)(D) = (df D,0) for f R and D W . Hence the symbol of ι is given by the formula ι(T) = (T, 0) for T g˜ .

Lemma 3.3.Suppose that M C1 and ^W is the first order prolongation of W with kernel g˜ . Given a differential operator φ : W M of order 1 , there exists a unique R -linear map φ˜ : ^ W M such that φ = φ˜ ι . Moreover, if φ is a cocycle then so is φ˜ as well. In this case the symbol φ : g˜ M of φ is also an R -linear cocycle. In order that a cocycle φ be R -linear, it is necessary and sufficient that φ˜ vanish on g˜ .

Proof. Every -linear map φ˜ : ^ W M satisfying φ = φ˜ ι can be written as

φ˜(T,D) = φ(T) + φ(D)forT g˜andD W.

where φ is a -linear map g ˜ M . In order that φ˜ be R -linear, it is necessary and sufficient that φ be R -linear and the equality φ(fD) fφ(D) = φ(df D) hold for all f R and D W . Thus φ = φ is the only choice which gives the desired property. Suppose that φ is a cocycle. Let N = M R be the direct sum of two R -modules. Define operators

ρN(D)(m,h) = (ρM(D)m + hφ(D),Dh) σN(T)(m,h) = (σM(T)m + hφ(T),0)

where D W , T g˜ , m M , h R . One checks straightforwardly that N is now an object of the category C˜1 . By Lemma 3.2 ρN extends to a representation ρ˜N of ^W satisfying (3.2), (3.3). In fact we have an exact sequence 0 M N R 0 in C ˜ 1 which is also an exact sequence of ^W -modules. Furthermore,

ρ˜N(D˜)(0, 1) = σN(T)(0, 1)+ρN(D)(0, 1) =(φ(T)+φ(D),0) =(φ˜(D˜),0)

for any D˜ = (T,D) ^ W . Applying the operator

ρ˜N([D˜,D˜]) = [ρ˜ N(D˜),ρ˜N(D˜)]

to (0, 1) , where D˜ ^ W is a second element, we get the cocycle condition for φ ˜ . Since the symbol ι : g˜ ^ W is the canonical embedding which is a homomorphism of Lie algebras, φ = φ˜ ι is a cocycle as well. Finally, φ is R -linear if and only if φ = 0 , i.e., if φ˜ ι = 0 .

4. Construction of universal cocycles

Among the 1-cocycles φ : W M of differential order 2 with values in the objects of C1 one can look for a one which satisfies the following universality property: for every object M C 1 and a 1-cocycle φ : W M of differential order 2 there exists a unique morphism ξ : M M in C1 such that the 1-cocycle φ ξ φ is a differential operator of order 1 . In fact proposition 4.1 gives such a cocycle φ : W S2Ω1 W which will be called the universal differential order 2 cocycle. When rkRW = 1 , proposition 4.5 describes a 1-cocycle W Ω1 Ω1 satisfying a similar universality property with respect to 1 -cocycles of differential order 3 . We call it the universal differential order 3 cocycle.

Definition 4.1. A -bilinear map : W × W W is a torsion-free connection on W if

(4.1) (fD,D) f(D,D) = 0,

(4.2) (D,fD) f(D,D) = (Df)D,

(4.3) (D,D) (D,D) = [D,D]

for all f R and D,D W . If only the first two identities are fulfilled then is a connection on W . A 1 -cocycle ψ : W R is a divergence on W if it satisfies the identity

(4.4) ψ(fD) fψ(D) = Df,f R,D W.

Proposition 4.1. There is a 1 -cocycle φ : W S2Ω1 W which is a differential operator of order 2 and whose symbol φ is the identity endomorphism of S2Ω1 W .

Proof. The Lie algebra W operates on the -bilinear maps W × W W in a natural way. If is a torsion-free connection on W , put

φ(D) = D ,D W.

One checks straightforwardly that D is a symmetric R -bilinear map for every element D . By the assumption (1.2) we may identify S2 Ω1 W with the group of such maps. Namely, given θ1,θ2 Ω1 and D W , the tensor θ1θ2 D determines the map

(D,D)(θ 1,Dθ 2,D + θ 1,Dθ 2,D)D,D,D W.

Thus φ takes values in S2Ω1 W . It is a cocycle since it comes from a coboundary in a larger module. Let f, g R . We have

(δfφ)(D)(D,D) = [fD,(D,D)] f[D,(D,D)] ([fD,D],D) + f([D,D],D) (D, [fD,D]) + f(D, [D,D]) = ((D,D)f)D + (Df)(D,D) (Df)[D,D] + (Df)(D,D) + (DDf)D,

and

(δgδfφ)(D)(D,D) =((Df)(Dg) + (Df)(Dg))D.

In other words, (δgδfφ)(D) = (df dg) D . We see that φ is a differential operator of order 2 with symbol φ = id . To complete the proof we need the Lemma below.

Lemma 4.2. A torsion-free connection on W does exist.

Proof. Let ^W be the first order prolongation of W with kernel g , as described in section 3. The projection ^W W is an epimorphism of R -modules. It splits by the projectivity of W over R . Thus there exists a map ξ : W g such that the assignment D(ξ(D),D) defines an R -module homomorphism W ^ W . That means that ξ(fD) = fξ(D) df D for all f R , D W . Now g operates on W by means of σW , and we have

ξ(fD)D = f ξ(D)D + (Df)D

for all f R , D, D W . Setting (D,D) = ξ(D)D , we get a connection on W . Let

τ(D,D) = (D,D) (D,D) [D,D].

