Cohomology of Lie algebras of vector fields is also known as Gelfand-Fuks cohomology owing to I. M. Gelfand and D. B. Fuks [Funkts. Anal. Prilozh. 3, No. 2, 87 (1969; Zbl 0194.24601), Funct. Anal. Appl. 3, 194–210 (1970); translation from Funkts. Anal. Prilozh. 3, No. 3, 32–52 (1969; Zbl 0216.20301)] who introduced cochain complexes of continuous cochains in order to remove infinite dimensionality of (some) cohomology spaces.
The article under review proposes a new setup: the object of study are Lie subalgebras \(W\) of the Lie algebra of derivations of a commutative associative unital ring \(R\), forming a projective module of constant finite rank \(n\geq 1\) under the natural \(R\)-module structure. While the traditional class of examples are Lie algebras of smooth vector fields \(\text{Vect}(X)\) on \({\mathcal C}^{\infty}\) manifolds \(X\), holomorphic fields on Stein manifolds or algebraic fields on smooth algebraic varieties and their respective subalgebras, the generality of the present approach allows to include for example Cartan type Lie algebras in positive characteristic (standing assumption is however \(2^{-1}\in R\)).
Amazingly, the author proves results on derivations and central extensions for \(W\), i.e. for degree \(1\) cohomology with values in some category of representations of \(W\) (a subcategory of \({\mathcal C}_1\) to be defined below) and degree \(2\) cohomology with trivial coefficients, using only algebraic (i.e. possibly non-continuous) cochains (in fact, the author works over \({\mathbb Z}\) instead of a field). This allows to take on cohomology of vector fields a nice algebraic point of view, though it seems to be not suited for higher degree cohomology.
The theorem on central extensions reads as follows: [Theorem 7.1] Assume that \(3R=R\). If \(\text{rk}_RW>1\), then every \({\mathbb Z}\)-split central extension of \(W\) splits. If \(\text{rk}_RW=1\), then the universal central extension of \(W\) has kernel \(H^1(\Omega)\), the latter space being the cohomology of the corresponding de Rham complex.
A similar result also follows from Gelfand-Fuchs theory, but only for the Lie algebra of all \({\mathcal C}^{\infty}\) vector fields on a smooth manifold, continuous cohomology and over \({\mathbb R}\) or \({\mathbb C}\). Applications abound: not only manifolds of real dimension \(1\) furnish examples (cf. the universal central extension of the Lie algebra \(Vect(S^1)\) of smooth vector fields on the circle), but also effectivly \(1\)-dimensional objects on higher dimensional manifolds: while the article cites the reviewer’s work on the universality of the central extension of the Lie algebra \(\text{Vect}_{(1,0)}(\Sigma)\) of complexified type \((1,0)\)-vector fields on compact Riemann surfaces \(\Sigma\) [F. Wagemann, J. Geom. Phys. 36, No. 1–2, 103–116 (2000; Zbl 0991.17017)], where the universal center is of dimension \(g\) the genus of \(\Sigma\), on the other hand Novikov’s conjecture [cf. D. V. Millionshchikov, Transl., Ser. 2, Am. Math. Soc. 179(33), 101–108 (1997; Zbl 0904.17020)] receives a positive answer: let \(p_1,\dots,p_k\in\Sigma\) be distinct points of a compact Riemann surface \(\Sigma\) and \(\text{Mer}_k(\Sigma)\) be the Lie algebra of meromorphic vector fields on \(\Sigma\) with possible poles only in \(p_1,\dots,p_k\). The center of the universal central extension of \(\text{Mer}_k(\Sigma)\) is then conjectured to be \(H^1(\Sigma\setminus\{p_1,\dots,p_k\})\). Up to now, this was only known for continuous cohomology [F. Wagemann, Lett. Math. Phys. 47, No. 2, 173–177 (1999; Zbl 0927.17009) and More remarks on the cohomology of Krichever-Novikov algebras (preprint)]. It is a nice exercice to show that the right cohomology space arises for the universal center using theorem \(7.1\) above.
Let me now explain how the above mentioned results are obtained: in the first section it is introduced a certain category of \(W\)-modules, modeled on the geometric context. Assume \(\Omega^1:=Hom_R(W,R)\) satisfies \(\Omega^1=RdR\) (standing assumption) and set \({\mathfrak g}\) to be the Lie algebra \(\Omega^1\otimes_RW\cong{\mathfrak gl}_R(W)\). The category \({\mathcal C}_1\) consists of abelian groups \(M\) which are simultaneously \(R\)-modules (via \(R\ni f\mapsto f_M\)), \(W\)-modules (via \(W\ni D\mapsto \rho_M(D)\)) and \({\mathfrak g}\)-modules (via \({\mathfrak g}\ni T\mapsto \sigma_M(T)\)) such that
(1.6) \([\rho_M(D),f_M]=(Df)_M\)
(1.7) \([\sigma_M(T),f_M]=0\)
(1.8) \(\rho_M(fD)=f_M\circ\rho_M(D)+\sigma_M(df\otimes D)\)
(1.9) \(\sigma_M(fT)=f_M\circ\sigma_M(T)\).
\({\mathcal C}_0\) denotes the full subcategory of objects \(M\) with \(\sigma_M=0\). Needless to say that in case \(W=Vect(X)\), \(\Omega^1=\Omega^1(X)\) and modules of tensor fields of any kind constitue objects of \({\mathcal C}_1\), \(\rho\) being the Lie derivative and \(\sigma\) being defined using contractions of tensors.
