Lemma 15.125.7. Let $(R, \mathfrak m)$ be a Noetherian local ring of dimension $d$, let $g_1, \ldots , g_ d$ be a system of parameters, and let $I = (g_1, \ldots , g_ d)$. If $e_ I/d!$ is the leading coefficient of the numerical polynomial $n \mapsto \text{length}_ R(R/I^{n+1})$, then $e_ I \leq \text{length}_ R(R/I)$.
Proof. The function is a numerical polynomial by Algebra, Proposition 10.59.5. It has degree $d$ by Algebra, Proposition 10.60.9. If $d = 0$, then the result is trivial. If $d = 1$, then the result is Lemma 15.125.1. To prove it in general, observe that there is a surjection
\[ \bigoplus \nolimits _{i_1, \ldots , i_ d \geq 0,\ \sum i_ j = n} R/I \longrightarrow I^ n/I^{n + 1} \]
sending the basis element corresponding to $i_1, \ldots , i_ d$ to the class of $g_1^{i_1} \ldots g_ d^{i_ d}$ in $I^ n/I^{n + 1}$. Thus we see that
\[ \text{length}_ R(R/I^{n + 1}) - \text{length}_ R(R/I^ n) \leq \text{length}_ R(R/I) {n + d - 1 \choose d - 1} \]
Since $d \geq 2$ the numerical polynomial on the left has degree $d - 1$ with leading coefficient $e_ I / (d - 1)!$. The polynomial on the right has degree $d - 1$ and its leading coefficient is $\text{length}_ R(R/I)/ (d - 1)!$. This proves the lemma. $\square$
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