Then τ : W × W W is a skewsymmetric R -bilinear map, the torsion of the connection. By a well known characterization of finitely generated projective modules, there exists a finite number of elements D1 ,,Ds W and θ1 , ,θs Ω1 such that every D W is expressed as θi,DDi . We get

τ(D,D) = i,j=1sθ i,Dθ j,Dτ(D i,Dj) = ν(D,D) ν(D,D)

where ν(D,D) = i<jθi,Dθ j,Dτ(D i,Dj) . Thus ν : W × W W is an R -bilinear map and = ν a torsion-free connection on W .

Lemma 4.3. There exists a -linear map ψ : W R satisfying identity (4.4). If rk RW = 1 then any such ψ is a divergence.

Proof. Let γ : g R be the contraction and ξ : W g the map considered in the proof of Lemma 4.2. Then ψ = γ ξ satisfies (4.4). Assume that ψ is an arbitrary -linear map satisfying (4.4). One checks straightforwardly that its coboundary ω : W × W R ,

ω(D,D) = D(ψ(D)) D(ψ(D)) ψ([D,D])forD,D W,

is R -bilinear and skewsymmetric. It corresponds therefore to a homomorphism of R -modules 2W R . If rkRW = 1 , then 2W = 0 , and ω = 0 . In other words, ψ is a cocycle.

Lemma 4.4. The composite φ : W R Ω1 of a divergence ψ with the differential d is a 1 -cocycle. Furthermore, φ is a differential operator of order 2 whose symbol φ is the epimorphism π : S2Ω1 W Ω1 in C1 defined by the rule

θθ Dθ,Dθ + θ,Dθ  for θ,θ Ω1andD W.

Proof. Clearly π is a morphism in C1 . Its surjectivity can be verified by passing to the reductions modulo the maximal ideals of R , where it becomes immediate. Since ψ is a cocycle and d a W -equivariant map, φ is a cocycle. Both ψ and d are differential operators of order 1 whose symbols are given by

ψ(θ D) = θ,Dandd(θ h) = hθ,

where θ Ω1 , D W , h R . Hence φ is a differential operator of order 2, and the computation of its symbol gives π , as required.

Proposition 4.5. Suppose that rk RW = 1 and 3 is invertible in R . Then there is a 1 -cocycle φ : W Ω1 Ω1 which is a differential operator of order 3 with symbol φ : S3Ω1 W Ω1 Ω1 an isomorphism in C1 .

Proof. The symmetric and the tensor powers of the R -module Ω1 coincide because rkRΩ1 = rk RW = 1 . To put it differently, every R -multilinear expression involving several arguments from Ω1 is symmetric in these arguments. The same observation applies to W . Identify Ω1 Ω1 with the group of R -bilinear maps W × W R so that a tensor θ1 θ2 with θ1,θ2 Ω1 corresponds to the map

(D,D)θ 1,Dθ 2,D,  where D,D W.

Let ψ : W R be a divergence. Take φ to be the composite

W ψ −−→ R d −−→ Ω1 ψ −−→ Ω1 Ω1,

where ψ(θ)(D,D) = ψ(D) , θ(D) D(θ,D) for θ Ω1 and  D , D W .

It is immediate that ψ(θ) is R -linear in D . Given f R , the expression

ψ(θ)(fD,D) f ψ(θ)(D,D) = (Df)θ,D (Df)θ,D

is skewsymmetric in D,D . So it has to vanish as well. Thus ψ is well defined. Now

ψ(fθ)(D,D) f ψ(θ)(D,D) = (Df)θ,D,

whence ψ(fθ) fψ(θ) = θ df . It follows that ψ is a differential operator of order 1 with symbol (ψ) minus identity transformation of S2Ω1Ω1 Ω1 . We saw in the proof of Lemma 4.4 that both ψ and d are differential operators of order 1 . The composite φ is therefore a differential operator of order 3 . Its symbol is computed as follows

φ(θ 1θ2θ3 D) = θi1,Dθi2θi3 = 6θ1,Dθ2θ3

for θ1,θ2,θ3 Ω1 and D W , where the sum is taken over all permutations of indices 1, 2, 3 and we use that the terms are symmetric in θ ’s. Since the natural pairing between Ω1 and W induces an isomorphism of R -modules Ω1 WR , again by the rank one assumption, φ : S3Ω1 W S2Ω1 is an isomorphism as well. To show that φ is a cocycle we embed S2Ω1 into the W -module of all -bilinear maps W × W R . Define ω : W × W R by the rule

ω(D,D) = ψ(D)ψ(D) D(ψ(D)) D(ψ(D))

for D,D W . As ψ is a cocycle, Dψ is the coboundary of ψ(D) for every D W . Hence

(Dω)(D,D) = (Dψ)(D) ψ(D) + ψ(D) (Dψ)(D) D((Dψ)(D)) D((Dψ)(D)) = D(ψ(D)) ψ(D) + ψ(D) D(ψ(D)) DD(ψ(D)) DD(ψ(D)).

On the other hand, computing φ(D) straightforwardly yields

φ(D)(D,D) = ψ(D) D(ψ(D)) DD(ψ(D)).

Since the left hand side is symmetric in D ,D , we get Dω = 2φ(D) . Thus φ is a coboundary in a larger module.

5. Determination of cohomology

Let M C1 . We introduce a filtration on the group of 1-cocycles Z1 (W,M) letting F i Z1(W,M) for i 0 denote its subgroup consisting of cocycles W M which are differential operators of order i . Set F1Z1(W,M) = 0 . We have seen in proposition 2.4 that F3Z1(W,M) (respectively F2Z1(W,M) when rk RW > 1 ) exhausts all the 1 -cocycles. Let FiH1(W,M) be the image of FiZ1(W,M) in the cohomology group H1(W,M) . First we are going to determine the factors

gr iH1(W,M) = F iH1(W,M)F i1H1(W,M).

