On finite dimensionality of homology of subalgebras of vector fields

We show that the tensor product of modules of tensor fields is a noetherian module as a module over any graded Lie subalgebra of finite codimension in the Lie algebra of polynomial vector fields on ${\mathbb{R}}^n$. As a corollary, we prove the conjecture of I.M.Gelfand announced at ICM’1970 at Nice on finite dimensionality of continuous cohomology of graded Lie subalgebras of formal vector fields $W_n$.