Gelfand-Fuks cohomology of vector fields on algebraic varieties

For an affine algebraic variety, we introduce algebraic Gelfand-Fuks cohomology of polynomial vector fields with coefficients in differentiable $AV$-modules. Its complex is given by cochains that are differential operators in the sense of Grothendieck. Using the jets of vector fields, we compute this cohomology for varieties with uniformizing parameters. We prove that in this case, Gelfand-Fuks cohomology with coefficients in a tensor module decomposes as a tensor product of the de Rham cohomology of the variety and the cohomology of the Lie algebra of vector fields on affine space, vanishing at the origin. We explicitly compute this cohomology for affine space, the torus, and Krichever-Novikov algebras.

💡 Research Summary

The paper develops an algebraic version of Gelfand‑Fuks cohomology for the Lie algebra V of polynomial vector fields on an affine algebraic variety X, with coefficients in differentiable AV‑modules. An AV‑module is a module equipped with compatible actions of the coordinate ring A and the Lie algebra V, satisfying the Leibniz rule; differentiability means that the V‑action is given by a Grothendieck differential operator (equivalently, the module is annihilated by a sufficiently high power of the jet ideal).

Starting from the Chevalley‑Eilenberg complex C⁎(V,M), the authors impose the Gelfand‑Fuks condition: a cochain φ must vanish after applying a finite‑order polynomial differential operator in each argument. The resulting subcomplex C⁎{GF}(V,M) defines the algebraic Gelfand‑Fuks cohomology H⁎{GF}(V,M).

A key technical step is to replace the ordinary Lie algebra cohomology H⁎(V,M) by the A‑linear cohomology of the Lie algebra of polynomial jets A#V. Using the weak Rinehart enveloping algebra A#U(V) and its strong version D(A,V), they show that for any strong AV‑module M there is an isomorphism H⁎(V,M) ≅ H⁎_A(A#V,M). Moreover, the lift of a Gelfand‑Fuks cochain to A#V is precisely a “finite” cochain, i.e. one that vanishes when any argument lies in the jet ideal Δ^p⊗AV for some p. Finite cochains extend uniquely to the completed jet algebra Â#V, allowing the authors to work with the infinite‑jet Lie algebra.

When X possesses a system of uniformizing parameters (x₁,…,x_n), the completed jet algebra admits a semi‑direct product decomposition (proved in earlier work): Â#V ≅ V ⋉ (Â ⊗ L₊), where L₊ is the Lie algebra of polynomial vector fields on affine space ℂⁿ that vanish at the origin. For a finite‑dimensional L₊‑module W, the tensor module A⊗W on X is considered. The authors establish a Künneth formula for this semi‑direct product (Theorem 5.2): \