Proposition 5.1. Let M C1 . Then there are isomorphisms

(5.1) gr0H1(W,M)H R1(W,Mg),

(5.2) gr1H1(W,M) Ker(H R1(g˜,M)W H R2(W,Mg)),

(5.3) gr2H1(W,M) Mor C1(S2Ω1 W,M),

(5.4) gr3H1(W,M) Mor C1(Ω1 Ω1,M)when rk RW = 1and3R = R.

Proof. Let ^W be the first order prolongation of W with kernel g˜ . The canonical embedding W ^ W is a Lie algebra homomorphism and also a differential operator of order 1. Hence the restriction to W of every R -linear cocycle ^W M is in F1Z1(W,M) . The resulting map ZR1( ^W,M) F 1Z1(W,M) is bijective by Lemma 3.3. Obviously the ^W -coboundaries correspond to the W -coboundaries. It follows F1H1(W,M)H R1( ^W,M). Again by Lemma 3.3

gr 0H1(W,M)F 0H1(W,M) Ker(H R1( ^W,M) H R1(g˜,M))

since the cohomology classes on the left are represented by the R -linear cocycles W M , while those on the right by the cocycles ^W M with zero restriction to g˜ . Lemma 3.1 gives now (5.1). Moreover, (5.2) also follows because

gr 1H1(W,M) Coker(F 0H1(W,M) F 1H1(W,M)) Coker(HR1(W,Mg) H R1( ^W,M)).

Next, by Lemma 2.2 we have a map

F2Z1(W,M) Mor C1(S2Ω1 W,M) ()

which assigns to a 1 -cocycle φ : W M in the F 2 term of the filtration its symbol φ . The kernel of this map is clearly F1Z1(W,M) . Since all coboundaries are in F1Z1(W,M) by Lemma 2.1, the map above induces an embedding of gr2H1(W,M) into Mor C1(S2Ω1 W,M) . Suppose now that ξ : S2Ω1 W M is a morphism in C1 . Let φ : W S2Ω1 W be the 1 -cocycle given by proposition 4.1. Then ξ φ is a 1 -cocycle W M which lies in F2Z1(W,M) and has symbol ξ φ = ξ as ξ is R -linear and φ = id . Thus () is surjective. The assertion about gr 3H1(W,M) is proved similarly, taken into account proposition 4.5.

Lemma 5.2. Suppose that 0 C0 C1 is a cochain complex of finitely generated projective R -modules and q > 0 an integer such that Hi(CmC) = 0 for all maximal ideals m of R in all degrees i < q . Then Hi(C M) = 0 for any R -module M and i < q .

Proof. The differential d : C0 C1 induces by passing to the reductions modulo any maximal ideal m an injective map C0mC0 C1mC1 because H0 (CmC) vanishes. It follows that d maps C0 isomorphically onto a direct summand of C1 . In fact, the localizations Cm0 and Cm1 at m are free modules of finite rank over the local ring Rm , and therefore Cm0 is mapped isomorphically onto a direct summand of Cm 1 by [1, II, § 3, Proposition 6]. Hence d is injective by [1, II, § 3, Theorem 1]. The cokernel of d is a finitely presented R -module because C0 and C1 are finitely generated and projective. By [1, II, § 3, Corollary 1 to Proposition 12] dC0 is a direct summand of C1 , as required.

Now C decomposes into a direct sum A B of two subcomplexes where A0 = C0 , A1 = dC0 and Ai = 0 in all other degrees, while B1 is any R -module complement of A1 in C1 . If M is an R -module then C M is a direct sum of complexes A M and B M . Therefore

Hi(C M)Hi(A M) Hi(B M)

for all i . Since the differential A0 M A1 M is an isomorphism, we have Hi(A M) = 0 for all i . Taking M = Rm gives Hi (BmB) = 0 for all maximal ideals m in degrees i < q . Since nonzero terms in B start from degree 1, we complete the proof reindexing B and applying induction on q .

Lemma 5.3. Suppose that Q C1 is an object satisfying (1.10), (1.11) and M C1 an object of type Q . If H1 (gmg,QmQ) = 0 for all maximal ideals m of R then HR1(g,M) = 0 as well.

Proof. Consider the standard cochain complex CR (g,Q) of R -multilinear alternating maps g × × g Q . Its components are finitely generated projective R -modules since so are Q and the exterior powers of g . Furthermore, if M0 is an R -module on which g operates trivially, then

CR(g,Q M 0)CR(g,Q) M 0.

In particular, C(gmg,QmQ)C R(g,QmQ)C R(g,Q) Rm for every m . The reduced modulo m complexes have zero cohomology in degree 1 according to the assumptions. Since gmg has a proper commutant, the first cohomology group of the trivial gmg -module is nonzero. Hence QmQ is not trivial, and therefore H0(gmg,QmQ) = 0 as well. By Lemma 1.3 MQ M0 for a suitable M0 . We complete the proof applying Lemma 5.2 with q = 2 .

Lemma 5.4. Let V be a vector space of dimension n over a field k of characteristic p2 . Denote by K¯ the kernel of the linear map S2V V V defined by the rule

θθ vθ,vθ + θ,vθforθ,θ V andv V .

The glV -modules V , V , V nV , and K¯ when n > 1 and n + 10 modp are nontrivial, absolutely irreducible, pairwise nonisomorphic and have zero cohomology in degree 1 (with the exceptions only for p = 3 , n 2 ).

The glV -module S2 V V is isomorphic with the direct sum K¯ V of two irreducible modules when n > 1 and n + 10 modp . It has a single irreducible factor module isomorphic to V otherwise. The glV -module K¯ has never V as its factor module.

