Lobachevskii Journal of Mathematics

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Vol. 14, 2004, 69–107 \begin{matrix}c \\ ◯ \end{matrix} 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 Der R. 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 𝔤𝔩_{n}(R/𝔪), 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 GL_{n}.

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 H^{1}(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_{ℝ} H^{1}(Ω) = 1. Another interesting example considered by Wagemann recently [27] is the Lie algebra Vect_{1,0} Σ of complexified C^{∞} vector fields of type (1, 0) on a Riemann surface Σ. The relative de Rham complex is here the complex 0 → Ω^{0,0} → Ω^{1,0} → 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 \overline{O} of antiholomorphic functions on Σ, its cohomology in degree 1 is isomorphic with H^{1}(Σ, \overline{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 ℤ-algebra of derivations such that RW = W, that is, W is an R-submodule in Der R. Put Ω^{1} = Hom_{R}(W, R) and define df ∈ Ω^{1} 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) Ω^{1} = R · dR. We furnish the R-module 𝔤 = Ω^{1} ⊗_{R} W with an R-bilinear Lie product setting

[θ ⊗ D, θ′ ⊗ D′] = ⟨θ′, D⟩ θ ⊗ D′ − ⟨θ, D′⟩ θ′ ⊗ D

for elements θ, θ′ ∈ Ω^{1} and D, D′ ∈ W, where ⟨·, ·⟩ stands for the natural pairing Ω^{1} × W → R. There is an isomorphism σ_{W} of 𝔤 onto the Lie algebra 𝔤𝔩_{R} W of all R-linear transformations of W defined by the rule

σ_{W} (θ ⊗ D)(D′) = −⟨θ, D′⟩D .

Definition 1.1. Denote by C_{1} the category whose objects are additive groups M together with a system of operators f_{M}, ρ_{M}(D), σ_{M}(T) defined for each f ∈ R, D ∈ W, T ∈ 𝔤 so that the following properties are satisfied:

(1.4) the operators f_{M} give M the structure of an R-module , (1.5) the operators ρ_{M}(D) give M the structure of a W-module , (1.6) [ρ_{M}(D), f_{M}] = (Df)_{M}, (1.7) [σ_{M}(T), f_{M}] = 0 , (1.8) ρ_{M}(fD) = f_{M} ∘ ρ_{M}(D) + σ_{M}(df ⊗ D) , (1.9) σ_{M}(fT) = f_{M} ∘ σ_{M}(T) .

The morphisms in C_{1} are the maps that commute with the actions of R and W. Denote by C_{0} the full subcategory of C_{1} consisting of objects M with σ_{M} = 0.

The category C_{1} is closed under several operations. If M, N are its objects then the R-modules M ⊗_{R} N and Hom_{R}(M, N) are in a natural way objects of C_{1} too. We agree to write M ⊗N suppressing the subscript in the tensor product. The corresponding operators are given by

\begin{aligned}ρ_{M⊗N}(D)(u ⊗ v) & = & ρ_{M}(D)u ⊗ v + u ⊗ ρ_{N}(D)v, \\ σ_{M⊗N}(T)(u ⊗ v) & = & σ_{M}(T)u ⊗ v + u ⊗ σ_{N}(T)v, \\ ρ_{Hom_{R}(M,N)}(D)ξ & = & ρ_{M}(D) ∘ ξ − ξ ∘ ρ_{N}(D) \\ σ_{Hom_{R}(M,N)}(T)ξ & = & σ_{M}(T) ∘ ξ − ξ ∘ σ_{N}(T), \end{aligned}

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 ⨂^{r}M of the underlying R-module of M is an object of C_{1}. The same is valid for the symmetric power S^{r}M and the exterior power ⋀^{r}M of the R-module M as these are factors of ⨂^{r}M 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 C_{0} letting ρ_{R} be the natural action of W on R and σ_{R} = 0. Similarly, W together with the adjoint representation ρ_{W} and σ_{W} defined earlier is an object of C_{1}. Hence Ω^{1} and 𝔤 are objects of C_{1} too. Most of the natural R-linear maps that we will happen to deal with are in fact morphisms in C_{1}. For instance, so is σ_{M} : 𝔤 → End_{R} M for any M ∈ C_{1}. That σ_{M} is a W-equivariant map is asserted in Lemma 1.1 below. Another example is the contraction γ : Ω^{1} ⊗ 𝔤 → R defined by the rule γ(θ ⊗ D) = ⟨θ, D⟩ for θ ∈ Ω^{1} and D ∈ W.

Note that (1.2) enables one to define the trace function tr : End_{R} W → R as follows. The R-module ⋀^{n}W is projective of rank 1. Therefore End_{R} ⋀^{n}W ≅ R. There is a natural representation of the Lie algebra 𝔤𝔩_{R} W in ⋀^{n}W via R-linear transformations. Each element T_{W} ∈ 𝔤𝔩_{R} W acts as a multiplication by a certain element of R called the trace of T_{W}. Now γ(T) = − tr σ_{W} (T) for T ∈ 𝔤. Similarly, γ(T) is the trace of the R-linear endomorphism σ_{Ω^{1}}(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 Ω^{1} is the module of linear differential forms. Tensor fields of any possible type constitute an object of the category C_{1} 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 C_{1}. Then the operators σ_{M}(T) with T ∈ 𝔤 define a representation of 𝔤 and [ρ_{M}(D), σ_{M}(T)] = σ_{M}(ρ_{𝔤}(D)T ) for all D ∈ W, T ∈ 𝔤. Every morphism in C_{1} 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 σ_{M}(T′) = ρ_{M}(gD′) − g_{M} ∘ ρ_{M}(D′), and

[ρ_{M}(D), σ_{M}(T′)] = ρ_{M}([D, gD′]) − g_{M} ∘ ρ_{M}([D, D′]) − (Dg)_{M} ∘ ρ_{M}(D′)

= σ_{M}(ρ_{𝔤}(D)T′)

since ρ_{𝔤}(D)T′ = dg ⊗ [D, D′] + d(Dg) ⊗ D′. Now for T = df ⊗ D with f ∈ R and D ∈ W we express σ_{M}(T) from (1.8) and get

[σ_{M}(T), σ_{M}(T′)] = σ_{M}(ρ_{𝔤}(fD)T′ − f · ρ_{𝔤}(D)T′) = σ_{M}(σ_{𝔤}(T)T′) .

One checks that

σ_{𝔤}(T)T′ = σ_{Ω^{1}}(T)(dg) ⊗ T′ + dg ⊗ σ_{W} (T)D′
= ⟨dg, D⟩ df ⊗ D′ − dg ⊗ ⟨df, D′⟩ D = [T, T′] .

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

If M ∈ C_{1} and 𝔪 is a maximal ideal of R then M/𝔪M is a module for the Lie algebra 𝔤/𝔪𝔤 over the field R/𝔪. We may identify 𝔤/𝔪𝔤 with the Lie algebra 𝔤𝔩(W/𝔪W) of all linear transformations of the vector space W/𝔪W. 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 ∈ C_{1}:

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

(1.11) for each maximal ideal 𝔪 of R the quotient Q/𝔪Q is an absolutely irreducible module for the Lie algebra 𝔤/𝔪𝔤.

Lemma 1.2. Suppose that Q ∈ C_{1} satisfies (1.10), (1.11). Then the associative R-algebra End_{R} Q is generated by the endomorphisms σ_{Q}(T) with T ∈ 𝔤. Proof. Put A = End_{R} Q, and let B be its subalgebra (containing the identity endomorphism) generated by all σ_{Q}(T). For each maximal ideal 𝔪 of R the image of B in A/𝔪A ≅ End_{R/𝔪}(Q/𝔪Q) is the associative subalgebra generated by all endomorphisms of an absolutely irreducible representation in Q/𝔪Q. It is therefore the whole A/𝔪A, 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 C_{1} have to be dealt with. Hence we are led to the following Definition 1.2. Suppose that Q ∈ C_{1} is an object satisfying (1.10), (1.11). We say that an object M ∈ C_{1} is of type Q if there is a homomorphism of associative R-algebras End_{R} Q → End_{R} M which takes σ_{Q}(T) to σ_{M}(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_{R} Q in End_{R} M is the subalgebra, say A_{M}, generated by the endomorphisms σ_{M}(T) with T ∈ 𝔤. Each σ_{M′}(T) is the restriction of σ_{M}(T). Therefore M′ is stable under A_{M}, and the resulting homomorphism of R-algebras A_{M} → End_{R} M′ takes σ_{M}(T) to σ_{M′}(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 ∈ C_{1} is of type R if and only if σ_{M}(T) = 0 for all T ∈ 𝔤, that is, M ∈ C_{0}. Lemma 1.3. Suppose that Q is an object of C_{1} satisfying (1.10), (1.11). Then the functor M_{0} ↦ M_{0} ⊗ Q is an equivalence between C_{0} and the full subcategory of C_{1} consisting of objects of type Q. Proof. Put A = End_{R} Q. Note that Q is a projective generator in the category of R-modules. By Morita theory the functor M_{0} ↦ M_{0} ⊗ 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 ∈ C_{1} is of type Q. Then the homomorphism A → End_{R} M afforded by the definition 1.2 makes M into an A-module. Let M_{0} = Hom_{A}(Q, M). Then M_{0} ⊂ H where H = Hom_{R}(Q, M) is an object of C_{1}. In view of Lemma 1.2 η ∈ H is in M_{0} if and only if η commutes with the action of 𝔤. Hence M_{0} is the kernel of the morphism φ : H → Hom_{R}(𝔤, H) in C_{1} defined by the rule

φ(η)(T) = σ_{H}(T)η = σ_{M}(T) ∘ η − η ∘ σ_{Q}(T)

for η ∈ H, T ∈ 𝔤. Thus M_{0} ∈ C_{1} and, since the induced action of 𝔤 in M_{0} is trivial, in fact M_{0} ∈ C_{0}. The canonical map M_{0} ⊗ Q → M is a morphism in C_{1}. It is bijective by Morita theory.

Conversely, suppose that M_{0} ∈ C_{0}. Then M = M_{0} ⊗ Q is an object of C_{1} and σ_{M}(T) = id ⊗ σ_{Q}(T) for all T ∈ 𝔤. The assignment ξ ↦ id ⊗ ξ defines a homomorphism of R-algebras A → End_{R} M which takes σ_{Q}(T) to σ_{M}(T). Thus M is of type Q. The canonical map M_{0} → Hom_{A}(Q, M) is a bijective morphism in C_{0}. □

Lemma 1.4. Let Q, Q′ be two objects of C_{1} satisfying (1.10), (1.11). Suppose that M, M′ ∈ C_{1} are objects of type Q and Q′ respectively, so that M ≅ M_{0} ⊗ Q and M′ ≅ M′_{0} ⊗ Q′ for some M_{0}, M′_{0} ∈ C_{0}. If the 𝔤/𝔪𝔤-modules Q/𝔪Q and Q′/𝔪Q′ are not isomorphic for every maximal ideal 𝔪 of R then Mor_{C_{1}}(M, M′) = 0. If Q = Q′ then Mor_{C_{1}}(M, M′) ≅ Mor_{C_{0}}(M_{0}, M′_{0}). Proof. Suppose there is a nonzero morphism M → M′ in C_{1}. Its image N is a factor object of M and a subobject of M′ in C_{1}. It is therefore of type Q and Q′ simultaneously. Put A = End_{R} Q, A′ = End_{R} Q′, and let B be the subalgebra of the associative R-algebra End_{R} N generated by all endomorphisms σ_{N}(T) with T ∈ 𝔤. There is a homomorphism of R-algebras A → B which takes σ_{Q}(T) to σ_{N}(T) for each T ∈ 𝔤. Clearly it is surjective. Since A is finitely generated over R, so is B as well. Furthermore, B ≠ 0 because N ≠ 0. By Nakayama’s Lemma there exists a maximal ideal 𝔪 of R such that B ≠ 𝔪B. Fix such an ideal.

