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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