Proof. The assertion about irreducibility is immediate for all modules stated, except for K¯ . In fact, S2V V is the component of degree 1 in the graded Lie algebra of general Cartan type G1 G0 G1 having G1 = V and G0 = glV . The structure of G1 as a G0 -module was investigated in [15, I, § 10]. If n > 1 and p does not divide n + 1 then K¯ is irreducible and G1K¯ V . This is clearly not affected by field extensions, so that we get the absolute irreducibility. Note that

dim K¯ = dim S2V V dim V = 1 2n2(n+1)n = 1 2n(n1)(n+2) > n.

However, the first three modules under consideration all have dimension n . It is easy to check that these three are nonetheless pairwise nonisomorphic with an exception for p = 3 , n = 1 .

The vanishing of cohomology is a general fact when p = 0 . If p > 0 it is achieved by inspection of weights with respect to a Cartan subalgebra h of glV . The weights ɛ1,,ɛn of V constitute a basis for the dual space h . The weights of V are now ɛi , the weights of V nV are ɛi δ , and those of K¯ are ɛl ɛi ɛj , where δ = ɛ1 + + ɛn and 1 i,j,l n . None of them is among the roots of glV , which are ɛl ɛi (except when p = 3 , n = 2 ).

Look now at the other possibilities for G1 and K¯ . If n = 1 then K¯= 0 and G1 V . Suppose that n > 1 but p divides n + 1 . By [15] K¯ is a single maximal submodule of G1 . Moreover, G1 contains a single irreducible submodule, say V , which satisfies V K¯ and V V . The factor module K¯V is irreducible of dimension greater than n . The only exception is the case n = 2 , p = 3 when glV acts trivially in K¯V . In any case V is not a factor module of K¯ , as asserted.

The category C1 contains objects Q satisfying (1.10) and (1.11) which give a certain canonical glueing of the gmg -modules of each type considered in Lemma 5.4. Three of these are W , Ω1 , and Ω1 Ωn where n = rk RW and Ωn =nΩ1 . Denote by K the kernel of the epimorphism π : S2Ω1 W Ω1 defined in Lemma 4.4. Since both Ω1 and S2Ω1 W are finitely generated projective R -modules, so is K as well. If m is a maximal ideal of R then KmK is the kernel of the induced linear map S2V V V of vector spaces over Rm where V = WmW and V is its dual. Suppose that (n + 1)R = R . Then it follows from Lemma 5.4 that KmK is an absolutely irreducible gmg -module, and so K satisfies (1.10), (1.11). Moreover, there is a canonical decomposition

S2Ω1 WK Ω1

in C1 . To see this consider the morhism μ : Ω1 g S2Ω1 W defined by the rule θ (θ D)θθ D for θ,θ Ω1 and D W . Then

π μ(θ T) = σΩ1(T)θ + γ(T)θforallθ Ω1andT g,

where γ : g R is the contraction. Since σΩ1 : g End RΩ1 is an isomorphism in C1 , there is a W -invariant element 1 g which corresponds to the identity endomorphism of Ω1 . The assignment θ μ(θ 1) defines obviously a morphism ν : Ω1 S2Ω1 W in C1 . Since γ(1) = tr 1Ω1 = rk RΩ1 = n , we deduce that π ν = (n + 1) 1Ω1 . If n + 1 is invertible in R then the restriction of π gives an isomorphism Im ν Ω1 . In this case Im ν is a subobject complementary to K in S2Ω1 W .

Now we have come to the final results on cohomology. Let Q C1 be an object satisfying conditions (1.10), (1.11) and M C1 an object of type Q . Recall that n = rk RW . In addition to (1.1) assume that 3 is invertible in R as well, at least when n = 1 .

Theorem 5.5. If for every maximal ideal m of R the gmg -module QmQ is isomorphic to neither the trivial module, nor WmW , nor (WmW) , nor KmK when n > 1 and char Rm does not divide n + 1 , nor 2(WmW) when n = 1 , then

H1(W,M)H R1(g,M)W .

If, moreover, H1(gmg,QmQ) = 0 for all m then H1(W,M) = 0 .

If R is an algebra over a field of characteristic 0 then char Rm = 0 for every maximal ideal m of R . In this case the reductive Lie algebra gmg over the field Rm has zero cohomology with coefficients in any nontrivial irreducible finite dimensional module, i.e., the condition H1 (gmg,QmQ) = 0 is fulfilled automatically. We will be proving theorem 5.5 simultaneously with the next result which treats exceptional modules.

Theorem 5.6. Suppose that M = M0 Q where M0 C0 , and let M0W be the subgroup of W -invariants in M0 .

Proof. Using proposition 5.1, we examine those objects Q for which the groups gr iH1(W,M) are not all zero. If a nonzero factor occurs for i = 0 , then Mg0 . Clearly Mg is a subobject of M , and in fact Mg C0 . The inclusion MgM is a nonzero morphism in C1 . By Lemma 1.4 QmQ is a trivial gmg -module for at least one maximal ideal m of R .

Next, gr 1H1(W,M)0 implies that HR1(g˜,M)W 0 . There is a surjective homomorphism g˜ g of Lie algebras over R whose kernel is J W , where J is the kernel of the canonical map Ω ˜1 Ω1 . Each R -linear cocycle φ : g˜ M determines by restriction an R -linear map J W M . Furthermore, the coboundaries restrict to J W trivially because g˜ operates in M via g . If the cohomology class of φ is W -invariant, then Dφ is a coboundary and therefore restricts to J W trivially for every D W . That means that the restriction J W M of φ is a W -invariant map and therefore a morphism in C1 . It is zero if and only if φ factors through g . Thus there is an exact sequence

0 HR1(g,M)W H R1(g˜,M)W Mor C1(J W,M). ()

If HR1(g,M)0 then H1 (gmg,QmQ)0 for at least one m by Lemma 5.3. On the other hand, J W C1 is an object of type W since J C0 according to Lemma 1.6. By Lemma 1.4 Mor C1(J W,M)0 implies QmQWmW as gmg -modules for at least one m .

