The Stacks project

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. $\square$

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