Lemma 10.69.4. Let $R$ be a Noetherian ring. Let $M$ be a finite $R$-module. Let $\mathfrak p$ be a prime. Let $x_1, \ldots , x_ c$ be a sequence in $R$ whose image in $R_{\mathfrak p}$ forms an $M_{\mathfrak p}$-quasi-regular sequence. Then there exists a $g \in R$, $g \not\in \mathfrak p$ such that the image of $x_1, \ldots , x_ c$ in $R_ g$ forms an $M_ g$-quasi-regular sequence.

Proof. Consider the kernel $K$ of the map (10.69.0.1). As $M/JM \otimes _{R/J} R/J[X_1, \ldots , X_ c]$ is a finite $R/J[X_1, \ldots , X_ c]$-module and as $R/J[X_1, \ldots , X_ c]$ is Noetherian, we see that $K$ is also a finite $R/J[X_1, \ldots , X_ c]$-module. Pick homogeneous generators $k_1, \ldots , k_ t \in K$. By assumption for each $i = 1, \ldots , t$ there exists a $g_ i \in R$, $g_ i \not\in \mathfrak p$ such that $g_ i k_ i = 0$. Hence $g = g_1 \ldots g_ t$ works. $\square$

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