Consider the case gr 2H1(W,M)0 . By (5.3) there exists a nonzero morphism ξ : S2Ω1 W M in C1 . Its image, say M , is a nonzero subobject of M . Then M C 1 is an object of type Q , and MM 0 Q with M0 C 0 by Lemma 1.3. Since W and Ω1 are finitely generated over R , so is M as well. Hence M mM for at least one maximal ideal m . The quotient M0mM 0 is a trivial gmg -module. The gmg -module M mMM 0mM 0 QmQ is therefore completely reducible with all irreducible submodules isomorphic to QmQ . On the other hand, M mM is an epimorphic image of S2(WmW) WmW . According to Lemma 5.4 either QmQ(WmW) or n > 1 , n + 1 is invertible in Rm and QmQKmK .

Finally, gr 3H1(W,M)0 implies that n = 1 and

QmQ 2(WmW)

for at least one m by similar reasons. Under the hypotheses of theorem 5.5, the groups griH1(W,M) vanish for all i1 . Both Mg and Mor C1(J W,M) are zero by Lemma 1.4. Hence (5.2) and () give

H1(W,M) gr 1H1(W,M)H R1(g˜,M)W H R1(g,M)W .

The final statement of theorem 5.5 follows from Lemma 5.3. We now check one by one all cases in theorem 5.6.

(i) By the above gr iH1(W,M) = 0 for i = 2, 3 . Hence an exact sequence

0 gr 0H1(W,M) H1(W,M) gr 1H1(W,M) 0.

The term on the left is retrieved from (5.1), where Mg = M . In view of (5.2) and () there is an embedding gr 1H1(W,M)H R1(g˜,M)W H R1(g,M)W . Since g annihilates M , we have

HR1(g,M) Hom R(g[g,g],M).

The commutant [g,g] consists of all elements of g that act in W with trace zero. Hence [g,g] is the kernel of the contraction γ : g R . It follows that γ induces an isomorphism g[g,g]R , and HR1(g,M) Hom R(R,M)M . We get thus the required exact sequence. Every 1 -cocycle φ : W M is a differential operator of order 1 and its symbol φ : g M is an R -linear cocycle. By the above φ(θ D) = θ,Dm for θ Ω1 , D W , where m MW is the element corresponding to the cohomology class of φ . If f R , D W then

φ(fD) fφ(D) = φ(df D) = df,Dm = (Df)m.

Suppose now that ψ : W R is a divergence. Given a W -invariant m MW , the assignment f fm defines a morphism ιm : R M in C1 . The composite φ = ιm ψ is then a cocycle W M . Furthermore, φ = ι m ψ . Since ψ = γ , the map H1(W,R) RW takes the cohomology class of ψ to 1 . As the map H1 (W,M) MW is natural in M , it takes the cohomology class of φ to ιm(1) = m .

(ii) Here gr iH1(W,M) vanishes for i1 . By lemmas 5.3 and 5.4 HR1(g,M) = 0 . From (5.2) and () we deduce now an embedding

H1(W,M) gr 1H1(W,M)H R1(g˜,M)W Mor C1(J W,M).

The exact sequence 0 J Ω ˜1 Ω1 0 in C˜1 splits as a sequence of R -modules since Ω1 is R -projective. It gives rise therefore to an exact sequence in C ˜ 1

0 Hom R(Ω1,M 0) Hom R(Ω ˜1,M 0) Hom R(J,M0) 0.

Here Hom R(Ω1,M 0)M0 Hom R(Ω1,R)M 0 W = M . On the other hand, Hom R(Ω ˜1,M 0) Der(R,M0) by the universality property of Kähler differentials. Thus M is embedded into Der(R,M0) and

NM =( Der(R,M0)M)W Hom R(J,M0)W = Mor C0(J,M0).

Consider the diagram

( Der(R,M0)M)W Mor C0(J,M0) H1(W,M) `−−→ Mor C1(J W,M)

where the left vertical arrow is the connecting map in cohomology arising from the short exact sequence of W -modules 0 M Der(R,M0) Der(R,M0)M 0 and the right one is the isomorphism of Lemma 1.4. Now take a W -invariant coset Δ + M where Δ N . Pushing it right in the diagram gives a morphism J M0 in C0 which is the restriction of the R -linear map ξ : Ω ˜1 M 0 defined by the rule ξ(df) = Δf for f R . Pushing Δ + M down gives the cohomology class of the cocycle φ : W M such that φ(D) = D Δ for D W . Pushing φ further right, we come to a morphism J W M in C1 which is the restriction of the symbol φ : Ω ˜1 W M of φ . If f R , D W then the derivation (fD) Δ f(D Δ) takes g R to

ρM0(fD)(Δg) f ρM0(D)(Δg) Δ(fDg) + f Δ(Dg) M0.

The first two terms cancel as σM0 = 0 . Hence the result is (Dg) (Δf) , which is the value at g of the element Δf D M0 W regarded as a derivation R M0 . In other words, φ(df D) = Δf D . Since φ = ξ 1 W , the diagram is anticommutative. It follows that the left arrow is an isomorphism.