The factor algebra B/𝔪B is a homomorphic image of a simple associative algebra A/𝔪A ≅ End_{R/𝔪}(Q/𝔪Q). It follows that B/𝔪B ≅ A/𝔪A. By symmetry we have B/𝔪B ≅ A′/𝔪A′ as well. Up to isomorphism, Q/𝔪Q is a unique simple module for A/𝔪A. Similarly, A′/𝔪A′ has a unique simple module Q′/𝔪Q′. If we let A/𝔪A operate in Q′/𝔪Q′ via the algebra isomorphism φ : A/𝔪A → ∼ A′/𝔪A′ constructed above, there has to be an isomorphism of A/𝔪A-modules ι : Q/𝔪Q → ∼ Q′/𝔪Q′. Denote by σ_{𝔪} : 𝔤/𝔪𝔤 → A/𝔪A the reduction modulo 𝔪 of the map σ_{Q} : 𝔤 → A and by σ′_{𝔪} : 𝔤/𝔪𝔤 → A′/𝔪A′ the reduction of σ_{Q′} : 𝔤 → A′. Then σ′_{𝔪} = φ ∘ σ_{𝔪} by the construction. Hence ι is an isomorphism of 𝔤/𝔪𝔤modules. 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 ∈ C_{1}. 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 ρ_{M}(D), D ∈ W, solely. It is worth keeping in mind that the χ_{M} defined below is an R-multilinear function of its arguments. For θ, θ′ ∈ Ω^{1} and D, D′ ∈ W put

χ_{M}(θ, θ′, D, D′) = (σ_{M}(θ ⊗ D) − ⟨θ, D⟩_{M}) σ_{M}(θ′ ⊗ D′)

+ (σ_{M}(θ′ ⊗ D) − ⟨θ′, D⟩_{M}) σ_{M}(θ ⊗ D′) .

Lemma 1.5. Let M ∈ C_{1}. Then for all f, g ∈ R and D, D′ ∈ W we have

− ρ_{M}(fgD) ρ_{M}(D′) + ρ_{M}(fD) ρ_{M}(gD′) + ρ_{M}(gD) ρ_{M}(fD′) − ρ_{M}(D) ρ_{M}(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 C_{1} in which the role of Ω^{1} is transferred to the R-module of Kähler differentials which we denote as Ω̃^{1}. Recall that it is defined together with a derivation d : R → Ω̃^{1}, universal in the class of derivations with values in R-modules (a ℤlinear map Δ : R → M with M an arbitrary R-module is a derivation if Δ(fg) = f Δg + g Δf for all f, g ∈ R). The universality property gives a unique R-linear map Ω̃^{1} → Ω^{1} rendering commutative the diagram

\xymatrix{ & R \ar[dl]_{d} \ar[dr]^{d} \\ Ω̃^{1} \ar[rr] &  & Ω^{1}}

This map is surjective in view of (1.3). It induces an R-bilinear pairing Ω̃^{1} × W → R. The same formula as in the case of 𝔤 defines now a Lie multiplication on 𝔤̃ = Ω̃^{1} ⊗ W, the tensor product being over R. Definition 1.3. Denote by C̃_{1} the category whose objects are additive groups M together with a system of operators f_{M}, ρ_{M}(D), σ_{M}(T) defined for each f ∈ R, D ∈ W, T ∈ 𝔤̃ subject to the conditions (1.4)–(1.9). The morphisms in C̃_{1} are the maps that commute with the actions of R and W.

The epimorphism of R-modules Ω̃^{1} → Ω^{1} induces a surjective homomorphism of Lie algebras 𝔤̃ → 𝔤 over R. The category C_{1} can be identified therefore with the full subcategory in C̃_{1} whose objects M satisfy σ_{M}(T) = 0 for all T in the kernel of 𝔤̃ → 𝔤. All natural operations in C_{1} have their analogues in C̃_{1}. If M ∈ C̃_{1} then its R-linear transformations χ_{M}(θ, θ′, D, D′) make sense for elements θ, θ′ ∈ Ω̃^{1} and D, D′ ∈ W. Lemmas 1.1 and 1.5 generalize to C̃_{1} as they are formal consequences of (1.4)–(1.9). Lemma 1.6. The R-module Ω̃^{1} can be in a unique way furnished with additional operators which make it an object of C̃_{1} and the universal derivation d : R → Ω̃^{1} a W-equivariant map. Furthermore,

σΩ̃^{1}(θ ⊗ D) θ′ = ⟨θ′, D⟩ θ for θ, θ′ ∈ Ω̃^{1} and D ∈ W.

The canonical map Ω̃^{1} → Ω^{1} is an epimorphism in C̃_{1} and its kernel J an object of the subcategory C_{0}. Proof. Put M = Ω̃^{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) for f ∈ R and θ, 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) − f_{M} ∘ 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) − f_{M} ∘ ρ(D))(dg) = d(f · Dg) − f · d(Dg) = (Dg) · df = ⟨dg, D⟩ df = σ(df ⊗ D)(dg)

where f, g ∈ R, D ∈ W and σ = σ_{M} 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 Ω̃^{1} → Ω^{1} is a W-equivariant homomorphism of R-modules, i.e., a morphism in the category C̃_{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 ∈ 𝔤̃. Thus J ∈ C_{0}. □

2. The differential order of a cocycle

A 1-cocycle φ : W → M with coefficients in a W-module M is a ℤ-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 ∈ C_{1}, every 1-cocycle is a differential operator of order at most 3. In general, given two R-modules M and N, a ℤ-linear map ξ : N → M and an element f ∈ R, define

δ_{f}ξ = ξ ∘ f_{N} − f_{M} ∘ ξ

where f_{M} and f_{N} are multiplication operators on M and N. We call ξ a differential operator of order ≤ r if δ_{f_{1}} ⋯ δ_{f_{r+1}}ξ = 0 for all f_{1}, …, f_{r+1} ∈ R. Denote by Diff_{r}(N, M) the group of all differential operators of order ≤ r. For f, g ∈ R one has

δ_{f}δ_{g}ξ = δ_{g}δ_{f}ξ and δ_{fg}ξ = (δ_{f}ξ) ∘ g_{N} + f_{M} ∘ (δ_{g}ξ) .

It follows that the map R × ⋯ × R (r times) → Hom_{R}(N, M) given by the rule (f_{1}, …, f_{r}) ↦ δ_{f_{1}} ⋯ δ_{f_{r}}ξ 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 Ω̃^{1}×⋯×Ω̃^{1} (r times) → Hom_{R}(N, M), hence also an R-linear map from the r-th symmetric power S^{r}Ω̃^{1} to Hom_{R}(N, M). There is therefore an R-linear map

ξ^{♭} : S^{r}Ω̃^{1} ⊗ N → M

such that

ξ^{♭}(df_{1} ⋯ df_{r} ⊗ u) = (δ_{f_{1}} ⋯ δ_{f_{r}}ξ)(u)

for f_{1}, …, f_{r} ∈ 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 ℤ-linear map then δ_{f}(ξ ∘ η) = δ_{f}ξ ∘η +ξ ∘δ_{f}η. 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_{1} ⋯ θ_{r+s} ⊗ u) = ∑ ξ^{♭}(θ_{i_{1}} ⋯ θ_{i_{r}} ⊗ η^{♭}(θ_{i_{r+1}} ⋯ θ_{i_{r+s}} ⊗ u)) for θ_{1}, …, θ_{r+s} ∈ Ω̃^{1} and u ∈ P, where the sum ranges over all permutations of indices 1, …, r + s such that i_{1} < … < i_{r} and i_{r+1} < … < i_{r+s}.

The exact sequence 0 → J → Ω̃^{1} → Ω^{1} → 0 in our settings induces a surjective homomorphism of symmetric algebras SΩ̃^{1} → SΩ^{1} whose kernel is the ideal of SΩ̃^{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 · S^{r−1}Ω̃^{1}) ⊗ N → S^{r}Ω̃^{1} ⊗ N then it induces an R-linear map S^{r}Ω^{1} ⊗ N → M which we still call the symbol of ξ and denote as ξ^{♭}.

Lemma 2.1. Let M ∈ C_{1}. Then every 1-coboundary φ : W → M is a differential operator of order ≤ 1. Proof. There is m ∈ M such that φ(D) = ρ_{M}(D)m for all D ∈ W. By (1.8) (δ_{f}φ)(D) = σ_{M}(df ⊗ D)m for f ∈ R, D ∈ W, whence δ_{f}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 C_{1}. If φ is a differential operator of order ≤ 2 then its symbol φ^{♭} induces a morphism S^{2}Ω^{1} ⊗W → M in C_{1}. If φ is a differential operator of order ≤ 3 and either rk_{R} W > 1 or 3 is invertible in R then φ^{♭} induces a morphism S^{3}Ω^{1} ⊗ W → M. Proof. Assume φ ∈ Diff_{3}(W, M). Its symbol φ^{♭} : S^{3}Ω̃^{1} ⊗ W → M is an R-module homomorphism. We will show that φ^{♭} is a W-module homomorphism as well, hence a morphism in C̃_{1}. The Lie algebra W operates on the ℤ-linear maps ξ : W → M in a natural way, and D·δ_{f}ξ = δ_{Df}ξ+δ_{f}Dξ for f ∈ R, D ∈ W. Now Dφ is the coboundary of φ(D) ∈ M, whence δ_{f}δ_{g}(Dφ) = 0 for all f, g ∈ R by Lemma 2.1. We deduce

D · δ_{f_{1}}δ_{f_{2}}δ_{f_{3}}φ = δ_{Df_{1}}δ_{f_{2}}δ_{f_{3}}φ + δ_{f_{1}}δ_{Df_{2}}δ_{f_{3}}φ + δ_{f_{1}}δ_{f_{2}}δ_{Df_{3}}φ

for f_{1}, f_{2}, f_{3} ∈ R and D ∈ W. Evaluating at D′ ∈ W yields

ρ_{M}(D)φ^{♭}(ω ⊗ D′) − φ^{♭}(ω ⊗ [D, D′]) = φ^{♭}(Dω ⊗ D′)

where ω = df_{1} · df_{2} · df_{3}. Since Ω̃^{1} = R · dR, the equality holds actually for all ω ∈ S^{3}Ω̃^{1} and gives the W-invariance of φ^{♭}.

Next we want to show that φ^{♭} factors through S^{3}Ω^{1} ⊗W. The kernel 𝔨 of the canonical homomorphism 𝔤̃ → 𝔤 annihilates M and W since both modules are in C_{1}. As φ^{♭} is 𝔤̃-equivariant by the C̃_{1} version of Lemma 1.1, it must vanish on the R-submodule K ⊂ S^{3}Ω̃^{1} ⊗ W spanned by the tensors ω ⊗ D with ω ∈ 𝔨 · S^{3}Ω̃^{1} and D ∈ W. We will show that 𝔨 · S^{3}Ω̃^{1} = J · S^{2}Ω̃^{1}, which means that K is the kernel of the canonical epimorphism S^{3}Ω̃^{1} ⊗W → S^{3}Ω^{1} ⊗W, as required. Recall that 𝔤̃ operates in the symmetric algebra SΩ̃^{1} via derivations. Given D ∈ W, let i_{D} denote the derivation of SΩ̃^{1} such that i_{D}f = 0 for f ∈ R and i_{D}θ′ = ⟨θ′, D⟩ for θ′ ∈ Ω̃^{1}. The composite θi_{D} of i_{D} with the multiplication by θ ∈ Ω̃^{1} is again a derivation. Hence θ ⊗ D acts in the symmetric algebra as θi_{D}, both derivations having the same values on elements of R and Ω̃^{1}. Now 𝔨 is spanned by the tensors θ ⊗ D with θ ∈ J, D ∈ W. Therefore 𝔨 · S^{3}Ω̃^{1} = JN, where N ⊂ S^{2}Ω̃^{1} is the R-submodule spanned by the elements i_{D}ω with D ∈ W, ω ∈ S^{3}Ω̃^{1}. It remains to verify that N = S^{2}Ω̃^{1}.

