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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