Gelfand-Fuchs cohomology in algebraic geometry and factorization algebras

TL;DR

This paper computes the Lie algebra cohomology of regular vector fields on smooth affine varieties over characteristic zero fields, using algebraic techniques of factorization algebras, extending classical topological results to an algebraic setting.

Contribution

It provides an algebraic analogue of Gelfand-Fuks cohomology for algebraic varieties using factorization algebra techniques, without relying on topology.

Findings

Computed Lie algebra cohomology of T(X) for smooth affine varieties

Established a topological interpretation relative to complex embeddings

Extended classical Gelfand-Fuks results to algebraic geometry context

Abstract

Let X be a smooth affine variety over a field k of characteristic 0 and T(X) be the Lie algebra of regular vector fields on X. We compute the Lie algebra cohomology of T(X) with coefficients in k. The answer is given in topological terms relative to any embedding of k into complex numbers and is analogous to the classical Gelfand-Fuks computation for smooth vector fields on a C-infinity manifold. Unlike the C-infinity case, our setup is purely algebraic: no topology on T(X) is present. The proof is based on the techniques of factorization algebras, both in algebro-geometric and topological contexts.