(iii) In this case gr iH1(W,M) = 0 for i2 . If ξ : K M is a morphism in C1 then its image K is a subobject of M . Hence K is of type Ω1 , and so KK 0 Ω1 with K0 C 0 by Lemma 1.3. For every maximal ideal m of R the gmg -module K mKK 0mK 0 (WmW) is completely reducible with all irreducible submodules isomorphic to (WmW) . On the other hand, K mK is an epimorphic image of KmK which does not have (WmW) as its factor module according to Lemma 5.4. Hence K = mK . Since K is finitely generated over R , it follows K = 0 by Nakayama’s Lemma. Thus every morphism S2 Ω1 W M vanishes on K and therefore factors through Ω1 . By (5.3)

H1(W,M) gr 2H1(W,M) Mor C1(S2Ω1 W,M) Mor C1(Ω1,M).

(iv) Again gr iH1(W,M) is nonzero for i = 2 only. Since Mor C1(Ω1,M) = 0 by Lemma 1.4, we have

H1(W,M) gr 2H1(W,M) Mor C1(K Ω1,M) Mor C1(K,M).

(v) Here gr iH1(W,M) = 0 for i3 , and

H1(W,M) gr 3H1(W,M) Mor C1(Ω1 Ω1,M).

The description of cocycles representing cohomology classes is immediate from the construction of isomorphisms. Lemma 1.4 shows also that in any of the cases

Mor C1(Q,M) Mor C0(R,M0) = Hom R(R,M0)W M 0W .

Corollary 5.7. The Lie algebra Der W of all -linear derivations of W is isomorphic with the normalizer N of W in Der R .

Proof. The adjoint representation of N in its ideal W induces a homomorphism of Lie algebras N Der W . It is an isomorphism because so is the induced map NW Der W ad WH1(W,W) by (ii).

This result was obtained earlier [22] under assumptions weaker than (1.2), (1.3). If k RW is a subring, then a derivation Δ N is k -linear if and only if so is the induced derivation ad Δ Der W . Therefore the subalgebra Der kW of k -linear derivations of W is isomorphic with N Der kR , where Der kR are the k -linear derivations of R . If R is the ring of functions on X , a smooth manifold, real analytic one or a Stein space, and k the field of real or complex numbers then Der kR is isomorphic with the Lie algebra of vector fields Vect X of respective class on X . It follows that all k -linear derivations of Vect X are inner.

6. The case of commuting derivations

We will specialize our assumptions on R , W . Suppose that R is an algebra over a field k and that W is a free R -module generated by a system of pairwise commuting k -linear derivations 1,,n . We still keep our basic assumptions (1.1)–(1.3). There is a very explicit construction of certain representations of W in this case. I would like to thank Naihong Hu who drew my attention to Shen’s paper [20] where this construction appeared under the name of mixed products. Accordingly, we are able to write down the 1-cocycles quite explicitly. Commuting derivations appear in many interesting situations. For instance, the Lie algebras of Witt type in positive characteristic fit into our present settings. Degree one cohomology in that special case was considered by Chiu and Shen [3] and Dzhumadil’daev [4].

Denote by D the k -linear span of 1 ,,n . This is an abelian subalgebra of W = RDR kD . We have

gglRWR k glkD.

Let Eij be the linear transformation of D such that Eijl = δjli where indices i,j,l are taken among 1,,n . If θ Ω1 then the element θ i g corresponds to jθ,j Eij under the isomorphism above. Given a representation σV of glkD in a vector space V over k , put

V = R kV

and define a representation of W in this space setting

ρV (gi) = gi id V j(jg)R σV (Eij)forg R,i = 1,,n,

where fR stands for the multiplication operator on R corresponding to an element f R . The operators fR id V give V a compatible R -module structure, so that (1.6) is fulfilled. Extend σV by R -linearity to a representation of g in V . If θ Ω1 then

σV (θ i) = jθ,jR σV (Eij).

Take f,g R . As

ρV (gi)(gRid V )ρV (i) = j(jg)RσV (Eij) = σV (dgi),

we have for D = gi W

ρV (fD) (fR id V ) ρV (D) = σV (d(fg) i) σV (f dg i) = σV (df D).

Thus (1.8) is fulfilled too. We see that V together with the three module structures we have described is an object of C1 . In fact is a functor from the category of glkD -modules to C1 . It takes the tensor product of two glkD -modules to the tensor product of the corresponding objects in C1 , the symmetric and exterior powers of a glkD -module to the symmetric and exterior powers in C1 . If dim V < and V is the contragredient glkD -module then V Hom R(V,R) in C1 .

Clearly V is free over R . It has finite rank provided dim V < . If V is an absolutely irreducible glkD -module, then so is V(m V )Rm kV as a module over gmgRm k glkD where m is any maximal ideal of R . In this case V satisfies (1.10) and (1.11). Therefore we can determine its cohomology applying theorems 5.5, 5.6 where we take M = Q = V . Note that Q corresponds to M0 = R under the equivalence of Lemma 1.3. Denote by RW the ring of W -invariant elements in R . In case char k = 3 assume n > 1 . There are several cases.

(i) If V = k is the trivial glkD -module then V R and

H1(W,R) = H1(Ω) RW cls(ψ)

where the first summand is the subgroup of cohomology classes represented by the R -linear cocycles and the second summand is a free cyclic module over the ring RW whose generator is the cohomology class of a divergence ψ : W R . One can take ψ with zero values on D . Then ψ(fi) = if for f R and i = 1,,n .

(ii) If V = D is the natural glkD -module, then we have V W and H1(W,W)NW , where N is the normalizer of W in Der R .

