Let \mathfrak{g} http://latex.mathoverflow.net/png?%5Cmathfrak%7Bg%7D$\mathfrak{g}$ be a finite dimensional Lie algebra over k$k$, let U$U$ be its enveloping algebra, and let M$M$ be a \mathfrak{g} http://latex.mathoverflow.net/png?%5Cmathfrak%7Bg%7D$\mathfrak{g}$-module (not necessarily finite dimensional). Call the invariant dimension of M$M$ the largest i$i$ such that Ext^i_U(k,M)\neq 0 http://latex.mathoverflow.net/png?Ext%5Ei%5FU%28k%2CM%29%5Cneq%200$\operatorname{Ext}^i_U(k,M)\neq 0$. 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$M$ is a cyclic \mathfrak{g} http://latex.mathoverflow.net/png?%5Cmathfrak%7Bg%7D$\mathfrak{g}$-module (that is, its generated by a single element as a U$U$-module), then is the number of relations of M$M$ always greater than the invariant dimension minus 1$1$?
- If I$I$ is a left ideal in U$U$, 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$U$ is a polynomial ring. In this case, the invariant dimension of an ideal I$I$ coincides with the height of I$I$, and so (2) becomes Krull's height theorem (and (1) follows immediately from (2)).