The local-to-global principle of all cohomology computations of global objects on a manifold expresses itself in the present context in viewing a projective \(M\in ob({\mathcal C}_1)\) as being glued from representations \(M/{\mathfrak m}M\) of \({\mathfrak g}/{\mathfrak m}{\mathfrak g}\cong{\mathfrak gl}(W/{\mathfrak m}W)\) for the maximal ideals \({\mathfrak m}\triangleleft R\). An important class of objects in \({\mathcal C}_1\) is given by objects of type \(Q\) where \(Q\) is a finitely generated projective \(R\)-module with nice gluing properties: \(M\) is of type \(Q\) iff there is a homomorphism of \(R\)-algebras \(\text{End}_R(Q)\to \text{End}_R(M)\) sending \(\sigma_Q(T)\) to \(\sigma_M(T)\) for all \(T\in{\mathfrak g}\). Using Morita theory, the author shows that tensoring by \(Q\) gives an equivalence between \({\mathcal C}_0\) and the full subcategory of \({\mathcal C}_1\) consisting of objects of type \(Q\).
Another well-known concept in Gelfand-Fukhs theory is the order of a cocycle: denote by \(\delta_f\xi\) for a \({\mathbb Z}\)-linear map \(\xi:N\to M\) between \(R\)-modules the default of being an \(R\)-module map with respect to \(f\): \(\delta_f\xi=\xi\circ f_N-f_M\circ\xi\). Then \(\xi\) is called a differential operator of order \(\leq r\) iff \(\delta_{f_1}\dots\delta_{f_{r+1}}\xi=0\) for all \(f_1,\dots,f_{r+1}\in R\). The main result of section \(2\) shows that for \(M\in ob({\mathcal C}_1)\), every \(1\)-cocycle on \(W\) with values in \(M\) is a differential operator of order \(\leq 3\). If \(rk_RW>1\), it is actually of order \(\leq 2\).
Infinite dimensionality of cohomology spaces in the cohomology theory of Lie algebras of vector fields on a manifold \(X\) is due to the fact that cochains are only \({\mathbb R}\) or \({\mathbb C}\)-linear, whereas \({\mathcal C}^{\infty}(X)\)-linear concepts correpond to differential forms. In order to deal with the \(R\)-linear part of the cohomology of \(W\), the author uses in section \(3\) a sort of Lie algebroid \(\tilde{W}\) (itself a Lie algebra over \({\mathbb Z}\)) called first order prolongation of \(W\), and relates then cohomologies of \(\tilde{W}\) and \(W\) by an exact sequence (being the initial terms of a Hochschild-Serre type spectral sequence). Then a prolongation mechanism is shown, constructing to a \(1\)-cocycle \(\varphi:W\to M\) which is a differential operator of order \(\leq 1\) a unique \(R\)-linear cocycle \(\tilde{\varphi}:\tilde{W}\to M\) such that \(\varphi=\tilde{\varphi} \circ i\) (where \(i:W\to\tilde{W}\) is not a Lie homomorphism), and characterizing when \(\varphi\) is itself \(R\)-linear.
In section \(4\) is investigated a universality property of \(1\)-cocycles \(\varphi:W\to M\) of order \(\leq 2\) with values in \(M\in ob({\mathcal C}_1)\). Namely, such a \(\varphi\) is universal if for any \(\varphi':W\to M'\) of the same kind, there is a unique morphism \(\xi:M\to M'\) such that \(\varphi'-\xi\circ\varphi\) is of order \(\leq 1\). Then the author constructs such a universal differential order \(2\) cocycle \(\varphi:W\to S^2\Omega^1\otimes W\) using (algebraic) torsion-free connections. In case \(rk_RW=1\) and \(3^{-1}\in R\), he constructs furthermore a universal differential order \(3\) cocycle (i.e. universal within order \(\leq 3\) cocycles) \(\varphi:W\to \Omega^1\otimes\Omega^1\) using the divergence \(1\)-cocycle.
Eventually, in section \(5\), the author computes the spaces of different graduation of \(H^1(W,M)\) corresponding to the order filtration using the previously mentioned results. With this description at hand, he can deduce the \(1\)-cohomology of an object \(M\) of type \(Q\), reducing it to the cases where \(M=M_0\otimes Q\), \(M_0\in {\mathcal C}_0\) and \(Q\) in a particular list, and a following case-by-case study of the different “exceptional” modules. As a corollary, the Lie algebra \(Der(W)\) of all \({\mathbb Z}\)-linear derivations of \(W\) is shown to be isomorphic to the normaliser \(N\) of \(W\) in \(Der(R)\), [cf. S. M. Skryabin, Mosc. Univ. Math. Bull. 43, No. 3, 56–58 (1988; Zbl 0654.17008)].
Section \(6\) treats the special case where \(R\) is an algebra over a field \({\mathbb K}\) and \(W\) is the the free \(R\)-module generated by \(n\) \({\mathbb K}\)-linear pairwise commuting derivations. Here, the \(1\)-cocycles can be written down explicitly.
Finally, section \(7\) proves the above mentioned theorem on central extensions. A key point is the embedding \[ H^2(W,V)\hookrightarrow H^1(W,Hom_{\mathbb Z}(W,V)), \] \(V\) being some \({\mathbb Z}\)-module (the center of the central extension!), because then the claim reduces to a computation of the \(1\)-cohomology with values in \(M=Hom_{\mathbb Z}(W,V)\), and the proof that \(M\in ob({\mathcal C}_1)\) and of type \(Q=\Omega^1\otimes\Omega^n\).