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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