If P ⊂ Ω̃^{1} is a finitely generated R-submodule then so is the span P^{2} ⊂ S^{2}Ω̃^{1} of all θθ′ with θ, θ′ ∈ P. By Nakayama’s Lemma the inclusion P^{2} ⊂ N holds if and only if P^{2} ⊂ N +𝔪P^{2} for all maximal ideals 𝔪 of R. It suffices to prove these inclusions only for those P that project onto the whole Ω^{1} since the latter is finitely generated over R. Fix P and 𝔪. Let θ ∈ P. If rk_{R} W > 1 then Ω^{1}/𝔪Ω^{1} ≅ (W/𝔪W)^{∗} has dimension > 1 over R/𝔪. We can find D′ ∈ W and θ′ ∈ P such that ⟨θ, D′⟩ ≡ 0, ⟨θ′, D′⟩ ≡ 1 modulo 𝔪. Then θ^{2} ≡ i_{D′}(θ^{2}θ′) modulo 𝔪P^{2}, so that θ^{2} ∈ N + 𝔪P^{2}. In view of (1.1) the squares θ^{2} span the whole P^{2}. Suppose now rk_{R} W = 1 but 3R = R. If θ ∉ 𝔪P we can find D ∈ W such that ⟨θ, D⟩ ≡ 1 modulo 𝔪. Then 3θ^{2} ≡ i_{D}(θ^{3}) modulo 𝔪P^{2}, whence again θ^{2} ∈ N + 𝔪P^{2}. If θ ∈ 𝔪P, take θ′ ∈ P such that θ′ ∉ 𝔪P. Since the squares of θ′, θ′ + θ, θ′ − θ are all in N + 𝔪P^{2}, so is θ^{2} as well. That completes the proof. The case when φ is a differential operator of order ≤ 2 is treated similarly. □

Lemma 2.3. The ℤ-linear span X of all endomorphisms χ_{W} (θ, θ′, D, D′) with θ, θ′ ∈ Ω^{1} and D, D′ ∈ W coincides with A = End_{R} W. 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

σ_{W} (θ ⊗ D) ∘ σ_{W} (θ′ ⊗ D′) = −⟨θ, D′⟩ σ_{W} (θ′ ⊗ D)

and therefore χ_{W} (θ, θ, D, D) = −4⟨θ, D⟩ σ_{W} (θ⊗D) where we take θ, θ′ ∈ Ω^{1} and D, D′ ∈ W. If ⟨θ, D⟩ ∉ 𝔪 then we can find f ∈ R such that −4⟨θ, D⟩f ≡ 1 modulo 𝔪. Multiplying the previous equality by f, we deduce immediately that σ_{W} (θ⊗D) ∈ X +𝔪A. Suppose that ⟨θ, D⟩ ∈ 𝔪 but D ∉ 𝔪W. Then ⟨θ′, D⟩ ∉ 𝔪 for some θ′ ∈ Ω^{1}. Since σ_{W} (θ′ ⊗ D) and σ_{W} ((θ + θ′) ⊗ D) are both in X + 𝔪A, so is σ_{W} (θ ⊗ D) as well. Suppose finally that D ∈ 𝔪W. Pick out D′ ∈ W such that D′ ∉ 𝔪W. Then σ_{W} (θ ⊗ D′) and σ_{W} (θ ⊗ (D + D′)) are both in X + 𝔪A, whence so is σ_{W} (θ ⊗ D). We proved that σ_{W} (𝔤) ⊂ X + 𝔪A, which gives the assertion because A = σ_{W} (𝔤). □

Proposition 2.4. Let M ∈ C_{1}. Then every 1-cocycle φ : W → M is a differential operator of order ≤ 3. If rk_{R} W > 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 = ρ_{M}(D) ∘ φ − φ ∘ ρ_{W} (D) ∈ Diff_{1}(W, M) .

In view of (1.6) ρ_{M}(D) ∈ Diff_{1}(M, M) and similarly ρ_{W} (D) ∈ Diff_{1}(W, W). Hence

ρ_{M}(D) ρ_{M}(D′) φ − φ ρ_{W} (D) ρ_{W} (D′) = ρ_{M}(D) (ρ_{M}(D′) φ − φ ρ_{W} (D′))

+ (ρ_{M}(D) φ − φ ρ_{W} (D)) ρ_{W} (D′)

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

χ_{M}(θ, θ′, D, D′) ∘ φ − φ ∘ χ_{W} (θ, θ′, D, D′) ∈ Diff_{2}(W, M)  (∗)

for all θ, θ′ ∈ dR ⊂ Ω^{1} and D, D′ ∈ W. The inclusion holds actually for all θ, θ′ ∈ Ω^{1} since χ_{M}(fθ, θ′, D, D′) = χ_{M}(θ, θ′, D, fD′), where f ∈ R, and similarly for χ_{W}. We get

(δ_{f}φ)∘χ_{W} (θ, θ′, D, D′) = φ∘f_{W} ∘χ_{W} (θ, θ′, D, D′)−f_{M}∘φ∘χ_{W} (θ, θ′, D, D′)

= f_{M} ∘ (χ_{M}(θ, θ′, D, D′) ∘ φ − φ ∘ χ_{W} (θ, θ′, D, D′))

− χ_{M}(fθ, θ′, D, D′) ∘ φ + φ ∘ χ_{W} (fθ, θ′, D, D′) ∈ Diff_{2}(W, M) .

It follows (δ_{f_{1}} ⋯ δ_{f_{4}}φ) ∘ χ = 0 for all f_{1}, …, f_{4} ∈ R and all χ in the subgroup X ⊂ End_{R} W described in Lemma 2.3. Since 1_{W} ∈ X, we deduce δ_{f_{1}} ⋯ δ_{f_{4}}φ = 0, i.e., φ is a differential operator of order ≤ 3.

Suppose rk_{R} W > 1 further on. Consider the symbol φ^{♭} : S^{3}Ω^{1} ⊗W → M. The map φ is a differential operator of order ≤ 2 if and only if φ^{♭} = 0. Put

N = { ω ∈ S^{3}Ω^{1} | ω ⊗ D ∈ Ker φ^{♭} for all D ∈ W } .

Applying δ_{f_{1}}δ_{f_{2}}δ_{f_{3}} to (∗), we get

χ_{M}(θ, θ′, D, D′) ∘ δ_{f_{1}}δ_{f_{2}}δ_{f_{3}}φ = δ_{f_{1}}δ_{f_{2}}δ_{f_{3}}φ ∘ χ_{W} (θ, θ′, D, D′), i.e.,

χ_{M}(θ, θ′, D, D′)(φ^{♭}(θ_{1}θ_{2}θ_{3} ⊗ D′′)) = φ^{♭}(θ_{1}θ_{2}θ_{3} ⊗ χ_{W} (θ, θ′, D, D′)D′′)

for all θ, θ′, θ_{1}, θ_{2}, θ_{3} ∈ dR and D, D′, D′′ ∈ W. In view of (1.3) this holds actually for all θ’s in Ω^{1}. In other words,

χ_{M}(θ, θ′, D, D′) ∘ φ^{♭} = φ^{♭} ∘ (id ⊗ χ_{W} (θ, θ′, D, D′)).  (∗∗)

It follows that all endomorphisms id ⊗ χ_{W} (θ, θ′, D, D′), hence by Lemma 2.3 all endomorphisms in id⊗End_{R} W, leave the kernel of φ^{♭} stable. The endomorphism id ⊗ σ_{W} (θ ⊗ D) decomposes as

S^{3}Ω^{1} ⊗ W ⟶^{−i_{θ}} S^{3}Ω^{1} ⟶^{j_{D}} S^{3}Ω^{1} ⊗ W

where the maps i_{θ}, j_{D} are given by

i_{θ}(ω ⊗ D′) = ⟨θ, D′⟩ω,  j_{D}(ω) = ω ⊗ D,  ω ∈ S^{3}Ω^{1}, D′ ∈ W.

As Ker φ^{♭} is stable under j_{D} ∘ i_{θ} for every D ∈ W and θ ∈ Ω^{1} by the above, we see that i_{θ}(Ker φ^{♭}) ⊂ N for every θ. On the other hand, φ^{♭} is a morphism in C_{1} by Lemma 2.2. In particular, φ^{♭} commutes with the actions of 𝔤. Therefore

χ_{M}(θ, θ′, D, D′) ∘ φ^{♭} = φ^{♭} ∘ χ_{S^{3}Ω^{1}⊗W} (θ, θ′, D, D′).

Comparing this with (∗∗), we see that the endomorphisms

χ_{S^{3}Ω^{1}⊗W} (θ, θ′, D, D′) − id ⊗ χ_{W} (θ, θ′, D, D′)

have images in Ker φ^{♭}.

Now we are ready to prove that N = S^{3}Ω^{1}, and so φ^{♭} = 0. By Nakayama’s Lemma it suffices to show that S^{3}Ω^{1} = N + 𝔪 · S^{3}Ω^{1} for all maximal ideals 𝔪 of R. Fix 𝔪. Let θ, θ′ ∈ Ω^{1}. We will check that θ^{2}θ′ ∈ N +𝔪·S^{3}Ω^{1}, whence our assertion. Since the vector space W/𝔪W and its dual Ω^{1}/𝔪Ω^{1} have dimension > 1 over R/𝔪, we can find D_{1} ∈ W, D_{1} ∉ 𝔪W, such that ⟨θ′, D_{1}⟩ ≡ 0, and then find D_{2} ∈ W, θ_{1}, θ_{2} ∈ Ω^{1} such that ⟨θ_{i}, D_{j}⟩ ≡ δ_{ij} modulo 𝔪 for i, j = 1, 2. Put

a = (χ_{S^{3}Ω^{1}⊗W} (θ, θ, D_{1}, D_{1})−id⊗χ_{W} (θ, θ, D_{1}, D_{1}))(θ_{1}^{2}θ′⊗D_{2}) ∈ S^{3}Ω^{1}⊗W.

As we noted above, a ∈ Ker φ^{♭}. Now compute a. Using Lemma 2.5 below, we get a = ω_{1} ⊗ D_{1} + ω_{2} ⊗ D_{2} where ω_{1} turns out to be irrelevant, while

ω_{2} = χ_{S^{3}Ω^{1}}(θ, θ, D_{1}, D_{1})(θ_{1}^{2}θ′) ≡ 4θ^{2}θ′ (mod 𝔪 · S^{3}Ω^{1}) .

It follows i_{θ_{2}}(a) ≡ ω_{2} ≡ 4θ^{2}θ′ modulo 𝔪 · S^{3}Ω^{1}. Since i_{θ_{2}}(a) ∈ N, we get the conclusion about θ^{2}θ′. □ Lemma 2.5. Suppose that θ ∈ Ω^{1}, D ∈ W and A, B ∈ C_{1}. Then the endomorphism χ_{A⊗B}(θ, θ, D, D) is equal to

χ_{A}(θ, θ, D, D) ⊗ 1_{B} + 4σ_{A}(θ, D) ⊗ σ_{B}(θ, D) + 1_{A} ⊗ χ_{B}(θ, θ, D, D) .

Denote by i_{D} the derivation of the symmetric algebra SΩ^{1} such that i_{D}f = 0 for f ∈ R and i_{D}θ′ = ⟨θ′, D⟩ for θ′ ∈ Ω^{1}. If A = SΩ^{1} and ω ∈ A then

χ_{A}(θ, θ, D, D) ω = 2θ^{2} · i_{D}^{2}ω .

Proof. One checks (i) straightforwardly using the definitions of operators. Under hypotheses of (ii), σ_{A}(θ, D)ω = θ · i_{D}ω. The conclusion of (ii) follows from the computation

σ_{A}(θ, D)^{2}ω = θ^{2} · i_{D}^{2}ω + ⟨θ, D⟩θ · i_{D}ω = θ^{2} · i_{D}^{2}ω + ⟨θ, D⟩ σ_{A}(θ, D)ω .

□

3. First order prolongations

If M is an object of the category C_{0} then the standard cochain complex C^{•}(W, M) of ℤ-multilinear alternating maps W ×⋯×W → M contains a subcomplex C_{R}^{•}(W, M) whose elements are R-multilinear maps. The cohomology H_{R}^{•}(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, C_{R}^{•}(W, M) ≅ M ⊗ Ω^{•} where Ω^{•} is the exterior algebra of the R-module Ω^{1}. In particular, C_{R}^{•}(W, R) ≅ Ω^{•} generalizes the classical de Rham complex. If now M ∈ C_{1} 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 ℤ 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) [D̃, fD̃′] = f [D̃, D̃′]+(π(D̃)f)D̃′, for all f ∈ R and D̃, D̃′ ∈ W. Such algebraic structures under different names were considered by many people [11, 14, 17, 19, 23]. In particular, the pair W, id_{W} satisfies these conditions. Suppose M is an additive group endowed with an R-module and a W -module structures. Denote by ρ̃_{M}(D̃) and f_{M} the operators on M corresponding to elements D̃ ∈ W and f ∈ R. If the identities

(3.2) [ρ̃_{M}(D̃), f_{M}] = (π(D̃)f)_{M} ,

(3.3) ρ̃_{M}(fD̃) = f_{M} ∘ ρ̃_{M}(D̃) 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 H_{R}^{•}(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 H_{R}^{•}(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 ≅ W /𝔨. In particular, the 𝔨-invariants in M form an R-submodule M^{𝔨} which has the induced structure of a W-module. As (3.2), (3.3) carry over to the induced representation of W in M^{𝔨}, the R-linear cohomology H_{R}^{•}(W, M^{𝔨}) 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 → H_{R}^{1}(W, M^{𝔨}) → H_{R}^{1}(W , M) → H_{R}^{1}(𝔨, M)^{W} → H_{R}^{2}(W, M^{𝔨}) .

