The Stacks project

Lemma 86.14.11. 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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