(iii) If V = D then V Ω1 and H1 (W, Ω1) is a free cyclic module over RW generated by the cohomology class of any cocycle φ : W Ω1 whose symbol is the canonical epimorphism S2Ω1 W Ω1 in C1 . By Lemma 4.4 one can take φ = d ψ where ψ : W R is a divergence. If ψ is the same as in (i), then

φ(fi)(j) = j(ψ(fi)) = jifforf Rand1 i,j n.

(iv) Suppose n > 1 and char k does not divide n + 1 . If V is the kernel of the canonical linear map S2D D D then V K is the kernel of the canonical epimorphism π : S2Ω1 W Ω1 in C1 . The group H1 (W,K) is a free cyclic module over RW generated by the cohomology class of the cocycle φ = πK φu where πK : S2Ω1 W K is the projection and φu : W S2Ω1 W the universal differential order 2 cocycle of proposition 4.1. We can take a torsion free connection : W × W R with zero restriction to D × D . Let f,g R and 1 i,j,j n . Then (fj,gj) = (fjg) j , and so

φu(fi)(j,j) = ([fi,j],j) (j,[fi,j]) = (jjf) i

We have πK = id 1 n+1ν π where ν : Ω1 S2Ω1 W is the canonical morphism in C1 such that π ν = (n + 1) 1Ω1 . Let ɛ1 , ,ɛn be the dual basis for the free R -module Ω1 , so that ɛi,j = δij . Since rɛr r g corresponds to the identity endomorphism of Ω1 under σΩ1 , the morphism ν takes θ Ω1 to rθɛr r . Now φu (fi) = 1 2 r,s(rsf) ɛrɛs i. Applying π to this, we get s(isf) ɛs in Ω1 . Applying next ν , we get r,s(isf) ɛsɛr r in S2Ω1 W . It follows

φ(fi)(j,j) = (jjf) i 1 n + 1(ijf) j 1 n + 1(ijf) j.

(v) Suppose that n = 1 . If V = D D then V Ω1 Ω1 and H1 (W, Ω1 Ω1) is a free cyclic module over RW generated by the cohomology class of the cocycle φ described in proposition 4.5. Recall that φ = ψ d ψ where ψ is a divergence. As an R -module, Ω1 Ω1 is free with one generator ɛ2 = ɛ ɛ , where ɛ Ω1 is specified by the relation ɛ,1 = 1 . We take ψ with zero value on 1 . Then ψ(fɛ)( 1,1) = 1f , and so ψ(fɛ) = ( 1f) ɛ2 , for f R . As dψ(f1) = (12f) ɛ , we get

φ(f1) = (13f) ɛ2.

(vi) Suppose that V is an absolutely irreducible finite dimensional glkD -module other than those considered in (i)–(v). Then H1 (W,V )H R1(g,V )W . The standard cochain complex for g admits the following identification:

CR(g,V ) Hom R( g,V )Rk Hom k( glkD,V ) = C(gl kD,V )

with the differential 1R dV where dV is the differential of the standard cochain complex C (gl kD,V ) of k -multilinear alternating maps glkD × × glkD V . What we get above is, moreover, an isomorphism in the category C1 . Since dV is a glkD -equivariant map, the differential 1R dV is a morphism in C1 . By passing to the cohomology we still get objects of C1 and an isomorphism in this category

HR(g,V )R kH(gl kD,V ) = H(gl kD,V ).

As glkD acts in the cohomology group of its module V trivially, the action of W in the corresponding object of C1 is given by the rule D (f ζ) = Df ζ for D W , f R and ζ H(gl kD,V ) . Taking the W -invariants, we conclude

H1(W,V )RW kH1(gl kD,V ).

We want to describe this isomorphism on the level of cocycles. Given a 1-cocycle φ : gl kD V , define a map φ : W V by the rule

φ(fi) = jjf φ(E ij)forf Randi = 1,,n.

Clearly, φ is a differential operator of order 1. Since φ(i) = 0 , we have

φ(df i) = φ(fi) = j(1Rφ)( jfEij) = (1Rφ)(σ W (dfi)).

Thus φ = (1 R φ) σ W , that is, φ : g V is an R -linear cocycle and cls(φ) corresponds to 1 cls(φ) , where cls stands for the cohomology class of a cocycle. Let us check that φ is itself a cocycle. Consider the first order prolongation ^W of W with kernel g ˜ . By Lemma 3.3 there is an R -linear map φ˜ : ^ W V whose composite with the canonical embedding ι : W ^ W gives φ . Denote by η : ^ W × ^ W V its coboundary. Then η(T,T) = 0 for all T,T g˜ since φ ˜ ι = φ is a cocycle. Observe now that φ is a D -equivariant map and vanishes on D . Hence φ˜ is also D -equivariant and vanishes on ι(D) . It follows that η has zero value whenever one of its arguments is in ι(D) . Since η is R -bilinear and g ˜ + ι(D) generates ^W as an R -module, η is identically zero. Thus φ˜ is a cocycle, and so is φ as well. We see that cls(φ) corresponds to 1 cls(φ) .

7. Universal central extensions

Recall that a -split central extension of W by a -module V is an exact sequence 0 V L W 0 where L W is a homomorphism of Lie algebras over whose kernel is a central ideal and also a -module direct summand of L . We call V the kernel of the central extension. An extension is said to be split if L contains a subalgebra mapped isomorphically onto W . The equivalence classes of -split central extensions of W by V are in a one-to-one correspondence with the cohomology classes of -bilinear 2 -cocycles W × W V , the coefficients being a trivial W -module. A -split central extension Lu of W by a -module U is universal if for every other -module V and a -split central extension L of W by V there is a unique -linear map U V which extends to a morphism between the two extensions:

0 -----// U-----//Lu  ----//W  -----//0
        |        |       |
        |        |       |id
        -----//  -----//  -----//   -----//
0       V       L       W        0.