Proof. The map on the left is obtained by taking the composites of the R-linear 1-cocycles W → M^{𝔨} 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 M^{𝔨}. Restricting the R-linear 1-cocycles W → M to 𝔨 gives the next map. Its images are W-invariant classes in H_{R}^{1}(𝔨, M) since W acts in H_{R}^{1}(W , M) trivially. If a cocycle W → M vanishes on 𝔨 then it has values in M^{𝔨} 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(D̃) = D̃φ′ is a coboundary for every D̃ ∈ W. We get thus a map η : W → B_{R}^{1}(𝔨, M) with values in the group of coboundaries 𝔨 → M. If f ∈ R, D̃ ∈ W, T ∈ 𝔨 then

η(fD̃)(T) = ρ̃_{M}(fD̃)φ′(T) − φ′([fD̃, T])

= f · ρ_{M}(D̃)φ′(T) − φ′(f[D̃, T]) = f · η(D̃)(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 φ̃ : W → M with the property that η(D̃) is the coboundary of φ̃(D̃) for every D̃ ∈ 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 → B_{R}^{1}(𝔨, M) is an epimorphism of R-modules, the restriction 𝔠 → B_{R}^{1}(𝔨, M) of η can be lifted to an R-linear map φ′′ : 𝔠 → M. Taking φ̃ to be φ′ on 𝔨 and φ′′ on 𝔠 fulfills our requirement. Let now ψ : W ×W → M be the coboundary of φ̃. Then ψ vanishes when one of its arguments is in 𝔨. It induces therefore a 2-cocycle \overline{ψ} : W ×W → M which takes values in M^{𝔨}. The last map in the statement of the Lemma takes the cohomology class of φ′ to that of \overline{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 ρ_{𝔤} 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 𝔤̃. The kernel of π is 𝔤 in the former case and 𝔤̃ 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 𝔤̃×W) with the Lie algebra and R-module structures just described the first order prolongation of W with kernel 𝔤 (respectively 𝔤̃). Lemma 3.2.Suppose that W is the first order prolongation of W with kernel 𝔤 or 𝔤̃. Then C_{1} (respectively C̃_{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

ρ̃_{M}(T, D) = σ_{M}(T) + ρ_{M}(D) , T ∈ 𝔤, D ∈ W,

establishes a one-to-one correspondence between the families of operators ρ̃_{M}(D̃) on M defined for each D̃ ∈ W and the families of operators ρ_{M}(D), σ_{M}(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 ρ̃_{M} is a Lie algebra representation then so is its restriction ρ_{M} to W. Conversely, if M ∈ C̃_{1} then it is immediate from Lemma 1.1 that ρ̃_{M} is a Lie algebra representation. The same Lemma shows also that the morphisms in C̃_{1} are precisely the maps that are homomorphisms of Rmodule and W -module structures. □

Now we can verify (3.1). Suppose that M is an object of C_{1} (respectively C̃_{1}) such that the induced representation ρ̃_{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 ρ̃_{M} ⊂ ker π = 𝔤. However 𝔤 acts in W faithfully. In case of kernel 𝔤̃ we can take M = Ω̃^{1}. Again (1.2) ensures the faithfulness of σ_{M} and ρ̃_{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

f_{M} ∘ [ρ̃_{M}(D̃), ρ̃_{M}(D̃′)] + (π(D̃)f)_{M} ∘ ρ̃_{M}(D̃′) .

Thus we can apply Lemma 3.1 to the situation where M is an object of C_{1} or even C̃_{1} and W is the first order prolongation of W with kernel 𝔤̃. In this case 𝔨 = 𝔤̃. 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 (δ_{f}ι)(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 ∈ 𝔤̃. □

Lemma 3.3.Suppose that M ∈ C_{1} and W is the first order prolongation of W with kernel 𝔤̃. Given a differential operator φ : W → M of order ≤ 1, there exists a unique R-linear map φ̃ : W → M such that φ = φ̃ ∘ ι. Moreover, if φ is a cocycle then so is φ̃ as well. In this case the symbol φ^{♭} : 𝔤̃ → M of φ is also an R-linear cocycle. In order that a cocycle φ be R-linear, it is necessary and sufficient that φ̃ vanish on 𝔤̃. Proof. Every ℤ-linear map φ̃ : W → M satisfying φ = φ̃ ∘ ι can be written as

φ̃(T, D) = φ′(T) + φ(D) for T ∈ 𝔤̃ and D ∈ W.

where φ′ is a ℤ-linear map 𝔤̃ → M. In order that φ̃ 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

\begin{matrix}ρ_{N}(D)(m, h) = (ρ_{M}(D)m + h φ(D), Dh) \\ σ_{N}(T)(m, h) = (σ_{M}(T)m + h φ^{♭}(T), 0) \end{matrix}

where D ∈ W, T ∈ 𝔤̃, m ∈ M, h ∈ R. One checks straightforwardly that N is now an object of the category C̃_{1}. By Lemma 3.2 ρ_{N} extends to a representation ρ̃_{N} of W satisfying (3.2), (3.3). In fact we have an exact sequence 0 → M → N → R → 0 in C̃_{1} which is also an exact sequence of W -modules. Furthermore,

ρ̃_{N}(D̃)(0, 1) = σ_{N}(T)(0, 1)+ρ_{N}(D)(0, 1) = (φ^{♭}(T)+φ(D), 0) = (φ̃(D̃), 0) for any D̃ = (T, D) ∈ W. Applying the operator

ρ̃_{N}([D̃, D̃′]) = [ρ̃_{N}(D̃), ρ̃_{N}(D̃′)]

to (0, 1), where D̃′ ∈ W is a second element, we get the cocycle condition for φ̃. Since the symbol ι^{♭} : 𝔤̃ → W is the canonical embedding which is a homomorphism of Lie algebras, φ^{♭} = φ̃∘ ι^{♭} is a cocycle as well. Finally, φ is R-linear if and only if φ^{♭} = 0, i.e., if φ̃ ∘ ι^{♭} = 0. □

4. Construction of universal cocycles

Among the 1-cocycles φ : W → M of differential order ≤ 2 with values in the objects of C_{1} 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 C_{1} 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 → S^{2}Ω^{1} ⊗ W which will be called the universal differential order 2 cocycle. When rk_{R} W = 1, proposition 4.5 describes a 1-cocycle W → Ω^{1} ⊗Ω^{1} 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 ℤ-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′′) = (D′f) 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) − f ψ(D) = Df, f ∈ R, D ∈ W. Proposition 4.1. There is a 1-cocycle φ : W → S^{2}Ω^{1} ⊗ W which is a differential operator of order 2 and whose symbol φ^{♭} is the identity endomorphism of S^{2}Ω^{1} ⊗ W. Proof. The Lie algebra W operates on the ℤ-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 S^{2}Ω^{1} ⊗W with the group of such maps. Namely, given θ_{1}, θ_{2} ∈ Ω^{1} and D ∈ W, the tensor θ_{1}θ_{2} ⊗ D determines the map

(D′, D′′) ↦ (⟨θ_{1}, D′⟩⟨θ_{2}, D′′⟩ + ⟨θ_{1}, D′′⟩⟨θ_{2}, D′⟩)D , D′, D′′ ∈ W.

Thus φ takes values in S^{2}Ω^{1} ⊗ W. It is a cocycle since it comes from a coboundary in a larger module. Let f, g ∈ R. We have

(δ_{f}φ)(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 + (D′f) ∇(D, D′′) − (D′f) [D, D′′]
+ (D′′f) ∇(D′, D) + (D′D′′f) D ,

and

(δ_{g}δ_{f}φ)(D)(D′, D′′) = ((D′f)(D′′g) + (D′′f)(D′g))D .

In other words, (δ_{g}δ_{f}φ)(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) = f ξ(D)− df ⊗ D for all f ∈ R, D ∈ W. Now 𝔤 operates on W by means of σ_{W}, and we have

ξ(fD)D′ = f · ξ(D)D′ + (D′f)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 D_{1}, …, D_{s} ∈ W and θ_{1}, …, θ_{s} ∈ Ω^{1} such that every D ∈ W is expressed as ∑ ⟨θ_{i}, D⟩ D_{i}. We get

τ(D′, D′′) = ∑_{i,j=1}^{s} ⟨θ_{i}, D′⟩⟨θ_{j}, D′′⟩ τ(D_{i}, D_{j}) = ν(D′, D′′) − ν(D′′, D′)

where ν(D′, D′′) = ∑_{i<j} ⟨θ_{i}, D′⟩⟨θ_{j}, D′′⟩ τ(D_{i}, D_{j}). Thus ν : W × W → W is an R-bilinear map and ∇′ = ∇ − ν a torsion-free connection on W. □ Lemma 4.3. There exists a ℤ-linear map ψ : W → R satisfying identity (4.4). If rk_{R} W = 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 ℤ-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′]) for D, D′ ∈ W,

is R-bilinear and skewsymmetric. It corresponds therefore to a homomorphism of R-modules ⋀^{2}W → R. If rk_{R} W = 1, then ⋀^{2}W = 0, and ω = 0. In other words, ψ is a cocycle. □

Lemma 4.4. The composite φ : W → R → Ω^{1} 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 : S^{2}Ω^{1} ⊗ W → Ω^{1} in C_{1} defined by the rule

θθ′ ⊗ D ↦ ⟨θ, D⟩θ′ + ⟨θ′, D⟩θ for θ, θ′ ∈ Ω^{1} and D ∈ W .

Proof. Clearly π is a morphism in C_{1}. 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, D⟩ and d^{♭}(θ ⊗ h) = h θ ,

where θ ∈ Ω^{1}, 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_{R} W = 1 and 3 is invertible in R. Then there is a 1-cocycle φ : W → Ω^{1}⊗Ω^{1} which is a differential operator of order 3 with symbol φ^{♭} : S^{3}Ω^{1} ⊗ W → Ω^{1} ⊗ Ω^{1} an isomorphism in C_{1}. Proof. The symmetric and the tensor powers of the R-module Ω^{1} coincide because rk_{R} Ω^{1} = rk_{R} W = 1. To put it differently, every R-multilinear expression involving several arguments from Ω^{1} is symmetric in these arguments. The same observation applies to W. Identify Ω^{1} ⊗ Ω^{1} with the group of R-bilinear maps W × W → R so that a tensor θ_{1} ⊗ θ_{2} with θ_{1}, θ_{2} ∈ Ω^{1} corresponds to the map

(D′, D′′) ↦ ⟨θ_{1}, D′⟩⟨θ_{2}, D′′⟩ , where D′, D′′ ∈ W.

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

W ⟶^{ψ} R ⟶^{d} Ω^{1} ⟶^{ψ^{∗}} Ω^{1} ⊗ Ω^{1},

where ψ^{∗}(θ)(D′, D′′) = ψ(D′), θ(D′′) − D′′(⟨θ, D′⟩) for θ ∈ Ω^{1} 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′′) = (D′f) ⟨θ, D′′⟩ − (D′′f) ⟨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′′) = −(D′′f) ⟨θ, D′′⟩,

whence ψ^{∗}(fθ) − f ψ^{∗}(θ) = −θ ⊗ df. It follows that ψ^{∗} is a differential operator of order 1 with symbol (ψ^{∗})^{♭} minus identity transformation of S^{2}Ω^{1} ≅ Ω^{1} ⊗ Ω^{1}. 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_{1}θ_{2}θ_{3} ⊗ D) = − ∑ ⟨θ_{i_{1}}, D⟩ θ_{i_{2}}θ_{i_{3}} = −6 ⟨θ_{1}, D⟩ θ_{2}θ_{3}

for θ_{1}, θ_{2}, θ_{3} ∈ Ω^{1} 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 Ω^{1} and W induces an isomorphism of R-modules Ω^{1} ⊗ W ≅ R, again by the rank one assumption, φ^{♭} : S^{3}Ω^{1} ⊗ W → S^{2}Ω^{1} is an isomorphism as well. To show that φ is a cocycle we embed S^{2}Ω^{1} into the W-module of all ℤ-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))−D′D′′(ψ(D))−D′′D′(ψ(D)) .

