Lemma 88.14.11. Let \varphi : A \to B be an arrow of \textit{WAdm}^{Noeth}. Assume \varphi is adic, topologically of finite type, flat, and A/I \to B/IB is étale for some (resp. any) ideal of definition I \subset A. Let \mathfrak q \subset B be rig-closed such that \mathfrak p = A \cap \mathfrak q is rig-closed as well. Then \mathfrak p B_\mathfrak q = \mathfrak q B_\mathfrak q.
Proof. Let \kappa be the residue field of the 1-dimensional complete local ring A/\mathfrak p. Since A/I \to B/IB is étale, we see that B \otimes _ A \kappa is a finite product of finite separable extensions of \kappa , see Algebra, Lemma 10.143.4. One of these is the residue field of B/\mathfrak q. By Algebra, Lemma 10.96.12 we see that B/\mathfrak p B is a finite A/\mathfrak p-algebra. It is also flat. Combining the above we see that A/\mathfrak p \to B /\mathfrak p B is finite étale, see Algebra, Lemma 10.143.7. Hence B/\mathfrak p B is reduced, which implies the statement of the lemma (details omitted). \square
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