Lemma 15.125.3. Let $(R,\mathfrak m)$ be a Noetherian local ring of dimension $d > 1$, let $f \in \mathfrak m$ be an element not contained in any minimal prime ideal of $R$, and let $k\in \mathbf{N}$. Then there exist elements $g_1, \ldots , g_{d - 1} \in \mathfrak m^ k$ such that $f, g_1, \ldots , g_{d - 1}$ is a system of parameters.
Proof. We have $\dim (R/fR) = d - 1$ by Algebra, Lemma 10.60.13. Choose a system of parameters $\overline{g}_1, \ldots , \overline{g}_{d - 1}$ in $R/fR$ (Algebra, Proposition 10.60.9) and take lifts $g_1, \ldots , g_{d - 1}$ in $R$. It is straightforward to see that $f, g_1, \ldots , g_{d - 1}$ is a system of parameters in $R$. Then $f, g_1^ k, \ldots , g_{d - 1}^ k$ is also a system of parameters and the proof is complete. $\square$
Post a comment
Your email address will not be published.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).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)