  We compute the local cohomology of vector fields on a manifold. In the smooth
case this recovers the diagonal cohomology studied in work of Losik, Guillemin,
Fuks and others. In the holomorphic case this cohomology has recently appeared
in work of Hennion and Kapranov in their study of the Lie algebra cohomology of
vector fields on a complex manifold. Additionally, we construct explicit
representatives for cocycles in Gelfand--Fuks cohomology via descent.
