Lemma 10.58.10. Let $k$ be a field. Suppose that $I \subset k[X_1, \ldots , X_ d]$ is a nonzero graded ideal. Let $M = k[X_1, \ldots , X_ d]/I$. Then the numerical polynomial $n \mapsto \dim _ k(M_ n)$ (see Example 10.58.9) has degree $< d - 1$ (or is zero if $d = 1$).

Proof. The numerical polynomial associated to the graded module $k[X_1, \ldots , X_ d]$ is $n \mapsto \binom {n - 1 + d}{d - 1}$. For any nonzero homogeneous $f \in I$ of degree $e$ and any degree $n >> e$ we have $I_ n \supset f \cdot k[X_1, \ldots , X_ d]_{n-e}$ and hence $\dim _ k(I_ n) \geq \binom {n - e - 1 + d}{d - 1}$. Hence $\dim _ k(M_ n) \leq \binom {n - 1 + d}{d - 1} - \binom {n - e - 1 + d}{d - 1}$. We win because the last expression has degree $< d - 1$ (or is zero if $d = 1$). $\square$

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