Proposition 10.58.5. Let $R$ be a Noetherian local ring. Let $M$ be a finite $R$-module. Let $I \subset R$ be an ideal of definition. The Hilbert function $\varphi _{I, M}$ and the function $\chi _{I, M}$ are numerical polynomials.

**Proof.**
Consider the graded ring $S = R/I \oplus I/I^2 \oplus I^2/I^3 \oplus \ldots = \bigoplus _{d \geq 0} I^ d/I^{d + 1}$. Consider the graded $S$-module $N = M/IM \oplus IM/I^2M \oplus \ldots = \bigoplus _{d \geq 0} I^ dM/I^{d + 1}M$. This pair $(S, N)$ satisfies the hypotheses of Proposition 10.57.7. Hence the result for $\varphi _{I, M}$ follows from that proposition and Lemma 10.54.1. The result for $\chi _{I, M}$ follows from this and Lemma 10.57.5.
$\square$

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