Gelfand-Fuks cohomology of vector fields on algebraic varieties

TL;DR

This paper introduces algebraic Gelfand-Fuks cohomology for polynomial vector fields on affine algebraic varieties, computes it explicitly for certain varieties, and reveals its decomposition relating to de Rham and Lie algebra cohomologies.

Contribution

It defines a new algebraic cohomology theory for vector fields on varieties and computes it explicitly for key examples, linking it to known cohomologies.

Findings

Cohomology decomposes as a tensor product of de Rham and Lie algebra cohomologies.

Explicit calculations for affine space, torus, and Krichever-Novikov algebras.

Cohomology vanishes at the origin for certain modules.

Abstract

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.