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

φ(D)(D′, D′′) = ψ(D′) · D′′(ψ(D)) − D′′D′(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 ∈ C_{1}. We introduce a filtration on the group of 1-cocycles Z^{1}(W, M) letting F_{i}Z^{1}(W, M) for i ≥ 0 denote its subgroup consisting of cocycles W → M which are differential operators of order ≤ i. Set F_{−1}Z^{1}(W, M) = 0. We have seen in proposition 2.4 that F_{3}Z^{1}(W, M) (respectively F_{2}Z^{1}(W, M) when rk_{R} W > 1) exhausts all the 1-cocycles. Let F_{i}H^{1}(W, M) be the image of F_{i}Z^{1}(W, M) in the cohomology group H^{1}(W, M). First we are going to determine the factors

gr_{i} H^{1}(W, M) = F_{i}H^{1}(W, M)/F_{i−1}H^{1}(W, M) .

Proposition 5.1. Let M ∈ C_{1}. Then there are isomorphisms (5.1) gr_{0} H^{1}(W, M) ≅ H_{R}^{1}(W, M^{𝔤}) , (5.2) gr_{1} H^{1}(W, M) ≅ Ker(H_{R}^{1}(𝔤̃, M)^{W} → H_{R}^{2}(W, M^{𝔤})) , (5.3) gr_{2} H^{1}(W, M) ≅ Mor_{C_{1}}(S^{2}Ω^{1} ⊗ W, M) , (5.4) gr_{3} H^{1}(W, M) ≅ Mor_{C_{1}}(Ω^{1}⊗Ω^{1}, M) when rk_{R} W = 1 and 3R =

R. Proof. Let W be the first order prolongation of W with kernel 𝔤̃. 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 F_{1}Z^{1}(W, M). The resulting map Z_{R}^{1}(W , M) → F_{1}Z^{1}(W, M) is bijective by Lemma 3.3. Obviously the W - coboundaries correspond to the W-coboundaries. It follows F_{1}H^{1}(W, M) ≅ H_{R}^{1}(W , M). Again by Lemma 3.3

gr_{0} H^{1}(W, M) ≅ F_{0}H^{1}(W, M) ≅ Ker(H_{R}^{1}(W , M) → H_{R}^{1}(𝔤̃, 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 𝔤̃. Lemma 3.1 gives now (5.1). Moreover, (5.2) also follows because

gr_{1} H^{1}(W, M) ≅ Coker(F_{0}H^{1}(W, M) → F_{1}H^{1}(W, M))

≅ Coker(H_{R}^{1}(W, M^{𝔤}) → H_{R}^{1}(W , M)) .

Next, by Lemma 2.2 we have a map

F_{2}Z^{1}(W, M) → Mor_{C_{1}}(S^{2}Ω^{1} ⊗ 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 F_{1}Z^{1}(W, M). Since all coboundaries are in F_{1}Z^{1}(W, M) by Lemma 2.1, the map above induces an embedding of gr_{2} H^{1}(W, M) into Mor_{C_{1}}(S^{2}Ω^{1} ⊗ W, M). Suppose now that ξ : S^{2}Ω^{1} ⊗ W → M is a morphism in C_{1}. Let φ : W → S^{2}Ω^{1} ⊗ W be the 1-cocycle given by proposition 4.1. Then ξ ∘φ is a 1-cocycle W → M which lies in F_{2}Z^{1}(W, M) and has symbol ξ ∘ φ^{♭} = ξ as ξ is R-linear and φ^{♭} = id. Thus (∗) is surjective. The assertion about gr_{3} H^{1}(W, M) is proved similarly, taken into account proposition 4.5. □ Lemma 5.2. Suppose that 0 → C^{0} → C^{1} → … is a cochain complex of finitely generated projective R-modules and q > 0 an integer such that H^{i}(C^{•}/𝔪C^{•}) = 0 for all maximal ideals 𝔪 of R in all degrees i < q. Then H^{i}(C^{•} ⊗ M) = 0 for any R-module M and i < q. Proof. The differential d : C^{0} → C^{1} induces by passing to the reductions modulo any maximal ideal 𝔪 an injective map C^{0}/𝔪C^{0} → C^{1}/𝔪C^{1} because H^{0}(C^{•}/𝔪C^{•}) vanishes. It follows that d maps C^{0} isomorphically onto a direct summand of C^{1}. In fact, the localizations C_{𝔪}^{0} and C_{𝔪}^{1} at 𝔪 are free modules of finite rank over the local ring R_{𝔪}, and therefore C_{𝔪}^{0} is mapped isomorphically onto a direct summand of C_{𝔪}^{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 C^{0} and C^{1} are finitely generated and projective. By [1, II, §3, Corollary 1 to Proposition 12] dC^{0} is a direct summand of C^{1}, as required.

Now C^{•} decomposes into a direct sum A^{•} ⊕ B^{•} of two subcomplexes where A^{0} = C^{0}, A^{1} = dC^{0} and A^{i} = 0 in all other degrees, while B^{1} is any R-module complement of A^{1} in C^{1}. If M is an R-module then C^{•} ⊗ M is a direct sum of complexes A^{•} ⊗ M and B^{•} ⊗ M. Therefore

H^{i}(C^{•} ⊗ M) ≅ H^{i}(A^{•} ⊗ M) ⊕ H^{i}(B^{•} ⊗ M)

for all i. Since the differential A^{0} ⊗ M → A^{1} ⊗ M is an isomorphism, we have H^{i}(A^{•} ⊗M) = 0 for all i. Taking M = R/𝔪 gives H^{i}(B^{•}/𝔪B^{•}) = 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 ∈ C_{1} is an object satisfying (1.10), (1.11) and M ∈ C_{1} an object of type Q. If H^{1}(𝔤/𝔪𝔤, Q/𝔪Q) = 0 for all maximal ideals 𝔪 of R then H_{R}^{1}(𝔤, M) = 0 as well. Proof. Consider the standard cochain complex C_{R}^{•}(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 M_{0} is an R-module on which 𝔤 operates trivially, then

C_{R}^{•}(𝔤, Q ⊗ M_{0}) ≅ C_{R}^{•}(𝔤, Q) ⊗ M_{0} .

In particular, C^{•}(𝔤/𝔪𝔤, Q/𝔪Q) ≅ C_{R}^{•}(𝔤, Q/𝔪Q) ≅ C_{R}^{•}(𝔤, Q) ⊗ R/𝔪 for every 𝔪. The reduced modulo 𝔪 complexes have zero cohomology in degree 1 according to the assumptions. Since 𝔤/𝔪𝔤 has a proper commutant, the first cohomology group of the trivial 𝔤/𝔪𝔤-module is nonzero. Hence Q/𝔪Q is not trivial, and therefore H^{0}(𝔤/𝔪𝔤, Q/𝔪Q) = 0 as well. By Lemma 1.3 M ≅ Q ⊗ M_{0} for a suitable M_{0}. 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 p ≠ 2. Denote by \overline{K} the kernel of the linear map S^{2}V^{∗} ⊗ V → V^{∗} defined by the rule

θθ′ ⊗ v ↦ ⟨θ, v⟩ θ′ + ⟨θ′, v⟩ θ for θ, θ′ ∈ V^{∗} and v ∈ V .

The 𝔤𝔩 V -modules V, V^{∗}, V^{∗} ⊗ ⋀^{n}V^{∗}, and \overline{K} when n > 1 and n + 1 ≢ 0 (mod p) are nontrivial, absolutely irreducible, pairwise nonisomorphic and have zero cohomology in degree 1 (with the exceptions only for p = 3, n ≤ 2).

The 𝔤𝔩 V -module S^{2}V^{∗} ⊗ V is isomorphic with the direct sum \overline{K} ⊕ V^{∗} of two irreducible modules when n > 1 and n + 1 ≢ 0 (mod p). It has a single irreducible factor module isomorphic to V^{∗} otherwise. The 𝔤𝔩 V - module \overline{K} has never V^{∗} as its factor module. Proof. The assertion about irreducibility is immediate for all modules stated, except for \overline{K}. In fact, S^{2}V^{∗} ⊗ V is the component of degree 1 in the graded Lie algebra of general Cartan type G_{−1} ⊕ G_{0} ⊕ G_{1} ⊕ ⋯ having G_{−1} = V and G_{0} = 𝔤𝔩 V. The structure of G_{1} as a G_{0}-module was investigated in [15, I, §10]. If n > 1 and p does not divide n + 1 then \overline{K} is irreducible and G_{1} ≅ \overline{K} ⊕ V^{∗}. This is clearly not affected by field extensions, so that we get the absolute irreducibility. Note that

dim \overline{K} = dim S^{2}V^{∗}⊗V −dim V^{∗} = \frac{1}{2}n^{2}(n+1)−n = \frac{1}{2}n(n−1)(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 𝔤𝔩 V. The weights ε_{1}, …, ε_{n} of V constitute a basis for the dual space 𝔥^{∗}. The weights of V^{∗} are now −ε_{i}, the weights of V^{∗} ⊗ ⋀^{n}V^{∗} are −ε_{i} − δ, and those of \overline{K} are ε_{l} − ε_{i} − ε_{j}, where δ = ε_{1} + ⋯ + ε_{n} and 1 ≤ i, j, l ≤ n. None of them is among the roots of 𝔤𝔩 V, which are ε_{l} − ε_{i} (except when p = 3, n = 2).

Look now at the other possibilities for G_{1} and \overline{K}. If n = 1 then \overline{K} = 0 and G_{1} ≅ V^{∗}. Suppose that n > 1 but p divides n + 1. By [15] \overline{K} is a single maximal submodule of G_{1}. Moreover, G_{1} contains a single irreducible submodule, say V′, which satisfies V′ ⊂ \overline{K} and V′ ≅ V^{∗}. The factor module \overline{K}/V′ is irreducible of dimension greater than n. The only exception is the case n = 2, p = 3 when 𝔤𝔩 V acts trivially in \overline{K}/V′. In any case V^{∗} is not a factor module of \overline{K}, as asserted. □

The category C_{1} contains objects Q satisfying (1.10) and (1.11) which give a certain canonical glueing of the 𝔤/𝔪𝔤-modules of each type considered in Lemma 5.4. Three of these are W, Ω^{1}, and Ω^{1} ⊗ Ω^{n} where n = rk_{R} W and Ω^{n} = ⋀^{n} Ω^{1}. Denote by K the kernel of the epimorphism π : S^{2}Ω^{1} ⊗ W → Ω^{1} defined in Lemma 4.4. Since both Ω^{1} and S^{2}Ω^{1} ⊗ W are finitely generated projective R-modules, so is K as well. If 𝔪 is a maximal ideal of R then K/𝔪K is the kernel of the induced linear map S^{2}V^{∗} ⊗ V → V^{∗} of vector spaces over R/𝔪 where V = W/𝔪W and V^{∗} is its dual. Suppose that (n + 1)R = R. Then it follows from Lemma 5.4 that K/𝔪K is an absolutely irreducible 𝔤/𝔪𝔤-module, and so K satisfies (1.10), (1.11). Moreover, there is a canonical decomposition

S^{2}Ω^{1} ⊗ W ≅ K ⊕ Ω^{1}

in C_{1}. To see this consider the morhism μ : Ω^{1} ⊗ 𝔤 → S^{2}Ω^{1} ⊗ W defined by the rule θ ⊗ (θ′ ⊗ D) ↦ θθ′ ⊗ D for θ, θ′ ∈ Ω^{1} and D ∈ W. Then

π ∘ μ(θ ⊗ T) = σ_{Ω^{1}}(T)θ + γ(T)θ for all θ ∈ Ω^{1} and T ∈ 𝔤 ,

where γ : 𝔤 → R is the contraction. Since σ_{Ω^{1}} : 𝔤 → End_{R} Ω^{1} is an isomorphism in C_{1}, there is a W-invariant element 1 ∈ 𝔤 which corresponds to the identity endomorphism of Ω^{1}. The assignment θ ↦ μ(θ⊗1) defines obviously a morphism ν : Ω^{1} → S^{2}Ω^{1} ⊗ W in C_{1}. Since γ(1) = tr 1_{Ω^{1}} = rk_{R} Ω^{1} = n, we deduce that π ∘ ν = (n + 1) · 1_{Ω^{1}}. If n + 1 is invertible in R then the restriction of π gives an isomorphism Im ν → Ω^{1}. In this case Im ν is a subobject complementary to K in S^{2}Ω^{1} ⊗ W.

