Let $R = \mathbf{C}[x_1, \ldots, x_n]$ and let $M$ be a graded $R$-module which is finite-dimensional over $\mathbf{C}$ and suppose $ 0 \leftarrow M \leftarrow R^g \leftarrow R^d \leftarrow \cdots $ is a minimal resolution of $M$. I believe that it follows that $d \geq gn$, but don't know how prove this, or derive it from standard results in commutative algebra.

The connection to Krull's PIT is that this theorem should cover the case $g = 1$.