On finite dimensionality of homology of subalgebras of vector fields

TL;DR

This paper proves that modules of tensor fields are noetherian over certain graded Lie subalgebras of polynomial vector fields and confirms Gelfand's conjecture on the finite dimensionality of cohomology for these subalgebras.

Contribution

It establishes the noetherian property of tensor modules and verifies Gelfand's long-standing conjecture on cohomology finiteness for graded Lie subalgebras of formal vector fields.

Findings

Tensor product modules are noetherian over graded Lie subalgebras.

Finite dimensionality of continuous cohomology for graded Lie subalgebras of formal vector fields.

Confirmation of Gelfand's conjecture from 1970.

Abstract

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 $W_n$.