TITLE: Relation between Lie Algebra Cohomology and Number of Relations of a Cyclic Module?
QUESTION [8 upvotes]: 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)).

REPLY [3 votes]: This is FALSE as stated, even for the augmentation ideal. For example, consider the 3-dimensional Heisenberg Lie algebra g and the trivial module k. Then it's easy to see that the third (=top) cohomology is non-zero, but due to the noncommutativity of g, the left ideal I=annihilator of k may be generated by only two elements. The inequality proposed by Simon, in fact, goes the other way: the number of generators for I is at most the number of generators of gr I.
Admittedly, this is a cheap shot, since the right notion of the number of generators for a left ideal I of U(g) is the minimal dimension of an ad(g)-invariant subspace of U(g) generating I - in particular, this number survives the passage to the associated graded. I don't know whether the modified statement is true.