Now we have come to the final results on cohomology. Let Q ∈ C_{1} be an object satisfying conditions (1.10), (1.11) and M ∈ C_{1} an object of type Q. Recall that n = rk_{R} W. 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 𝔤/𝔪𝔤-module Q/𝔪Q is isomorphic to neither the trivial module, nor W/𝔪W, nor (W/𝔪W)^{∗}, nor K/𝔪K when n > 1 and char R/𝔪 does not divide n + 1, nor ⨂^{2}(W/𝔪W)^{∗} when n = 1, then

H^{1}(W, M) ≅ H_{R}^{1}(𝔤, M)^{W} .

If, moreover, H^{1}(𝔤/𝔪𝔤, Q/𝔪Q) = 0 for all 𝔪 then H^{1}(W, M) = 0.

If R is an algebra over a field of characteristic 0 then char R/𝔪 = 0 for every maximal ideal 𝔪 of R. In this case the reductive Lie algebra 𝔤/𝔪𝔤 over the field R/𝔪 has zero cohomology with coefficients in any nontrivial irreducible finite dimensional module, i.e., the condition H^{1}(𝔤/𝔪𝔤, Q/𝔪Q) = 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 = M_{0} ⊗ Q where M_{0} ∈ C_{0}, and let M_{0}^{W} be the subgroup of W-invariants in M_{0}.

i) If Q = R, so that M ∈ C_{0}, then there is an exact sequence

0 → H_{R}^{1}(W, M) → H^{1}(W, M) → M^{W} .

The final map is surjective provided that there exists a divergence W → R. If φ : W → M is a 1-cocycle and m ∈ M^{W} the image of its cohomology class then φ(fD) = f φ(D) + (Df)m for all f ∈ R and D ∈ W.

ii) If Q = W then M is canonically embedded into the W-module Der(R, M_{0}) of ℤ-linear derivations R → M_{0}. Denote by N the group of those Δ ∈ Der(R, M_{0}) which satisfy D · Δ ∈ M for all D ∈ W. Then H^{1}(W, M) ≅ N/M.

iii) If Q = Ω^{1} then

H^{1}(W, M) ≅ Mor_{C_{1}}(Ω^{1}, M) ≅ M_{0}^{W} .

Every cohomology class is represented by a cocycle which is the composite of the universal differential order 2 cocycle W → S^{2}Ω^{1}⊗ W, the canonical morphism S^{2}Ω^{1} ⊗ W → Ω^{1} and a morphism Ω^{1} → M in C_{1}.

iv) Suppose n > 1 and (n + 1)R = R. If Q = K then

H^{1}(W, M) ≅ Mor_{C_{1}}(K, M) ≅ M_{0}^{W} .

Every cohomology class is represented by a cocycle which is the composite of the universal differential order 2 cocycle W → S^{2}Ω^{1}⊗ W, the canonical projection S^{2}Ω^{1} ⊗ W → K and a morphism K → M in C_{1}.

v) Suppose n = 1. If Q = Ω^{1} ⊗ Ω^{1} then

H^{1}(W, M) ≅ Mor_{C_{1}}(Ω^{1} ⊗ Ω^{1}, M) ≅ M_{0}^{W} .

Every cohomology class in this group is represented by a cocycle which is the composite of the universal differential order 3 cocycle W → Ω^{1} ⊗ Ω^{1} and a morphism Ω^{1} ⊗ Ω^{1} → M in C_{1}.

Proof. Using proposition 5.1, we examine those objects Q for which the groups gr_{i} H^{1}(W, M) are not all zero. If a nonzero factor occurs for i = 0, then M^{𝔤} ≠ 0. Clearly M^{𝔤} is a subobject of M, and in fact M^{𝔤} ∈ C_{0}. The inclusion M^{𝔤} ↪ M is a nonzero morphism in C_{1}. By Lemma 1.4 Q/𝔪Q is a trivial 𝔤/𝔪𝔤-module for at least one maximal ideal 𝔪 of R.

Next, gr_{1} H^{1}(W, M) ≠ 0 implies that H_{R}^{1}(𝔤̃, M)^{W} ≠ 0. There is a surjective homomorphism 𝔤̃ → 𝔤 of Lie algebras over R whose kernel is J ⊗ W, where J is the kernel of the canonical map Ω̃^{1} → Ω^{1}. Each R-linear cocycle φ : 𝔤̃ → M determines by restriction an R-linear map J ⊗ W → M. Furthermore, the coboundaries restrict to J ⊗ W trivially because 𝔤̃ operates in M via 𝔤. If the cohomology class of φ is Winvariant, 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 C_{1}. It is zero if and only if φ factors through 𝔤. Thus there is an exact sequence

0 → H_{R}^{1}(𝔤, M)^{W} → H_{R}^{1}(𝔤̃, M)^{W} → Mor_{C_{1}}(J ⊗ W, M) .  (∗)

If H_{R}^{1}(𝔤, M) ≠ 0 then H^{1}(𝔤/𝔪𝔤, Q/𝔪Q) ≠ 0 for at least one 𝔪 by Lemma 5.3. On the other hand, J ⊗ W ∈ C_{1} is an object of type W since J ∈ C_{0} according to Lemma 1.6. By Lemma 1.4 Mor_{C_{1}}(J ⊗ W, M) ≠ 0 implies Q/𝔪Q ≅ W/𝔪W as 𝔤/𝔪𝔤-modules for at least one 𝔪.

Consider the case gr_{2} H^{1}(W, M) ≠ 0. By (5.3) there exists a nonzero morphism ξ : S^{2}Ω^{1} ⊗ W → M in C_{1}. Its image, say M′, is a nonzero subobject of M. Then M′ ∈ C_{1} is an object of type Q, and M′ ≅ M′_{0} ⊗Q with M′_{0} ∈ C_{0} by Lemma 1.3. Since W and Ω^{1} are finitely generated over R, so is M′ as well. Hence M′ ≠ 𝔪M′ for at least one maximal ideal 𝔪. The quotient M′_{0}/𝔪M′_{0} is a trivial 𝔤/𝔪𝔤-module. The 𝔤/𝔪𝔤-module M′/𝔪M′ ≅ M′_{0}/𝔪M′_{0} ⊗ Q/𝔪Q is therefore completely reducible with all irreducible submodules isomorphic to Q/𝔪Q. On the other hand, M′/𝔪M′ is an epimorphic image of S^{2}(W/𝔪W)^{∗} ⊗ W/𝔪W. According to Lemma 5.4 either Q/𝔪Q ≅ (W/𝔪W)^{∗} or n > 1 , n + 1 is invertible in R/𝔪 and Q/𝔪Q ≅ K/𝔪K.

Finally, gr_{3} H^{1}(W, M) ≠ 0 implies that n = 1 and

Q/𝔪Q ≅ ⨂^{2}(W/𝔪W)^{∗}

for at least one 𝔪 by similar reasons. Under the hypotheses of theorem 5.5, the groups gr_{i} H^{1}(W, M) vanish for all i ≠ 1. Both M^{𝔤} and Mor_{C_{1}}(J ⊗ W, M) are zero by Lemma 1.4. Hence (5.2) and (∗) give

H^{1}(W, M) ≅ gr_{1} H^{1}(W, M) ≅ H_{R}^{1}(𝔤̃, M)^{W} ≅ H_{R}^{1}(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_{i} H^{1}(W, M) = 0 for i = 2, 3. Hence an exact sequence

0 → gr_{0} H^{1}(W, M) → H^{1}(W, M) → gr_{1} H^{1}(W, M) → 0 .

The term on the left is retrieved from (5.1), where M^{𝔤} = M. In view of (5.2) and (∗) there is an embedding gr_{1} H^{1}(W, M) ↪ H_{R}^{1}(𝔤̃, M)^{W} ≅ H_{R}^{1}(𝔤, M)^{W}. Since 𝔤 annihilates M, we have

H_{R}^{1}(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 H_{R}^{1}(𝔤, M) ≅ Hom_{R}(R, M) ≅ M. We get thus the required exact sequence. Every 1cocycle 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, D⟩m for θ ∈ Ω^{1}, D ∈ W, where m ∈ M^{W} is the element corresponding to the cohomology class of φ. If f ∈ R, D ∈ W then

φ(fD) − f φ(D) = φ^{♭}(df ⊗ D) = ⟨df, D⟩m = (Df)m .

Suppose now that ψ : W → R is a divergence. Given a W-invariant m ∈ M^{W}, the assignment f ↦ fm defines a morphism ι_{m} : R → M in C_{1}. The composite φ = ι_{m} ∘ ψ is then a cocycle W → M. Furthermore, φ^{♭} = ι_{m} ∘ ψ^{♭}. Since ψ^{♭} = γ, the map H^{1}(W, R) → R^{W} takes the cohomology class of ψ to 1. As the map H^{1}(W, M) → M^{W} is natural in M, it takes the cohomology class of φ to ι_{m}(1) = m.

(ii) Here gr_{i} H^{1}(W, M) vanishes for i ≠ 1. By lemmas 5.3 and 5.4 H_{R}^{1}(𝔤, M) = 0. From (5.2) and (∗) we deduce now an embedding

H^{1}(W, M) ≅ gr_{1} H^{1}(W, M) ↪ H_{R}^{1}(𝔤̃, M)^{W} ↪ Mor_{C_{1}}(J ⊗ W, M) .

The exact sequence 0 → J → Ω̃^{1} → Ω^{1} → 0 in C̃_{1} splits as a sequence of R-modules since Ω^{1} is R-projective. It gives rise therefore to an exact sequence in C̃_{1}

0 → Hom_{R}(Ω^{1}, M_{0}) → Hom_{R}(Ω̃^{1}, M_{0}) → Hom_{R}(J, M_{0}) → 0 .

Here Hom_{R}(Ω^{1}, M_{0}) ≅ M_{0} ⊗ Hom_{R}(Ω^{1}, R) ≅ M_{0} ⊗ W = M. On the other hand, Hom_{R}(Ω̃^{1}, M_{0}) ≅ Der(R, M_{0}) by the universality property of Kähler differentials. Thus M is embedded into Der(R, M_{0}) and

N/M = (Der(R, M_{0})/M)^{W} ≅ Hom_{R}(J, M_{0})^{W} = Mor_{C_{0}}(J, M_{0}) .

Consider the diagram

(Der(R, M_{0})/M)^{W}  ≅  Mor_{C_{0}}(J, M_{0})
↓  ↓≀
H^{1}(W, M)  ⟶ Mor_{C_{1}}(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, M_{0}) → Der(R, M_{0})/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 → M_{0} in C_{0} which is the restriction of the R-linear map ξ : Ω̃^{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 C_{1} which is the restriction of the symbol φ^{♭} : Ω̃^{1} ⊗ W → M of φ. If f ∈ R, D ∈ W then the derivation (fD) · Δ − f(D · Δ) takes g ∈ R to

ρ_{M_{0}}(fD)(Δg) − f · ρ_{M_{0}}(D)(Δg) − Δ(fDg) + f · Δ(Dg) ∈ M_{0} .

The first two terms cancel as σ_{M_{0}} = 0. Hence the result is −(Dg) · (Δf), which is the value at g of the element −Δf ⊗ D ∈ M_{0} ⊗ W regarded as a derivation R → M_{0}. 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_{i} H^{1}(W, M) = 0 for i ≠ 2. If ξ : K → M is a morphism in C_{1} then its image K′ is a subobject of M. Hence K′ is of type Ω^{1}, and so K′ ≅ K′_{0} ⊗ Ω^{1} with K′_{0} ∈ C_{0} by Lemma 1.3. For every maximal ideal 𝔪 of R the 𝔤/𝔪𝔤-module K′/𝔪K′ ≅ K′_{0}/𝔪K′_{0} ⊗ (W/𝔪W)^{∗} is completely reducible with all irreducible submodules isomorphic to (W/𝔪W)^{∗}. On the other hand, K′/𝔪K′ is an epimorphic image of K/𝔪K which does not have (W/𝔪W)^{∗} 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 S^{2}Ω^{1} ⊗ W → M vanishes on K and therefore factors through Ω^{1}. By (5.3)

H^{1}(W, M) ≅ gr_{2} H^{1}(W, M) ≅ Mor_{C_{1}}(S^{2}Ω^{1} ⊗ W, M) ≅ Mor_{C_{1}}(Ω^{1}, M) .

