Gel′fand-Fuchs cohomology in algebraic geometry and factorization algebras

Abstract

Let X X be a smooth affine variety over a field k \mathbf k of characteristic 0 0 and T ( X ) T(X) be the Lie algebra of regular vector fields on X X . We compute the Lie algebra cohomology of T ( X ) T(X) with coefficients in k \mathbf k . The answer is given in topological terms relative to any embedding k ⊂ C \mathbf k\subset \mathbb {C} and is analogous to the classical Gel′fand-Fuchs computation for smooth vector fields on a C ∞ C^\infty -manifold. Unlike the C ∞ C^\infty -case, our setup is purely algebraic: no topology on T ( X ) T(X) is present. The proof is based on the techniques of factorization algebras, both in algebro-geometric and topological contexts.