Processing math: 100%

The Stacks project

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


Comments (0)


Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.