(iv) Again gr_{i} H^{1}(W, M) is nonzero for i = 2 only. Since Mor_{C_{1}}(Ω^{1}, M) = 0 by Lemma 1.4, we have

H^{1}(W, M) ≅ gr_{2} H^{1}(W, M) ≅ Mor_{C_{1}}(K ⊕ Ω^{1}, M) ≅ Mor_{C_{1}}(K, M) .

(v) Here gr_{i} H^{1}(W, M) = 0 for i ≠ 3, and

H^{1}(W, M) ≅ gr_{3} H^{1}(W, M) ≅ Mor_{C_{1}}(Ω^{1} ⊗ Ω^{1}, 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_{C_{1}}(Q, M) ≅ Mor_{C_{0}}(R, M_{0}) = Hom_{R}(R, M_{0})^{W} ≅ M_{0}^{W} .

□

Corollary 5.7. The Lie algebra Der W of all ℤ-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 N/W → Der W/ ad W ≅ H^{1}(W, W) by (ii). □

This result was obtained earlier [22] under assumptions weaker than (1.2), (1.3). If k ⊂ R^{W} 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_{k} W of k-linear derivations of W is isomorphic with N ∩ Der_{k} R, where Der_{k} R 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_{k} R 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 = R𝔇 ≅ R ⊗_{k} 𝔇. We have

𝔤 ≅ 𝔤𝔩_{R} W ≅ R ⊗_{k} 𝔤𝔩_{k} 𝔇 .

Let E_{ij} be the linear transformation of 𝔇 such that E_{ij}∂_{l} = δ_{jl}∂_{i} where indices i, j, l are taken among 1, …, n. If θ ∈ Ω^{1} then the element θ⊗∂_{i} ∈ 𝔤 corresponds to − ∑_{j} ⟨θ, ∂_{j}⟩ ⊗ E_{ij} under the isomorphism above. Given a representation σ_{V} of 𝔤𝔩_{k} 𝔇 in a vector space V over k, put

RV = R ⊗_{k} V

and define a representation of W in this space setting

ρ_{RV} (g∂_{i}) = g∂_{i} ⊗ id_{V} − ∑_{j} (∂_{j}g)_{R} ⊗ σ_{V} (E_{ij}) for g ∈ R, i = 1, …, n,

where f_{R} stands for the multiplication operator on R corresponding to an element f ∈ R. The operators f_{R} ⊗ id_{V} give RV a compatible Rmodule structure, so that (1.6) is fulfilled. Extend σ_{V} by R-linearity to a representation of 𝔤 in RV. If θ ∈ Ω^{1} then

σ_{RV} (θ ⊗ ∂_{i}) = − ∑_{j} ⟨θ, ∂_{j}⟩_{R} ⊗ σ_{V} (E_{ij}) .

Take f, g ∈ R. As

ρ_{RV} (g∂_{i})−(g_{R} ⊗id_{V})∘ρ_{RV} (∂_{i}) = − ∑_{j} (∂_{j}g)_{R} ⊗σ_{V} (E_{ij}) = σ_{RV} (dg⊗∂_{i}) ,

we have for D = g∂_{i} ∈ W

ρ_{RV} (fD) − (f_{R} ⊗ id_{V}) ∘ ρ_{RV} (D) = σ_{RV} (d(fg) ⊗ ∂_{i}) − σ_{RV} (f · dg ⊗ ∂_{i}) = σ_{RV} (df ⊗ D) .

Thus (1.8) is fulfilled too. We see that RV together with the three module structures we have described is an object of C_{1}. In fact R is a functor from the category of 𝔤𝔩_{k} 𝔇-modules to C_{1}. It takes the tensor product of two 𝔤𝔩_{k} 𝔇-modules to the tensor product of the corresponding objects in C_{1}, the symmetric and exterior powers of a 𝔤𝔩_{k} 𝔇-module to the symmetric and exterior powers in C_{1}. If dim V < ∞ and V^{∗} is the contragredient 𝔤𝔩_{k} 𝔇-module then RV^{∗} ≅ Hom_{R}(RV, R) in C_{1}.

Clearly RV is free over R. It has finite rank provided dim V < ∞. If V is an absolutely irreducible 𝔤𝔩_{k} 𝔇-module, then so is RV/(𝔪 · RV) ≅ R/𝔪⊗_{k}V as a module over 𝔤/𝔪𝔤 ≅ R/𝔪⊗_{k}𝔤𝔩_{k} 𝔇 where 𝔪 is any maximal ideal of R. In this case RV satisfies (1.10) and (1.11). Therefore we can determine its cohomology applying theorems 5.5, 5.6 where we take M = Q = RV. Note that Q corresponds to M_{0} = R under the equivalence of Lemma 1.3. Denote by R^{W} 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 𝔤𝔩_{k} 𝔇-module then RV ≅ R and

H^{1}(W, R) = H^{1}(Ω) ⊕ R^{W}· 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 R^{W} whose generator is the cohomology class of a divergence ψ : W → R. One can take ψ with zero values on 𝔇. Then ψ(f∂_{i}) = ∂_{i}f for f ∈ R and i = 1, …, n.

(ii) If V = 𝔇 is the natural 𝔤𝔩_{k} 𝔇-module, then we have RV ≅ W and H^{1}(W, W) ≅ N/W, where N is the normalizer of W in Der R.

(iii) If V = 𝔇^{∗} then RV ≅ Ω^{1} and H^{1}(W, Ω^{1}) is a free cyclic module over R^{W} generated by the cohomology class of any cocycle φ : W → Ω^{1} whose symbol is the canonical epimorphism S^{2}Ω^{1} ⊗ W → Ω^{1} in C_{1}. 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(f∂_{i})(∂_{j}) = ∂_{j}(ψ(f∂_{i})) = ∂_{j}∂_{i}f for f ∈ R and 1 ≤ 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 S^{2}𝔇^{∗} ⊗ 𝔇 → 𝔇^{∗} then RV ≅ K is the kernel of the canonical epimorphism π : S^{2}Ω^{1} ⊗ W → Ω^{1} in C_{1}. The group H^{1}(W, K) is a free cyclic module over R^{W} generated by the cohomology class of the cocycle φ = π_{K} ∘ φ_{u} where π_{K} : S^{2}Ω^{1} ⊗ W → K is the projection and φ_{u} : W → S^{2}Ω^{1} ⊗ 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 ∇(f∂_{j}, g∂_{j′}) = (f∂_{j}g) · ∂_{j′}, and so

φ_{u}(f∂_{i})(∂_{j}, ∂_{j′}) = −∇([f∂_{i}, ∂_{j}], ∂_{j′}) − ∇(∂_{j}, [f∂_{i}, ∂_{j′}]) = (∂_{j}∂_{j′}f) · ∂_{i}

We have π_{K} = id −\frac{1}{n+1} ν ∘ π where ν : Ω^{1} → S^{2}Ω^{1} ⊗ W is the canonical morphism in C_{1} such that π ∘ ν = (n + 1) · 1_{Ω^{1}}. Let ε_{1}, …, ε_{n} be the dual basis for the free R-module Ω^{1}, so that ⟨ε_{i}, ∂_{j}⟩ = δ_{ij}. Since ∑_{r} ε_{r}⊗∂_{r} ∈ 𝔤 corresponds to the identity endomorphism of Ω^{1} under σ_{Ω^{1}}, the morphism ν takes θ ∈ Ω^{1} to ∑_{r} θε_{r} ⊗ ∂_{r}. Now φ_{u}(f∂_{i}) = \frac{1}{2} ∑_{r,s} (∂_{r}∂_{s}f) · ε_{r}ε_{s} ⊗ ∂_{i} . Applying π to this, we get ∑_{s} (∂_{i}∂_{s}f) · ε_{s} in Ω^{1}. Applying next ν, we get ∑_{r,s} (∂_{i}∂_{s}f) · ε_{s}ε_{r} ⊗ ∂_{r} in S^{2}Ω^{1} ⊗ W. It follows

φ(f∂_{i})(∂_{j}, ∂_{j′}) = (∂_{j}∂_{j′}f) · ∂_{i} − \frac{1}{n + 1}(∂_{i}∂_{j}f) · ∂_{j′} − \frac{1}{n + 1}(∂_{i}∂_{j′}f) · ∂_{j} .

(v) Suppose that n = 1. If V = 𝔇^{∗} ⊗ 𝔇^{∗} then RV ≅ Ω^{1} ⊗ Ω^{1} and H^{1}(W, Ω^{1} ⊗ Ω^{1}) is a free cyclic module over R^{W} 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, Ω^{1} ⊗ Ω^{1} is free with one generator ε^{2} = ε⊗ε, where ε ∈ Ω^{1} is specified by the relation ⟨ε, ∂_{1}⟩ = 1. We take ψ with zero value on ∂_{1}. Then ψ^{∗}(fε)(∂_{1}, ∂_{1}) = −∂_{1}f, and so ψ^{∗}(fε) = −(∂_{1}f) · ε^{2}, for f ∈ R. As dψ(f∂_{1}) = (∂_{1}^{2}f) · ε, we get

φ(f∂_{1}) = −(∂_{1}^{3}f) · ε^{2}.

(vi) Suppose that V is an absolutely irreducible finite dimensional 𝔤𝔩_{k} 𝔇-module other than those considered in (i)–(v). Then H^{1}(W, RV) ≅ H_{R}^{1}(𝔤, RV)^{W}. The standard cochain complex for 𝔤 admits the following identification:

C_{R}^{•}(𝔤, RV) ≅ Hom_{R}(⋀ 𝔤, RV) ≅ R⊗_{k}Hom_{k}(⋀ 𝔤𝔩_{k} 𝔇, V) = RC^{•}(𝔤𝔩_{k} 𝔇, V)

with the differential 1_{R} ⊗ d_{V} where d_{V} is the differential of the standard cochain complex C^{•}(𝔤𝔩_{k} 𝔇, V) of k-multilinear alternating maps 𝔤𝔩_{k} 𝔇 × ⋯×𝔤𝔩_{k} 𝔇 → V. What we get above is, moreover, an isomorphism in the category C_{1}. Since d_{V} is a 𝔤𝔩_{k} 𝔇-equivariant map, the differential 1_{R} ⊗d_{V} is a morphism in C_{1}. By passing to the cohomology we still get objects of C_{1} and an isomorphism in this category

H_{R}^{•}(𝔤, RV) ≅ R ⊗_{k} H^{•}(𝔤𝔩_{k} 𝔇, V) = RH^{•}(𝔤𝔩_{k} 𝔇, V) .

As 𝔤𝔩_{k} 𝔇 acts in the cohomology group of its module V trivially, the action of W in the corresponding object of C_{1} is given by the rule D · (f ⊗ ζ) = Df⊗ζ for D ∈ W, f ∈ R and ζ ∈ H^{•}(𝔤𝔩_{k} 𝔇, V). Taking the W-invariants, we conclude

H^{1}(W, RV) ≅ R^{W} ⊗_{k} H^{1}(𝔤𝔩_{k} 𝔇, V).

We want to describe this isomorphism on the level of cocycles. Given a 1-cocycle φ′ : 𝔤𝔩_{k} 𝔇 → V, define a map φ : W → RV by the rule

φ(f∂_{i}) = − ∑_{j} ∂_{j}f ⊗ φ′(E_{ij}) for f ∈ R and i = 1, …, n .

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

φ^{♭}(df⊗∂_{i}) = φ(f∂_{i}) = − ∑_{j} (1_{R}⊗φ′)(∂_{j}f⊗E_{ij}) = (1_{R}⊗φ′)(σ_{W} (df⊗∂_{i})) .