In other words, a universal central extension corresponds to an isomorphism of functors H2(W,V ) Hom (U,V ) in V . Recall that Ω = Ω1 is the de Rham complex relative to W .

Theorem 7.1. Assume that 3R = R . If rk RW > 1 then every -split central extension of W splits. If rk RW = 1 then the universal central extension of W has kernel H1(Ω) and is determined by the 2 -cocycle φ : W × W H1(Ω) such that φ(D,D) for D, D W is the cohomology class of the 1 -form ψ(D) dψ(D) , where ψ : W R is a divergence.

Without assumption 3R = R the same proof shows that W has no nontrivial central extensions provided rkRW > 2 .

Proof. Put n = rk RW . There is an obvious embedding

H2(W,V )H1(W, Hom (W,V )) ()

obtained by separating the two arguments of the cocycles W × W V . The image of H2(W,V ) consists precisely of the cohomology classes represented by the 1-cocycles φ : W Hom (W,V ) which satisfy φ(D)(D) = 0 for all D W . Consider the W -module M = Hom (W,V ) and denote by ρM the corresponding representation of W . In a natural way M is a right module for the ring End RW , hence a left module for the opposite ring (End RW)op . In particular, M is also an R -module. Let f R , D W , ξ M . As ρM (D)ξ = ξ ρW (D) and fM ξ = ξ fW , we get

[ρM(D),fM]ξ = ξ(fW ρW (D)+ρW (D)fW ) = ξ(Df)W = (Df)Mξ,

which verifies (1.6). For T g define an endomorphism σM(T) End RM by the rule

σM(T)ξ = ξ (γ(T) 1W σW (T)),ξ M,

where γ : g R is the contraction. As

(ρM(fD)ξ f ρM(D)ξ)(D) = ξ( [fD,D] + [D,fD]) = ξ((Df)D σ W (df D)D)

for f R and D,D W , we get (1.8). The other identities in the definition 1.1 are immediate. Thus M is now an object of C1 . Put Q = Ω1 Ωn . This is an object of C1 satisfying (1.10), (1.11). Since Ωn is a projective R -module of rank 1 , its endomorphism algebra can be identified with R , which yields isomorphisms of R -algebras

End RQ End RΩ1 End RΩn End RΩ1(End RW)op.

Let T g . The endomorphism σΩn(T) is just the multiplication by the trace of σΩ1(T) , that is, by γ(T) . Hence

σQ(T) = σΩ1(T)1Ωn +1Ω1 σΩn(T) = σΩ1(T)1Ωn +γ(T)1Ω1 1Ωn.

Under the isomorphisms above it is sent to σΩ1(T) + γ(T) 1Ω1 in End RΩ1 , and then to γ(T) 1W σW (T) in (End RW)op . Thus M is of type Q .

We apply theorems 5.5, 5.6 to compute H1 (W,M) . If m is a maximal ideal of R then QmQ(WmW)n(WmW) . By Lemma 5.4 this gmg -module is isomorphic to neither the trivial module, nor WmW , nor (WmW) , nor KmK . Furthermore, H1 (gmg,QmQ) = 0 . It follows H1(W,M) = 0 provided n > 1 . If n = 1 then

H1(W,M) Mor C1(Ω1 Ω1,M).

The right hand side can be described as the W -invariant elements in

Hom R(Ω1 Ω1,M) Hom (Ω1 Ω1 W,V ) Hom (Ω1,V )

because Ω1 WR . Taking the W -invariants, we get

Mor C1(Ω1 Ω1,M) Hom W (Ω1,V ) Hom (Ω1ρ Ω1(W)(Ω1),V ).

Since Ω2 = 0 , the classical formula relating the action of W and the interior product on differential forms gives ρΩ1 (D)θ = d(θ,D) for θ Ω1 , D W . Since the elements θ,D span the whole R , again by the projectivity assumption, the group ρΩ1 (W)(Ω1) consists of all exact 1 -forms. Hence Ω1ρ Ω1(W)(Ω1)H1(Ω) .

Now take a -linear map η : H1(Ω) V and write out the corresponding 1 -cocycle φ : W M . Tracing back the isomorphisms above, we first find the morphism ζ : Ω1 Ω1 M in C1 . It is related to η as follows:

ζ(ω)(D) = η( cls(iDω))forω Ω1 Ω1,D W,

where iDω Ω1 is given by (iDω)(D) = ω(D,D) for D W (recall that Ω1 Ω1 can be identified with the group of R -bilinear maps W × W R ) and cls refers to the cohomology class of a 1 -form. Then φ is the composite of ζ and the cocycle of proposition 4.5, that is, φ = ζ ψ d ψ where ψ : W R is a divergence. Note that

iD(ψ(θ)) = ψ(D)θ d(θ,D)forθ Ω1,D W.

We have φ(D) = ζ(ω) where ω = ψ(θ) and θ = dψ(D) . As iDω ψ(D)θ modulo the exact 1-forms, it follows

φ(D)(D) = η( cls(ψ(D) dψ(D)))forD,D W.

Since ψ(D) dψ(D) = 1 2dψ(D)2 is an exact form, we have φ(D)(D) = 0 . This shows that () is an isomorphism and so

H2(W,V ) Hom (H1(Ω),V ).

Thus the central extension corresponding to V = H1(Ω) and the identity map η : H1(Ω) V is universal. The corresponding cocycle is determined as well.

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Research Institute of Mathematics and Mechanics, Kazan State University, Universitetskaya str. 17, Kazan:420008, RUSSIA

E-mail address: sskryabi@vub.ac.be

Received December 20, 2003