Timeline for A question about strict henselian local rings

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Nov 8, 2019 at 7:35	comment	added	Lao-tzu		Anyway, this argument is rather indirect, there should be a more direct way to show it. I will just leave it there in case others may give a direct proof.
Nov 8, 2019 at 7:34	comment	added	Lao-tzu		I can confirm it's correct now by passing to limit using [Milne / Étale Cohomology, Chapter II, Lemma 3.3], which will give another decomposition of $X({\bar{y}})$ of the same form, but the terms in the coproduct are now $\mathscr{O}_{X, \bar{x}}$, forcing $\mathscr{O}_{X(\bar{y}), a}\cong\mathscr{O}_{X, \bar{x}}$.
Nov 7, 2019 at 21:48	comment	added	Minseon Shin		I think it's correct, see e.g. Tag 05WR. In your setup, $R \to S$ is a finite ring map, and $R_{\mathfrak{p}}^{sh} \otimes _{R_{\mathfrak{p}}} S_{\mathfrak{q}}$ is a localization of $R_{\mathfrak{p}}^{sh} \otimes _{R_{\mathfrak{p}}} (R \setminus \mathfrak{p})^{-1}S$, which is finite over $R_{\mathfrak{p}}^{sh}$, hence a finite product of (strictly henselian) local rings.
Nov 6, 2019 at 14:48	history	asked	Lao-tzu	CC BY-SA 4.0