Let $\mathfrak{g}$ be a finite dimensional Lie algebra over $k$, let $U$ be its enveloping algebra, and let $M$ be a $\mathfrak{g}$-module (not necessarily finite dimensional). Call the *invariant dimension* of $M$ the largest $i$ such that $\operatorname{Ext}^i_U(k,M)\neq 0$. This is the same as the degree of the largest non-vanishing Lie algebra cohomology group.

Here are two equivalent statements of my question.

- If $M$ is a cyclic $\mathfrak{g}$-module (that is, its generated by a single element as a $U$-module), then is the number of relations of $M$ always greater than the invariant dimension minus $1$?
- If $I$ is a left ideal in $U$, then is the number of generators of I always greater than the invariant dimension?

The reason why I would suspect such a thing is that it is true in the case of an abelian Lie algebra; that is, when $U$ is a polynomial ring. In this case, the invariant dimension of an ideal $I$ coincides with the height of $I$, and so (2) becomes Krull's height theorem (and (1) follows immediately from (2)).