Thus φ^{♭} = (1_{R} ⊗φ′)∘σ_{W}, that is, φ^{♭} : 𝔤 → RV 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 𝔤̃. By Lemma 3.3 there is an R-linear map φ̃ : W → RV whose composite with the canonical embedding ι : W → W gives φ. Denote by η : W × W → RV its coboundary. Then η(T, T′) = 0 for all T, T′ ∈ 𝔤̃ since φ̃ ∘ ι^{♭} = φ^{♭} is a cocycle. Observe now that φ is a 𝔇-equivariant map and vanishes on 𝔇. Hence φ̃ 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 𝔤̃ + ι(𝔇) generates W as an R-module, η is identically zero. Thus φ̃ 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 ℤ-split central extension of W by a ℤ-module V is an exact sequence 0 → V → L → W → 0 where L → W is a homomorphism of Lie algebras over ℤ whose kernel is a central ideal and also a ℤ-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 ℤ-split central extensions of W by V are in a one-to-one correspondence with the cohomology classes of ℤ-bilinear 2-cocycles W × W → V, the coefficients being a trivial W-module. A ℤ-split central extension L^{u} of W by a ℤ-module U is universal if for every other ℤ-module V and a ℤ-split central extension L of W by V there is a unique ℤ-linear map U → V which extends to a morphism between the two extensions:

\xymatrix{0 \ar[r] & U \ar[r] \ar[d] & L^{u} \ar[r] \ar[d] & W \ar[r] \ar[d]^{id} & 0 \\ 0 \ar[r] & V \ar[r] & L \ar[r] & W \ar[r] & 0.}

In other words, a universal central extension corresponds to an isomorphism of functors H^{2}(W, V) ≅ Hom_{ℤ}(U, V) in V. Recall that Ω^{•} = ⋀ Ω^{1} is the de Rham complex relative to W. Theorem 7.1. Assume that 3R = R. If rk_{R} W > 1 then every ℤsplit central extension of W splits. If rk_{R} W = 1 then the universal central extension of W has kernel H^{1}(Ω) and is determined by the 2cocycle φ : W × W → H^{1}(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 rk_{R} W > 2.

Proof. Put n = rk_{R} W. There is an obvious embedding

H^{2}(W, V) ↪ H^{1}(W, Hom_{ℤ}(W, V))  (∗)

obtained by separating the two arguments of the cocycles W × W → V. The image of H^{2}(W, V) consists precisely of the cohomology classes represented by the 1-cocycles φ : W → Hom_{ℤ}(W, V) which satisfy φ(D)(D) = 0 for all D ∈ W. Consider the W-module M = Hom_{ℤ}(W, V) and denote by ρ_{M} the corresponding representation of W. In a natural way M is a right module for the ring End_{R} W, hence a left module for the opposite ring (End_{R} W)^{op}. In particular, M is also an R-module. Let f ∈ R, D ∈ W, ξ ∈ M. As ρ_{M}(D)ξ = −ξ ∘ ρ_{W} (D) and f_{M}ξ = ξ ∘ f_{W}, we get

[ρ_{M}(D), f_{M}] ξ = ξ∘(−f_{W} ∘ρ_{W} (D)+ρ_{W} (D)∘f_{W} ) = ξ∘(Df)_{W} = (Df)_{M}ξ ,

which verifies (1.6). For T ∈ 𝔤 define an endomorphism σ_{M}(T) ∈ End_{R} M by the rule

σ_{M}(T) ξ = ξ ∘ (γ(T) · 1_{W} − σ_{W} (T)) , ξ ∈ M,

where γ : 𝔤 → R is the contraction. As

(ρ_{M}(fD)ξ − f · ρ_{M}(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 C_{1}. Put Q = Ω^{1} ⊗ Ω^{n}. This is an object of C_{1} satisfying (1.10), (1.11). Since Ω^{n} is a projective R-module of rank 1, its endomorphism algebra can be identified with R, which yields isomorphisms of R-algebras

End_{R} Q ≅ End_{R} Ω^{1} ⊗ End_{R} Ω^{n} ≅ End_{R} Ω^{1} ≅ (End_{R} W)^{op}.

Let T ∈ 𝔤. The endomorphism σ_{Ω^{n}}(T) is just the multiplication by the trace of σ_{Ω^{1}}(T), that is, by γ(T). Hence

σ_{Q}(T) = σ_{Ω^{1}}(T) ⊗ 1_{Ω^{n}} + 1_{Ω^{1}} ⊗ σ_{Ω^{n}}(T) = σ_{Ω^{1}}(T) ⊗ 1_{Ω^{n}} + γ(T) · 1_{Ω^{1}} ⊗ 1_{Ω^{n}} .

Under the isomorphisms above it is sent to σ_{Ω^{1}}(T)+γ(T)·1_{Ω^{1}} in End_{R} Ω^{1}, and then to γ(T) · 1_{W} − σ_{W} (T) in (End_{R} W)^{op}. Thus M is of type Q.

We apply theorems 5.5, 5.6 to compute H^{1}(W, M). If 𝔪 is a maximal ideal of R then Q/𝔪Q ≅ (W/𝔪W)^{∗} ⊗ ⋀^{n}(W/𝔪W)^{∗}. By Lemma 5.4 this 𝔤/𝔪𝔤-module is isomorphic to neither the trivial module, nor W/𝔪W, nor (W/𝔪W)^{∗}, nor K/𝔪K. Furthermore, H^{1}(𝔤/𝔪𝔤, Q/𝔪Q) = 0. It follows H^{1}(W, M) = 0 provided n > 1. If n = 1 then

H^{1}(W, M) ≅ Mor_{C_{1}}(Ω^{1} ⊗ Ω^{1}, M) .

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

Hom_{R}(Ω^{1} ⊗ Ω^{1}, M) ≅ Hom_{ℤ}(Ω^{1} ⊗ Ω^{1} ⊗ W, V) ≅ Hom_{ℤ}(Ω^{1}, V)

because Ω^{1} ⊗ W ≅ R. Taking the W-invariants, we get

Mor_{C_{1}}(Ω^{1} ⊗ Ω^{1}, M) ≅ Hom_{W} (Ω^{1}, V) ≅ Hom_{ℤ}(Ω^{1}/ρ_{Ω^{1}}(W)(Ω^{1}), V) .

Since Ω^{2} = 0, the classical formula relating the action of W and the interior product on differential forms gives ρ_{Ω^{1}}(D) θ = d(⟨θ, D⟩) for θ ∈ Ω^{1}, D ∈ W. Since the elements ⟨θ, D⟩ span the whole R, again by the projectivity assumption, the group ρ_{Ω^{1}}(W)(Ω^{1}) consists of all exact 1-forms. Hence Ω^{1}/ρ_{Ω^{1}}(W)(Ω^{1}) ≅ H^{1}(Ω).

Now take a ℤ-linear map η : H^{1}(Ω) → V and write out the corresponding 1-cocycle φ : W → M. Tracing back the isomorphisms above, we first find the morphism ζ : Ω^{1} ⊗ Ω^{1} → M in C_{1}. It is related to η as follows:

ζ(ω)(D) = η(cls(i_{D}ω)) for ω ∈ Ω^{1} ⊗ Ω^{1}, D ∈ W,

where i_{D}ω ∈ Ω^{1} is given by (i_{D}ω)(D′) = ω(D, D′) for D′ ∈ W (recall that Ω^{1} ⊗ Ω^{1} 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

i_{D′}(ψ^{∗}(θ)) = ψ(D′) θ − d(⟨θ, D′⟩) for θ ∈ Ω^{1}, D′ ∈ W.

We have φ(D) = ζ(ω) where ω = ψ^{∗}(θ) and θ = dψ(D). As i_{D′}ω ≡ ψ(D′)θ modulo the exact 1-forms, it follows

φ(D)(D′) = η(cls(ψ(D′) · dψ(D))) for D, D′ ∈ W.

Since ψ(D) · dψ(D) = \frac{1}{2}dψ(D)^{2} is an exact form, we have φ(D)(D) = 0. This shows that (∗) is an isomorphism and so

H^{2}(W, V) ≅ Hom_{ℤ}(H^{1}(Ω), V ) .

Thus the central extension corresponding to V = H^{1}(Ω) and the identity map η : H^{1}(Ω) → V is universal. The corresponding cocycle is determined as well. □

References

[1] N. Bourbaki Commutative Algebra Springer, Berlin, (1989)

[2] A. Campillo J. Grabowski G. Müller Derivation algebras of toric varieties, Compos. Math. 116 (1999) 119-132.

[3] S. Chiu G.Yu. Shen Cohomology of graded Lie algebras of Cartan type of characteristic p, Abh. Math. Sem. Univ. Hamburg 57 (1987) 139–156.

106 S. SKRYABIN

[4] A.S. Dzhumadil’daev On the cohomology of modular Lie algebras (in Russian), Mat. Sbornik 119 (1982) 132–149.

[5] A.S. Dzhumadil’daev Central extensions of the Zassenhaus algebra and their irreducible representations (in Russian), Mat. Sbornik 126 (1985) 473–489.

[6] D.B. Fuks Cohomology of infinite dimensional Lie algebras, Consultants Bureau, New York, 1986.

[7] J. Grabowski Isomorphisms and ideals of the Lie algebras of vector fields, Invent. Math. Vol. 50, 1978. 13–33.

[8] J. Grabowski Derivations of the Lie algebras of analytic vector fields, Compositio Math. Vol. 43, 1981. 239–252.

[9] J. Grabowski Ideals of the Lie algebras of vector fields revisited, Rend. Circ. Mat. Palermo Vol. 32, 1993. 89–95.

[10] H. Hauser G. Müller On the Lie algebra Θ(X) of vector fields on a singularity, J. Math. Sci. Uni. Tokyo Vol. 1, 1994. 239–250.

[11] J. Herz Pseudo-algbre de Lie, Compt. Rend. Acad. Sci. Paris Vol. 236, 1953. 1935–1937.

[12] N. Jacobson Classes of restricted Lie algebras of characteristic p II, Duke Math. J. Vol. 10, 1943. 107–121.

[13] D.A. Jordan On the ideals of a Lie algebra of derivations, J. London Math. Soc. Vol. 33, 1986. 33–39.

[14] D. Kastler R. Stora Lie-Cartan pairs, J. Geom. Phys. Vol. 2, 1985. 1–31.

[15] A.I. Kostrikin I.R. Shafarevich Graded Lie algebras of finite characteristic (in Russian), Izv. Akad. Nauk USSR Ser. Mat. Vol. 33, 1969. 251–322.

[16] K. Mackenzie Lie groupoids and Lie algebroids in differential geometry, London Math. Soc. Lecture Note Series, Vol. 124, Cambridge Univ. Press, Cambridge, 1987.

[17] R.S. Palais The cohomology of Lie d-rings, In Proc. Sympos. Pure Math., Vol. III, American Mathematical Society, 1961. pp. 130–137.

[18] R. Ree Note on generalized Witt algebras, Canad. J. Math., Vol. 11, 1959. pp. 345–352.

[19] G.S. Rinehart Differential forms on general commutative algebras, Trans. Amer. Math. Soc., Vol. 108, 1963. pp. 195–222.

[20] G. Shen Graded modules of graded Lie algebras of Cartan type, I. Mixed products of modules, Scientia Sinica Ser. A, Vol. 29, 1986. pp. 570–581.

[21] Th. Siebert Lie algebras of derivations and affine algebraic geometry over fields of characteristic 0, Math. Ann., Vol. 305, 1996. pp. 271–286.

[22] S. Skryabin Regular Lie rings of derivations (in Russian), Vestnik Moscov. Univ. Ser. I. Mat. Mekh., no. 3 1988. pp. 59–62.

[23] S. Skryabin An algebraic approach to the Lie algebras of Cartan type, Comm. Algebra, Vol. 21, 1993. pp. 1229–1336.

[24] S. Skryabin Independent systems of derivations and Lie algebra representations, In the Proceedings of the Chebotarev Centennial Conference on Algebra and Analysis (Kazan, 1994), Eds. M.M. Arslanov, A.N. Parshin, I.R. Shafarevich, Walter de Gruyter, Berlin, 1996. pp. 115–150.

[25] H. Strade R. Farnsteiner Modular Lie Algebras and their Representations, Marcel Dekker Textbooks and Monographs, Vol. 116, Marcel Dekker, New York, 1988.

DEGREE ONE COHOMOLOGY 107

[26] F. Takens Derivations of vector fields, Compositio Math., Vol. 26, 1973. pp. 151–158.

[27] F. Wagemann A two-dimensional analogue of the Virasoro algebra, J. Geom. Phys., Vol. 36, 2000. pp. 103–116.

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