Lemma 16.13.4. Let R be a Noetherian ring. Let \mathfrak p \subset R be a prime ideal. Let f_1, \ldots , f_ m \in R[x_1, \ldots , x_ n]. Suppose that (a_1, \ldots , a_ n) \in ((R_\mathfrak p)^\wedge )^ n is a solution. If R_\mathfrak p is a G-ring, then for every integer N there exist
an étale ring map R \to R',
a prime ideal \mathfrak p' \subset R' lying over \mathfrak p
a solution (b_1, \ldots , b_ n) \in (R')^ n in R'
such that \kappa (\mathfrak p) = \kappa (\mathfrak p') and a_ i - b_ i \in (\mathfrak p')^ N(R'_{\mathfrak p'})^\wedge .
Proof.
By Theorem 16.13.2 we can find a solution (b'_1, \ldots , b'_ n) in some ring R'' étale over R_\mathfrak p which comes with a prime ideal \mathfrak p'' lying over \mathfrak p such that \kappa (\mathfrak p) = \kappa (\mathfrak p'') and a_ i - b'_ i \in (\mathfrak p'')^ N(R''_{\mathfrak p''})^\wedge . We can write R'' = R' \otimes _ R R_\mathfrak p for some étale R-algebra R' (see Algebra, Lemma 10.143.3). After replacing R' by a principal localization if necessary we may assume (b'_1, \ldots , b'_ n) come from a solution (b_1, \ldots , b_ n) in R'. Setting \mathfrak p' = R' \cap \mathfrak p'' we see that R''_{\mathfrak p''} = R'_{\mathfrak p'} which finishes the proof.
\square
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